Extensions of the Stability Theorem of the Minkowski Space in General Relativity 9780821848234, 2009008908

Table of contents :
Cover
Title page
General table of contents
General introduction
Solutions of the Einstein vacuum equations, by Lydia Bieri
Dedication
Abstract
Acknowledgments
Contents
Introduction
Preliminary tools
Main theorem
Comparison
Error estimates
Second fundamental form ?: estimates for the components of ?
Second fundamental form ?: estimating ? and ?
Uniformization theorem
? on the surfaces ?-changes in ? and ?
The last slice
Curvature tensor-components
Uniformation theorem: standard situation, cases 1 and 2
Bibliography
Index
Solutions of the Einstein-Maxwell equations, by Nina Zipser
Abstract
Acknowledgments
Contents
Introduction
Norms and notation
Existence theorem
The electromagnetic field
Error estimates for ?
Interior estimates for ?
Comparison theorem for the Weyl tensor
Error estimates for ?
Second fundamental form
The lapse function
Optical function
Conclusion
Bibliography
Back Cover

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Extensions of the Stability Theorem of the Minkowski Space in General Relativity Lydia Bieri Nina Zipser

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AMS/IP

https://doi.org/10.1090/amsip/045

Studies in Advanced Mathematics Volume 45

Extensions of the Stability Theorem of the Minkowski Space in General Relativity Part I: Solutions of the Einstein Vacuum Equations By Lydia Bieri Part II: Solutions of the Einstein-Maxwell Equations By Nina Zipser

American Mathematical Society))))1))))International Press

Shing-tung Yau, General Editor 2000 Mathematics Subject Classification. Primary 83C05; Secondary 58J45, 53C80.

For additional information and updates on this book, visit www.ams.org/bookpages/amsip-45

Library of Congress Cataloging-in-Publication Data Bieri, Lydia, 1972– Extensions of the stability theorem of the Minkowski space in general relativity / Lydia Bieri, Nina Zipser. p. cm. — (AMS/IP studies in advanced mathematics ; v. 45) Includes bibliographical references and index. ISBN 978-0-8218-4823-4 1. Generalized spaces. 2. Stability. 3. General relativity (Physics)—Mathematics. I. Zipser, Nina, 1972– II. Title. QA689.B44 2009 516.3 ′74—dc22

2009008908

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society and International Press. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c ⃝2009 by the American Mathematical Society and International Press. All rights reserved. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines ⃝

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the International Press home page at URL: http://www.intlpress.com/ 10 9 8 7 6 5 4 3 2 1

14 13 12 11 10 09

General Table of Contents Part I: Solutions of the Einstein Vacuum Equations, Lydia Bieri

1

Chapter 1.

Introduction

11

Chapter 2.

Preliminary Tools

43

Chapter 3.

Main Theorem

51

Chapter 4.

Comparison

69

Chapter 5.

Error Estimates

105

Chapter 6.

Second Fundamental Form k: Estimates for the Components of k

167

Chapter 7.

Second Fundamental Form χ: Estimating χ and ζ

189

Chapter 8.

Uniformization Theorem

221

Chapter 9.

χ on the Surfaces S – Changes in r and s

241

Chapter 10.

The Last Slice

247

Appendix A.

Curvature Tensor – Components

283

Appendix B.

Uniformization Theorem: Standard Situation, Cases 1 and 2

285

Bibliography

291

Index

293

Part II: Solutions of the Einstein-Maxwell Equations, Nina Zipser

297

Chapter 1.

Introduction

307

Chapter 2.

Norms and Notation

321

Chapter 3.

Existence Theorem

337

Chapter 4.

The Electromagnetic Field

343

Chapter 5.

Error Estimates for F

363 iii

iv

GENERAL TABLE OF CONTENTS

Chapter 6.

Interior Estimates for F

413

Chapter 7.

Comparison Theorem for the Weyl Tensor

425

Chapter 8.

Error Estimates for W

439

Chapter 9.

Second Fundamental Form

453

Chapter 10.

The Lapse Function

465

Chapter 11.

Optical Function

471

Chapter 12.

Conclusion

485

Bibliography

491

General Introduction

1. Overview This book consists of two independent works that prove different extensions of D. Christodoulou and S. Klainerman’s stability theorem of the Minkowski space in General Relativity. The first part, by Lydia Bieri, discusses solutions of the Einstein vacuum equations (obtained in her Ph.D. thesis [2] in 2007), and the second part, by Nina Zipser, discusses solutions of the Einstein-Maxwell equations (obtained in her Ph.D. thesis [14] in 2000). To the authors’ present knowledge, these are the only extensions of the celebrated results in ‘The global nonlinear stability of the Minkowski space’ [8]. In the first part of the book, Lydia Bieri solves the Cauchy problem for the Einstein vacuum (EV) equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behaviour. In particular, she assumes one less decay in the power of r and one less derivative than in [8]. She proves that also in this case, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. Contrary to the situation in [8], certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. The main results of this work are stated in the ‘Introduction’, section 1.3, in Theorem 2, and in the chapter ‘Main Theorem’ in Theorem 3. In ‘The global nonlinear stability of the Minkowski space’ [8], D. Christodoulou and S. Klainerman proved the following result: ‘Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic.’ It is still an open question, what the optimal conditions are for nontrivial asymptotically flat initial data sets to give rise to a maximal complete development. L. Bieri’s work contributes to answering this question, stating sharp conditions on the decay at spatial infinity. It addresses the global, nonlinear stability of solutions of the Einstein vacuum (EV) equations in General Relativity. Solutions of the EV equations Rµν = 0 v

vi

GENERAL INTRODUCTION

are spacetimes (M, g), where M is a four-dimensional, oriented, differentiable manifold and g is a Lorentzian metric obeying the EV equations. As a consequence of imposing fewer conditions on her data, the spacetime curvature in Bieri’s case is no longer in L∞ (M ). She only controls one derivative of the curvature (Ricci) in L2 (H). By the trace lemma, the Gauss curvature K in the leaves of the u-foliation S is only in L4 . Contrary to that, in [8], the Ricci curvature is in L∞ (H), and in L∞ (S). Christodoulou and Klainerman control two derivatives of the curvature (Ricci) in L2 (H). The situation in Bieri’s case is both a disadvantage and an advantage. First, as she does not have the curvature bounded in L∞ , certain steps of the proof become more subtle. On the other hand, she does not have to control the second derivatives of the curvature, which simplifies the proof. The fact that she does not use any rotational vectorfields in her proof is a major simplification. She gains control of the angular derivatives of the curvature directly from the Bianchi equations, whereas in [8], a difficult construction of rotational vectorfields was necessary. Another major difference to the situation studied in [8] by Christodoulou and Klainerman, and which arises from Bieri’s relaxed assumptions, is the fact that she encounters borderline cases in view of decay in the power of r, indicating that the conditions in her main theorem on the decay at infinity on the initial data are sharp. Any further relaxation would make the corresponding integrals diverge and the argument would not close any more. Also in Bieri’s situation, energy and linear momentum are well-defined and conserved, whereas the (ADM) angular momentum is not defined. This is different to the situation investigated in [8], where all these quantities are well-defined and conserved. In the second part of this book, Nina Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell (EM) equations. A nontrivial solution of the EM equations is a nontrivial Lorentzian manifold – or curved spacetime – with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced in [8] and outlined below. To generalize Christodoulou and Klainerman’s results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field F ; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of F . In particular the Ricci curvature is a constant multiple of the stress-energy tensor for F . Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field. To produce estimates for the electromagnetic field, Zipser uses the Maxwell equations together with the stress-energy tensor much like Christodoulou and Klainerman use the Bianchi equations and

1. OVERVIEW

vii

Bel-Robinson tensor to produce global energy estimates. Also as in [8], she uses modified Lie derivatives of the electromagnetic field to obtain higherorder estimates. Once she produces good estimates for the electromagnetic field, she can bound the extra terms that appear in (i) the inhomogeneous equations for the Weyl tensor, (ii) the elliptic system for the parameters of the time foliation, and (iii) the Hessian of the optical function. After the extra terms are controlled, the results follow from generalizations of the proofs in [8]. In proving the stability of Minkowski space, Christodoulou and Klainerman rely on the invariant formulation of the EV equations. They cite two primary difficulties to overcome. The first difficulty is that a general spacetime has no symmetries, and therefore the conformal isometry group is trivial. Hence, the vectorfields needed to construct conserved quantities do not exist. The second is the highly non-linear nature of the hyperbolic equations, which makes it difficult to bound the asymptotic behavior of solutions. To tackle these issues, Christodoulou and Klainerman rely on geometric constructions that are analogous to structures existing on Minkowski spacetime. These include the following: a time-foliation, whose hypersurfaces are defined as the level-sets of a time function; an optical function, whose level sets define the null-structure of the spacetime; and the definition of the action of the subgroup of the conformal isometry group of Minkowski spacetime corresponding to time translations, scale transformations, inverted time translations and the spatial rotation group O(3). Whereas in the second part of the book, all these vectorfields play a crucial role, the first part relies on the first three but does not use rotational vectorfields. These geometric constructions have three key applications. First, the structure equations of the time-foliation together with the condition that the surfaces are maximal give rise to an elliptic system of equations for the parameters of the foliation. Once Christodoulou and Klainerman produce good estimates for the spacetime Riemann curvature tensor, these parameters are fully determined by this elliptic system. Second, Christodoulou and Klainerman use the optical function to construct quasi-conformal Killing fields. Third, these vectorfields can be used to produce higher-order energy estimates of the curvature. In particular, instead of estimating derivatives of the curvature directly from the Bianchi identities, they apply modified Lie derivatives with respect to the quasi-conformal Killing fields. The modified Lie derivatives applied to the traceless part of the Riemann curvature tensor retain the attributes of a Weyl tensor and satisfy inhomogeneous equations derived from the Bianchi equations. The norms for the global energy estimates are then constructed from the Bel-Robinson tensor, which is defined as a quadratic of a Weyl tensor. To prove the main theorem in their work, Christodoulou and Klainerman employ a continuity argument. In essence, once they show the existence of a spacetime slab with appropriate conditions on the initial slice, they use

viii

GENERAL INTRODUCTION

the global energy estimates discussed above to control the geometry of the spacetime slab. This allows them to prove that the geometry on the last slice can be bounded by the initial data and thus extend their solution. These concepts are explained in detail in the introduction to the first part of this book. Both works use the main approach of Christodoulou and Klainerman from [8]. This method has been elaborated and combined with new ideas by Christodoulou in order to study the formation of black holes in GR [6]. Our notation stays close to the notation in [8]. However, the two cases investigated here differ fundamentally from the original one and so do the details of the proofs. The differences are pointed out in the two introductions. As for the generalization in the EV case, the details of the proof are very different from the original result and require new ideas. In the EM case, the energymomentum tensor is equal to the stress-energy tensor of an electro-magnetic field, and therefore, additional curvature terms have to be controlled. 2. Former Work In the framework of the Cauchy problem for the EV equations Rµν = 0 the following question had been the subject of investigations by many authors for a long time: Is there any non-trivial, asymptotically flat initial data whose maximal development is complete? In 1952, Y. Choquet-Bruhat focussed the question of local existence and uniqueness of solutions, in GR. In [3] she treated the Cauchy problem for the Einstein equations, locally in time, she showed existence and uniqueness of solutions, reducing the Einstein equations to wave equations, introducing harmonic (or wave) coordinates. We recall that for a Riemannian manifold (M, g) a function Φ is called harmonic if △g Φ = 0 with △g Φ = g µν ∇µ (∂ν Φ), where ∇ is the covariant derivative on M associated to g. If the metric g is Lorentzian, then the equation △g Φ = 0 is the wave equation. She proved the well-posedeness of the local Cauchy problem in these coordinates. The local result led to a global theorem proved by Y. Choquet-Bruhat and R. Geroch in [4], stating the existence of a unique maximal future development for each given initial data set. In a next step, it is natural to ask whether this maximal future development is complete. R. Penrose gave a negative answer in his incompleteness theorem [13]. See also [5]. The said theorem tells us that, if in the initial data set (H, g¯, k), H is noncompact but complete, if the positivity condition on the energy holds, and H contains a closed trapped surface S, the boundary of a compact domain in H, then the corresponding maximal future development is incomplete. An exposition of this theorem is given in [5]. Definition 1. A closed trapped surface S in a non-compact Cauchy hypersurface H is a two-dimensional surface in H, bounding a compact domain such that trχ < 0 on S.

3. MATHEMATICAL AND PHYSICAL STRUCTURES

ix

Note that an infinitesimal displacement of S in M towards the future along the outgoing null geodesic congruence results in a pointwise decrease of the area element. The theorem of Penrose and its extensions by S. Hawking and R. Penrose led directly to the question formulated above: Is there any non-trivial asymptotically flat initial data whose maximal development is complete? The answer was given in the joint work of D. Christodoulou and S. Klainerman [8], ‘The global nonlinear stability of the Minkowski space’. A rough version of the theorem is stated in the first monograph at the end of subsection 1.2, whereas a more precise version is given in theorem 1. The problem studied by Christodoulou and Klainerman in [8] was suggested by S.-T. Yau to Klainerman in 1978. (Personal communication Yau, Christodoulou, 2008.) Lately, a proof under stronger conditions for the global stability of Minkowski space for the EV equations and asymptotically flat Schwarzschild initial data was given by H. Lindblad and I. Rodnianski [11, 12], the latter for EV (scalar field) equations. They worked with a wave coordinate gauge, showing the wave coordinates to be stable globally. Concerning the asymptotic behaviour, the results are less precise than the ones of Christodoulou and Klainerman in [8]. Moreover, there are more conditions to be imposed on the data than in [8]. There is a variant for the exterior part of the proof from [8] using a double-null foliation by S. Klainerman and F. Nicol` o in [10]. Also a semiglobal result was given by H. Friedrich [9] with initial data on a spacelike hyperboloid. A still open question is: What is the sharp criteria for non-trivial asymptotically flat initial data sets to give rise to a maximal development that is complete? Or, to what extent can the result of [8] be generalized? The results of [8] as well as of [14] and [2] are much more general than the others cited above, as all the other works place stronger conditions on the data. In this book, we give the results of [2] and [14], proving two different generalizations of [8]. 3. Mathematical and Physical Structures In this section, we expose some fundamental mathematical and physical structures. 3.1. Spacetime and Curvature. Let (M, g) denote our spacetime, represented by a 3 + 1 dimensional manifold M with Lorentzian metric g. The tangent space at each point of (M, g) is isomorphic to Minkowski spacetime. We recall the following facts about the Riemannian curvature tensor R. For any given vectorfields X, Y, Z on (M, g), it is (1)

R(X, Y )Z = DX DY Z − DY DX Z − D[X,Y ] Z.

x

GENERAL INTRODUCTION

Or in arbitrary local coordinates: (2)

α = ∂λ Γαµν − ∂ν Γαµλ + Γαβλ Γβµν − Γαβν Γβµλ . Rµλν

Also, recall the Christoffel symbols to be:

1 Γµαβ = g µν (∂α gβν + ∂β gαν − ∂ν gαβ ). 2 The Ricci tensor then reads: (3)

α Rµν = Rµαν = ∂α Γαµν − ∂ν Γαµα + Γαβα Γβµν − Γαβν Γβµα .

The Riemannian curvature tensor R of the spacetime M fulfills the following Bianchi identities: (4)

D[α Rβγ]δϵ := Dα Rβγδϵ + Dβ Rγαδϵ + Dγ Rαβδϵ = 0. Now, the traceless part of the curvature tensor reads:

1 Cαβγδ = Rαβγδ − (gαγ Rβδ + gβδ Rαγ − gβγ Rαδ − gαδ Rβγ ) 2 1 (5) + (gαγ gβδ − gαδ gβγ )R, 6 with R denoting the scalar curvature. Actually, C is called the conformal curvature tensor of the spacetime. This is a particular example of a Weyl tensor. In general, these tensors are defined as follows. Definition 2. A Weyl tensor W is a 4-tensor satisfying all the symmetry properties of the curvature tensor and in addition being traceless. W is said to fulfill the Bianchi equation, if it is: (6)

D[α Wβγ]δϵ = 0.

We mainly work with the Weyl tensor and call it W . Note that, in general, a Weyl field is not required to satisfy the Bianchi equation. However, in our situation, W does indeed obey it, and it even plays a major role in the proof of our main result. To see it, we remark that splitting the Riemannian curvature tensor into a part given by the Ricci tensor and a part represented by the Weyl tensor, the Bianchi identities (4) then yield differential relations between the Ricci and the Weyl tensor. One takes the first and second contractions of (4) and rewrites this first contraction for the Weyl tensor, obtaining an equation which in dimension n = 4 is equivalent to the Bianchi identities. In the EV case, the Weyl tensor satisfies the homogeneous equations Dα Wαβγδ = 0, and in the EM case the inhomogeneous equations 1 Dα Wαβγδ = (Dγ Rβδ − Dδ Rβγ ). 2

3. MATHEMATICAL AND PHYSICAL STRUCTURES

xi

Given a Weyl field W , we can define the left ∗ W and right W ∗ Hodge duals to be: 1 ∗ (7) Wαβγδ = ϵαβµν W µν γδ 2 1 (8) W ∗αβγδ = Wαβ µν ϵµνγδ , 2 where ϵαβγδ are the components of the volume element of M . One can think of (7) as freezing the second pair of indices and considering W as a 2-form relative to the first pair, correspondingly of (8) as freezing the first pair of indices and considering W as a 2-form in the second pair. Note that these definitions of left and right Hodge duals are equivalent. It can easily be checked that ∗ W = W ∗ is also a Weyl tensorfield. Further, it is ∗ ∗

( W ) = −W.

As the volume element of M comes into play in defining the Hodge duals right above, and as it will be involved in future parts of this work, let us write down the multiplication properties of the coefficients. The second up to the fifth of the subsequent equations are obtained by corresponding contractions. ϵµ1 µ2 µ3 µ4 ϵν1 ν2 ν3 ν4 = −det(δνµji )i,j=1,...,4

ϵµ1 µ2 µ3 µ4 ϵµ1 ν2 ν3 ν4 = −det(δνµji )i,j=2,3,4

ϵµ1 µ2 µ3 µ4 ϵµ1 µ2 ν3 ν4 = −2det(δνµji )i,j=3,4

ϵµ1 µ2 µ3 µ4 ϵµ1 µ2 µ3 ν4 = −6δνµ44

ϵµ1 µ2 µ3 µ4 ϵµ1 µ2 µ3 µ4 = −24.

Next, we define the electric-magnetic decomposition of W to be the following contractions with X, where X is an arbitrary given vectorfield. The decomposition consists of the two 2-tensors: (9)

iiX (W )αβ = Wµανβ X µ X ν

(10)

iiX (∗ W )αβ =∗ Wµανβ X µ X ν .

These tensors are symmetric, traceless and orthogonal to X. It can be shown that they completely determine W , if X is not null. (See also [7].) As in our spacetime manifold (M, g), the metric g is Lorentzian, there exists a vector V in Tp M such that gp (V, V ) < 0. Then, its gp -orthogonal complement is defined as ΣV = {X : gp (X, V ) = 0} and gp restricted to ΣV is positive definite. Then we can choose a positive orthonormal frame (e0 , e1 , e2 , e3 )p at each p in M continuously. That is, we obtain the positive orthonormal frame field consisting of e0 , e1 , e2 , e3 with: V (11) e0 = ! −g(V, V ) and e1 , e2 , e3 being an orthonormal basis for ΣV .

xii

GENERAL INTRODUCTION

Any given vector X in Tp M can be expanded as X = X 0 e0 + X 1 e1 + X 2 e2 + X 3 e3 " = X i ei (i = 0, 1, 2, 3). i

Consequently, one has

g(ei , ej ) = ηij = diag(−1, +1, +1, +1). g(X, X) = −(X 0 )2 + (X 1 )2 + (X 2 )2 + (X 3 )2 " = ηij X i X j ij

At a point p in M , we can distinguish three types of vectors. Namely, null, timelike and spacelike vectors. The vectors of the first type form a double cone at p, while the vectors of the second type form an open set of two connected components, that is, the interior of this cone, and the vectors of the third type a connected open set being the exterior of the cone. They are defined as follows. Definition 3. The null cone (or light cone) at p in M is Np = {X ̸= 0 ∈ Tp M : gp (X, X) = 0}.

The double cone consists of Np+ and Np− : Np = Np+ ∪ Np− .

Denote by Ip+ the interior of Np+ and by Ip− the interior of Np− . Definition 4. The set of timelike vectors at p in M is given by Ip := Ip+ ∪ Ip− = {X ∈ Tp M : gp (X, X) < 0}.

Definition 5. The set of spacelike vectors at p in M is defined to be Sp := {X ∈ Tp M : gp (X, X) > 0}.

Thus, Sp is the exterior of Np . For physical reasons, the spacetime should be time-orientable. Therefore, one assumes that it is possible to choose continuously a vector V ∈ Ip+ at each point p in M . That is, one has a continuous timelike future directed a continuous future-directed timelike vectorfield vectorfield. Denote by e0 ! on M at unit magnitude −g(e0 , e0 ) = 1. Thus, one is able to say what the causal future and past of any event (point) in spacetime means. To do so, we first give the definition of a causal curve. Definition 6. A causal curve in M is a differentiable curve γ whose tangent vector γ˙ at each point p in M belongs to Ip ∪ Np , i.e. is either timelike or null. Remark: This means that either γ˙p ∈ Ip+ ∪ Np+ at each p along γ in which case γ is called future-directed, or γ˙p ∈ Ip− ∪ Np− at each p along γ in which case γ is called past-directed.

3. MATHEMATICAL AND PHYSICAL STRUCTURES

xiii

Definition 7. The causal future of a point p in M , denoted by J + (p), is the set of all points q ∈ M for which there exists a future-directed causal curve initiating at p and ending at q. Correspondingly, we can define J − (p), the causal past of p. To be more general, we also need the causal future of a set S in M : Definition 8. The causal future J + (S) of any set S ⊂ M , in particular in the case that S is a closed set, is J + (S) = {q ∈ M : q ∈ J + (p)for some p ∈ S}.

Similarly, the definition is given for J − (S). The boundaries ∂J + (S) and ∂J − (S) of J + (S) and J − (S), respectively, for closed sets S are null hypersurfaces. They are generated by null geodesic segments. The null geodesics generating J + (S) have past end points only on S. These null hypersurfaces ∂J + (S) and ∂J − (S) are realized as level sets of functions u satisfying the eikonal equation g µν ∂µ u∂ν u = 0. Let us come back to a causal curve (definition 6) and say how distances are measured. For this, we define the arc length of this curve and the temporal distance of two points as follows: Definition 9. The arc length of a causal curve γ between the points corresponding to the parameter values λ = a, λ = b is # b! −g(γ(λ), ˙ γ(λ))dλ. ˙ L[γ](a, b) = a

If q ∈ J + (p), we define the temporal distance of q from p by τ (q, p) =

sup

all future-directed causal curves from p to q

L[γ].

Note that the arc length is independent of the parametrization. Recall that in Riemannian geometry the following statement about minimizing geodesics holds. Theorem 1. (Hopf-Rinow): For a complete Riemannian manifold any 2 points can be joined by a minimizing geodesic. In Lorentzian geometry the analogous statement is, in general, false. However, it is true, if the spacetime admits a Cauchy hypersurface. If the supremum is achieved and the metric is C 1 , the maximizing curve is a causal geodesic; after suitable reparametrization the tangent vector is parallelly transported along the curve. For the next statement, let us first introduce another important quantity: The deformation tensor of X, namely (X) π, is given as (12) (13)

(X)

παβ = (LX g)αβ

−(X) π αβ = (LX g −1 )αβ .

xiv

GENERAL INTRODUCTION

Given a Weyl field W and a vectorfield X, the Lie derivative of W with respect to X is not, in general, a Weyl field. For, it has trace. In fact, it is: (14)

g αγ (LX Wαβγδ ) = π αγ Wαβγδ .

In view of this, we define the following modified Lie derivative: 1 (X) 3 LˆX W := LX W − [W ] + tr(X) πW 2 8

(15) with (16)

(X)

[W ]αβγδ := παµ Wµβγδ + πβµ Wαµγδ + πγµ Wαβµδ + πδµ Wαβγµ .

To a Weyl field one can associate a tensorial quadratic form, a 4-covariant tensorfield which is fully symmetric and trace-free; a generalization of one found previously by Bel and Robinson [1]. As in [8] it is called the BelRobinson tensor: 1 (17) Qαβγδ = (Wαργσ Wβ ρ δ σ + ∗ Wαργσ ∗ Wβ ρ δ σ ). 2 It satisfies the following positivity condition: (18)

Q(X1 , X2 , X3 , X4 ) ≥ 0

where X1 , X2 , X3 and X4 are future-directed timelike vectors. Moreover, if W satisfies the Bianchi equations then Q is divergence-free: (19)

Dα Qαβγδ = 0.

Equation (19) is a certain property of the Bianchi equations. It is connected with their conformal behaviour. In fact, they are covariant under conformal isometries. To be precise, let Ω be a scalar. Then, if Φ : M → M is a conformal isometry of the spacetime, i.e., Φ∗ g = Ω2 g, and if W is a solution, also Ω−1 Φ∗ W is a solution. The Bel-Robinson tensor Q is an important tool in our work. We shall come back to it later in the two parts. In view of the principal part of the Ricci curvature, let us say a few words about the symbol. The principal part of the Ricci curvature is 1 αβ g {∂µ ∂α gβν + ∂ν ∂α gβµ − ∂µ ∂ν gαβ − ∂α ∂β gµν }. 2 If we substitute in the principal part ∂µ ∂ν gαβ by ξµ ξν g˙ αβ , where ξµ are the components of a covector and g˙ αβ the components of a possible variation of g, then we obtain the symbol σξ at a point p ∈ M and a covector ξ ∈ Tp∗ M for a given metric g: 1 ˙ µν = g αβ (ξµ ξα g˙ βν + ξν ξα g˙ βµ − ξµ ξν g˙ αβ − ξα ξβ g˙ µν ). (σξ · g) 2

3. MATHEMATICAL AND PHYSICAL STRUCTURES

xv

More generally, for a given metric g the symbol σξ at a point p in M and a covector ξ at p is the linear operator on the space of 2-covariant, symmetric tensors h at p, defined by: 1 (σξ · h) = {ξ ⊗ iξ h + iξ h ⊗ ξ − trhξ ⊗ ξ − (ξ, ξ)h}. 2 We use the following notation:

(20)

(iξ h)ν = g αβ ξα hβν , (ξ, ξ) = g αβ ξα ξβ , (ξ ⊗ l)µν = ξµ lν , g αβ hαβ = trh.

One observes that for any given covector ξ and any l ∈ Tp∗ M , belongs to the null space.

h=ξ⊗l+l⊗ξ σξ · h = 0.

This mirrors the general covariance of the EV equations. Whenever g is a solution of the EV equations, then the pullback of a diffeomorphism of the manifold onto itself of g is also a solution. For X being a vectorfield on M generating a 1-parameter group of diffeomorphisms on M , the symbol for the Lie derivative (LX g)µν = Dµ Xν + Dν Xµ is ξµ X˙ ν + ξν X˙ µ , whith X˙ µ being the components of an arbitrary covector. Consider the following equivalence relation h1 ∼ h2 ⇔ h2 − h 1 = l ⊗ ξ + ξ ⊗ l

for l in Tp∗ M , which gives a quotient space Q. There are two possibilities in view of the null space of the symbol σξ . It depends on the choice of the covector ξ with (ξ, ξ) ̸= 0 or (ξ, ξ) = 0, whether it is trivial or nontrivial. Let us consider the two situations: First, (ξ, ξ) ̸= 0: If ξ is not null, then σξ has only trivial null space on Q. Second, (ξ, ξ) = 0: If ξ is null, we can choose ξ in the same component of the null cone Np∗ in Tp∗ M such that (ξ, ξ) = −2. Then select a unique representative h out of each equivalence class {h} ∈ Q such that iξ h = 0.

Then it follows that the null space of σξ can be identified with the space of trace-free quadratic forms on the 2-dimensional spacelike plane Π, the g-orthogonal complement of the linear span of ξ and ξ. This is the space of gravitational degrees of freedom at a point.

xvi

GENERAL INTRODUCTION

If we suppose to be given a t-foliation and the EV equations, as explained in the introduction of the first monograph. Then the electric-magnetic decomposition of the curvature tensor R with respect to the future-oriented unit normal to the time foliation is denoted by E, H. They are symmetric, traceless 2-tensors tangent to the foliation. In terms of these quantities the equations (1.19) and (1.20) read: (21) (22)

∇i kjm − ∇j kim = ϵij l Hlm

¯ ij + trkkij − kim k m = Eij . R j

4. Preliminary Tools 4.1. Hodge Theory. Throughout this book, we use many analytic and geometric tools. A major one is the Hodge theory. We mainly apply it to obtain estimates in 2 and 3 dimensions. Therefore, let us introduce Hodge systems and derive estimates for them. Most of the presented results have been proven by D. Christodoulou and S. Klainerman in [8] for their setting. We give the proofs for certain of the following results, and refer to [8] for the others stating the difference of our proofs from theirs. 4.1.1. Hodge Systems on S. In the sequel we assume (S, γ) to be a compact, 2-dimensional Riemannian manifold. In the first monograph, L. Bieri shows that the Gauss-Bonnet theorem as well as the uniformization theorem hold in this case with L4 bounds on the curvature in S. In the second monograph, N. Zipser works with the correspondingly same assumptions on the curvature as D. Christodoulou and S. Klainerman in [8]. By K we denote the Gauss curvature of S. Also, let S have strictly positive curvature, that is km > 0 with km = minS r2 K. We shall first introduce different types of Hodge systems, and afterwards we will state the corresponding theorems for these systems. So, recall now the following definition of a Hodge dual. ∗ξ

Definition 10. Let ξ be a given vectorfield on S, then its Hodge dual is defined by ∗ ξA = ϵAB ξ B ,

where ϵAB denote the components of the area element relative to an arbitrary frame eA with A = 1, 2. If ξ is a symmetric, traceless 2-tensor, its left, ∗ ξ, and right, ξ ∗ , Hodge duals are defined as ∗

C ξAB = ϵAC ξB ,

Remark: Note that the tensors traceless. Also, one easily verifies that ∗

∗ C ξAB = ξA ϵCB . ∗ξ

ξ = −ξ ∗ .

and ξ ∗ are also symmetric and

4. PRELIMINARY TOOLS

xvii

We denote by div / the divergence operator on S and by curl / the curl operator on S. For any (k + 1)-dimensional tensor ξ these are given by / B ξA1 ···Ak B div / ξA1 ···Ak = ∇

(23) (24)

/ B ξA1 ···Ak C . curl / ξA1 ···Ak = ϵBC ∇

Also, recall that the trace operator on S is

trξA1 ···Ak−1 = γ BC ξA1 ···Ak−1 BC .

(25)

Next, we are going to present the types of Hodge systems, that we will always use throughout this work. H1 Let ξ be a vector on S that verifies (26)

div / ξ=f

(27)

curl / ξ=g with f and g being given scalar functions on S. H2 Let ξ be a symmetric, traceless 2-tensor on S that verifies

(28)

div /ξ=f with f being a given vector. H0 This is a special case of H1. Namely, we consider the scalar Poisson equation on S. So, let Φ be a scalar function on S that verifies

(29)

△ /Φ=f

with f being an arbitrary scalar function on S. H(k+1) Let ξ be a symmetric, traceless (k+1)-tensor that verifies (30)

div /ξ=f

(31)

curl / ξ=g with f and g being given k covariant, symmetric tensors on S.

Further on, we will study properties of results of the Hodge systems above, and derive corresponding estimates. First, we state the following: Proposition 1. Let (S, γ) be a 2-dimensional, compact Riemannian manifold. 1. Assume that the vectorfield ξ is a solution of H1. Then it is # # 2 2 |∇ /ξ| +K|ξ| = | f |2 + | g |2 . (32) S

S

2. Assume that the symmetric, traceless 2-tensor ξ is a solution of H2. Then it is # # (33) |∇ / ξ |2 + 2K | ξ |2 = 2 | f |2 . S

S

xviii

GENERAL INTRODUCTION

3. This is a particular case of the first item of this proposition. Assume that Φ is a solution of H0. Then it is # # 2 2 2 |∇ / Φ | + K| Φ | = | f |2 . (34) S

S

Proof. This proof follows from the proof of proposition 2. Proposition 2. Assume that ξ is an arbitrary, (k+1) covariant, totally symmetric tensor that verifies the following generalized Hodge system: H’(k+1): (35)

div /ξ=f

(36)

curl / ξ=g

(37)

trξ = h

with f and g given k covariant, symmetric tensors and h a covariant symmetric tensor of rank (k − 1). Then it is # # |∇ / ξ |2 + (k + 1)K | ξ |2 = | f |2 + | g |2 + kK | h |2 . (38) S

S

Note that if k = 0, then we take trξ = 0. That is # # |∇ / ξ |2 + K | ξ |2 = | f |2 + | g |2 . (39) S

S

Proof of Proposition 2: Writing out the curl equation (36), we obtain (40)

/ B ξA1 ···Ak C − ϵBC gA1 ···Ak . ∇ / C ξA1 ···Ak B = ∇

Now, differentiate (40) and recall the fact that / D ξA1 ···Ak C − ∇ /D∇ / B ξA1 ···Ak C ∇ /B∇ =

k " j=1

M RAj M BD ξA ˆ ···A 1

j ···Ak C

M + RCM BD ξA , 1 ···Ak

the commutator for the corresponding tensors ξ being zero. We obtain ∇ /D∇ / C ξA1 ···Ak B = ∇ /D∇ / B ξA1 ···Ak C − ϵBC ∇ / D gA1 ···Ak =∇ /B∇ / D ξA1 ···Ak C +

k " j=1

M RAj M BD ξA ˆ ···A 1

M + RCM BD ξA − ϵBC ∇ / D gA1 ···Ak . 1 ···Ak

j ···Ak C

4. PRELIMINARY TOOLS

xix

Taking the trace relative to C and D yields △ / ξA1 ···Ak B = ∇ / B (∇ / C ξA1 ···Ak C ) − ϵBC ∇ / C gA1 ···Ak +

k "

M RAj M B C ξA ˆ ···A

k "

γM C γAj B γ Aj B RAj CB C ξ M A1 ···Aˆj ···Ak C − RξA1 ···Ak B

k "

γAj B RhA1 ···Aˆj ···Ak − RξA1 ···Ak B .

j=1

j ···Ak C

1

M − RM B ξA 1 ···Ak

=∇ / B fA1 ···Ak − ϵBC ∇ / C gA1 ···Ak +

j=1

=∇ / B fA1 ···Ak − ϵBC ∇ / C gA1 ···Ak +

j=1

In view of the formula relating the Gauss curvature K to the Riemannian curvature Rabcd , multiplying the last equation by ξ A1 ···Ak B and integrating on S, we deduce that # # # ξ A1 ···Ak B △ / ξA1 ···Ak B = ξ A1 ···Ak B ∇ / B fA1 ···Ak − ξ A1 ···Ak B ϵBC ∇ / C gA1 ···Ak S S S # A1 ···Ak B + (k + 1)Kξ ξA1 ···Ak B S

−

# " k S j=1

That is # # # 2 2 |∇ /ξ| = |f | + S

S

S

γAj B Kξ A1 ···Ak B hA1 ···Aˆi ···Ak .

2

|g| −

#

S

2

(k + 1)K | ξ | +

This closes the proof. Then the next proposition follows directly.

#

S

kK | h |2 .

Proposition 3. Under the same assumptions as in proposition 2, there exists a constant c such that # # 2 2 |∇ / ξ | +K| ξ | ≤ c | f |2 + | g |2 + K | h |2 . (41) S

S

We immediately deduce the following proposition.

Proposition 4. Let ξ be as in proposition 2 and traceless in addition. Then there are constants C1 and C2 such that # # 2 |∇ / ξ | ≤ C1 | f |2 + | g |2 #S #S 2 K | ξ | ≤ C2 | f |2 + | g |2 . S

S

xx

GENERAL INTRODUCTION

As a consequence of these, one has the following. Proposition 5. Assume that km > 0. Let ξ be a solution of either H1 or H2. Then the next statements are valid. 1. There exists a constant c1 (K), such that # # (42) | f |2 . {| ∇ / ξ |2 + r−2 | ξ |2 } ≤ c1 S

S

2. There exists a constant c2 (K), such that # # 2 2 −2 2 (43) {| ∇ / ξ | + r | ξ | } ≤ c2 {| ∇ / f |2 + r−2 | f |2 }. S

S

estimates, can be shown for any 2 ≤ p < ∞. The analogue, namely We will make use of them as well. The corresponding proof applies the classical Calderon-Zygmund inequalities and the uniformization theorem. As shown in a separate chapter of the first monograph, the uniformization theorem is also valid for the case with K ∈ Lp for any 2 ≤ p < ∞. Now, we have the following result. Lp

Proposition 6. Let (S, γ) be a 2-dimensional, compact Riemannian manifold. Assume that km > 0, and let ξ be a solution of either H1 or H2. Then the following estimates hold: 1. There exists a constant c(K, p) such that for all 2 ≤ p < ∞, it is # # (44) {| ∇ / ξ |p + r−p | ξ |p } ≤ c {| f |p + | g |p }. S

S

2. There exists a constant c(K, p) such that for all 2 ≤ p < ∞, it is # # 2 p |∇ / ξ | ≤ c {| ∇ / f |p + r−p | f |p + | ∇ / g |p + r−p | g |p }. (45) S

S

4.1.2. Hodge Systems on H. Throughout this chapter, we will denote by (H, g) a 3-dimensional Riemannian manifold diffeomorphic to R3 , on which there exists a generalized radial function u with second fundamental form θ and Gaussian curvature K. We will require that u is quasiconvex, which means that trθ > 0,

K > 0.

We shall work with Hodge systems on Ht in the first monograph in the Chapter 6.2 ‘Estimating the Components δ, ϵ, ηˆ of the Second Fundamental Form’, where we will state and prove the estimates for the considered Hodge systems in detail. In the second monograph, we shall use the Lp theory for 2-d Hodge systems developed in [8] to produce estimates for the second fundamental form of the time foliation. Here, let us consider the following: Starting from a Hodge system for a 1-form A on H, we shall now give results for A and its corresponding

4. PRELIMINARY TOOLS

xxi

derivatives. So, let us introduce the following Hodge system for a smooth, compactly supported symmetric 1-tensor A on H: div A = ∇j Aj = f 1 (47) ∇j Ak = g. (curlA)i = ϵjk 2 i Now, we state the following proposition. (46)

Proposition 7. Let A be a smooth, compactly supported, symmetric 1-tensor on H. Then the following identity holds: # # # # 2 2 2 | ∇A | = |f | + |g| − R | A |2 , (48) H

H

H

H

where R is the magnitude of the Ricci curvature of H. Proof. We have Then, we calculate

1 ∇ j Ak . (curlA)i = ϵjk 2 i

1 | curlA |2 = (∇j Ak − ∇k Aj )(∇j Ak − ∇k Aj ) 2 = | ∇A |2 − (∇j Ak )(∇k Aj ).

(49)

Integrating this on H, we obtain for the last term, using integration by parts and commuting derivatives: # # # k j k j (∇j Ak )(∇ A ) = − Ak ∇j ∇ A = − Aj ∇k ∇j Ak H H H # $ j % =− A ∇j divA + Aj Rkjkl Al #H # =− Aj ∇j divA − Rjl Aj Al H # H # 2 (50) = | divA | − R | A |2 . H

Thus, we now see that # # 2 (51) | curlA | = H

H

H

2

| ∇A | −

which proves the proposition.

#

H

2

| divA | +

#

H

R | A |2 ,

Remark: Take equation (48) and write it as follows: # # # # 2 2 2 | ∇A | + R|A| = |f | + | g |2 . (52) H

H

H

H

Note that, if the right-hand side of (1.52) is finite, that is if & & ∥ curlA ∥L2 < ∞ and ∥ divA ∥L2 < ∞, then it is H | ∇A |2 + H R | A |2 < ∞.

xxii

GENERAL INTRODUCTION

Next, we will state two propositions concerning the first and second derivatives of a symmetric, 2-tensor on H. So, let V be a symmetric 2tensor on H. Consider the Hodge system (53)

div V = ρ

(54)

curl V = σ

(55)

tr V = 0,

where ρ is a given 1-form and σ a given symmetric, traceless, 2-covariant tensor. We can now formulate the following. Proposition 8. Let V be a smooth, compactly supported 2-symmetric tensor on H, that verifies the Hodge system ((53)–(55)). Then it is ( # ' ( # ' 1 1 2 im n 2 2 2 | ∇V | + 3Rmn V Vi − R | V | = |σ| + |ρ| . (56) 2 2 H H We can also derive the next statement, estimating the second derivative of V . Proposition 9. Let the assumptions of proposition 8 hold. Then there exists a constant c such that (57) #

H

| ∇2 V |2 ≤ c

#

H

$

% | ∇σ |2 + | ∇ρ |2 + | Ric || ∇V |2 + | Ric |2 | V |2 . Bibliography

[1] L. Bel. Introduction d’un tenseur du quatrieme ordre. C.R.Acad.Sci. Paris 247. (1959). 1094. [2] L. Bieri. An Extension of the Stability Theorem of the Minkowski Space in General Relativity. ETH Zurich, Ph.D. thesis. 17178. Zurich. (2007). [3] Y. Choquet-Bruhat. Th´ eor`eme d’existence pour certain syst` emes d’equations aux d´eriv´ees partielles nonlin´eaires. Acta Math. 88. (1952). 141–225. [4] Y. Choquet-Bruhat, R. Geroch. Global Aspects of the Cauchy Problem in General Relativity. Comm.Math.Phys. 14. (1969). 329–335. [5] D. Christodoulou. Mathematical problems of general relativity theory I and II. Volume 1: EMS publishing house ETH Z¨ urich. (2008). Volume 2 to appear: EMS publishing house ETH Z¨ urich. [6] D. Christodoulou. The Formation of Black Holes in General Relativity. EMS publishing house ETH Z¨ urich. (2009). ISBN 978-3-03719-067-8. [7] D. Christodoulou, S. Klainermann. Asymptotic properties of linear field equations in Minkowski space. Comm. Pure Appl. Math. 43. (1990). 137–199. [8] D. Christodoulou, S. Klainerman. The global nonlinear stability of the Minkowski space. Princeton Math.Series 41. Princeton University Press. Princeton. NJ. (1993). [9] H. Friedrich. On the Existence of n-Geodesically Complete or Future Complete Solutions of Einstein’s Field Equations with Smooth Asymptotic Structure. Comm.Math.Phys. 107. (1986). 587–609.

BIBLIOGRAPHY

xxiii

[10] S. Klainerman, F. Nicol` o. The Evolution Problem in General Relativity. Progress in Math.Phys. 25. Birkh¨ auser. Boston. (2003). [11] H. Lindblad, I. Rodnianski. Global Existence for the Einstein Vacuum Equations in Wave Coordinates. Comm.Math.Phys. 256. (2005). 43–110. [12] H. Lindblad, I. Rodnianski. The global stability of Minkowski space-time in harmonic gauge. To appear. [13] R. Penrose. Gravitational collapse and space-time singularities. Phys. Rev. Letters. 14. (1965). 57–59. [14] N. Zipser. The Global Nonlinear Stability of the Trivial Solution of the EinsteinMaxwell Equations. Ph.D. thesis. Harvard Univ. Cambridge MA. (2000).

Part I Solutions of the Einstein Vacuum Equations Lydia Bieri

To my family

Abstract This work addresses the global, nonlinear stability of solutions of the Einstein vacuum (EV) equations in General Relativity. Solutions of the EV equations Rµν = 0 are spacetimes (M, g), where M is a four-dimensional, oriented, differentiable manifold and g is a Lorentzian metric obeying the EV equations. In the pioneering work ‘The global nonlinear stability of the Minkowski space’, D. Christodoulou and S. Klainerman proved the following result: ‘Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic.’ However, it is still an open question, what the optimal conditions are for non-trivial asymptotically flat initial data sets to give rise to a maximal complete development. This monograph, containing the main results of the present author’s Ph.D. thesis, contributes to answering this question by generalizing the results of D. Christodoulou and S. Klainerman. In particular, we solve the Cauchy problem with more general, asymptotically flat initial data, and we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in L∞ (M ) anymore. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned.

3

Acknowledgments This monograph contains the main results of my Ph.D. thesis [6]. I would like to express my deepest gratitude to my advisor Demetrios Christodoulou. Without his constant guidance and support this work would never have been possible. I very much appreciate that he gave me the opportunity to enlarge my knowledge in mathematical physics and to share with me many beautiful insights in General Relativity. It has always been a great pleasure to discuss with him different topics. Especially, I would like to thank him for his ideas that enter this monograph, on estimating χ, uniformization and for the idea of not using rotational vectorfields at all. The proof of the uniformization theorem with K ∈ L4 (S) follows suggestions by personal communication with Demetrios Christodoulou. I thank very much Michael Struwe for having spent a lot of time in reading my thesis. For discussions with him and his suggestions I am very grateful. I would like to thank especially Shing-Tung Yau for encouraging me to publish this book and for his interest in my research. Moreover, I would like to thank him for having acquainted me with the work of his former Ph.D. student Nina Zipser. Special thanks belong to Mihalis Dafermos for his early interest in my work as well as for discussions. I thank Robert Bartnik, speaking with him about my work was very inspiring. Cliff Taubes deserves special thanks for his interest in my work. In this sense, my thanks go also to Eugene Trubowitz. I am also grateful to my dear friends and colleagues from Swiss Federal Institute of Technology (ETH) and Harvard University as well as from Massachusetts Institute of Technology (MIT) and the Boston area. My very special thanks go to my family and my friends who have always supported me, and with whom I have spent a very good time together. Finally, I thank Harvard University and ETH, and especially the two Departments of Mathematics which provided a good framework. Moreover, I thank the Swiss National Science Foundation (SNF) for having partially supported the original thesis.

5

Contents Chapter 1.

Introduction 1.1. Statement of the Result in a Rough Version 1.2. Setting 1.3. Statement of the Result in a More Precise Version 1.4. Discussion and Outline of the Proof

11 11 12 23 24

Chapter 2.

Preliminary Tools 2.1. 3D-Results 2.1.1. Notation and Basic Results 2.1.2. Inequalities: Sobolev, Poincar´e and others

43 43 43 46

Chapter 3.

Main Theorem 3.1. Norms 3.1.1. Norms for the Curvature Tensor R 3.1.2. Norms for the Second Fundamental Form k of the t-Foliation 3.1.3. Norms for the Lapse Function Φ 3.1.4. Norms for the Hessian of the Optical Function u 3.2. Statement of the Main Theorem 3.2.1. Most Important Achievements and Main Differences to Former Work 3.3. Crucial Steps of the Proof of the Main Theorem – Bootstrap Argument 3.3.1. Local Existence Theorem 3.3.2. Bootstrap Argument

51 51 52

Chapter 4.

Comparison 4.1. Curvature R, Weyl Tensor W and Bel-Robinson Tensor Q 4.2. Bianchi Equations and Identities Involving W and Q 4.3. Null Decomposition 4.3.1. Null Frame 4.3.2. Technicalities 4.3.3. Null Decomposition of the Weyl Tensor W 4.3.4. Ricci Coefficients 4.4. Bianchi Equations Relative to a Null Frame: Derivation 7

55 57 59 61 62 63 63 63 69 69 72 77 77 79 81 84 85

8

CONTENTS

4.5.

Bianchi Equations Relative to a Null Frame: Statement 94 4.6. Assumptions for the Comparison Argument 96 4.7. The Comparison Argument and the Controlling Quantities Q0 and Q1 in the Comparison Argument 98 Chapter 5.

Error Estimates 5.1. Setting 5.2. Weyl Current and its Null Decomposition 5.3. Boundedness Theorem: Statement and Proof 5.3.1. Assumptions 5.3.2. Consequences 5.3.3. Statement of the Boundedness Theorem 5.3.4. Proof of the Boundedness Theorem

Chapter 6.

Second Fundamental Form k: Estimates for the Components of k 6.1. Decomposing the Equations for k Relative to the Radial Foliation 6.2. Estimating the Components δ, ϵ, ηˆ of the Second Fundamental Form 6.2.1. Estimates for δ, ϵ and ηˆ – Preliminary Discussion 6.2.2. Estimates for δ, ϵ and ηˆ – Proof 6.3. Estimates and Recovering the Bootstrap Assumptions

Chapter 7.

Chapter 8.

105 106 110 118 120 121 125 126 167 167 168 171 174 187

Second Fundamental Form χ: Estimating χ and ζ 7.1. Statement of the Main Theorem for χ and ζ – Sketch of the Proof 7.2. Proof of the Main Theorem: Estimates for χ 7.3. Proof of the Main Theorem: The Torsion (ζ) System 7.3.1. The div − curl System for 1-Forms on a 2-Dimensional, Compact, Riemannian Manifold 7.3.2. Treatment of the Propagation Equation for µ

189

202

Uniformization Theorem 8.1. Standard Situation 8.2. Uniformization with K ∈ L4 (S) 8.2.1. Gauss-Bonnet Theorem for K ∈ L4 (S) 8.2.2. Uniformization Theorem for K ∈ L4 (S)

221 221 228 229 234

189 191

207 209

CONTENTS

9

Chapter 9. χ on the Surfaces S – Changes in r and s 9.1. Setting 9.2. Second Fundamental Form χ on the Surfaces S – Changes in r and s 9.2.1. Surfaces Ss and St,u 9.2.2. Estimates for χ in St,u

243 244 244

Chapter 10.

247

The Last Slice 10.1. Inverse Lapse Problem – Equation of Motion of Surfaces 10.2. Setting 10.3. Comparing Null Frames 10.4. Main Theorem 10.4.1. Integrating over u in Ht∗ 10.5. Higher Derivatives

241 241

247 248 250 259 274 276

Appendix A.

Curvature Tensor – Components

283

Appendix B.

Uniformization Theorem: Standard Situation, Cases 1 and 2

285

Bibliography

291

Index

293

CHAPTER 1

Introduction 1.1. Statement of the Result in a Rough Version The laws of General Relativity (GR) are the Einstein equations linking the curvature of the spacetime to its matter content. 1 (1.1) Gµν := Rµν − gµν R = 2Tµν , 2 (rationalized units 4πG = 1), where, for µ, ν = 0, 1, 2, 3, Gµν is called the Einstein tensor, Rµν is the Ricci curvature tensor, R the scalar curvature tensor, g the metric tensor and Tµν denotes the energy-momentum tensor. Definition 1. A spacetime manifold is a 4-dimensional, oriented, differentiable manifold M with a Lorentzian metric g. Definition 2. A Lorentzian metric g is a continuous assignment of a non-degenerate quadratic form gp , of index 1, in Tp M at each p ∈ M .

This work addresses the global, nonlinear stability of solutions of the Einstein vacuum (EV) equations in General Relativity. Solutions of the EV equations in GR: (1.2)

Rµν = 0

are spacetimes (M, g), where M is a four-dimensional, oriented, differentiable manifold and g is a Lorentzian metric obeying the EV equations. In this monograph, we study these equations for asymptotically flat systems. These are solutions where M looks like flat Minkowski space with diagonal metric η = (−1, +1, +1, +1) outside of spatially compact regions. Many physical cases require to study the Einstein equations in vacuum. Isolated gravitating systems such as binary stars, clusters of stars, galaxies etc. can be described in GR by asymptotically flat solutions of these equations. For, they can be thought of as having an asymptotically flat region outside the support of the matter. Einstein’s equations may be thought of as second-order differential equations for the metric tensor field gµν . In fact, they form a set of ten coupled nonlinear partial differential equations for the metric involving its first and second derivatives. There are ten unknown functions of the metric components facing ten equations. But the Bianchi identity gives four constraints; so in fact the Einstein equations only provide six independent differential equations, which is the appropriate number of equations to determine 11

12

1. INTRODUCTION

the spacetime. Using the four degrees of freedom to make coordinate transformations, we give arbitrary values to four of the components of the metric. Or, we can say that two metrics on a manifold define the same spacetime if there is a diffeomorphism which takes one metric into the other. That is why the Einstein equations should define the metric only up to an equivalence class under diffeomorphisms. In fact there are four degrees of freedom corresponding to the general covariance of these equations. As mentioned in the ‘General Introduction’, in view of the EV equations (1.2), it is an open problem, what is the sharp criteria for non-trivial asymptotically flat initial data sets to give rise to a maximal development that is complete. We generalize the results of D. Christodoulou and S. Klainerman in their joint work ‘The global nonlinear stability of the Minkowski space’ [19]. Every strongly asymptotically flat, maximal, initial data which is globally close to the trivial data gives rise to a solution which is a complete spacetime tending to the Minkowski spacetime at infinity along any geodesic. We solve the Cauchy problem with more general, asymptotically flat initial data. In particular, we assume one less decay in the power of r and one less derivative than in [19]. We prove that also in this case, the initial data, obeying appropriate smallness conditions, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. But, in accordance with the initial data, the asymptotical flatness is correspondingly weaker. Contrary to the situation in [19], certain estimates in our proof are borderline in view of decay, indicating that the conditions in our main theorem on the decay at infinity on the initial data are sharp. Our main results are stated in the ‘Introduction’, section 1.3, in Theorem 2, and in the chapter ‘Main Theorem’ in Theorem 3. Before that, we review the result of [19] in Theorem 1. 1.2. Setting The spacetime manifold (M, g) is defined above. We show in the ‘General Introduction’, how to choose a positive orthonormal frame field, and we introduce basic definitions. Now, we refer to the definitions of null, timelike, spacelike vectors and of a causal curve. Let us continue here with the following crucial object. Definition 3. H is called a null hypersurface if at each point x in H the induced metric gx | Tx H is degenerate. This means that there exists a L ̸= 0 ∈ Tx H such that gx (L, X) = 0 ∀X ∈ Tx H.

Let u be a function for which each of its level sets is a null hypersurface. Then, we have g(L, L) = 0, gµν Lµ Lν = 0.

1.2. SETTING

13

The same reads in terms of du: g µν ∂µ u∂ν u = 0, which is the eikonal equation. In fact, L is a geodesic vectorfield, that is, the integral curves of L are null geodesics. It can be seen by a short computation: (1.3)

(DL L)µ = Lν Dν Lµ

(1.4)

gµλ (DL L)µ = Lν Dν Lλ where

Lλ = gµλ Lµ = −∂λ u.

As the Hessian of a function is symmetric: Dν (∂λ u) = Dλ (∂ν u), it follows (1.5)

Lν Dν Lλ = Lν Dλ Lν ( ' 1 g(L, L) = 0. = ∂λ 2

(1.6)

A null hypersurface is generated by null geodesic segments. There are two kinds of hypersurfaces that play a crucial role here. These are the null and the spacelike hypersurfaces. Definition 4. A hypersurface H is called spacelike if at each x in H, the induced metric gx | Tx H =: g¯x is positive definite.

We observe that, (H, g¯) is a proper Riemannian manifold. And the g-orthogonal complement of Tx H is a 1-dimensional subspace of Tx M on which gx is negative definite. Thus there exists a vector Nx ∈ Ix+ of unit magnitude gx (Nx , Nx ) = −1 whose span is this 1-dimensional subspace. We refer to N (the so-defined vectorfield along H) as the future-directed unit normal to H. Next, the second fundamental form k of the hypersurface H is given as follows: Definition 5. The second fundamental form k of H is a 2-covariant symmetric tensorfield on H, defined by (1.7)

k(X, Y ) = g(DX N, Y )

∀X, Y ∈ Tx H.

k is symmetric. A very important notion in GR and in this work is a Cauchy hypersurface, being defined with the help of causal curves: Definition 6. A Cauchy hypersurface is a complete spacelike hypersurface H in M (i.e. (H, g¯) is a complete Riemannian manifold) such that if γ is any causal curve through any point p ∈ M , then γ intersects H at exactly one point.

14

1. INTRODUCTION

Remark: Not all spaces admit Cauchy hypersurfaces. A spacetime admitting a Cauchy hypersurface is called globally hyperbolic. Assuming the spacetime to be globally hyperbolic, a time function t can be defined. Definition 7. Let the spacetime (M, g) be globally hyperbolic. A time function is then a differentiable function t such that (1.8)

dt · X > 0

for all X ∈ Ip+ and for all p ∈ M.

The foliation given by the level surfaces Ht of t is called t-foliation. Denote by T the following future-directed normal to this foliation: T µ = −Φ2 g µν ∂ν t. It is T t = T µ ∂µ t = 1. Now, a spacetime foliated in this ¯ where M ¯ is a 3-manifold, fashion, is diffeomorphic to the product R × M ¯ each level set Ht of t being diffeomorphic to M . The integral curves of T are the orthogonal curves to the Ht -foliation. They are parametrized by t. Relative to this representation of M , the metric g reads: (1.9)

g = −Φ2 dt2 + g¯

with g¯ = g¯(t) denoting the induced metric on Ht . Note that g¯ is positive definite. Here, Φ is the lapse function corresponding to the time function t. It is defined as follows: (1.10)

1

Φ := (−g µν ∂µ t∂ν t)− 2 .

This lapse function measures the normal separation of the leaves Ht . By N (already given above) denote the unit normal N = Φ−1 T . Its integral curves are the same as for T , but parametrized by arc length s. In view of the first variational formula, which we are going to present, consider a frame field e1 , e2 , e3 for Ht , Lie transported along the integral curves of T . That is, we have [T, ei ] = 0 for i = 1, 2, 3. Denote g¯ij = g¯(ei , ej ) = g(ei , ej ). Then the first variational formula is: (1.11)

kij = k(ei , ej ) gij 1 ∂¯ (1.12) . = 2Φ ∂t One can choose a time function t the level sets Ht of which are maximal spacelike hypersurfaces. This eliminates the indeterminacy of the evolution equations (see below). That is, one imposes in addition the condition tr k = 0.

Any compact perturbation of Ht then decreases the volume. The covariant differentiation on the spacetime M is denoted by D. For that on H write ∇. Whenever a different notation is used, it is indicated.

1.2. SETTING

15

In the sequel, denote by R the Riemannian curvature tensor of M , and by ¯ the one of H. We shall work with the Weyl tensor W , not directly with R the Riemannian curvature, for reasons explained below. Next, we are going to state the structure equations with respect to the t-foliation. They relate the spacetime curvature Rαβγδ to the Ricci curva¯ ij of Ht , the second fundamental form k and the lapse function Φ. ture R ¯ ij completely Note that in the 3-dimensional leaf Ht , its Ricci curvature R ¯ is ¯ determines the induced Riemannian curvature tensor Rijkl as follows (R ij ¯ the scalar curvature g¯ Rij ): (1.13)

¯ jl + g¯jl R ¯ ik − g¯il R ¯ jk − 1 (¯ ¯ ¯ ijkl = g¯ik R gik g¯jl − g¯jk g¯il )R. R 2

Before that, let us give the following formulas for the frame field (e0 , e1 , e2 , e3 ) for M , where e0 = Φ1 T denotes the future-directed unit normal to the Ht and (e1 , e2 , e3 ) is an orthonormal frame tangent to the leaves of the foliation: (1.14)

Di e0 = kij ej

(1.15)

Di ej = ∇i ej + kij e0

(1.16) (1.17)

D0 e0 = (Φ−1 ∇i Φ)ei ¯ 0 ei + (Φ−1 ∇i Φ)e0 , D0 ei = D

¯ 0 ei denoting the projection of D0 ei to the tangent space of the foliation. D ¯ 0 ei = 0. Note that in a so-called Fermi propagated frame, it is D Here, note that g00 = −1, g0i = 0 and gij = g¯ij = g(ei , ej ) for i, j = 1, 2, 3. The structure equations of the foliation are: (1.18) (1.19) (1.20)

∂kij = ∇i ∇j Φ − (Ri0j0 − kim kjm )Φ ∂t ∇i kjm − ∇j kim = Rm0ij

¯ ij + trk kij − kim kjm = Rij + Ri0j0 . R

The latter, (1.20), is the trace of the Gauss equations: (1.21)

¯ imjn + kij kmn − kin kmj = Rimjn . R

The second, (1.19), are the Codazzi equations. Whereas (1.18) are the second variation equations. In the sequel, we are going to decompose the EV equations with respect to the foliation by the level sets Ht of a time function t. We start from the structure equations just introduced. Consider (1.20) and substitute for Ri0j0 in the second variation equations (1.18). Then, the part Rij = 0 of the EV equations is equivalent to (1.22)

∂kij ¯ ij + kij trk − 2kim kjm )Φ. = ∇i ∇j Φ − (R ∂t

16

1. INTRODUCTION

Take the Codazzi equations (1.19) and compute the trace (1.23)

∇i kij − ∂j trk = R0j .

Thus the Einstein equation R0j = 0 is equivalent to (1.24)

∇i kij − ∂j trk = 0.

In view of (1.21), the double trace of the Gauss equations is (1.25)

ˆ 00 , ¯ + (trk)2 − | k |2 = R + 2R00 = 2R R

i k m and R ˆ µν = Rµν − 1 gµν R. Therefore, the Einstein where |k|2 = km i 2 ˆ 00 = 0 is equivalent to equation R ¯ + (trk)2 − | k |2 = 0. (1.26) R

That is, we have: the evolution equations (1.12) and (1.22):

∂¯ gij = 2Φkij ∂t ∂kij ¯ ij + kij trk − 2kim kjm )Φ = ∇i ∇j Φ − (R ∂t as well as the constraint equations (1.24) and (1.26): ∇i kij − ∇j trk = 0

¯ + (trk)2 − | k |2 = 0 R

Also, taking the trace of (1.18) yields

∂trk = △Φ − (R00 + | k |2 )Φ. ∂t Then, in view of the EV equations and (1.26), the equation (1.27) is (1.27)

∂trk ¯ + (trk)2 )Φ. = △Φ − (R ∂t Next, let us say what an initial data set in our framework is.

(1.28)

Definition 8. An initial data set is a triplet (H, g¯, k) with (H, g¯) being a three-dimensional complete Riemannian manifold and k a two-covariant symmetric tensorfield on H, satisfying the constraint equations: ∇i kij − ∇j trk = 0

¯ | k |2 +(trk)2 = 0. R−

The constraint equations constrain the initial data. We recall that a development of an initial data set is an EV spacetime (M, g) together with an imbedding i : H → M such that g and k are the induced first and second fundamental forms of H in M . As motivated at the beginning of the ‘Introduction’, we are studying asymptotically flat solutions of the EV equations (1.2): Rµν = 0.

1.2. SETTING

17

Therefore, we are going to explain what an asymptotically flat initial data set is. In view of the different types of asymptotic flatness, we first give the general definitions. Then, we will see what kind is used in [19], and we shall say, in what sense exactly our data is asymptotically flat. We will state our theorem in a more precise version below and discuss our asymptotically flat data. Definition 9. A general asymptotically flat initial data set (H, g¯, k) is an initial data set such that • the complement of a compact set in H is diffeomorphic to the complement of a closed ball in R3 • and there exists a coordinate system (x1 , x2 , x3 ) in this complement relative to which the metric components g¯ij → δij kij → 0 ) 1 sufficiently rapidly as r = ( 3i=1 (xi )2 ) 2 → ∞.

Generally, one defines ‘strong’ asymptotic flatness as follows: Definition 10. A strongly asymptotically flat initial data set is an initial data set (H, g¯, k) with: 1. M is Euclidean at infinity. 2. There exists a chart on the neighbourhood of infinity in which the following holds: ( ' 2M δij + o2 (r−1 ). (1.29) g¯ij = 1 + r 3. It is: kij = o1 (r−2 ).

(1.30) M denotes the mass.

In [19], Christodoulou and Klainerman consider the following strongly asymptotically flat initial data set: Definition 11. (SAFCK) We define a strongly asymptotically flat initial data set in the sense of [19] (studied by Christodoulou and Klainerman) and in the following denoted by SAFCK initial data set, to be an initial data set (H, g¯, k), where g¯ and k are sufficiently smooth and there exists a coordinate system (x1 , x2 , x3 ) defined in a neighbourhood of infinity such that, ) 1 as r = ( 3i=1 (xi )2 ) 2 → ∞, g¯ij and kij are: ( ' 3 2M δij + o4 (r− 2 ) (1.31) g¯ij = 1 + r (1.32)

5

kij = o3 (r− 2 ),

where M denotes the mass.

18

1. INTRODUCTION

In our work, we consider the asymptotically flat initial data set of the following form: Definition 12. (AFB) We define an asymptotically flat initial data set to be a AFB initial data set, if it is an asymptotically flat initial data set (H0 , g¯, k), where g¯ and k are sufficiently smooth and for which there exists a coordinate system (x1 , x2 , x3 ) in a neighbourhood of infinity such that with ) 1 r = ( 3i=1 (xi )2 ) 2 → ∞, it is: 1

(1.33)

g¯ij = δij + o3 (r− 2 )

(1.34)

kij = o2 (r− 2 ).

3

Note that this initial data is more general than the one in ((1.31), (1.32)) in the sense that in (1.33) and in (1.34) we have one less derivative and less fall-off than in ((1.31), (1.32)). We shall see later on that, as we are assuming less on our initial data, the description of the asymptotic behaviour of the curvature components is less precise as in [19] (with (1.31), (1.32)). However, it is as precise as it can be with these relaxed assumptions. The case we study does not tend as fast to Minkowski as the situation in [19]. In our proof, we use the main structure as in the proof of [19], namely a bootstrap argument. However, the proof itself and the techniques differ very much from the original one. Our more general case requires more subtle and different treatment of the most delicate estimates. In fact, we encounter borderline estimates in view of decay for the most difficult terms, which means, that any further relaxation of our assumptions would yield to divergence. This indicates that our conditions on the decay at infinity on our initial data are sharp. On the other hand as less derivatives are assumed, less derivatives have to be controlled globally. In this sense, the new proof simplifies considerably. One of the fundamental differences of the two situations is the following fact: Whereas in [19], the curvature is pointwise bounded, this is not true in our case, where the curvature only lies in W 1,2 . We shall see later in this introduction, how this influences the new proof. For certain asymptotically flat data sets, the ADM definitions of energy E, linear momentum P and angular momentum J are well defined and finite. The ADM definitions are as follows: Definition 13. (Arnowitt, Deser, Misner (ADM)) Let Sr = {| x |= r} be the coordinate sphere of radius r and dSj the Euclidean oriented area element of Sr . Then we define • Total Energy (1.35)

1 E = lim 4 r→∞

#

"

Sr i,j

(∂i g¯ij − ∂j g¯ii )dSj ,

1.2. SETTING

• Linear Momentum (1.36)

1 P = − lim 2 r→∞ i

#

Sr

19

(kij − g¯ij trk)dSj ,

• Angular Momentum # 1 ϵijm xj (kmn − g¯mn trk)dSn . (1.37) J i = − lim 2 r→∞ Sr

For strongly asymptotically flat initial data (definition 10), total energy, linear and angular momentum are well defined and conserved. Thus, also in the work [19] of Christodoulou and Klainerman (definition 11), all these quantities are well defined and conserved. Positive energy theorems have been proven in [32, 38, 39]. In our case, having only asymptotically flat initial data with less decay (definition 12), the total energy and the linear momentum are shown to be well defined and conserved. We are still within the frame for which R. Bartnik’s positive mass theorem applies [3]. Generally, total energy and linear momentum are well defined and conserved for asymptotically flat data sets such that there exists a coordinate system in the neighbourhood of infinity in which the following holds (1.38)

g¯ij = δij + o2 (r−α ),

1 α> . 2 In view of conserved quantities, Noether’s theorem comes into play in a generalized version. We recall the fundamental theorem of Noether, which says: In the framework of a Lagrangian theory to each continuous group of transformations leaving the Lagrangian invariant there corresponds a quantity which is conserved. In particular, energy corresponds to time translations, linear momentum corresponds to space translations, angular momentum corresponds to space rotations. A crucial role in this book play the foliations by a time function t and an optical function u. Let us define the maximal time function t right away, whereas the second foliation shall be introduced subsequently. The definition 7 of a time function implies a freedom of choice. In fact, t being subject only to dt · X > 0 + for all X ∈ Ip and for all p ∈ M , is arbitrary. We now fix our time function t by the condition to be maximal. This means, we require the level sets Ht of the time function t to be maximal spacelike hypersurfaces. It describes the fact that any compact perturbation of Ht decreases the volume. Thus, Ht satisfies the maximal hypersurface equation (1.39)

kij = o1 (r−1−α ),

(1.40)

trk = 0.

20

1. INTRODUCTION

The existence of maximal surfaces in asymptotically flat spacetimes under slightly more general conditions, but for data with the same fall-off as ours has been proven by R. Bartnik, P.T. Chru´sciel and N. O’Murchadha in [4]. It was first proven by R. Bartnik for stronger fall-off in [2]. Definition 14. A maximal time function is a time function t whose level sets are maximal spacelike hypersurfaces, being complete and tending to parallel spacelike coordinate hyperplanes at spatial infinity. We also require that the associated lapse function Φ tends to 1 at spatial infinity. There is one such function up to an additive constant for each choice of family of parallel spacelike hyperplanes in Minkowski spacetime. These families are connected by the action of elements of the Lorentz group. One can single out one family by choosing P i = 0.

(1.41)

Then the time function t is unique up to an additive constant. The constraint ((1.24), (1.26)) and evolution equations ((1.12), (1.22)) then take the following form for the foliation by a maximal time function (maximal foliation): Constraint equations for a maximal foliation: (1.42) (1.43) (1.44)

trk = 0 ∇i kij = 0 ¯ = | k |2 . R

Evolution equations for a maximal foliation: (1.45) (1.46)

∂¯ gij = 2Φkij ∂t ∂kij ¯ ij − 2kim kjm )Φ. = ∇i ∇j Φ − (R ∂t

In view of the maximality condition, taking the trace of the second variation equations (1.46), yields the lapse equation: (1.47)

△Φ = | k |2 Φ.

Let us now discuss the second foliation. The optical function u is a solution of the Eikonal equation: (1.48)

g αβ

∂u ∂u = 0. ∂xα ∂xβ

This equation tells us that the level sets Cu of u are null hypersurfaces. The (t, u) foliations of the spacetime define a codimension 2 foliation by 2-surfaces (1.49)

St,u = Ht ∩ Cu ,

1.2. SETTING

21

the intersection between Ht (foliation by t) and a u-null-hypersurface Cu (foliation by u). The area radius r(t, u) of St,u is then defined as: * Area (St,u ) (1.50) r(t, u) = . 4π To construct this optical function u, we first choose a 2-surface S0,0 , diffeomorphic to S 2 , in H0 . We assume the spacetime to have been constructed. Then the boundary ∂J + (S0,0 ) of the future of S0,0 consists of an outer and an inner component. They are generated by the congruence of outgoing, respectively incoming, null geodesic normals to S0,0 . Now, the zero-level set C0 of u is defined to be this outer component. In order to construct all the other level sets Cu for u ̸= 0, we start on the last slice Ht∗ of a spacetime slab. We solve on Ht∗ an equation of motion of surfaces. This is done in a special chapter and we sketch it later in this introduction. It forms a crucial part of our work. These level sets Cu are also outgoing null hypersurfaces. By construction, u is a solution of the eikonal equation (1.48). Important structures of the spacetime used in the proof are coming from a comparison with the Minkowski spacetime. Crucial are the canonical spacelike foliation, the null structure and the conformal group structure. As the situation to be studied here is ‘close’ to Minkowksi spacetime, which shall be specified below, we can use part of its conformal isometry group. In [19], the authors defined the action of the subgroup of the conformal group of Minkowski spacetime corresponding to the time translations, the scale transformations, the inverted time translations and the spatial rotation group O(3). For our present proof, we also define the actions for the first three of these, but not for O(3). In contrast to [19], where the construction of the rotational vectorfields is a major part of the proof, we do not work with rotational vectorfields at all. Recalling the construction in [19], once the functions t and u have been fixed, the rotation group O(3) takes any given hypersurface Ht onto itself. The orbit of O(3) through a given point p is the corresponding surface St,u through p. The surfaces (1.49) are the orbits of the rotation group O(3) on Ht . In our situation, the vectorfields for the time and inverted time translations as well as for the scalings supply everything that is needed to obtain the estimates. The group of time translations has already been defined. This corresponds to the choice of a canonical time function t. The integral curves of the generating vectorfield T are the timelike curves orthogonal to the hypersurfaces Ht , and are parametrized by t. For the corresponding group {fτ } holds, that fτ is a diffeomorphism of Ht onto Ht+τ . Further, the vectorfields for the scaling and inverted time translations, that is, S and K, respectively, are also constructed with the help of the function u. We have now introduced the most important quantities and structures, in order to state our main results. Before that, let us consider the result of Christodoulou and Klainerman [19] in more details.

22

1. INTRODUCTION

In our work, we consider asymptotically flat initial data. (See the definitions 9 and 10 for the different types of asymptotic flatness as well as the definitions 11 (SAFCK) and 12 (AFB).) In [19] Christodoulou and Klainerman study SAFCK initial data sets (definition 11). Their strongly asymptotically flat initial data set has to satisfy a certain smallness assumption. Therefore, in [19] the authors introduce a quantity QCK (x(0) , b) that has to be controlled by a small positive ϵ. It is $ % QCK (x(0) , b) = sup b−2 (d20 + b2 )3 | Ric |2 H

+b (1.51)

+

−3

'# " 3

# " 1 H l=0

H l=0

(d20

(d20 + b2 )l+1 | ∇l k |2 2 l+3

+b )

l

2

|∇B|

(

where d0 (x) = d(x(0) , x) is the Riemannian geodesic distance between the point x and a given point x(0) on H, and b denotes a positive constant, whereas ∇l are the l-covariant derivatives, and B (Bach tensor) is the following symmetric, traceless 2-tensor ' ( 1 ab Bij = ϵj ∇a Rib − gib R . 4 QCK (x(0) , b) is used to formulate the following global smallness assumption in [19]. Global Smallness Assumption CK: A strongly asymptotically flat initial data set is said to satisfy the global smallness assumption CK if the metric g¯ is complete and there exists a sufficiently small positive ϵ such that (1.52)

inf

x(0) ∈H,b≥0

QCK (x(0) , b) < ϵ.

Then, one version of the main theorem in [19], ‘The global nonlinear stability of the Minkowski space’, by Christodoulou and Klainerman is stated as follows: Theorem 1. (D. Christodoulou and S. Klainerman, [19], p. 17, Theorem 1.0.3) Any strongly asymptotically flat, maximal, initial data set that satisfies the global smallness assumption CK (1.52), leads to a unique, globally hyperbolic, smooth and geodesically complete solution of the EV equations foliated by a normal, maximal time foliation. This development is globally asymptotically flat. The full version of this theorem, Christodoulou and Klainerman provide in [19], p. 298, Theorem 10.2.1. There is no additional restriction on the data. The authors do not use a preferred coordinate system, but their proof relies on the invariant

1.3. STATEMENT OF THE RESULT IN A MORE PRECISE VERSION

23

formulation of the EV equations. Moreover, they obtain a precise description of the asymptotic behaviour at null infinity. In the next subsection, we are going to state our new result with more general asymptotically flat initial data. 1.3. Statement of the Result in a More Precise Version We consider an AFB initial data set (definition 12). Let us re-emphasize that this initial data is more general than the one in ((1.31), (1.32)) in the sense that in (1.33) and in (1.34) we have one less derivative and less fall-off than in ((1.31), (1.32)). Our initial data set (H0 , g¯, k) has to satisfy the global smallness assumption below. We introduce Q(a, 0) for the terms that have to be controlled by a small positive ϵ. At a later point in the proof, ϵ has to be taken suitably small, depending on other quantities. '# $ % −1 | k |2 +(a2 + d20 ) | ∇k |2 + (a2 + d20 )2 | ∇2 k |2 dµg¯ Q(a, 0) = a H0 ( # $ 2 % 2 2 2 2 2 2 (1.53) + (a + d0 ) | Ric | + (a + d0 ) | ∇Ric | dµg¯ , H0

where a is a positive scale factor. From now on, let Q(a, 0) denote the infimum over all a of the quantity defined in (1.53). We consider asymptotically flat initial data sets for which the metric g¯ is complete and there exists a small positive ϵ such that (1.54)

Q(a, 0) < ϵ.

A more precise version of our main theorem is the following: Theorem 2. Any asymptotically flat, maximal initial data set, with complete metric g¯, satisfying inequality (1.54), where the ϵ has to be taken sufficiently small, leads to a unique, globally hyperbolic, smooth and geodesically complete solution of the EV equations, foliated by the level sets of a maximal time function. This development is globally asymptotically flat. For later reference, we state the global smallness assumption B as follows. Global Smallness Assumption B: An asymptotically flat initial data set satisfies the global smallness assumption B, if the metric g¯ is complete and there exists a sufficiently small positive ϵ such that (1.55)

Q(a, 0) < ϵ.

The global smallness assumption B has to be considered together with the main theorem 3 in chapter 3.2. Then, ϵ in (1.55) has to be taken suitably small such that the inequalities stated in the main theorem 3 hold. This main theorem 3 is the most precise statement of our results.

24

1. INTRODUCTION

To prove this result (theorem 2, respectively theorem 3), we do not need any preferred coordinate system, but we rely on the invariant formulation of the EV equations. Also, the asymptotic behaviour is given in a precise way. We remark that by geodesically complete is denoted what in GR is called g-complete which means that every causal geodesic can be extended for all parameter values. For a complete and fast orientation, we now collect the references to our main results and to the results of Christodoulou and Klainerman. In the ‘Introduction’, we state our main theorem in subsection 1.3, theorem 2. Our initial data is specified AFB in definition 12, and the global smallness assumption B is given through inequality (1.55). The full version of our main theorem is stated in subsection 3.2, theorem 3. Crucial steps of the proof of the main theorem, in particular the bootstrap argument, as well as a suggestion, how to read the proof, are given in subsection 3.3 and in the ‘Introduction’. As far as the statement of the main theorem of [19] by Christodoulou and Klainerman is concerned, it is given in our ‘Introduction’, subsection 1.2, theorem 1. Their initial data is specified SAFCK in definition 11, and their global smallness assumption CK in inequality (1.52). In the original work [19], the main theorem is stated on p. 17, theorem 1.0.3. The full version of the main theorem of [19] by Christodoulou and Klainerman is given in [19], p. 298, main theorem 10.2.1. 1.4. Discussion and Outline of the Proof There are several important properties of our spacetime which play a crucial role in our proof. We are going to discuss them now and also give an outline of the proof of our main theorem 2, respectively of the full version of our main theorem 3. In order to obtain the precise estimates, it is necessary to work with appropriate foliations of the spacetime (M, g). These are the foliation into hypersurfaces Ht given by the time function t and the one into nullhypersurfaces Cu by the optical function u, as they were introduced earlier in this chapter. Thus, the (t, u) foliations of the spacetime (M, g) define a codimension-2-foliation by 2-surfaces (1.56)

St,u = Ht ∩ Cu .

The asymptotic behaviour of the curvature tensor R and the Hessian of t and u can only be fully described by decomposing them into components tangent and normal to St,u . One achieves this by introducing null pairs consisting of two futuredirected null vectors e4 and e3 orthogonal to St,u with e4 tangent to Cu and + , (1.57) e4 , e3 = −2. A null pair together with an orthonormal frame e1 , e2 on St,u forms a null frame.

1.4. DISCUSSION AND OUTLINE OF THE PROOF

25

The null decomposition of a tensor relative to a null frame e4 , e3 , e2 , e1 is obtained by taking contractions with the vectorfields e4 , e3 . Also define τ−2 := 1 + u2 .

With respect to this frame, we obtain the following null decomposition of the Riemann curvature tensor of an EV spacetime: (1.58)

RA3B3 = αAB

(1.59)

RA334 = 2β A

(1.60) (1.61)

R3434 = 4ρ

∗

R3434 = 4σ

(1.62)

RA434 = 2βA

(1.63)

RA4B4 = αAB

with

α, α : S-tangent, symmetric, traceless tensors β, β : S-tangent 1-forms ρ, σ : scalars .

We show, as a part of our main result, that these components are controlled in the sense of the main theorem. And our estimates yield the decay behaviour: ( ' 3 −1 − 2 α = O r τ− ( ' 1 −2 − 2 β = O r τ− - 5. ρ, σ, α, β = o r− 2

At this point, we cite the decay properties of the corresponding null components in [19]: ( ' 5 −1 − 2 α = O r τ− ( ' 3 −2 − 2 β = O r τ− ρ = O(r−3 ) ( ' 1 −3 − 2 σ = O r τ− - 7. α, β = o r− 2

With our relaxed assumptions we control the following. Recall (1.53), the global smallness assumption (1.55) and our main theorem 2. We control

26

1. INTRODUCTION

one derivative of the curvature (Ricci) in H. For Ric including corresponding weights according to (1.53): Ric ∈ W 1,2 (H).

The trace lemma gives for the Gauss curvature K in the leaves of the u-foliation S: K ∈ L4 (S).

We see that the curvature is not in L∞ (H), but only in W 1,2 (H). Thus, our assumptions yield a spacetime curvature which is not pointwise bounded. Whereas in [19] the curvature is pointwise bounded. They have two derivatives of the curvature in L2 (H). For Ric including weights as in (1.51), in view of their global smallness assumption (1.52) and their main theorem 1: Ric ∈ L∞ (H).

This yields for the Gauss curvature K in the surfaces S that K ∈ L∞ (S). In our case, we also control two derivatives of the second fundamental form k. Therefore, by Sobolev inequalities, it is k ∈ L∞ (H).

Then, also in the surfaces S, the second fundamental form k lies in L∞ (S). Working with this approach, there are two main difficulties to be discussed. They have been stated and solved by Christodoulou and Klainerman in [19]. As these concepts are also crucial in our work, let us now say what they are. However, employing these concepts in our setting requires fundamentally new ideas in our proofs. It shall be explained below. The said difficulties are: 1) In view of obtaining ‘energy estimates’, if one wants to define the energy-momentum tensor for gravitation analogously to classical field theories, where the field equations are derived from an action A, and the variation of this action with respect to the underlying metric yields the energymomentum tensor, then everything would become trivial. For, the action A vanishes for the EV equations. 2) A general spacetime has no symmetries. Thus the conformal isometry group is trivial. Hence, the vectorfields needed to construct conserved quantities do not exist. 1) As the goal is to find estimates for the spacetime curvature to give control on regularity, one way to attack the problem could be to focus on the definition of the energy-momentum tensor appropriate to a geometric Lagrangian, namely considering the variation of the action A with respect to the underlying metric. Generally, for a domain D with compact closure in M and Lagrangian L the action A is defined as: # Ldµg . (1.64) A[D] = D

1.4. DISCUSSION AND OUTLINE OF THE PROOF

27

Variations supported in D of the action, with respect to the underlying metric, yield the energy-momentum tensor as follows: # 1 ˙ T µν g˙ µν dµg . (1.65) A[D] = − 2 D But this approach would not work here (or in [19]), because the variation (1.65) vanishes for the gravitational Lagrangian: L = − 14 Rdµg , that is for the Einstein-Hilbert action: # 1 (1.66) A[D] = − Rdµg . 4 D This vanishing is stated in the Euler-Lagrange equations for gravitation, that is in the EV equations. An alternative, one could think, could be Noether’s theorem after subtracting an appropriate divergence relative to a background metric. But the energy could give control on the solutions only after the isoperimetric constant is controlled. Therefore, the energy alone could not help to prove regularity. For a discussion of this problem, see also [16]. However, the right way to resolve the first difficulty is the following: Consider the Bianchi identities: (1.67)

D[α Rβγ]δϵ = 0

as differential equations for the curvature and the Einstein equations: Rµν = 0, as algebraic conditions on the curvature. At this point, breaking the connection between the metric and the curvature, one introduces the Weyl tensor W of a given spacetime (M, g), which has all the symmetry properties of the curvature tensor and in addition is traceless (1.68)

g αβ Wαµβν = 0,

which is the analogue of the Einstein equations, and satisfies the Bianchi equations, (1.69)

D[ϵ Wαβ]γδ = 0.

We remark that the Bianchi equations are linear. Note that the Riemann curvature tensor has 20 independent components, whereas the conformal curvature and Ricci tensors have 10 components each. Next, we are going to refer to the Bel-Robinson tensor Q. It is defined out of the Weyl tensor W and given above in (17). The main properties of Q are stated after its definition (17). This quantity Q can then be thought of as the ‘energy-momentum tensor’ in our setting. The Bel-Robinson tensor Q, in fact, plays the same role for solutions of the Bianchi equations as the energy-momentum tensor of an electricmagnetic field plays for the solutions of the Maxwell equations. Assume to be given three vectorfields X, Y, Z, each of which generating a 1-parameter group of conformal isometries of the spacetime (M, g). Then the 1-form P = −Q(·, X, Y, Z)

28

1. INTRODUCTION

is divergence-free. It follows thus that the integral on a Cauchy hypersurface H # ∗ P H

is conserved (and is positive definite, if all of the vectorfields X, Y, Z are timelike future-directed), recalling that ∗ P is the dual 3-form: ∗ Pµαβ = P ν ϵνµαβ . To investigate the Einstein equations, being hyperbolic and nonlinear, we use energy estimates of the type in [19]. Aiming at global results, the classical energy estimates could not be used, as they are only applied for solutions being local in time. Instead, we introduce energies Q0 (t) and Q1 (t) (see definition below), being integrals over Ht involving the Bel-Robinson tensor Q of the spacetime curvature W and of the Lie derivatives of W , which serve to estimate the curvature components by a comparison argument. This procedure shall be outlined subsequently. This is one of the core parts of our work, and it is different from the work of D. Christodoulou and S. Klainerman in a fundamental way, which will also be explained below. The quantities Q0 (t) and Q1 (t) themselves are estimated by a continuity argument to be bounded by a multiple of the initial value Q1 (0), as performed in a separate chapter called ‘Error Estimates’. More precisely, we show the error terms, that are generated while estimating the growth of Q0 and Q1 , to be controlled. In this procedure it is important to assess the structure of these nonlinear terms. It turns out that the most troublesome terms cancel by identities that are consequences of the covariance and algebraic properties of the Einstein equations. As a major result of this chapter emerges the fact that the estimates for the most delicate of these error terms are borderline. This means, that any further relaxation of the assumptions would lead to divergence and the argument would not close anymore. This needs further explanation at this point. Contrary to many problems in analysis, where the principal terms, that is the terms containing the highest derivatives, are the most sensitive ones to estimate, whereas the non-principal terms (containing less or no derivatives) are usually easier to handle, throughout this book and especially in the said chapter, the most difficult terms to be estimated are of higher order with respect to asymptotic behaviour (that is they have less decay), but they are non-principal in view of derivatives. On the other hand the expressions, which are principal with respect to derivatives behave better asymptotically, and therefore can be controlled easier. Thus, by ‘borderline’ we always mean borderline in view of decay (asymptotic behaviour). It is an essential difference between the situation investigated by D. Christodoulou and S. Klainerman in [19] and ours that their worst terms still being of lower order in asymptotic behaviour than ours, the borderline case does not appear, whereas in our setting the estimates for the highest order terms in view of asymptotic behaviour are really borderline.

1.4. DISCUSSION AND OUTLINE OF THE PROOF

29

2) The second difficulty is, that a general spacetime has no symmetries, that is the conformal isometry group is trivial. Then one could not construct integral conserved quantities by using vectorfields in conjunction with energy-momentum tensors. The solution is as follows: As a spacetime arising from arbitrary asymptotically flat initial data is itself supposed to be asymptotically flat at spacelike infinity in general, and also, with corresponding smallness assumptions of the initial data, as the time tends to infinity, one could expect the spacetime to approach Minkowski spacetime. Now, Minkowski spacetime has a large conformal isometry group. The idea is, to use part of it in the following way. One defines in the limit an action of a subgroup. Next, one extends this action backwards in time up to the initial hypersurface in a manner as to obtain an action of the said subgroup globally. This has to be done in a way such that the deviation from conformal isometry is globally small and goes to zero at infinity sufficiently rapidly. It is described by the circumstance that the deformation tensors π ˆ of the generating vectorfields are globally small and approach zero sufficiently fast at infinity. In order to derive a complete system of estimates, we define the action of the subgroup of the conformal group of Minkowski spacetime corresponding to the time translations, the scaling and the inverted time translations. We recall that, contrary to the work [19] of D. Christodoulou and S. Klainerman, our proof does not involve any rotational vectorfields. The action of the group of time translations is the easiest to define. Having chosen a canonical maximal time function t, the corresponding time translation vectorfield T generates the action, taking the maximal hypersurfaces into each other, as described above. This and the optical function u, from which the action of the other groups are defined, are introduced above, where we discuss the (t, u)-foliation of the spacetime. Let us define the vectorfields S for the scaling and K for the inverted time translation. First, we introduce the function u to be (1.70)

u = u + 2r.

The time translation vectorfield T has already been defined. We only remark here, that it can be written as in the subsequent formula. Let L and L be the outgoing, respectively incoming, null normals to the surface St,u given by (1.49), for which the component along T is equal to T . Also, the integral curves of L are the null geodesic generators of the null hypersurfaces Cu parametrized by t. Then T is expressed as % 1$ (1.71) T = L+L . 2 The generator S of scalings is defined to be: % 1$ (1.72) S = uL + uL . 2

30

1. INTRODUCTION

And the generator K of inverted time translations is defined as: % 1$ (1.73) K = u 2 L + u2 L . 2 ¯ Then the vectorfield K = K + T reads as: % $ ¯ = 1 τ+2 L + τ−2 L . K 2 We shall see below, how these vectorfields are used to construct conserved quantities. More precisely, we will construct a set of quantities whose growth can be controlled in terms of the quantities themselves. This procedure is in the spirit of the theorem of Noether. We observe that it is crucial to work with a characteristic foliation of the spacetime in order to define the actions of groups in spacetime which are called quasi-conformal isometries. We find that, similarly as in [19], also in our work, the global geometry of a characteristic (null) hypersurface as well as, in addition, the foliation of spacetime by such hypersurfaces play an important role. The geometry of null hypersurfaces has already been employed earlier by R. Penrose in his incompleteness theorem [34]. Next, we give the main steps of the proof of our main result. Let us first give only the steps in order to obtain an overview, and discuss them further subsequently. The whole proof relies on an overall bootstrap argument, which includes more smaller bootstrap arguments. The main steps can be summarized as follows:

(1.74)

(1.75)

(1.76) (1.77)

1. Estimate an appropriate quantity Q1 (W ), which is an integral over Ht involving the Bel-Robinson tensor Q of the spacetime curvature W and of the Lie derivatives of W as below. At time t, this quantity Q1 (W ) can be calculated by its value at t = 0 and an integral from 0 to t, which both are controlled. Q1 (W ) is given by: Q1 (W ) = Q0 + Q1

with Q0 and Q1 being the following integrals, # ¯ T, T, T ) Q0 (t) = Q(W )(K, Ht # # ˆ ¯ ¯ K, ¯ T, T ). Q1 (t) = Q(LS W )(K, T, T, T ) + Q(LˆT W )(K, Ht

Ht

2. The Weyl tensor W verifying the Bianchi equations, is shown to be controlled through Q1 (W ) by a comparison argument. 3. The geometric quantities are estimated from curvature assumptions using the optical structure equations, elliptic estimates, evolution equations, and tools like Sobolev inequalities. The bootstrapping allows us to go from local to global. The whole procedure can mainly be split into three parts.

1.4. DISCUSSION AND OUTLINE OF THE PROOF

31

• Bootstrap assumptions: Initial assumptions on the main geometric quantities of the two foliations, i.e. {Ht } and {Cu }. • Local existence theorem: It guarantees the local existence of a unique solution and the preservation of the asymptotic behaviour in space of the metric, the second fundamental form and the curvature. • Bootstrap argument: Together with the evolution equations it yields the global existence of a unique solution as well as it shows the asymptotic behaviour to be preserved. As it plays a crucial role in our work, let us remind ourselves here, how the bootstrap argument works: The initial assumptions remain true in an open interval containing 0 by the local existence theorem and continuity. Then, we have two possibilities: either the problem has a global solution and the assumptions hold for all t ≥ 0, or there exists a t1 > 0 such that they are true for all t ∈ [0, t1 ) but that they are false at t = t1 . The second case is shown to be impossible. Therefore, the properties hold true for all time t ∈ [0, ∞). In order to estimate the geometric quantities of point three above in the framework of a bootstrap argument, one assumes the curvature components to be bounded. To see how this is done, let us give the idea for the case of the second fundamental form χ of S relative to C χ(X, Y ) = g(DX L, Y ) for any pair of vectors X, Y ∈ Tp S and L generating vectorfield of C. To estimate χ, we first split it into its trace trχ and traceless part χ. ˆ Then, one estimates χ from the propagation equation ∂trχ 1 ˆ |2 = 0 + (trχ)2 + | χ ∂s 2 and the elliptic system on each section Ss of C (the Codazzi equations)

(1.78)

(1.79)

1 div /χ ˆa = da trχ + fa 2

where fa involves curvature. Assuming estimates for the spacetime curvature on the right-hand side of (1.79) yields estimates for the quantities controlling the geometry of C as described by its foliation {Ss }. This is discussed in more details below when summarizing the corresponding chapter. Explanation of the Bootstrap Argument: The bootstrap argument is explained in details in chapter 3.3.2. The method of bootstrapping constitutes the core of the proof of the main theorem. Several steps are needed to close the bootstrap loop. They are given precisely in subsection 3.3.2. In what follows, let us outline the general procedure. In our case, we impose smallness conditions on the initial slice

32

1. INTRODUCTION

H0 on the norms of the components of the curvature, the second fundamental form k, the lapse function, the Hessian of the optical function, up to corresponding derivatives, as they are defined in 3.1. To simplify the notation, let f (t) denote any of these quantities. Let A1 (t) and A2 (t) be two norms of f (t). Thus, f (t) is a solution of an evolution equation. The quantities considered here, are continuous. In several places of our work, we show statements and inequalities of the following type to hold. Consider the solution f (t) on an interval [0, t], t > 0. There exist constants C1 and C2 such that for any solution f (t) of the evolution equation on [0, t] that obeys A1 (t) ≤ C1

(1.80) the following is true: (1.81)

If the bootstrap assumption (1.82)

!t

A1 (t) ≤ C2 A1 (0)e

0

A2 (t′ )dt′

.

A2 (t) ≤ C3 (1 + t)−α A1 (t)

holds for some α > 1, then, using (1.80), inequality (1.81) becomes (1.83)

A1 (t) ≤ C2 A1 (0)B(C1 , C3 , α),

where B(C1 , C3 , α) denotes a bounded expression involving C1 , C3 , α. In this last inequality, we estimate A1 (0) by (1.80). Then, we choose a positive ϵ < C1 sufficiently small such that 1 C2 ϵB(C1 , C3 , α) ≤ C1 . 2 One fixes initial data with

(1.84)

A1 (0) < ϵ. In chapter 3.3.2, we/define S to be the set of all t ≥ 0 such that there exists a spacetime slab t′ ∈[0,t] Ht′ for which a set of bootstrap assumptions holds (see (3.80)–(3.83)). This is the case because of the local existence theorem and continuity. We define t∗ := sup S as the supremum of S. The aim is then to prove that t∗ = ∞. This means that the assumptions BA0BA2 in subsection 3.3.2 hold globally. (See (3.80)–(3.83).) This then proves the global result. To do so, one uses a contradiction argument, assuming t∗ < ∞. Then, t∗ ∈ S, implying that the inequalities must be saturated at t∗ . The goal of the proof by contradiction is to show that this is not possible, that is, we have strict inequalities at t∗ . The fact, that the inequalities are not saturated at t∗ , is seen from the following. From the above it is clear that in the interval [0, t∗ ] inequality (1.80) holds. Then, choosing ϵ above sufficiently small, we derive (1.85)

1 A1 (t∗ ) ≤ C1 , 2

1.4. DISCUSSION AND OUTLINE OF THE PROOF

33

contradicting the definition of t∗ . Thus, again by local existence and continuity, the solution extends to a larger time interval up to t∗ + δ, for a small positive δ. This δ can be chosen such that A1 (t) < C1 on the extended interval, contradicting the definition of t∗ . Therefore, it follows that t∗ = ∞, and that the solution with the above properties exists globally. Let us now say, where in our proof, the main steps of the bootstrap argument are shown. Mainly using the bootstrap assumptions ((3.80)– (3.82)), we check all the assumptions of the comparison theorem 4 in the chapter 4 ‘Comparison’ as well as of the main theorem 5 of the chapter 5 ‘Error Estimates’. According to the main theorem 5 of the chapter 5 ‘Error Estimates’, the main quantities at t∗ are estimated by the corresponding ones at t = 0. Thus by the comaprison theorem 4 in the chapter 4 ‘Comparison’, we conclude that, for all t ∈ [0, t∗ ], inequality (3.88) holds: R[1] (t) ≤ cR[1] (0).

From (3.88) and from the fact that we can bound R[1] (0) by c · ϵ, we deduce (3.89) R[1] (t) ≤ cϵ. Choosing ϵ sufficiently small, yields (3.90) 1 R[1] (t) ≤ ϵ0 . 2 The corresponding proofs are done for the geometrical quantities in chapters 6, 7 and 10. At the end, the overall bootstrap argument closes and yields the desired results. Now, we give an outline of the monograph. In chapter 2, we present theory that is used throughout this work. Mainly it concerns Riemannian geometry in 2 and 3 dimensions, in addition to what is already presented in the ‘General Introduction’. We discuss in the ‘General Introduction’ results in dimension 2 for compact, Riemannian manifolds with positive Gauss curvature. Estimates in Lp for p > 2 and in L2 are discussed. At several points of this monograph, we work with Lp estimates on the surfaces S. In order to obtain them, we need the uniformization theorem so that we can use the Calderon-Zygmund theory for the corresponding Hodge systems on the standard sphere. We prove the uniformization theorem in chapter 8.2, theorem 11 for our setting, where the Gauss curvature K is in L4 (S). The slices Ht of our spacetime are 3-dimensional, complete, Riemannian manifolds, diffeomorphic to R3 and Euclidean at infinity. Each slice carries a structure induced by the level hypersurfaces of the optical function u. Also for these manifolds, symmetric Hodge systems are studied in the ‘General Introduction’. In chapter 2, the most important inequalities, such

34

1. INTRODUCTION

as Sobolev, etc. are presented. Moreover, in view of our work, the most important general results are given, such as the trace lemma, Lp and L2 estimates. The elliptic tools are used in many situations. In particular, they are crucial in deriving inequalities for the components of the second fundamental form. Note that in the estimates we have to make a difference between angular and normal components and derivatives relative to the radial foliation. This is a consequence of the decay behaviour of the spacetime curvature (in the wave zone). Following the notation of [19], we call such estimates degenerate, and the usual type non-degenerate. As we use two main foliations, the ideas and properties are presented. The one is given by a time function t slicing the spacetime into hypersurfaces Ht and the other one by an optical function u, yielding null-hypersurfaces Cu . In chapter 3, we state the main theorem 3 precisely, define all the required norms, and we explain the bootstrap argument (subsection 3.3.2) and the different steps of the proof of the main theorem. As a fundamental part of the whole argument, we make use of the local existence theorem, which is stated and proven in [19], statement theorem 10.2.2: p. 299/300, proof 10.2.2: p. 304–310, for their problem, and which also holds in our case. It requires the second fundamental form to be in L∞ , which is satisfied in our situation. The initial data set is discussed and the assumptions on the initial data are written down explicitly. As a central part of this work, we state and prove the comparison theorem 4. This is carried out in chapter 4. The comparison theorem 4 estimates the components of the Weyl curvature by the quantity Q1 (W ) introduced above. In a separate chapter 5, this quantity Q1 (W ) is shown ¯ their deformation to be bounded. As we need the vectorfields T , S, K, tensors are calculated. Within this whole part, the Bianchi equations play a crucial role. Thus, they are studied for our 4-dimensional spacetime. Let us recall here, that the Weyl tensorfield verifies the Bianchi equations. In fact, the Bianchi equations allow us to obtain the estimates of the angular derivatives of our curvature components directly. We re-emphasize that no rotational vectorfields are needed in the present proof. In [19], the authors introduced rotational vectorfields to obtain the corresponding angular derivatives. While this is different in our work, another fact is used similarly, that is the principle of conservation of signature. The chapter 5 ‘Error Estimates’ shows the quantity Q1 (W ), that is Q0 (t) and Q1 (t) to be bounded. In view of estimating Q1 (W ) from (1.77), note that Q0 (t) follows to be controlled directly, once Q1 (t) is estimated. The integral Q1 (t) for t∗ can be split into

Q1 (t∗ ) =

#

H0

¯ T, T, T ) + Q(LˆS W )(K,

#

H0

¯ K, ¯ T, T ) + E1 (W, t∗ ), Q(LˆT W )(K,

1.4. DISCUSSION AND OUTLINE OF THE PROOF

35

where E1 (W, t∗ ) is the integral on /Vt∗ of the absolute values of the error terms. Vt∗ denotes the spacetime slab t∈[0,t∗ ] Ht , for which the bootstrap assumptions hold. That is, the corresponding quantities are bounded by a small positive constant ϵ0 . These error terms are generated because of the fact that the integral (1.77) on Ht differs from an integral over H0 , as the vectorfields T, S, K are not exact conformal Killing fields (only quasi-conformal). The expressions in E1 (W, t∗ ) to be integrated are quadratic in the Weyl fields W and linear in the deformation tensors π ˆ of the vectorfields. Generally, for the integral on Ht , we have the following formula, proven in the chapter ‘Comparison’, for three arbitrary vectorfields X, Y, Z and T denoting the unit normal to the foliation by a time function t: # # Q(W )(X, Y, Z, T )dµg = Q(W )(X, Y, Z, T )dµg Ht

H0

+

# t-# 0

1 + 2 +

Ht′

#

(Z)

(divQ)βγδ X β Y γ Z δ Qαβγδ

Ht′

π

αβ

γ

X Y

δ

$(X)

%

π αβ Y γ Z δ +

(Y ) αβ

π

ZγXδ

.

Φdµg dt′ .

We also need the following integral on the cones Cu (1.86) ˜ 1 (W, u, t) = Q

#

Cu (t0 ,t)

¯ T, T, e4 ) + Q(LˆS W )(K,

#

Cu (t0 ,t)

¯ K, ¯ T, e4 ) Q(LˆT W )(K,

with Cu (t0 , t) denoting the part of the cone Cu between Ht0 and Ht for 0 ≤ t0 ≤ t. And what we have just said for (1.77), also holds for (1.86). In fact, we estimate 0 1 ˜ 1 (W, u, t) . sup Q1 (W, t), sup sup Q (1.87) Q∗ = max 1

t∈[0,t∗ ]

t∈[0,t∗ ] u∗ ≥u0 (t)

Thus, as shown in the said chapter, we have the inequality Q∗1 ≤ Q1 (0) + E1 (t∗ ).

In the main theorem 5 of this chapter 5, we prove that (1.88)

Q∗1 ≤ Q1 (0) + cϵ0 Q∗1 ,

which for ϵ0 chosen to be sufficiently small, becomes (1.89)

Q∗1 ≤ 2Q1 (0).

Therefore, we have to estimate the error terms E1 (t∗ ). We have already briefly sketched above, what has to be paid attention to in doing so, and that we encounter borderline estimates for the most delicate terms.

36

1. INTRODUCTION

We now give such an example of a borderline estimate, which will be treated in details in the chapter 5 ‘Error Estimates’. There are several borderline cases appearing in E1 (W, t∗ ). One of them we encounter in the integral (5.256): # τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | . Vte∗

The quantities we have to pay special attention to are α and (S) i. Here, α is ˆAB the null curvature component (1.58) and (S) i is the null component (S) π of the deformation tensor (see (12)) tangent to the surface. The null decomposition of the deformation tensor is given in (5.23). In view of the coarea formula (lemma 6), we write the integral (5.256) over Vte∗ as an integral on Cu and u in (5.257). After the calculations in ((5.258)–(5.259)), the problem reduces to estimating the integral (5.260): 11 0# 2 t∗ 2 2 (S) ∥ ˆi ∥ 4 ∥α∥ 4 dt L (St,u )

0

respectively (5.262): 0# t∗

0

≤

∥

(S)ˆ i ∥2L4 (St,u ) ∥

−3 cτ− 2

0#

t∗

0

α

L (St,u )

∥2L4 (St,u )

11 2

dt

(1 + t)−1 ∥ (S)ˆi ∥2L4 (St,u ) dt

11 2

.

The problem is rigorously solved in the chapter ‘Error Estimates’ in ((5.256)– (5.272)). Here, we give an abbreviated discussion in order to point out the main difficulties that appear and how these are solved. The curvature assumptions give 2 1 3 −3 sup r 2 ∥ α ∥L4 (St,u ) ≤ cτ− 2 t,onCu

and the assumption (5.51) on

∥r

(S) i 1 2

is

(S) i

∥∞,e ≤ ϵ0 .

Thus the integral in (5.262) is borderline. That is, we show that it is bounded. Any relaxation of our assumptions on the data involved would make this integral diverge. Now, in view of the definition of (S)ˆi and writing out the Lie derivative 1 1 LˆS g = uLˆL g + uLˆL g, 2 2 (S) ˆi involves the terms rχ the ˆ and uˆ χ. Whereas the latter has order of decay 1

1

O(r−1 τ−2 ), the first one is only of order o(r− 2 ). Therefore, by estimating rχ ˆ (S) ˆ i. Using the Codazzi equations and the assumption we obtain bounds for

1.4. DISCUSSION AND OUTLINE OF THE PROOF

37

that β ∈ L4 (St,u ), we show by elliptic estimates in chapter 10 to control rχ ˆ in terms of β. (1.90) 5

/χ ˆ ∥L4 (St,u ) ≤ c ∥ r2 β ∥L4 (St,u ) = cr 2− ∥ β− ∥ ∥ rχ ˆ ∥L4 (St,u ) + ∥ r2 ∇

L4 (St,u ) .

As we want to integrate (1 + t)−1 ∥ (S)ˆi ∥2L4 (St,u ) in (5.262), we study the 3

integral for ∥ r 2 β ∥2L4 (St,u ) : #

0

t∗

3

∥ r 2 β ∥2L4 (St,u ) dt ≤ c

#

t∗

∥ rβ ∥L2 (St,u ) · ∥ rβ ∥L2 (St,u ) dt

0

+c

#

0#

≤c

t∗

0 t∗

∥ rβ

0

0#

+c

×

∥ rβ ∥L2 (St,u ) · ∥ r2 ∇ / β ∥L2 (St,u ) dt

t∗

0

0#

t∗

0

0#

=c

Cu

0#

∥2L2 (St,u )

∥ rβ

dt

0

dt

2

Cu

r |β|

∥ rβ

∥2L2 (St,u )

11 2

dt

2

11 0 # 2

t∗

11

∥ r2 ∇ / β ∥2L2 (St,u ) dt

2

Cu

2

∥2L2 (St,u )

r2 | β |2

+c

11 0 #

11 0 #

11 2

r2 | β |2

2

11

4

Cu

2

2

r |∇ /β|

11 2

The right-hand side of the last inequality is bounded by the curvature assumptions on β. In order to prove the first inequality, we apply the isoperiolder inequality, metric inequality on St,u to | β |2 . Then, by mainly using H¨ 3 2 we derive an estimate for ∥ r 2 β ∥L4 (St,u ) on St,u . Finally, an estimate for the integral over t is deduced and the Cauchy-Schwarz inequality then gives the right-hand side of the last inequality which is bounded. These calculations are carried out in ((5.266)–(5.272)). What we have shown is 11 0 # 11 0# 2 2 1 r2 | β |2 r2 | β |2 ∥ r− 2 (S)ˆi ∥L2 ([0,t ],L4 (S )) ≤ c ∗

t,u

Cu

Cu

0#

11 0 #

+c

Cu

≤

cϵ0 Q∗1 .

r2 | β |2

2

Cu

r4 | ∇ / β |2

11 2

38

1. INTRODUCTION

From this it directly follows # τ+2 Φ | (ρ, σ)(LˆS W ) || trχ∥ (S)ˆi || α | ≤ cϵ0 Q∗1 . Vte∗

We observe that only the estimates of (S) i in terms of β yield the required bounds for this integral. Any further relaxation of our assumptions would lead to divergence of this integral. Therefore, this is indeed a borderline case. The estimates for many quantities in our work are borderline. Let us point out here, that this is true also for χ. We shall stress this fact when encountering the borderline cases in this monograph. They will require a more careful treatment. All the borderline cases are studied and computed in the corresponding chapters. Chapters 6, 7, 9, 10 of the monograph consist in estimating the geometric quantities with respect to the foliations {Ht } and {Cu }. The core part in here is the treatment of the components of the second fundamental form k and the Hessian of u. It is first explained, how the optical function u is constructed. The lapse functions for t and u are discussed. The components of the Hessian of u are estimated. In view of the results mentioned in the previous paragraph, the estimates derived here, have to be appropriate. Then, this makes it possible to close the proof of the main theorem and to derive the main results. To make this more precise, let us discuss in the following the procedure, and give the different quantities that are estimated. In chapter 3, we introduce the basic norms of our geometric quantities, that is in particular, of the components of k and of the components of the Hessian of u. We then state the definitive version of our main theorem. With all this comes a concrete description of the asymptotic behaviour of the geometric quantities. To estimate the components of the second fundamental form k is the content of chapter 6. k decomposes into the scalar kN N = δ, the S-tangent 1-form kAN = ϵA and the S-tangent, symmetric 2-tensor kAB = ηAB . The worst decay properties have ηˆAB , that is, the traceless part of ηAB . Having good estimates for the curvature, we can then use standard tools. In this section, the elliptic estimates from above are applied to derive the desired results. The main part is to prove them in the wave zone, where the behaviour of the components and of their derivatives depends on the direction. The elliptic system on Ht for k is given by: (1.91)

tr k = 0

(1.92)

curl k = H

(1.93)

div k = 0.

Also, recall from before that H is the magnetic part of the spacetime curvature relative to the time foliation. We also have: ¯ ij = kia k a j + Eij , (1.94) R

1.4. DISCUSSION AND OUTLINE OF THE PROOF

39

where E denotes the electric part of the spacetime curvature relative to the time foliation. This elliptic system on Ht for k is decomposed relative to the radial foliation. From the fact, that the nonlinear terms in these equations behave better in view of decay than the worst linear ones, follows that the estimates are essentially linear but nevertheless yield control of the full nonlinear problem. The estimates, being essentially linear in the present situation in contrast to [19], simplify the proof considerably. In chapter 7, the components of the Hessian of u are studied. To do so, we apply the basic method from the original proof in [19], namely the method of treating propagation equations along the cones Cu coupled to elliptic systems on the surfaces St,u . However, our estimates differ fundamentally from the ones in [19]. The reason for that is the fact that we do not have any L∞ bounds on the curvature, but we only control one derivative of the curvature in the hypersurfaces Ht and the Gauss curvature K in the surfaces St,u lies in L4 , as explained above. Our situation yields borderline estimates for χ (see below), whereas in [19], there are no borderline estimates. In fact, our assumptions on the fall-off cannot be relaxed. Thus, they are sharp in view of decay. Relative to a null frame (introduced before) the Hessian of u decomposes into χAB , ζA , ω, satisfying the following equations: dχAB = −χAC χCB + αAB ds dζA = −χAC ζB + χAB ζ B − βA ds dω = 2ζ · ζ− | ζ |2 −ρ. ds As α is traceless, the trace of χ, fulfills an equation without curvature terms, namely (7.12): dtrχ 1 ˆ |2 = − (trχ)2 − | χ ds 2 with χ ˆ denoting the traceless part of χ. For χ ˆ the null-Codazzi equations form an elliptic system on the surfaces St,u . Recall (1.79), which in details read: 1 1 (1.95) div /χ ˆa = −βa + ∇ / a trχ − χ ˆba ζb + trχζa . 2 2 ∇ / is the induced covariant differentiation on the surfaces St,u . As a main goal of this chapter, we show that χ is one degree of differentiability smoother than the curvature. This is achieved by a bootstrap argument with a certain assumption on the curvature term β on the right-hand side of (1.95), as sketched above. The divergence equation (1.95) is a Hodge system discussed in the ‘General Introduction’. Thus, the bootstrap argument together with (7.12), (1.95), the method of treating such coupled systems of propagation and elliptic equations together with the Hodge theory yield the estimates.

40

1. INTRODUCTION

In the same line, we obtain the results for ζ and χ. For ζ, we have the Hodge system 1 ˆ·χ ˆ div / ζ = −µ − ρ + χ 2 1 curl / ζ =σ− χ ˆ∧χ ˆ 2 with χ being the second fundamental form χ(X, Y ) = g(DX L, Y ) for X, Y ∈ Tp S and L being the inward null normal whereas µ denoting the mass aspect function 1 µ = K + trχtrχ − div / ζ. 4 The propagation equation for µ is derived to be

(1.96)

dµ 3 1 1 ˆ |2 + trχ | ζ |2 + trχ µ = − trχ | χ ds 2 4 2 ˆ + 2div / χ·ζ +χ ˆ·∇ / ⊗ζ,

ˆ ζ)ab = ∇ / a ζb + ∇ / b ζa − γab div / ζ. with (∇ / ⊗ Wihtin this framework, to obtain our estimates, the uniformization theorem for K ∈ L4 (S) is needed, as it is mentioned above and proven in the corresponding section. A further crucial part of this work is the chapter 10 about the last slice Ht∗ of the spacetime. Here, we study the construction of the optical function u on this last slice. It is done by solving an inverse lapse problem for the function u on Ht∗ . That is, starting from St∗ ,u = Ht∗ ∩ C0 , the intersection of the last slice with the initial cone C0 , the function u is defined to be the solution of: u |St∗ ,0 = 0, | ∇u |−1 = a, where a on each St∗ ,u fulfills with

△ / log a = f − f¯ − div / ϵ,

log a = 0

1 f = K − (trχ)2 . 4 This method of solving an inverse lapse problem for u is the same as in [19], whereas the proof itself differs fundamentally from the one in [19]. Again the fact that we do not have any L∞ bounds on the curvature is crucial, but the curvature only lies in H 1 of the hypersurface Ht∗ . As explained at the beginning of chapter 10, other methods to construct u, would not match our requirements on this function. The obvious choice of u on Ht∗ , namely minus the signed distance function from St∗ ,0 , is inappropriate because this distance function is only as smooth as the induced metric g¯t∗ . It is not one order better, which would be the maximal possible.

1.4. DISCUSSION AND OUTLINE OF THE PROOF

41

Thus, there would be a loss of one order of differentiability. But the problem does not allow to lose derivatives. That is, with this loss of one order of differentiability, the estimates would fail to close. Moreover, the use of other methods like the inverse mean curvature flow (IMCF) would not work here, neither. Using IMCF, the problem could be solved in the outward direction only. Whereas the equation of motion of surfaces can be solved in both directions, which is what we need. The equation, we use here, has to have the smoothing property described above as well as it has to be solvable in both directions. This excludes the IMCF and similar methods. It turns out that the equation of motion of surfaces yields exactly what we need. Let us explain now the principal ideas in the proof of the main theorem of this chapter and also show, in which sense our situation requires a special treatment. The main theorem 14 of this part is again obtained by a bootstrap argument. We have to assume estimates for the spacetime curvature on the last slice Ht∗ . To be precise, these assumptions have to be made with respect to the background foliation, as the level surfaces of u on the last slice do not yet exist, but have to be constructed. Note that, at the beginning, only St∗ ,0 is given. We have to use the estimates on the background foliation to control the curvature and geometric components of the foliation by u. The main theorem 14 estimates trχ, χ, ˆ ζ and their first angular derivatives, as well as K in dimensionless L4 -norms on the surfaces St∗ ,u . We show this by bootstrapping. A crucial part of the proof is solved by the trace lemma, yielding L4 -bounds on the curvature components in St∗ ,u . Relying on them, we then apply elliptic theory to obtain the estimates of the theorem. By a straightforward argument, the said quantities are shown to be controlled correspondingly in Ht∗ . Next, integrating in the last slice over u ∈ [0, ∞), yields the estimates for the second angular derivatives of trχ, χ ˆ and ζ in 2 L -norms in Ht∗ . Yet, in order to apply the trace lemma, as said above, one has to work more. Let u′ be a smooth function without critical points, defined in a tubular neighbourhood U ′ of S0 , and let Su′ ′ be the level sets of u′ . The function u′ is introduced and specified in more details in the main theorem 14 of the chapter 10 ‘The Last Slice’. For the present purpose, we only remark that U′ =

4

Su′ ′ .

u′ ∈(u′0 ,∞)

This is what we refer to as the background foliation. On the other hand, for the foliation in the bootstrap argument, with respect to the function u, we denote 4 Su . U= u∈[0,u1 )

42

1. INTRODUCTION

As the curvature components of the background foliation are not in L∞ , but only in H 1 , there is no straightforward procedure to bound the curvature components in the surfaces of the foliation directly, as it is done in [19]. We have to proceed in the following different way: Having, by assumption, the curvature components of the background foliation in H 1 (U ′ ), and the transformation formulas between the two foliations shown to be bounded, we derive that the curvature components relative to the u-foliation lie in H 1 (U ). Only now, the trace lemma can be applied to obtain the curvature components with respect to the u-foliation to lie in L4 (Su ). We re-emphasize, that, as a consequence from controlling one less derivative, the derivatives of these components are not bounded in the surfaces Su . Finally, all the bootstrap arguments close and so does the overall bootstrap argument, which finishes the proof of the main theorem. Concluding, we find that the estimates for some of our main quantities are borderline in view of decay, which means that it is not possible to relax further our assumptions. This indicates that the conditions in our theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned.

CHAPTER 2

Preliminary Tools 2.1. 3D-Results 2.1.1. Notation and Basic Results. In the sequel, let (H, g) be a 3-dimensional Riemannian manifold diffeomorphic to R3 with a generalized radial function u. This generalized radial function u is a differentiable, real function defined on all points of H, outside a center point O, taking values onto the interval (u0 , ∞) and verifying the assumptions: 1. u has no critical points. 2. The level surfaces of u, denoted by Su , are diffeomorphic to the 2-dimensional spheres S 2 . Further, denote by IntSu the part H\Su containing O, and by ExtSu the other one. We require that 5 3. u∈(u0 ,∞)IntSu = {O}, / 4. u∈(u0 ,∞)IntSu = H.

Note that in our case, where we only assume L4 -bounds on the curvature in Su , for these surfaces Su the isoperimetric constant is bounded. This we show in the chapter ‘Uniformization with K ∈ L4 (S)’. It is also proven that the Gauss-Bonnet and the uniformization theorems are therefore valid. First, we are going to introduce some notation and basics concerning (H, g) and u. This will follow the notation of [19]. Consider the metric g. Its canonical form, relative to the foliation induced by the level surfaces of u, is given by " γAB dΦA dΦB (2.1) ds2 = a2 du2 + A,B=1,2

with a =| ∇u |−1

measuring the normal separation of the surfaces Su . This a is called the lapse function of the foliation. And γ is the metric induced by g on the spheres Su . Further, let N = a−1 ∂u be the unit exterior normal to the foliation. Define Π = Π(u) to be the projection operator from the tangent space of H to the tangent space of Su . In detail, Π(u) is a 2-tensor, expressed relative to arbitrary coordinates 43

44

2. PRELIMINARY TOOLS

on H by: Πij = δji − N i Nj ,

(2.2)

where δji denotes the Kronecker unit tensor. j j The projection operator Π satisfies Πm i Πm = Πi . The induced metric γ n is the restriction of Πm i Πj gmn to the space of vectors tangent to Su . Remark: The projection operator can be extended to arbitrary covariant tensors Ui1 ···iM on H by forming the contractions Πij11 · · · ΠijM Ui1 ···iM . M

The new tensor is tangent to Su . Further on, there will only be considered tensors defined on H, that are tangent, at any point p, to the spheres Su passing through p. By ∇ / we denote the intrinsic covariant differentiation on Su and by K = K(u) its Gauss curvature. Given a covariant tensorfield U , of rank M , defined on all of H and at every point in Su tangent to Su , the derivative ∇ /U can be regarded as an (M + 1) covariant tensorfield defined on all of H, obtained by projecting to Su the H-covariant derivative of U . In detail, this is (2.3)

j

M +1 ∇jM +1 Uj1 ···jM . ∇ /iM +1 Ui1 ···iM = Πji11 · · · ΠiM +1

And by ∇ /N U we denote the projection of ∇N U to Su . That is M ∇N Uj1 ···jM . ∇ /N Ui1 ···iM = Πji11 · · · ΠjiM

(2.4)

The second fundamental form θ to the foliation is given by (2.5) (2.6)

n θij = Πm i Πj ∇ m N n = ∇ i N j − N i ∇ N N j

In particular, it is (2.7)

/j a. = ∇i Nj + a−1 Ni ∇

trθ = div N.

Let us express the derivatives ∇A and ∇N in an orthonormal frame in /N . So, if N , {eA }A=1,2 is an orthonormal frame in H, terms of ∇ /A and ∇ then it is (2.8) (2.9) (2.10) (2.11)

/N eA + a−1 (∇ /A a)N ∇N eA = ∇ ∇A N = θAB eB /B eA − θAB N ∇B eA = ∇ ∇N N = −a−1 (∇ /A a)eA .

Also, note that a special local orthonormal frame on H can be obtained by propagating a local orthonormal frame (eA )A=1,2 on a given sphere + , Su such that ∇N eA = fA N and eA , N = 0. Then, it is ∇ /N eA = 0 and −1 /A a). fA = a (∇

2.1. 3D-RESULTS

45

We go back to the second fundamental form. Relative to arbitrary coordinates on Su , the second fundamental form θ is given by θAB = (2a)−1 ∂u γAB .

(2.12) Thus, we have

1 ∂u log detγ 2a with detγ being the determinant of the metric γ relative to the given coordinates. The total area of Su is denoted by A(u). We now have # d atrθdµγ = 4πr2 atrθ A(u) = (2.14) du Su (2.13)

trθ = (2a)−1 γ AB ∂u γAB =

with dµγ being the area element of Su . For a scalar function f , we denote by f its average on Su . Now, computing the derivative relative to u of the integral over Su of an arbitrary scalar function f , yields the formula: # # d f dµγ = a(∇N f + trθf )dµγ . (2.15) du Su Su Remark that r is the function of u defined by A = 4πr2 .

(2.16)

Sometimes, we shall call λ the derivative (2.17)

λ :=

dr du :

dr r = atrθ. du 2

In view of (2.14) it is r atrθ = a−1 λ. 2a Let f be a continuous function on H. Then the coarea formula holds: ( # # ∞'# (2.19) f dµg = af dµγ du ∇N r =

(2.18)

H

u0

Su

with dµg being the volume element of H. This is a well-known result. Next, we give the structure equations of the u-foliation: (2.20) (2.21) (2.22)

/ a − ∇N trθ− | θ |2 RN N = N i N j Rij = −a−1 △ j Πm /l θil − ∇ /i trθ i N Rmj = ∇

n −1 Πm /i ∇ /j a − ∇N θij − θim θmj − Kγij . i Πj Rmn = −a ∇

These equations tell us, how the second fundamental form θ, the lapse function a and the Gauss curvature K of the surface Su are connected to the Ricci curvature Rij of the ambient space H. Considering (2.20) and taking the trace of (2.22) yields (2.23)

R − 2RN N = 2K − (trθ)2 + | θ |2 .

46

2. PRELIMINARY TOOLS

In the following statements and proofs, we shall use some constants related to the considered foliation. Therefore, we introduce now, the fundamental constants of the u-foliation: am = inf am (u); aM = sup aM (u) u

u

hm = inf hm (u); hM = sup hM (u) u

u

km = inf kK (u); kM = sup kK (u) u

u

h = sup h(u); ζ = sup ζ(u) u

u

κ = sup κ(u) u

with am (u) = inf a; aM (u) = sup a Su

Su

hm (u) = inf rtrθ; hM (u) = sup rtrθ Su 2

∥ K− ∥ kK (u) = r −

Su

4,Su

h(u) = sup | rtrθ |; ζ(u) = sup r | θˆ | Su

Su

κ(u) = sup | κ | Su

with θˆ being the traceless part of θ, and κ = ra−1 (atrθ − atrθ).

(2.24)

2.1.2. Inequalities: Sobolev, Poincar´ e and Others. Here, we shall treat results with respect to Su and H that will be used frequently in other chapters. The most important ones are the inequalities of Sobolev, Poincar´e and the trace lemma. First, we give the notation. Let I = sup I(u), u

Λ(u) = sup Λ(u), u

where I(u) is the isoperimetric constant of Su and Λ(u) is the dimensionless number 1 Λ(u) = 2 , r λ1 (u) where λ1 (u) is the first eigenvalue of −△ on Su . As mentioned above, the constants with respect to the surfaces Su are shown to be bounded in the corresponding chapter. So, we can formulate the isoperimetric and the Poincar´e inequalities on each Su . The isoperimetric inequality reads (2 '# # 2 (2.25) (Φ − Φ) ≤ I(u) |∇ /Φ | dµγ , Su

Su

2.1. 3D-RESULTS

47

where

# 1 Φ= Φdµγ A(u) Su denotes the average of Φ on Su . And the Poincar´e inequality is # # 1 (Φ − Φ)2 ≤ |∇ /Φ |2 . (2.26) λ(u) Su Su

The scalar function Φ being sufficiently smooth. On our 3-dimensional Riemannian manifold H, we can now prove some estimates of Sobolev type, that we shall use frequently in the whole work. The subsequent propositions 1 and 3 are shown in [19] in their setting with L∞ -bounds on the curvature. We are giving them here, as they are crucial for many proofs in our case with curvature in H 1 (H), respectively K ∈ L4 (S). The said propositions can be proven in the same way as in [19], as we have already shown the necessary constants (isoperimetric) to be bounded. (In [19] the authors use that I(u) and Λ(u) can be estimated by supSu r2 | K |.) 1 In what follows, let τ− = (1 + u2 ) 2 . Proposition 1. Assume the foliation given by the radial function u to have positive mean curvature, that is, trθ ≥ 0.

Let F be an arbitrary tensor on H, tangent to Su at each point. Then the following statements hold: ) such that 1. There exists a constant cS = c(I, aaM m '# (1 '# (1 6 2 6 6 2 2 2 r |F | ≤ cS | F | + r | ∇F | , H

H

if the right-hand side is finite. 2. There exists a constant cS = c(I, aM , am ) such that '# (1 '# (1 6 2 4 2 6 2 2 2 2 2 r τ− | F | ≤ cS | F | +r | ∇ /F | + τ− | ∇ /N F | , H

H

if the right-hand side is finite.

Proof. Now, we prove the first inequality. To do so, let Φ = | F |3 in the isoperimetric inequality (2.25) for Su . (2 '# # 3 2 3 3 (| F | −| F | ) ≤ I(u) |∇ / | F | | dµγ . Su

Su

That is

(2.27) '# # 6 | F | ≤ I(u) Su

Su

3

|∇ / | F | | dµγ

(2

+

#

Su

$

2 | F |3 | F |3 − (| F |3 )2

%

48

2. PRELIMINARY TOOLS

Using H¨ older inequality for the terms on the right-hand side, yields (' # ( '# # (2.28) | F |6 ≤ c(I) r−2 | F |4 | F |2 + r2 | ∇ /F |2 . Su

Su

Su

r6 aM (u)

and integrate with respect to u We multiply the inequality by to obtain # # ∞ 6 r aM (u) | F |6 du Su u0 (' # ( '# # ∞ (2.29) ≤ r4 | F |4 | F |2 + r2 | ∇ /F |2 du. aM (u)c(I) Su

Su

u0

Applying the coarea formula, we deduce: (2.30) '# (' # ( # aM 6 6 4 4 2 2 2 r |F | ≤ c(I) sup r |F | | F | +r | ∇ /F | . am u H Su H & Focusing on Su r4 | F |4 , we compute # # 4 4 r | F | =− div(r4 | F |4 N ) Su #ExtSu $ (divN )r4 | F |4 =− ExtSu % +4r3 (∇N r | F |4 + r | F |2 F · ∇ /N F ) ( ' # # 4 4 4 =− (2.31) r | F | trθ + λ − 4 r4 | F |2 F · ∇ /N F, ar ExtSu ExtSu where in the last equation we used (2.7) and (2.18). Now, the assumption trθ ≥ 0 yields # # 4 4 r |F | ≤ r4 | F |2 F · ∇ /N F ExtSu

Su

(2.32)

≤4

'#

ExtSu

6

6

r |F |

(1 ' # 2

2

ExtSu

2

r |∇ /N F |

(1 2

,

having applied H¨ older inequality for the last step. Next, we substitute (2.32) in (2.30) and obtain (2.33) ('# (' # (2 ' # aM 6 6 2 2 2 2 2 r | F | ≤ c I, r |∇ /N F | | F | +r | ∇ /F | . am H H H

Thus, we have (1 '# '# 6 6 6 (2.34) r |F | ≤ cS H

H

2

2

2

| F | + r | ∇F |

proving the first statement in the proposition.

(1 2

,

2.1. 3D-RESULTS

49

The second, degenerate, statement is shown as follows. First, we go back to inequality (2.28) and multiply it by r4 τ−2 aM (u), which yields (2.35) '# (' # ( # aM 4 2 6 2 2 4 2 2 2 r τ− | F | ≤ c(I) sup r τ− | F | | F | +r | ∇ /F | . am u H Su H & For the term Su r2 τ−2 | F |4 on the right-hand side, we calculate: # # 2 2 4 r τ− | F | = − div(r2 τ−2 | F |4 N ) Su #ExtSu 2 =− (trθ + λ)r2 τ−2 | F |4 ar ExtSu # # 2 2 2 (2.36) −4 r τ− | F | F · ∇ /N F − 2 a−1 ur2 | F |4 . ExtSu

ExtSu

As trθ ≥ 0, this becomes # # r2 τ−2 | F |4 ≤ −4 r2 τ−2 | F |2 F · ∇ /N F Su ExtSu # −2 (2.37) a−1 ur2 | F |4 . ExtSu

Observe that we have '# # −1 2 4 a ur | F | ≤ c(am ) ExtSu

2

ExtSu

as well as '# # r2 τ−2 | F |2 F · ∇ /N F ≤ ExtSu

ExtSu

τ−2

|F |

(1 ' # 2

ExtSu

( 1'# 2

2

|∇ /N F |

ExtSu

Therefore, (2.37) reads as (2.38) (1 ' # '# # 2 2 2 4 4 2 6 r τ− | F | ≤ 4 r τ− | F | Su

ExtSu

r4 τ−2

2

ExtSu

|F |

|F |

r4 τ−2

+ τ−2

6

(1 2

6

(1

2

(1

|F |

|∇ /N F |

Substituting the last inequality in (2.35), we deduce ( '# # r4 τ−2 | F |6 ≤ c(I, am , aM ) | F |2 + τ−2 | ∇ /N F |2 H

'# ×

(2.39)

H

ExtSu

2

2

| F | +r | ∇ /F |

which, finally, yields (1 '# '# 6 4 2 6 r τ− | F | ≤ cS (2.40) H

2

H

2

2

(2

2

,

2

| F | +r | ∇ /F |

proving the second statement of the proposition.

2

+ τ−2

2

|∇ /N F |

(1 2

,

.

.

50

2. PRELIMINARY TOOLS

Before we turn, in proposition 3, to another consequence of the proof above, let us also cite in proposition 2 the well-known general trace inequality for a compact, Riemannian manifold (M, g) with boundary. Proposition 2. (Trace Inequality) Let (M, g) be a compact, ndimensional, Riemannian manifold with boundary. If 1 ≤ p < n and p∗ := p(n−1) n−p , then the following holds: 1. For 1 ≤ q < p∗ the Sobolev trace imbedding H 1,p (M ) 4→ Lq (∂M )

is compact. 2. The Sobolev trace imbedding H 1,p (M ) 4→ Lp (∂M ) ∗

is continuous. And there exist positive constants A, B such that '# ( 1∗ (1 (1 '# '# p p p p∗ p p |u| ≤A |u| +B | ∇u | ∂M

holds for all u in

M 1,p H (M ).

M

The next result has already been shown in the proof of proposition 1 above. Proposition 3. Let (H, g) be the manifold introduced at the beginning of this chapter. Assume that the foliation given by the radial function u has positive mean curvature. Let F be an arbitrary tensor on H, tangent to Su at every point, and cS respectively cS denote the same constants as in proposition 1. Then the following holds: 1. '# (1 '# (1 4 2 4 4 2 2 2 r |F | ≤ cS | F | + r | ∇F | . sup u∈H

Su

2. sup u∈H

H

'#

≤ cS

Su

r2 τ−2

'#

H

4

|F | 2

(1 4

2

2

| F | +r | ∇ /F |

+ τ−2

2

|∇ /N F |

(1 2

.

CHAPTER 3

Main Theorem In this section, we state the main theorem and sketch the most important steps of the proof. In order to do so, we have to introduce the different norms, that we shall make use of. The chapter splits into • Norms • Statement of the Main Theorem • Crucial Steps of the Proof of the Main Theorem – Bootstrap Argument 3.1. Norms / We consider a spacetime slab t∈[0,t∗ ] Ht . In this section, we assume that this slab is foliated by a maximal time function t and by an optical function u. In what follows, we shall introduce the basic norms for the curvature R, the second fundamental form k, the lapse function φ and the components χ, ζ, ω of the Hessian of u. In most of the definitions below, we follow the notation of [19]. Let V be a vectorfield tangent to S. Then we define the norms on S: (1 '# p p (3.1) | V | dµγ , for 1 ≤ p < ∞ ∥ V ∥p,S (t, u) = St,u

(3.2)

= sup | V |, for p = ∞. St,u

Sometimes, we also denote these norms by | V |p,S (t, u). The following norms are stated for the interior and the exterior regions of each hypersurface Ht . The interior region I, also denoted by Hti , are all the points in Ht for which r0 (t) r≤ . 2 And the exterior region U , also denoted by Hte , are all the points in Ht for which r0 (t) r≥ . 2 Here, r0 (t) is the value of r corresponding to the area of St,0 , the surface of intersection between C0 and Ht . And u1 (t) is the value of u corresponding to r0 (t)/2. 51

52

3. MAIN THEOREM

Now, we introduce (3.3)

∥ V ∥p,i =

(3.4)

'#

p

(1

,

for 1 ≤ p < ∞

p

(1

,

for 1 ≤ p < ∞

|V |

Hti

∥ V ∥∞,i = sup | V | Hti

(3.5)

∥ V ∥p,e (t) =

(3.6)

'#

|V |

Hte

∥ V ∥∞,e (t) = sup | V | .

p

p

Hte

Sometimes, we denote the norms ((3.3)–(3.6)) also by | V |p,I , | V |p,U (t) or ∥ V ∥p,I , ∥ V ∥p,U (t) or | V |p,i , | V |p,e (t), respectively. 3.1.1. Norms for the Curvature Tensor R. We are now going to define the norms R0 (W ) and R1 (W ) as well as R0 (W ) and R1 (W ) for the null curvature components. The norms are stated for the interior and exterior regions of the hypersurface Ht . Norms R0 (W) and R1 (W): On each slice Ht , we denote by Rq (W ) the following maximum: (3.7)

Rq (W ) = max

$

%

e R (W ), i R (W ) q q

,

q = 0, 1.

By e Rq (W ) we denote the exterior and by i Rq (W ) the interior L2 -norms of the curvature given as follows: In the interior region I, we define for q = 0, 1, (3.8)

iR q

= r01+q ∥ Dq W ∥2,I .

Then, one sets (3.9)

iR [0]

(3.10)

iR [1]

= i R0

= i R[0] + i R1 .

We define the exterior norms e R0 (W ) and e R1 (W ) to be e R (W )2 0

(3.11)

=

#

U

+

τ−2

#

U

2

|α| +

#

2

U

r2 | β |2 +

#

2

r |β| +

U

r2 | α |2

#

U

2

2

r |ρ| +

#

U

r2 | σ |2

3.1. NORMS

and e R (W )2 1

(3.12)

=

#

#

2

U

4

2

#

4

2

#

|∇ /α| + r |∇ /β| + r |∇ /ρ| + r4 | ∇ / σ |2 U U U # # # 4 2 4 2 + r |∇ /β| + r |∇ /α| + τ−4 | ∇ / N α |2 U U U # # # 2 2 2 4 2 + τ− r | ∇ / Nβ | + r |∇ / Nρ | + r4 | ∇ / N σ |2 U #U # U 4 2 4 2 + r |∇ / Nβ | + r |∇ / Nα | . U

τ−2 r2

53

U

We refer to the norms of the components of R by the formulas: for q = 0, 1: e R (α) q

e R (α) q

···

=∥ τ− rq ∇ / q α ∥2,e =∥ rq+1 ∇ / q α ∥2,e

and correspondingly for the remaining components. / N α, αN = ∇ / N α and correspondingly for the other Denote αN = ∇ curvature components. Then, we set e R [α] = e R (α) 0 0

(3.13)

e R [α] = 1

-

e R (α)2 1

+ e R0 (αN )2

.1 2

Similarly as in (3.13), we proceed with all the other null components of the curvature. This allows us now to define for q = 0, 1 the following: .1 e R = e R [α]2 + e R [β]2 + · · · + e R [α]2 2 . (3.14) q q q q Then, one sets

(3.15)

eR [0]

(3.16)

eR [1]

= e R0

= e R[0] + e R1 .

Norms R0 (W) and R1 (W): Similarly as above, we define on each slice Ht the quantity Rq (W ) as the maximum: $ % (3.17) Rq (W ) = max e Rq (W ), i Rq (W ) , q = 0, 1.

By e Rq (W ) we denote the exterior and by i Rq (W ) the interior L2 -norms of the curvature given as follows: In the interior region I, we define for q = 0, 1, (3.18)

i

Rq = r01+q ∥ Dq W ∥2,I .

54

3. MAIN THEOREM

Then, one sets (3.19)

i

(3.20)

i

R[0] = i R0

R[1] = i R[0] + i R1 .

The exterior norms e R0 and e R1 we define as follows: # # # # e 2 2 2 2 2 2 2 τ− | α | + r |β| + r |ρ| + r2 | σ |2 R0 (W ) = U U U U # # (3.21) + r2 | β |2 + r2 | α |2 U

and e

2

R1 (W ) =

(3.22)

#

U

#

2

U

4

#

U

2

4

2

#

|∇ /α| + r |∇ /β| + r |∇ /ρ| + r4 | ∇ / σ |2 U U U # # # + r4 | ∇ / β |2 + r4 | ∇ / α |2 + τ−4 | α3 |2 #U #U #U # 4 2 2 2 2 + τ− | α 4 | + τ− r | β 3 | + r4 | β 4 |2 + r4 | ρ3 |2 #U #U # U # U + r4 | ρ4 |2 + r4 | σ3 |2 + r4 | σ4 |2 + r4 | β3 |2 U U U U # # # 4 2 4 2 + r | β4 | + r | α3 | + r4 | α4 |2 . U

τ−2 r2

U

We refer to the norms of the components of R by the formulas: for q = 0, 1: e e

Rq (α) =∥ τ− rq ∇ / q α ∥2,e Rq (α) =∥ rq+1 ∇ / q α ∥2,e ···

and correspondingly for the remaining components. Then, we set R0 [α] = e R0 (α) .1 2 e R1 [α] = e R1 (α)2 + e R0 (α3 )2 + e R0 (α4 )2 e

(3.23)

Similarly, this is done for all the other null components of the curvature. Thus, we define the following for q = 0, 1: .1 2 e Rq = e Rq [α]2 + e Rq [β]2 + · · · + e Rq [α]2 . (3.24) Then, one sets

(3.25)

e

(3.26)

e

R[0] = e R0

R[1] = e R[0] + e R1 .

3.1. NORMS

55

3.1.2. Norms for the Second Fundamental Form k of the t-Foliation. First, let us define i Kqp and e Kqp to be the interior and exterior weighted Lp -norms of the q-covariant derivatives of the components of the second fundamental form k. The quantity Kqp we then define as follows: (3.27)

$ % Kqp = max i Kqp , e Kqp ,

q = 0, 1, 2 and 1 ≤ p < ∞.

Correspondingly, we have

% $i K0∞ = max K0∞ , e K0∞

(3.28)

The interior norms i Kqp in (3.27) are given by: (3.29)

i

1+q− p2

Kqp = r0

∥ Dq k ∥p,i .

In view of the exterior norms e Kqp , let us remind ourselves that the second fundamental form k relative to the radial foliation of u on Ht decomposes into kN N = δ kAN = ϵA kAB = ηAB .

(3.30)

In addition, η decomposes into its trace trη = −δ and its traceless part ηˆ. Let us also introduce the following notation: δ4 = D / 4δ ϵ4 = D / 4ϵ

1 η ηˆ4 = D / 4 ηˆ + trχˆ 2 δ3 = D / 3δ / 3ϵ ϵ3 = D 1 ηˆ3 = D / 3 ηˆ + trχˆ η 2

(3.31) Then, we set e

Kqp (δ) =∥ r

e

(3.32)

e

Kqp (ϵ) =∥ r

Kqp (ˆ η ) =∥ r

( 32 − p3 +q) ( 32 − p3 +q) ( 32 − p3 +q)

∇ / q δ ∥p,e ∇ / q ϵ ∥p,e

∇ / q ηˆ ∥p,e

56

3. MAIN THEOREM

(3.33)

e

p Kq+1 (δ4 ) =∥ r

e

p Kq+1 (δ3 ) =∥ r

e

p Kq+1 (ϵ4 ) =∥ r

e

p Kq+1 (ϵ3 ) =∥ r

e

p Kq+1 (ˆ η4 ) =∥ r

e

p Kq+1 (ˆ η3 ) =∥ r

( 52 − p3 +q)

∇ / q δ4 ∥p,e

3 1 ( 32 − p3 +q) ( 2 − p ) q τ− ∇ / δ3

( 52 − p3 +q)

∇ / q ϵ4 ∥p,e

1 3 ( 32 − p3 +q) ( 2 − p ) q τ− ∇ / ϵ3

( 52 − p3 +q)

∥p,e ∥p,e

∇ / q ηˆ4 ∥p,e

3 1 (1− p2 +q) ( 2 − p ) q τ− ∇ / ηˆ3

∥p,e

Next, one sets K0p [δ] = e K0p (δ) e p K1 [δ] = e K1p (δ) + e K1p (δ3 ) + e K1p (δ4 ) e p K2 [δ] = e K2p (δ) + e K2p (δ3 ) + e K2p (δ4 ), e

(3.34)

and correspondingly we do this for ϵ and ηˆ. Then it is for all q = 0, 1, 2: (3.35)

e

Kqp = e Kqp [δ] + e Kqp [ϵ] + e Kqp [ˆ η ].

For the case p = 2, that will be used later on, we write simply Kq = Kq2 .

Thus, we define the basic spacetime norms for k as follows: (3.36)

i

(3.37)

i

(3.38)

i

(3.39)

e

(3.40)

e

(3.41)

e

(3.42)

K[0] = i K0

K[1] = i K[0] + i K1 K[2] = i K[1] + i K2 K[0] = e K0

K[1] = e K[0] + e K1 K[2] = e K[1] + e K2

K[q] = i K[q] + e K[q]

Correspondingly, we define i Kqp and e Kqp to be the interior and exterior weighted Lp -norms as given below. Then Kqp we define as follows: $ % (3.43) Kqp = max i Kqp , e Kqp , q = 0, 1, 2 and 1 ≤ p < ∞. Correspondingly, we have

(3.44)

$ % K0∞ = max i K0∞ , e K0∞

Now, in the interior region I we define the following norms for k: (3.45)

i Kp q

1+q− p2

= r0

∥ Dq k ∥p,i .

3.1. NORMS

57

On the other hand, in the exterior region U , we define the following norms: ⎫1 ⎧ ⎬p ⎨ 3 3 " 2 2 (2− p +i) 1− p +j−1 i j q p p e Kp (δ) = ∥ r ( 2 − p +q) ∇ / δ ∥p,e + ∥r (τ− ) ∇ / ∇N δ ∥p,e q ⎭ ⎩ i+j=q,j≥1

⎧ ⎨ 3 3 " e Kp (ϵ) = ∥ r ( 2 − p +q) ∇ / q ϵ ∥pp,e + q ⎩

∥r

(2− p2 +i)

i+j=q,j≥1

⎧ ⎨ 3 3 ( − +q) q e Kp (ˆ η ) = / ηˆ ∥pp,e + ∥r 2 p ∇ q ⎩

"

i+j=q,j≥1

(3.46)

∥r

1− p2 +j−1

(τ− )

(1− p2 +i)

(τ− )

Then we set

(3.47)

e Kp q

1− p2 +j

⎫1 ⎬p

∇ / i∇ / jN ϵ ∥pp,e ⎭

⎫1 ⎬p

/ jN ηˆ ∥pp,e ∇ / i∇ ⎭

= e Kqp (δ) + e Kqp (ϵ) + e Kqp (ˆ η)

similarly as above. For the case p = 2, we write simply Kq = Kq2 .

Putting this together into the basic norms of the second fundamental form: (3.48)

K[0] = K0

(3.49)

K[1] = K[0] + K1

(3.50)

K[2] = K[1] + K2 .

3.1.3. Norms for the Lapse Function Φ. Analogously to the previous cases, we first define i Lpq and e Lpq to be the interior and exterior weighted Lp -norms of the (q + 1)-covariant derivatives of the logarithm ϕ of the lapse function φ. Then we also define: $ % (3.51) Lpq = max i Lpq , e Lpq .

for q = 0, 1, 2. Correspondingly, we define $i ∞ e ∞ % (3.52) L∞ 0 = max L0 , L0 .

The interior norms i Lpq in (3.51) are given by:

(3.53)

i

1+q− p2

Lpq = r0

∥ Dq+1 ϕ ∥p,i .

In order to state the norms for e Lpq , we first decompose ∇ϕ as follows:

(3.54) (3.55)

/o A = ∇ / Aϕ ϕN = ∇N ϕ.

58

3. MAIN THEOREM

That is, we have 1 ∇ / φ φ A 1 ϕN = ∇N ϕ = ∇N φ. φ / Aϕ = /o A = ∇

Next, we set e

(3.56)

e

e e

(3.57)

(3.59)

)=∥ r

Lpq (ϕN ) = ∥ r

Lpq+1 (D / 4 /o

)=∥ r

Lpq+1 (D / 3 /o ) = ∥ r

e

Lpq+1 (D4 ϕN ) = ∥ r

e

Lpq+1 (D3 ϕN ) = ∥ e

(3.58)

Lpq (o/

e

r

Lpq+1 (D / S /o ) = ∥ r

Lpq+1 (D / S ϕN ) = ∥ r

e

Lpq+2 (D /SD / 4 /o ) = ∥ r

e

Lpq+2 (D /SD / 3 /o ) = ∥ r

e

Lpq+2 (DS D4 ϕN ) = ∥ r

e

Lpq+2 (DS D3 ϕN ) = ∥

r

$

$

$ $

$ $

$

$

$ $

$ $

3 − p3 +q 2 3 − p3 +q 2 5 − p3 +q 2 5 − p3 +q 2 5 − p3 +q 2 3 − p3 +q 2 3 − p3 +q 2 3 − p3 +q 2 5 − p3 +q 2 5 − p3 +q 2 5 − p3 +q 2 3 − p3 +q 2

%

∇ / q /o ∥p,e

%

∇ / qD / 4 /o ∥p,e

% % % % % % % % % %

∇ / q ϕN ∥p,e

∇ / qD / 3 /o ∥p,e ∇ / q D4 ϕN ∥p,e 3

τ−2

− p1

∇ / q D3 ϕN ∥p,e

∇ / q /o ∥p,e ∇ / q ϕN ∥p,e ∇ / qD / 4 /o ∥p,e ∇ / qD / 3 /o ∥p,e ∇ / q D4 ϕN ∥p,e 3

τ−2

− p1

∇ / q D3 ϕN ∥p,e .

Then, we set Lp0 [o/ ] = e Lp0 (o/ ) e p L1 [o/ ] = e Lp1 (o/ ) + e Lp1 (D / 3 /o ) + e Lp1 (D / 4 /o ) e p e p e p e p L2 [o/ ] = L2 (o/ ) + L2 (D / 3 /o ) + L2 (D / 4 /o ) e

(3.60)

and correspondingly for ϕN . Then, we define for q = 0, 1, 2: (3.61)

e

Lpq = e Lpq [o/ ] + e Lpq [ϕN ].

These norms we use for p = 2 and we write e

Lq = e L2q .

3.1. NORMS

59

Finally, we define i i i e e

(3.62)

e

L[0] = i L0

L[1] = i L[0] + i L1 L[2] = i L[1] + i L2 L[0] = e L0

L[1] = e L[0] + e L1 L[2] = e L[1] + e L2

and (3.63)

L[q] = i L[q] + e L[q] .

3.1.4. Norms for the Hessian of the Optical Function u. Here, we state the L2 -norms for the Hessian D2 u of the optical function u. Let i O , respectively e O , denote the interior, respectively exterior, norms. We q q set $ % (3.64) Oq = max i Oq , e Oq .

for q = 0, 1, 2. Correspondingly, we define $ % (3.65) O0∞ = max i O0∞ , e O0∞ .

In the interior region, we express the components of D2 u with respect to the standard null frame in terms of χ′ = aχ, ζ ′ = ζ, ω ′ = a−1 ω. Let us define in the interior: " i Oq (trχ′ − trχ′ ) = r0q ∥∇ / i D3j D4k (trχ′ − trχ′ ) ∥2,i i+j+k=q

i

Oq (χ ˆ′ ) = r0q

i

(3.66)

i

Oq (ζ ′ ) = r0q

Oq (ω ′ ) = r0q

"

i+j+k=q

"

i+j+k=q

"

i+j+k=q

∥∇ / iD / j3 D / k4 χ ˆ′ ∥2,i ∥∇ / i D3j D4k ζ ′ ∥2,i ∥∇ / i D3j D4k ω ′ ∥2,i .

Then, we set (3.67)

i

Oq = i Oq (trχ′ − trχ′ ) + i Oq (χ ˆ′ ) + i Oq (ζ ′ ) + i Oq (ω ′ ).

In the exterior region, we work with the l-pair and the null frame related to it. The components of the Hessian of u in the exterior region with respect to the l-null frame, that is trχ, χ, ˆ ζ, ω, behave differently.

60

3. MAIN THEOREM

Let us also introduce 1 trχ4 = D4 trχ + trχtrχ 2 1 trχ3 = D3 trχ + trχtrχ 2 / 4χ ˆ + trχχ ˆ χ ˆ4 = D / 3χ ˆ + trχχ ˆ χ ˆ3 = D 1 ζ4 = D / 4 ζ + trχζ 2 1 ζ3 = D / 3 ζ + trχζ 2 ω4 = D4 ω ω3 = D3 ω.

(3.68) Then we define e

Oq (trχ − trχ) = ∥ rq ∇ / q (trχ − trχ) ∥2,e e Oq (χ) ˆ = ∥ rq ∇ / qχ ˆ ∥2,e q e q Oq (ζ) = ∥ r ∇ / ζ ∥2,e e

e e

1

1

Oq (ω) = ∥ r− 2 +q τ−2 ∇ / q ω ∥2,e 2 3 e Oq (χ) = max e Oq (trχ − trχ), e Oq (χ) ˆ

(3.69)

Oq+1 (χ ˆ4 ) = ∥ r1+q ∇ / qχ ˆ4 ∥2,e Oq+1 (χ ˆ3 ) = ∥ r1+q ∇ / qχ ˆ3 ∥2,e

Oq+1 (trχ4 ) = ∥ r1+q ∇ / q trχ4 ∥2,e e Oq+1 (trχ3 ) = ∥ rq τ− ∇ / q trχ3 ∥2,e e

e

e e

(3.70)

e

Oq+1 (ζ4 ) = ∥ r1+q ∇ / q ζ4 ∥2,e Oq+1 (ζ3 ) = ∥ r1+q ∇ / q ζ3 ∥2,e 1

1

Oq+1 (ω4 ) = ∥ r 2 +q τ−2 ∇ / q ω4 ∥2,e 1

3

Oq+1 (ω3 ) = ∥ r− 2 +q τ−2 ∇ / q ω3 ∥2,e .

Next, we set O0 [trχ] = e O0 (trχ) e O1 [trχ] = e O1 (trχ) + e O1 (trχ3 ) + e O1 (trχ4 ) e O2 [trχ] = e O2 (trχ) + e O2 (trχ3 ) + e O2 (trχ4 ). e

(3.71)

Correspondingly, we proceed for χ, ˆ ζ, ω. Now, for q = 0, 1, 2 one sets (3.72)

e

Oq = e Oq2 = e Oq [trχ] + e Oq [χ] ˆ + e Oq [ζ] + e Oq [ω].

3.2. STATEMENT OF THE MAIN THEOREM

61

Then we set 2 | + sup | a − 1 | r I I 1 2 e ∞ e ∞ O[0] = O0 + sup r 2 | trχ − | + sup | a − 1 | . r r r r≥ 0 r≥ 0 i

(3.73)

1

∞ O[0] = i O0∞ + r02 sup | trχ −

2

2

Finally, we define i i i e e

(3.74)

e

and

∞ O[0] = i O0 + i O[0]

O[1] = i O[0] + i O1 O[2] = i O[1] + i O2

∞ O[0] = e O0 + e O[0]

O[1] = e O[0] + e O1 O[2] = e O[1] + e O2

O[q] = i O[q] + e O[q] .

(3.75)

3.2. Statement of the Main Theorem We recall (1.53) and the global smallness assumption B (1.55) from the ‘Introduction’. The small positive ϵ on the right and side of (1.55) has to be chosen suitably small later in the proof, depending on other quantities, such that the inequalities in the main theorem 3 hold. Theorem 3. (Main Theorem) Any asymptotically flat, maximal initial data set (AFB) of the form given in definition 12 in the ‘Introduction’ satisfying the global smallness assumption B stated in the ‘Introduction’, inequality (1.55), leads to a unique, globally hyperbolic, smooth and geodesically complete solution of the EV equations, foliated by the level sets of a maximal time function t, defined for all t ≥ −1. Moreover, there exists a global, smooth optical function u, that is a solution of the Eikonal equation defined everywhere in the exterior region r ≥ r20 , with r0 (t) denoting the radius of the 2-surface St,0 of intersection between the hypersurfaces Ht and a fixed null cone C0 with vertex at a point on H−1 . With respect to this foliation the following holds: (3.76)

e

(3.77)

e

R[1] , e K[2] , e O[2] , e L[2] ≤ ϵ0 K0∞ , e O0∞ , e L∞ 0 ≤ ϵ0 .

Moreover, in the complement of the exterior region, the following holds: (3.78) (3.79)

i

R[1] ,i K[2] ,i L[2] ≤ ϵ0 i

K0∞ ,i L∞ 0 ≤ ϵ0 .

The strict inequalities hold for t = 0 with ϵ on the right-hand sides.

62

3. MAIN THEOREM

The proof of this main theorem was sketched in the ‘Introduction’. In the next subsection, we are going to state the main steps of the proof and to explain the bootstrap argument in details. The different steps are carried out in separate chapters. We recall that in the global smallness assumption B (1.55) (see ‘Introduction’) the initial data has to be smaller than a sufficiently small positive ϵ. Later in the proof, this ϵ, has to be taken suitably small, depending on other quantities. In the bootstrap assumptions BA0-BA2 ((3.80)–(3.82)) and in (3.83) in the next subsection 3.3.2, the considered quantities have to be smaller than a small positive ϵ0 . As explained in subsection 3.3.2 and during the actual proof, we estimate the main quantities at times t by their values at t = 0, which are controlled by inequalities with ϵ on their righthand sides. Then, choosing ϵ sufficiently small, the right-hand sides of these inequalities can be made strictly smaller than ϵ0 from the bootstrap assumptions. The bootstrap argument is explained in details in the remaining part of this chapter. Our main theorem 3 provides existence and uniqueness of solutions under the relaxed assumptions (AFB, see (1.55)) as well as it describes the asymptotic behaviour as precisely as it is possible under these relaxed assumptions. Compared to the result [19], main theorem 10.2.1, p. 298, by Christodoulou and Klainerman, (we give one version of their main theorem in the ‘Introduction’, see theorem 1, and definition 11 under their global smallness assumption CK (1.52)), we impose less on our initial data. That is, we assume one less decay in the power of r of the data at infinity and one less derivative to be controlled. 3.2.1. Most Important Achievements and Main Differences to Former Work. • We show existence and uniqueness of solutions with less assumptions on the initial data, that is, we assume one less decay in the power of r and one less derivative than in [19]. Moreover, we describe the asymptotic behaviour. (See previous paragraph.) • As a consequence of imposing fewer conditions on our data, the spacetime curvature is not in L∞ (M ). We only control one derivative of the curvature (Ricci) in L2 (H). By the trace lemma, the Gauss curvature K in the leaves of the u-foliation S is only in L4 . Contrary to that, in [19], the Ricci curvature is in L∞ (H), and in L∞ (S). The authors control two derivatives of the curvature (Ricci) in L2 (H). Thus, in our situation, this is a disadvantage and an advantage. First, as we do not have the curvature bounded in L∞ , certain steps of the proof become more subtle. On the other hand, we do not have to control the second derivatives of the curvature, which simplifies the proof.

3.3. CRUCIAL STEPS OF THE PROOF OF THE MAIN THEOREM

63

• A major simplification is the fact, that we do not use any rotational vectorfields in our proof. We gain control on the angular derivatives of the curvature directly from the Bianchi equations. Whereas in [19], a difficult construction of rotational vectorfields was necessary. • Another major difference to the situation studied in [19] by Christodoulou and Klainerman, and which arises from our relaxed assumptions, is the fact that we encounter borderline cases in view of decay in the power of r, indicating that the conditions in our main theorem on the decay at infinity on the initial data are sharp. Any further relaxation would make the corresponding integrals diverge and the argument would not close any more. • We also remark that in our situation, energy and linear momentum are well-defined and conserved, whereas the (ADM) angular momentum is not defined. This is different to the situation investigated in [19], where all these quantities are well-defined and conserved. 3.3. Crucial Steps of the Proof of the Main Theorem – Bootstrap Argument In the ‘Introduction’, we gave an outline of our proof. There we sketched that it consists of one large bootstrap argument, containing other arguments of the same type but at different levels. Now, we are going to state and to explain the crucial steps of the proof of the main theorem, that is, of the bootstrap argument. 3.3.1. Local Existence Theorem. Relying on the local existence theorem, [19], statement theorem 10.2.2: p. 299/300, proof 10.2.2: p. 304–310, we will show global existence of a unique, globally hyperbolic, smooth and geodesically complete solution of the Einstein-vacuum equations, coming from initial data stated in the ‘Introduction’, definition 12 (AFB) and inequality 1.55 (global smallness assumption B). The local existence theorem, is stated and proven in [19] for their problem, and it also holds in our case. It requires the second fundamental form k to be in L∞ , which is satisfied in our situation. The proof of the local existence theorem in [19] mainly uses the ideas developed in the proof of the well-known existence result of Choquet-Bruhat [8] and modifies them simply. In [19], the authors formulate the conditions in the local existence theorem, according to their situation, imposing Ric(¯ g0 ) ∈ Hs,δ (H, g¯0 ) and k ∈ Hs+1,δ (H, g¯0 ) with s = 2 and δ = 1. We impose these conditions with s = 1 and for k also δ = 0. The proof still holds in the same way, as it only requires k in L∞ , which in our situation is true. 3.3.2. Bootstrap Argument. Now, we define / S to be the set of all t ≥ 0 such that there exists a spacetime slab t′ ∈[0,t] Ht′ endowed with

64

3. MAIN THEOREM

a canonical optical function with respect to which the following bootstrap assumptions hold: • BA0: For all t′ ∈ [0, t], assume that (3.80)

(3.81) (3.82)

3 1 (1 + t′ ) ≤ r0 (t′ ) ≤ (1 + t′ ). 2 2 ′ • BA1: For all t ∈ [0, t], assume that e

O0∞ , K0∞ , L∞ 0 ≤ ϵ0 .

• BA2: For all t′ ∈ [0, t], assume that

R[1] , e O[2] , K[2] , L[2] ≤ ϵ0 .

Additional Assumption on the last slice Ht : 3 2 3 (3.83) sup r 2 | ∇ / log a ≤ ϵ0 . Su

First, we show that the set S is not empty. Of course, the bootstrap assumptions hold at t = 0. Afterwards, we define the supremum of S as t∗ . If t∗ = ∞, then the global existence is proved. We use a continuity argument to derive that indeed t∗ = ∞. This goes by contradiction, assuming that t∗ < ∞. Then, it is t∗ ∈ S, and the inequalities must be saturated at t∗ . It will be shown that this cannot happen. This yields that t∗ = ∞. Step 1: We show that the set S is not empty. That is, it contains at least t = 0. / In view of the local existence theorem we can construct the past slab t′ ∈[−1,0] Ht′ and the initial cone C0 with vertex at a point on H−1 . Then an optical function u on t = 0 will be constructed by solving the inverse lapse problem, starting on the 2-surface S0,0 = C0 ∩ H0 , according to the results of the chapter ‘The Last Slice’. Using the results of the chapter ‘The Last Slice’, we proceed analogously for a background radial function u′ on H0 . One then shows that and how the norms R[1] , K[2] , L[2] , e O[2] as well as e O0∞ , K0∞ , L∞ 0 can be made arbitrarily small, that is the assumptions BA1 and BA2 are fulfilled. The additional assumption (3.83) follows from the main theorem 14 of ‘The Last Slice’ (chapter 10). Step 2: Let t∗ = sup S. If t∗ = ∞, then the global existence is proved. Now, we assume t∗ < ∞. Then, it is t∗ ∈ S. From/the following inequalities for the optical function u of the spacetime slab t∈[0,t∗ ] Ht in the interior region: $ % i O[2] ≤ c R[1] + K[2] + L[2] $ % i ∞ O0 ≤ c R[1] + K0∞ + L∞ (3.84) 0 , one derives that (3.85)

i

O[2] , i O0∞ ≤ cϵ0 .

Step 3: This is the core part of the proof. It splits into three sections, namely, obtaining estimates for

3.3. CRUCIAL STEPS OF THE PROOF OF THE MAIN THEOREM

65

a) R[1] , b) K[2] , L[2] , respectively, K0∞ , L∞ 0 , c) e O[2] , respectively, e O0∞ .

Using step 2 and the bootstrap assumptions BA0, BA1, BA2, we show e that the size of the norms e O0∞ , K0∞ , L∞ 0 and R[1] , K[2] , L[2] , O[2] cannot exceed a constant multiple of the respective size of the data at t = 0. Therefore, one can choose ϵ and ϵ0 sufficiently small such that 1 O0∞ , K0∞ , L∞ 0 ≤ ϵ0 2 1 e O[2] , R[1] , K[2] , L[2] ≤ ϵ0 . 2 e

(3.86)

This is achieved as follows: a) In this part, we use the bootstrap assumptions BA0, BA1, BA2 and inequality (3.85) to check all the assumptions of the comparison theorem 4 in the chapter 4 ‘Comparison’ as well as of the main theorem 5 of the chapter 5 ‘Error Estimates’. More precisely, the Assumptions 0, 1, 2 of the comparison theorem 4 in ‘Comparison’ 4 follow directly from the bootstrap assumptions BA0, BA1, BA2 and inequality (3.85). In the main theorem 5 in the ‘Error Estimates’ 5, the Interior and Exterior Assumptions on the deformation tensors for the ¯ are checked in a straightforward manner, writing vectorfields T , S and K out the calculations for the deformation tensors relative to the null frame. These components can also be found in [19], chapter 7, pages 172–175. According to the main theorem 5 of the chapter 5 ‘Error Estimates’, we have (3.87)

Q1∗ + Q0 ∗ ≤ c(Q1 (0) + Q0 (0)).

Thus by the comparison theorem 4 in the chapter 4 ‘Comparison’, we conclude that, for all t ∈ [0, t∗ ], (3.88)

R[1] (t) ≤ cR[1] (0).

We recall the remark from the ‘Introduction’ that in our work, we do not need rotational vectorfields at all to derive the estimates for R[1] (t). Contrary to [19], where rotational vectorfields were used in a crucial way, ¯ in conjunction with the we only work with the vectorfields T , S and K Bianchi equations to deduce the required estimates for R[1] (t). That is, we define the quantities Q0 and Q1 as given also in the ‘Introduction’ and which ¯ Then, are estimated in the ‘Error Estimates’ 5, with the help of T , S and K. we estimate R[1] (t) in terms of Q0 and Q1 by the comparison argument. From (3.88) and from the fact that we can bound R[1] (0) by c · ϵ, we deduce (3.89)

R[1] (t) ≤ cϵ.

66

3. MAIN THEOREM

Choosing ϵ sufficiently small, yields 1 R[1] (t) ≤ ϵ0 . 2 b) We show that the bootstrap assumptions BA0, BA1, BA2 and the inequality (3.85) imply the following:

(3.90)

K[2] , L[2] ≤ cR[1] .

Then, by the Sobolev inequalities, we deduce that K0∞ , L∞ 0 ≤ cR[1] ,

and therefore, choosing ϵ sufficiently small, we conclude 1 K0∞ , L∞ (3.91) 0 ≤ ϵ0 2 1 (3.92) K[2] , L[2] ≤ ϵ0 . 2 This is done in details for K in the chapter 6 ‘Second Fundamental Form k: Estimates for the Components of k’. The results for L then follow immediately. c) Here, we show that the bootstrap assumptions BA0, BA1, BA2 imply the following: O0∞ ≤ cϵ e O[2] ≤ cϵ. e

Therefore, if ϵ is sufficiently small, this yields 1 e ∞ (3.93) O0 ≤ ϵ0 2 1 e (3.94) O[2] ≤ ϵ0 . 2 This is carried out in details in the chapter 7 ‘Second Fundamental Form χ: Estimating χ and ζ’ and 10 ‘The Last Slice’. Step 4: This step is to be considered together with the previous one. However, as it is a crucial point within the whole procedure of the bootstrap argument, we formulate it separately. We show that we can extend our spacetime beyond a time t∗ . In particular, we use the result of the previous step together with the local existence theorem, with initial data at t∗ , to extend the spacetime up / from t∗ to t∗ + δ. Also, the optical function u′ of the spacetime slab t∈[0,t∗ ] Ht is extended by continuing the null geodesic generators of the hypersurfaces Cu′ into the future up to t∗ + δ. We choose δ to be sufficiently small, such ′∞ e ′∞ that the size of the norms R′ [1] , K′ [2] , L′ [2] , e O′ [2] and K′ ∞ 0 , L 0 , O 0 re2 ′3 3 mains strictly smaller than ϵ0 . Moreover, supS ′ ′ r 2 | ∇ / ′ log a′ in (3.83) u is strictly smaller than ϵ0 . Now, we start with Ht∗ +δ as last slice. The cut St∗ +δ,0 = Ht∗ +δ ∩ C0 of this last slice with C0 is the initial 2-surface, from which we start, to solve

3.3. CRUCIAL STEPS OF THE PROOF OF THE MAIN THEOREM

67

the propagation equation and construct a new optical function u on Ht∗ +δ . This is explained and done in details in the chapter 10 ‘The Last Slice’. We recall from the ‘Introduction’ that the function u′ gives the background foliation here. As the level surfaces of u in Ht∗ +δ do not exist yet, but have to be constructed, we use the bootstrap assumptions on the quantities of the background foliation and the comparison between the two foliations induced by u′ and u to control the curvature and geometric quantities of the foliation by u. Thus, using a bootstrap argument, we construct the new optical function u on Ht∗ +δ by solving an equation of motion of surfaces on Ht∗ +δ starting from St∗ +δ,0 . This is then extended to the past. We refer to the chapters 10 ‘The Last Slice’ and 7 ‘Second Fundamental Form χ: Estimating χ and ζ’. Using the continuity properties of the propagation equations, we deduce e ∞ that the new norms R[1] , K[2] , L[2] , e O[2] and K0∞ , L∞ 0 , O0 can be made arbitrarily close to the previous ones. This is done and explained in the chapter 10 ‘The Last Slice’. One therefore checks that the bootstrap assumptions BA1 and BA2 as well as BA0 and inequality (3.83) still hold. Thus, we obtain that t∗ + δ ∈ S, which contradicts the assumption that t∗ < ∞. Step 5: To complete the proof of the main / theorem, one shows that (t) the optical function u defined on the slab t′ ∈[0,t] Ht′ , starting from the last slice Ht in the exterior approaches a global exterior optical function u as t → ∞. Let (t) Cy denote the y-level set of (t) u./The 0-level set of (t) u is the part of the fixed cone C0 contained in the slab t′ ∈[0,t] Ht′ . Next, for t2 > t1 > 0, we have the functions (t2 ) u and (t1 ) u on Ht1 , and we remark that their 0-level sets coincide. This enables us to apply the comparison results of the chapter ‘The Last Slice’, from which difference of the functions - /it follows that.the 5 (t ) 1 in any region of the form Cy Ht1 with y1 , y2 being fixed y∈[y1 ,y2 ] but arbitrary, tends / to zero as t1 → ∞. Thus, we then obtain that in any spacetime region y∈[y1 ,y2 ] (t1 ) Cy the difference of (t2 ) u and (t1 ) u tends to zero as t1 → ∞. Therefore, we have proven that (t) u → u.

CHAPTER 4

Comparison Goal of this chapter is to state and to prove the comparison argument. It allows us to estimate our curvature components by quantities Q1 and Q0 that shall be introduced and shown to be controlled later in this section (see also ‘Introduction’). These quantities are integrals over Ht involving the Bel-Robinson tensor of the Weyl tensor and of the corresponding Lie derivatives of the Weyl tensor. Contrary to the work of D. Christodoulou and S. Klainerman in [19], where the authors introduced rotational vectorfields and where the construction of these as well as their application in the proof were fundamentally important, we do not work with rotational vectorfields at all. We only use the vectorfields T and S together with the Bianchi equations to derive estimates on the curvature components and their derivatives. That is, as given in the formulas (4.160) and (4.161) for the energies Q0 and Q1 , we take the Lie derivatives with respect to the time T and scaling S vectorfields, and we ¯ contract with T and the inverted time vectorfield K. In the following subsection, we are going to introduce the Weyl tensor W , given by the curvature. We will work with this tensor in connection with the Bianchi identities to control the curvature. In this part, we shall present how this is done. To begin with, we are going to introduce quantities that will be needed in the further procedure of this chapter. 4.1. Curvature R, Weyl Tensor W and Bel-Robinson Tensor Q Let us now, focus on the definitions and the setting up for the core part of this section. We consider the spacetime (M, g) described in the introduction. In the following, we recollect the most important definitions given in the ‘General Introduction’. First, we denote by (X) π the deformation tensor of X, which is given as follows: (X)

(4.1) (4.2)

παβ = (LX g)αβ

(X) αβ

−

π

= (LX g −1 )αβ .

The Riemann curvature tensor R of the spacetime (M, g) satisfies the following Bianchi identities: (4.3)

1 D[ϵ Rαβ]γδ = (Dϵ Rαβγδ + Dα Rβϵγδ + Dβ Rϵαγδ ) = 0. 3 69

70

4. COMPARISON

One computes the traceless part of the curvature tensor to be 1 Cαβγδ = Rαβγδ − (gαγ Rβδ + gβδ Rαγ − gβγ Rαδ − gαδ Rβγ ) 2 1 + (gαγ gβδ − gαδ gβγ )R, (4.4) 6 with Rαβ denoting the Ricci tensor and R the scalar curvature of the spacetime. Actually, C is called the conformal curvature tensor of the spacetime. This is a particular example of a Weyl tensor. We recall from the definition in the ‘General Introduction’ that the Weyl tensors W are 4-tensors that verify all the symmetry properties of the Riemann curvature tensor and, in addition, are traceless. That is, they fulfill the identities (4.5) (4.6)

Wαβγδ = −Wβαγδ = −Wαβδγ

Wαβγδ + Wαγδβ + Wαδβγ = 0

(4.7)

Wαβγδ = Wγδαβ

together with the additional condition on the trace (4.8)

Wβδ = W α βαδ = 0.

We remind ourselves of the fact that (4.7) is a consequence of the first two properties. Equation (4.6) is called the cylcic condition and is often written in the form Wα[βγδ] = 0, where [βγδ] stands for the cyclic permutation. A Weyl field is subject to the Bianchi equations, if it is: (4.9)

D[ϵ Wαβ]γδ = 0.

Given a Weyl field W , we can define the right W ∗ and left duals to be: 1 ∗ (4.10) Wαβγδ = ϵαβµν W µν γδ 2 1 ∗ (4.11) = Wαβ µν ϵµνγδ , Wαβγδ 2

∗W

Hodge

where ϵαβγδ are the components of the volume element of M . One can think of (4.10) as freezing the second pair of indices and considering W as a 2form relative to the first pair, correspondingly of (4.11) as freezing the first pair of indices and considering W as a 2-form in the second pair. Note that these definitions of left and right Hodge duals are equivalent. It can easily be checked that ∗ W = W ∗ is also a Weyl tensorfield. Further, it is ∗ ∗

( W ) = −W.

Given a Weyl field W and a vectorfield X, the Lie derivative of W with respect to X is not, in general, a Weyl field, for, it has trace. In fact, it is: (4.12)

g αγ (LX Wαβγδ ) = π αγ Wαβγδ .

4.1. CURVATURE R, WEYL TENSOR W AND BEL-ROBINSON TENSOR Q

71

In view of this, we defined the following modified Lie derivative: 1 3 LˆX W := LX W − (X) [W ] + tr(X) πW 2 8

(4.13) with (4.14)

(X)

[W ]αβγδ := π µ α Wµβγδ + π µ β Wαµγδ + π µ γ Wαβµδ + π µ δ Wαβγµ .

The deformation tensor of X, namely (X) π, has already been given. We also recall the electric-magnetic decomposition of W to be the following contractions with X, where X is an arbitrary given vectorfield. The decomposition consists of the two 2-tensors: (4.15)

iiX (W )αβ = Wµανβ X µ X ν

(4.16)

iiX (∗ W )αβ = ∗ Wµανβ X µ X ν .

These tensors are symmetric, traceless and orthogonal to X. It can be shown that they completely determine W , if X is not null. (See also [18].) Now, we recall the Bel-Robinson tensor and discuss its properties. So, let Q(W ) be the Bel-Robinson tensor associated to W . We write: (4.17)

Qαβγδ = (Wαργσ Wβ ρδ σ + ∗ Wαργσ ∗ Wβ ρδ σ ).

Let us state the most important facts about this tensor in the first lemma of the next subsection. They can be verified by computations using appropriate identities for W and its Hodge duals. The main steps of the proof can be found in [19], lemma 7.1.1, p. 136/137. The Bianchi equations (4.9) play an important role in the whole work. Nevertheless, for certain situations, it will be better to work with the following more general inhomogeneous equations. For, (4.9) does not commute with Lie derivatives, that is, error terms occur. This has already been done in [19] by Christodoulou and Klainerman. So, here, we follow their notation. We consider (4.18)

Dα Wαβγδ = Jβγδ

(4.19)

∗ Dα∗ Wαβγδ = Jβγδ

∗ = 12 Jβµν ϵµν γδ and J, J ∗ fulfilling the symmetries required by the with Jβγδ equations. One can then write

(4.20)

1 D[σ Wγδ]αβ = J˜σγδαβ = ϵµσγδ J ∗µ αβ = ϵµσγδ J µ st ϵst αβ . 2

Note that (4.18) and (4.19) are equivalent, as ∗ W = W ∗ . And in addition to (4.20), we also derive (4.21)

∗ D[σ ∗ Wγδ]αβ = J˜σγδαβ = −ϵµσγδ J µ αβ .

72

4. COMPARISON

4.2. Bianchi Equations and Identities Involving W and Q As the tensors W and Q play an important role throughout this monograph, we shall discuss here their properties, being fundamental in our proofs. We follow the notation of [19], where the authors discussed and proved the most important facts in their chapter 7. First, we state the lemma announced above. Lemma 1. Let W be an arbitrary Weyl tensor. Then the following statements are valid: 1. Q(W ) is symmetric in all pairs of indices. 2. Q(W ) is traceless in all pairs of indices. 3. Positivity condition: (4.22)

Q(X1 , X2 , X3 , X4 ) ≥ 0 for any quadruplet X1 , X2 , X3 , X4 of vectors at a point, all of which are future-directed timelike, equality holding if and only if W vanishes at this point.

Moreover, as W satisfies the Bianchi equations, Q is divergence-free: (4.23)

Dα Qαβγδ = 0.

Coming back to the modified Lie derivative in (4.13) and (4.14), we want to write this in a way, where we split π into its trace trπ and traceless part π ˆ . We have from (4.13) and (4.14): 3 LˆX Wαβγδ = LX Wαβγδ + tr(X) πWαβγδ 8 1 µ − (π α Wµβγδ + π µ β Wαµγδ + π µ γ Wαβµδ + π µ δ Wαβγµ ) . 2 Now, we decompose π according to 1 ˆαβ + trπ gαβ , παβ = π 4 which yields for the modified Lie derivative: (4.24)

1 LˆX Wαβγδ = LX Wαβγδ − tr(X) πWαβγδ 8 1 µ − (ˆ (4.25) ˆ µ β Wαµγδ + π ˆ µ γ Wαβµδ + π ˆ µ δ Wαβγµ ) π α Wµβγδ + π 2 As a tool for further calculations, let us recall, here, some identities, that can be verified by direct computation. Let X be a vectorfield and V a

4.2. BIANCHI EQUATIONS AND IDENTITIES INVOLVING W AND Q

73

1-form on M . Then the following hold: (4.26) Dα Dβ Xγ − Dβ Dα Xγ = Rαβγµ X µ (4.27) Dα Dβ Vγ − Dβ Dα Vγ = Rαβγµ V µ (4.28) (4.29)

Dβ Dα Xγ = Rαβγµ X µ + (X) Γαβγ LX Vα = X λ Dλ Vα + (Dα X λ )Vλ

Dσ (LX Vα ) = (Dσ X λ )Dλ Vα + X λ Dσ Dλ Vα (4.30)

+ (Dσ Dα X λ )Vλ + (Dα X λ )Dσ Vλ

(4.31)

LX (Dσ Vα ) = X λ Dλ Dσ Vα + (Dα X λ )Dσ Vλ + (Dσ X λ )Dλ Vα

where in the third equation ' ( 1 (X) (X) Dβ παγ + Dα (X) πβγ − Dγ(X) παβ . Γαβγ = (4.32) 2

In the sequel, we are going to formulate some well-known facts about the Weyl fields, that shall be used in this whole work. For some of the identities, whenever we consider it as necessary, we will give the proof in detail, whereas for others we shall sketch it or refer to corresponding literature. Most of the proofs go by straightforward calculation using the properties of the Weyl tensor and the volume element of the spacetime as well as of their derivatives. Many of the next statements are derived in [19], chapter 7, and we therefore refer to this monograph for further information. For a Weyl field W in view of the modified Lie derivative, we state the following lemma. Lemma 2. If W is a Weyl field and X an arbitrary vectorfield, then the next statements hold: 1. LˆX W is again a Weyl field. 2. Lˆ and ∗ commute: LˆX ∗ W = ∗ LˆX W.

Proof (Sketch). The proof uses identities with respect to the Weyl tensor as mentioned above. In particular, the first statement of the lemma follows by showing that (X) [W ] obeys the equations (4.5), (4.6) and therefore also (4.7), and by taking into account that the trace-free condition (4.8) is naturally true. To prove the second part of the lemma, one should recall the Lie derivative of the volume element, namely: 1 (LX ϵ)αβγδ = (tr(X) π)ϵαβγδ . 2 Then, the Hodge dual of the Lie derivative of W can be calculated and the second statement shown to hold. The computations of the main steps are carried out in [19], lemma 7.1.2, p. 139–141.

74

4. COMPARISON

Let us formulate next, how the divergence of Q reads in terms of W and J. The equation is given and proven as a proposition in [19] (proposition 7.1.1, p. 137–138). One can derive it by straightforward computation. Thus, if W is a Weyl field satisfying equations (4.18) and (4.19), then it is: (4.33)

∗ ∗ + ∗ Wβ µγ ν Jµδν . Dα Qαβγδ = Wβ µδ ν Jµγν + Wβ µγ ν Jµδν + ∗ Wβ µδ ν Jµγν

Using this equation, we can now write the integral of Q over the hypersurface Ht in terms / of the integral over H0 and the corresponding integral on a spacetime slab t′ ∈[0,t] Ht′ . To do so, we consider a spacetime (M, g) foliated by a time function t, where T denotes the unit normal to the foliation. Let X, Y , Z be three arbitrary vectorfields. We define (4.34)

Pα = Q(W )αβγδ X β Y γ Z δ .

Then, it is div P = X β Y γ Z δ (div Q)βγδ 1 (4.35) + Qαβγδ ((X) π αβ Y γ Z δ + (Y ) π αβ Z γ X δ + (Z) π αβ X γ Y δ ). 2 Therefore, one can write down the following proposition. Proposition 4. Let (M, g) be a spacetime foliated by a time function t. Let W be a solution of the inhomogeneous Bianchi equations (4.18) and (4.19), and let Q(W ) be its Bel-Robinson tensor. Further, Φ denotes the lapse function and g the induced metric of the time foliation. Then, under suitable assumptions at spacelike infinity on each slice Ht′ , for any vectorfields X, Y , Z, it is: # # Q(W )(X, Y, Z, T )dµg = Q(W )(X, Y, Z, T )dµg Ht H0 # 0# +

(4.36)

t

0

(div Q)βγδ X β Y γ Z δ

Ht′

$ 1 + Qαβγδ (X) π αβ Y γ Z δ + (Y ) π αβ Z γ X δ 2 1 % + (Z) π αβ X γ Y δ Φ dµg dt′ .

We are now going to prove two results about taking the derivative of the Lie, respectively modified Lie derivative of the Weyl tensorfield. The first one, we state in a lemma: Lemma 3. Let V be an arbitrary k-covariant tensorfield, X a vectorfield and π its deformation tensor. Also, let Γ be given by (4.32).

4.2. BIANCHI EQUATIONS AND IDENTITIES INVOLVING W AND Q

75

Then the commutation formula with respect to D and LX reads as follows: (4.37)

Dσ (LX Vα1 ···αk ) − LX (Dσ Vα1 ···αk ) =

k " i=1

(X)

Γαi σλ Vαi ···λ···αk .

Proof. The lemma follows directly from the equations ((4.26)–(4.32)). In what follows, we are going to derive an expression for Dα (LˆX Wαβγδ ). We first consider Dα (LX Wαβγδ ) − LX (Dα Wαβγδ ) < => ? =Jβγδ

=g

=g which yields

(4.38)

ασ

Dσ LX Wαβγδ

$ ασ

−LX (g ασ Dσ LX Wαβγδ ) < => ?

=+(X) π ασ Dσ Wαβγδ −g ασ LX Dσ Wαβγδ

% Dσ LX Wαβγδ − LX Dσ Wαβγδ + (X) π ασ Dσ Wαβγδ ,

Dα (LX Wαβγδ ) = LX Jβγδ + (X) π ασ Dσ Wαβγδ $ % + g ασ Dσ LX Wαβγδ − LX Dσ Wαβγδ .

The last term in (4.38) can be written as follows: $ % $ g ασ Dσ LX Wαβγδ − LX Dσ Wαβγδ = g ασ (X) Γασλ W λ βγδ + (X) Γβσλ Wαλ γδ % + (X) Γγσλ Wαβ λ δ + (X) Γδσλ Wαβγ λ (4.39)

With g ασ(X) Γασλ = then reads:

(X) Γ λ

and applying the lemma 3, equation (4.38)

Dα (LX Wαβγδ ) = LX Jβγδ + (X) π ασ Dσ Wαβγδ + (X) Γλ W λ βγδ

(4.40)

+ (X) Γβαλ W αλ γδ + (X) Γγαλ W α β λ δ + (X) Γδαλ W α βγ λ .

ˆ one Now, just using the definition for the modified Lie derivative L, writes 1 Dα (LˆX Wαβγδ ) = Dα (LX Wαβγδ ) − Dα(X) [W ]αβγδ 2 3 α (X) 3 + (D (tr π))Wαβγδ + tr(X) π Dα Wαβγδ 8 8 < => ? =Jβγδ

(4.41)

1 3 = Dα (LX Wαβγδ ) − Dα(X) [W ]αβγδ + tr(X) πJβγδ 2 8 3 α (X) + (D tr π)Wαβγδ . 8

76

4. COMPARISON

Next, in view of the second term on the right-hand side of (4.41), we compute: $ Dα(X) [W ]αβγδ = g ασ (Dσ π µ α )Wµβγδ + (Dσ π µ β )Wαµγδ % + (Dσ π µ γ )Wαβµδ + (Dσ π µ δ )Wαβγµ $ + g ασ π µ α Dσ Wµβγδ + π µ β Dσ Wαµγδ % + π µ γ Dσ Wαβµδ + π µ δ Dσ Wαβγµ (4.42) = (Dα πµα )W µ βγδ + (Dα πµβ )W αµ γδ

+ (Dα πµγ )W αβ µδ + (Dα πµδ )W α βγ µ

(4.43)

+ π µα Dα Wµβγδ + π µ β Jµγδ + π µ γ Jβµδ + π µ δ Jβγµ .

Then, with (4.41), (4.40) and (4.43), we derive the following expression for Dα (LˆX Wαβγδ ): % 1$ µ π β Jµγδ + π µ γ Jβµδ + π µ δ Jβγµ 2 3 1 + trπJβγδ + π µα Dα Wµβγδ 8 2 ( ' 1 3 (X) α + Γµ − D πµα W µ βγδ + (Dα trπ)Wµβγδ 2 8 ( ' 1 + (X) Γβαµ − Dα πµβ W αµ γδ 2 ( ' 1 (X) + Γγαµ − Dα πµγ W αβ µ δ 2 ( ' 1 (X) + Γδαµ − Dα πµδ W α βγ µ . 2

Dα (LˆX Wαβγδ ) = LX Jβγδ −

(4.44)

It is by the definition of (X) Γ: ' ( % 1 1$ (X) Γβαµ − Dα πµβ W αµ γδ = Dβ παµ − Dµ παβ . 2 2

The following term only depends on the traceless part of π, that is, the expressions involving the trace of π cancel. $ $ % % Dβ παµ − Dµ παβ W αµ γδ + Dγ παµ − Dµ παγ W αγ µ δ $ % + Dδ παµ − Dµ παδ W αβγ µ Also, we calculate ' ( 1 α 3 1 (X) Γµ − D πµα W µ βγδ + (Dµ trπ)Wµβγδ = (Dα π ˆµα )W µ βγδ , 2 8 2

4.3. NULL DECOMPOSITION

77

as well as we compute % % 1$ 1$ µ − π µ β Jµγδ + π µ γ Jβµδ + π µ δ Jβγµ = − π ˆ µ γ Jβµδ + π ˆ µ δ Jβγµ ˆ β Jµγδ + π 2 2 3 − trπJβγδ 8 1 1 µσ 1 Dσ Wµβγδ = π ˆ Dσ Wµβγδ + trπ Jβγδ . 2 2 8 Thus, we have proven the following proposition: Proposition 5. Let W be a Weyl field fulfilling Dα Wαβγδ = Jβγδ . And let X be any arbitrary vectorfield with deformation tensor π. Then the following holds:

with

1 µσ 1 ˆαµ W µ βγδ Dα (LˆX Wαβγδ ) = LˆX Jβγδ + π ˆ Dσ Wµβγδ + Dα π 2 2 1$ + (Dβ π ˆαµ − Dµ π ˆαβ )W αµ γδ 2 + (Dγ π ˆαµ − Dµ π ˆαγ )W αβ µ δ % + (Dδ π ˆαµ − Dµ π ˆαδ )W α βγ µ

1 1 LˆX Jβγδ := LX Jβγδ − (ˆ πβ µ Jµγδ + π ˆγ µ Jβµδ + π ˆδ µ Jβγµ ) + trπJβγδ . 2 8 4.3. Null Decomposition 4.3.1. Null Frame. We assume the spacetime (M, g) to be foliated by t and u as indicated in the introduction. We consider the surfaces St,u . Definition 15. A null pair compatible with t and u consists of two future-directed null vectors e4 and e3 , defined in some domain of the spacetime, if they are at any given point orthogonal to St,u passing through that point, and , + (4.45) e4 , e3 = −2.

A null pair together with an orthonormal frame e1 , e2 on St,u forms a null frame. In what follows, if nothing else specified, we shall use a null pair consisting of two future-directed null vectors e4 and e3 orthogonal to St,u with e4 tangent to Cu . Now, let e0 be the future-directed unit normal to the level sets of t, and let N be the outward unit normal to St,u with respect to the corresponding level set of t.

78

4. COMPARISON

One easily observes that the vectors e4 = e+ = e0 + N and e3 = e− = e0 − N form a null pair. As, it is directly verified that +

, e0 + N, e0 − N = −2.

It is called the standard null pair. Another null pair is the following: Let u be an optical function whose level sets are outgoing null hypersurfaces. Then one can define the null pair ∂u −1 (e + N ) and l = a(e − N ), which is called consisting of lµ = −g µν ∂x ν = a 0 0 the l-null pair of t and u. Next, let us define the projection from the tangent space of M to the tangent space of St,u . Definition 16. If we are given a null pair e+ , e− , then we define the tensor of projection from the tangent space of M to the tangent space of St,u to be Π(t, u) with Πµν = g µν +

% 1$ ν µ e− e+ + eν+ eµ− . 2

Remark: Note that Π satisfies Πki Πjk = Πji . Moreover, the induced metric γ on St,u is the restriction of Πµα Πνβ gµν to the space of vectors tangent to St,u . The projection operator is extended to arbitrary tensors Tαi ···αm on M by forming the contractions T . Πβα11 · · · Πβαm m βi ···βm The null decomposition of a tensor relative to a null frame e4 , e3 , e2 , e1 is obtained by taking contractions with the vectorfields e4 , e3 and projections to St,u . We have the following definition. Definition 17. Let T be a covariant tensor at a point of the spacetime. Then we define a null component of T to be any tensor tangent to the surface St,u at that point, obtained from T by contracting either with e4 or e3 and projecting to St,u . The null decomposition of the Weyl tensor W shall be given in chapter 4.3.3. It will allow us to control the properties of W along null directions. In what follows, we assume to be given a null pair e4 , e3 . In view of such a null pair, we defined the projection from the tangent space of M to the tangent space of St,u to be the tensor Π(t, u) given in definition 16.

4.3. NULL DECOMPOSITION

79

Also, let e1 , e2 , e3 , e4 be a null frame as introduced above. For this null frame, the Ricci rotation coefficients are defined as follows: (4.46)

g(DA e3 , eB ) = H AB

(4.47)

g(DA e4 , eB ) = HAB

(4.48)

g(D3 e3 , eA ) = 2Y A

(4.49)

g(D4 e4 , eA ) = 2YA

(4.50)

g(D4 e3 , eA ) = 2Z A

(4.51)

g(D3 e4 , eA ) = 2ZA

(4.52)

g(D3 e3 , e4 ) = 4Ω

(4.53)

g(D4 e4 , e3 ) = 4Ω

(4.54)

g(DA e4 , e3 ) = 2VA .

We can then write: (4.55)

DA e3 = H AB eB + VA e3

(4.56)

DA e4 = HAB eB − VA e4

(4.57) (4.58)

D3 e3 = 2Y A eA − 2Ωe3 D3 e4 = 2ZA eA + 2Ωe4

(4.59)

D4 e3 = 2Z A eA + 2Ωe3

(4.60)

D4 e4 = 2YA eA − 2Ωe4 .

And the remaining derivatives read: (4.61) (4.62) (4.63)

1 1 /B eA + HAB e3 + H AB e4 DB eA = ∇ 2 2 /3 eA + ZA e3 + Y A e4 D3 eA = D /4 eA + YA e3 + Z A e4 . D4 eA = D

4.3.2. Technicalities. In this short subsection, we are going to remark on certain technicalities which will be useful in the remaining part of this chapter. In the estimates for the components of the Weyl and the Bel-Robinson tensor, only the structure of these terms is important. Thus, the exact numerical coefficients are not relevant in this context. Therefore, one focuses on the actual null components. There is the principle of conservation of signature, which we are going to explain in the next two paragraphs, allowing us to conclude, which terms might show up in a decomposition. That is, it excludes certain terms by signature considerations. This principle was introduced by D. Christodoulou and S. Klainerman in [19]. And we are going to state it here. Signature: For any given covariant tensor T at a point of the spacetime M , one can define a null component of T to be any tensor tangent to St,u

80

4. COMPARISON

at that point (definition 17). We call such tensors S-tangent. Now, to any such component one assigns a signature. Definition 18. (Signature) The signature of such a component is defined to be the difference between the total number of contractions with e4 and the total number of contractions with e3 . This gives us the tool to state the said principle. Principle of conservation of signature: Let T be an arbitrary covariant tensor, expressed as a multilinear form in an arbitrary number of covariant tensors T1 · · · Tp , where the coefficients depend only on the spacetime metric and its volume form. Then the signature of any null term of T , expressed in terms of the null components T1 · · · Tp is equal to the sum of the signatures of each constituent in the decomposition. This is explained by the fact that the only non-vanishing null components of the metric and the volume form have signature zero. Moreover, considering any null component ξ of signature s and a covariant spacetime tensorfield T , denote its covariant derivative intrinsic to /ξ and assign to it the same signature s. Let us define D /3 ξ and St,u by ∇ D /4 ξ as the projections to St,u of D3 ξ and D4 ξ, respectively. We assign to them the signatures s − 1 and s + 1, respectively. One observes that ∇ /ξ, D /3 ξ and D /4 ξ are connected with the null components of the tensor DT through expressions involving the frame coefficients ((4.46)–(4.54)) above. For the null curvature components the signatures are given in subsection 4.3.3. Coming back to the Ricci rotation coefficients from above, they are being assigned the following signatures: s(Y ) = 2 s(Y ) = −2

s(H, Ω) = 1 (4.64)

s(H, Ω) = −1

s(Z, Z, V ) = 0.

Then, for a covariant spacetime tensorfield T , the equations giving the /4 -derivatives of the null comnull components of DT in terms of the ∇ /, D /3 , D ponents of T and the Ricci coefficients satisfy the principle of conservation of signature. Furthermore, besides using the signature, in the calculations for the components of the Weyl and the Bel-Robinson tensor, we shall make use of the following identities. Let X and Y be any vectors tangent to St,u . Then it is: (4.65)

XA XB + XB XA +∗XA ∗XB +∗XB ∗XA = 2δAB X · Y.

4.3. NULL DECOMPOSITION

81

Let ξ and η be any traceless, symmetric forms tangent to St,u . Then it is: (4.66)

ξAC η C B + ξBC η C A = δAB ξ · η.

4.3.3. Null Decomposition of the Weyl Tensor W. Now, we are going to present the decomposition of the Weyl tensor W in (M, g) with respect to the null frame e4 , e3 , e2 , e1 , introduced before. That is, e4 and e3 form a null pair which is supplemented by eA , A = 1, 2, a local frame field for St,u . Definition 19. We define the null components of W as follows: (4.67) (4.68) (4.69) (4.70) (4.71) (4.72) are

αµν (W ) = Πµ ρ Πν σ Wργσδ eγ3 eδ3 1 β µ (W ) = Πµ ρ Wρσγδ eσ3 eγ3 eδ4 2 1 ρ(W ) = Wαβγδ eα3 eβ4 eγ3 eδ4 4 1 σ(W ) = ∗Wαβγδ eα3 eβ4 eγ3 eδ4 4 1 βµ (W ) = Πµ ρ Wρσγδ eσ4 eγ3 eδ4 2 αµν (W ) = Πµ ρ Πν σ Wργσδ eγ4 eδ4 .

With other words, the components of a Weyl field W in such a frame αAB = W (eA , e4 , eB , e4 ), 1 βA = W (eA , e4 , e3 , e4 ), 2

αAB = W (eA , e3 , eB , e3 ), 1 β A = W (eA , e3 , e3 , e4 ), 2

1 ρ = W (e3 , e4 , e3 , e4 ), 4

1 σϵ(eA , eB ) = W (eA , eB , e3 , e4 ). 2

We observe that α, α are symmetric trace-free tensorfields on St,u , β, β are 1-forms on St,u , ρ, σ are functions on St,u , Here, ϵ is the area 2-form of St,u . Now let us consider the algebraically independent components of these: α, α, β, β : 2 algebraically independent components each, (ρ, σ) : 2 functions. Thus, there are 10 component-functions.

82

4. COMPARISON

The components (4.67)–(4.72) have the following signatures: s(α) = 2 s(α) = −2 s(β) = 1

s(β) = −1 s(ρ) = 0

s(σ) = 0. Remark: Observe that the null components (4.67)–(4.72) completely determine the Weyl tensor W and its dual ∗ W . They are called the null decomposition of W . In the sequel, we shall use the following notation for the curvature components: αAB = WA4B4 2βA = WA434 4ρ = W3434 ∗

−ϵAB βC = WABC4

−ϵAB ϵCD ρ = WABCD

For the spacetime dual of W it is: −∗ αAB = ∗ WA4B4 −2∗ βA = ∗ WA434

−2ρϵAB = ∗ WAB34

−ϵAB βC = ∗ WABC4

ϵAB ϵCD σ = ∗ WABCD

αAB = WA3B3 2β A = WA334

2σϵAB = WAB34

ϵAB ∗ β C = WABC3

−ρδAB + σϵAB = WA3B4 ∗

αAB = ∗ WA3B3

2∗ β A = ∗ WA334 4σ = ∗ W3434

−ϵAB β C = ∗ WABC3

−σδAB − ρϵAB = ∗ WA3B4

In view of these components, the electric-magnetic decomposition of a Weyl field reads as follows: 1 1 1 HAB = − ∗ αAB + ∗ αAB − σδAB (4.73) 4 4 2 1∗ 1∗ HAN = β A − βA (4.74) 2 2 (4.75) HN N = σ 1 1 1 (4.76) EAB = αAB + αAB − ρδAB 4 4 2 1 1 (4.77) EAN = β A + βA 2 2 (4.78) EN N = ρ. Let us now give the components of the Bel-Robinson tensor Q(W ) associated to W relative to e+ , e− . In fact, they are given correspondingly for any null pair e4 , e3 . In the (timelike) plane spanned by e+ , e− , i.e.

4.3. NULL DECOMPOSITION

83

(Tp St,u ) ⊥ they are: (4.79) (4.80) (4.81) (4.82) (4.83)

Q(W )(e− , e− , e− , e− ) = 2|α|2 ,

Q(W )(e+ , e− , e− , e− ) = 4|β|2 ,

Q(W )(e+ , e+ , e− , e− ) = 4|ρ2 + σ 2 |,

Q(W )(e+ , e+ , e+ , e− ) = 4|β|2 ,

Q(W )(e+ , e+ , e+ , e+ ) = 2|α|2 .

Note that e0 = 12 (e+ + e− ) is the unit future-directed normal to Ht . Thus, with this notation, we have L = Φe+ , L = Φe− ,

1 T = Φe0 = Φ(e+ + e− ), 2 ¯ = K + T = 1 Φ(τ+2 e+ + τ−2 e− ). K 2 ! √ We recall from before that τ+ = 1 + u2 and τ− = 1 + u2 . In the comparison theorem, we shall use the contractions (4.84) and (4.85). As the exact coefficients in these terms are of no importance in what will follow, we only perform the curvature null components with corresponding weights. To do so, we introduce the notation f ∼ g: Let f and g be positive functions. Then f ∼ g denotes the fact that there exists a constant C > 0 such that C −1 f ≤ g ≤ Cf. By direct computation, we obtain:

(4.84)

¯ T, T, T ) ∼ τ−2 | α |2 Q(W )(K,

+ τ+2 (| β |2 + | ρ |2 + | σ |2 + | β |2 + | α |2 )

¯ K, ¯ T, T ) ∼ τ−4 | α |2 +τ−2 τ+2 | β |2 Q(W )(K,

(4.85)

+ τ+4 (| ρ |2 + | σ |2 + | β |2 + | α |2 )

Correspondingly, straightforward calculations yield for the components of Q relative to any null pair e3 , e4 : Here we denote by {, } any quadratic

84

4. COMPARISON

form with coefficients depending only on the induced metric and area form on St,u . QA444 = {α, β}

QA344 = {β, ρ} + {∗ β, σ} QA433 = {β, ρ} + {∗ β, σ}

QAB44 = {β, β} + {α, ρ} + {∗ α, σ}

QAB34 = {β, β} + {(ρ, σ), (ρ, σ)}

QAB33 = {β, β} + {α, ρ} + {∗ α, σ} QA333 = {α, β}.

Here, only the structure of the terms is important, the exact coefficients of the quadratic terms on the right-hand side are irrelevant. One can either calculate the terms exactly, which is not necessary, or do the following: In a first step, using the principle of conservation of signature, we can explore which terms that are present on the right-hand side. This might still include certain terms, which do not show up in reality. However, by appealing to the identities (4.65) and (4.66) in a second step, these terms do, in fact, cancel. Let us do this for the components QA444 and QAB34 . For all the other components, it works analogously. QA444 : One sees directly, that QA444 has the signature 3. Therefore, the only quadratic term yielding this signature is αβ. Here, of course, we neglect the coefficients. QAB34 : Correspondingly, QAB34 has signature 0. This means, that in principle the following quadratic terms could show up (of course, neglecting coefficients): ρ2 , σ 2 , ρσ, ββ, αα. Now, the cancellation of the αα part is explained by the identity (4.65) and the fact that the 2-tensor QAB34 − 2δAB (ρ2 + σ 2 ), thought of as a tensor on St,u , is traceless. In view of the other components, let us mention additionally that the αβ term in QA344 as well as the αβ term in QA433 cancel in view of the equations α · β − ∗α · ∗β = 0

α · β − ∗ α · ∗ β = 0. 4.3.4. Ricci Coefficients. In the ‘Comparison’ as well as in the ‘Error Estimates’ the Ricci coefficients play a crucial role. They were introduced in (4.46) ff. Now, we calculate them explicitly for a standard null frame. We assume the spacetime (M, g) to be foliated by a maximal time function t with lapse Φ, second fundamental form k and a radial function u. The lapse function of the u-foliation in each Ht is a =| ∇u |−1 . Moreover, θ denotes the second fundamental form of the surfaces St,u with respect to Ht . Consider an orthonormal frame e1 , e2 , N on Ht . Assume that u is a solution of the Eikonal equation, which we refer to as u being optical, in the

4.4. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: DERIVATION

85

exterior region of the spacetime. From the introduction, one knows that the second fundamental form k is decomposed as follows: ηAB = kAB ϵA = kAN (4.86)

δ = kN N

In the chapter ‘Second Fundamental Form k’ we derive the estimates for these components. The Ricci coefficients are now calculated in the following proposition. Proposition 6. Assume the spacetime (M, g) to be foliated as explained above. Let e4 = T +N , e3 = T −N be the standard null pair corresponding to the (t, u) foliations. And let e1 , e2 , e3 = e− , e4 = e+ denote a corresponding null frame. Then the Ricci rotation coefficients are given by the formulas ((4.46)– (4.54)) with =χ =χ =ζ =ζ =0 =ξ 1 Ω= ν 2 1 Ω= ν 2 V =ϵ

H H Z Z Y Y

(4.87) where

χAB = θAB − ηAB χAB = −θAB − ηAB

ζA = a−1 ∇ /A a + ϵA

ζ A = Φ−1 ∇ /A Φ − ϵA

ξ A = Φ−1 ∇ /A Φ − a−1 ∇ /A a

(4.88)

ν = −Φ−1 ∇N Φ + δ

ν = Φ−1 ∇N Φ + δ.

4.4. Bianchi Equations Relative to a Null Frame: Derivation We want to express the Bianchi equations relative to a null frame, as we use them in this form in our proof. In this section, we derive the needed expressions, that is we perform equations for the corresponding derivatives

86

4. COMPARISON

of the components of the Weyl tensor W . The actual statement of the Bianchi equations relative to a null frame shall then be given in section 4.5. As we use some basic facts about commuting Lie derivatives of W with its null components, let us first give the commutation formulas. In the next lemma, we give the commutators of a vectorfield X with a null frame, and in the proposition right afterwards, the commutation formulas for W are performed. They had been proven by D. Christodoulou and S. Klainerman in [19]. Lemma 4. Let X be an arbitrary vectorfield. Then the commutators of X with the null frame e1 , e2 , e3 , e4 are: [X, e3 ] = (4.89)

(X)

P A eA + (X) M e3 + (X) N e4

PA eA + (X) N e3 + (X) M e4 1 1 [X, eA ] = Π[X, eA ] + (X) QA e3 + (X) QA e4 2 2 [X, e4 ] =

(X)

with the Lie coefficients of X relative to the null frame: (X) (X)

PA = g(DX e4 , eA ) − D4 XA

P A = g(DX e3 , eA ) − D3 XA 1 1 (X) M = − g(DX e4 , e3 ) + D4 X3 2 2 1 1 (X) M = − g(DX e3 , e4 ) + D3 X4 2 2 1 (X) N = D4 X4 2 1 (X) N = D3 X3 2 (X) QA = g(DX e4 , eA ) + DA X4 (4.90)

(X)

QA = g(DX e3 , eA ) + DA X3 .

Proof. To prove this, we only compute in a straightforward manner. To show the formula for [X, e4 ], we observe that 1 DX e4 = g(DX e4 , eA )eA − g(DX e4 , e3 )e4 2 as well as 1 1 D4 X = D4 XA eA − D4 X4 e3 − D4 X3 e4 . 2 2

4.4. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: DERIVATION

87

Thus, it is 1 [X, e4 ] = g(DX e4 , eA )eA − g(DX e4 , e3 )e4 2 1 1 − D4 XA eA + D4 X4 e3 + D4 X3 e4 . 2 2 And this is exactly the formula from the lemma. The other two identities hold correspondingly. Note that ˆ44 4(X) N = (X) π 4(X) N = (X) π ˆ33 (X) (X)

QA − (X) PA = (X) π ˆ4A

QA − (X) P A = (X) π ˆ3A

1 2((X) M + (X) M ) = (X) π34 = (X) π ˆ34 − tr(X) π. 2 Proposition 7. Let X be an arbitrary vectorfield and W an arbitrary Weyl tensor. Consider the null components of W , that is α(W ), . . . , α(W ), and the null components of LˆX W , that is α(LˆX W ), . . . , α(LˆX W ). Further, / X α denote the projection of LX α, . . . , LX α on St,u . Moreolet L / X α, . . . , L ˆ/ α be the traceless parts of the 2-tensors L ˆ ever, let L /X α, L /X α, LX α. X Then it is: ' ( 1 (X) (X) ˆ ˆ / X αAB + −2 M − tr π αAB α(LX W )AB = L 8 $(X) $ % % (X) − PA + QA βB − (X) PB + (X) QB βA % $ (4.91) + δAB (X) P + (X) Q · β ' ( 1 (X) 1 (X) (X) ˆ / X βA − π ˆAB βB + − M − tr π βA β(LX W )A = L 2 8 % $ % 3 $(X) 3 − PA + (X) QA ρ − ϵAB (X) PB + (X) QB σ 4 4 % 1 $(X) (4.92) P B + (X) QB αAB − 4 1 ρ(LˆX W ) = L / X ρ − tr(X) πρ 8 % % 1$ 1 $(X) (4.93) P A + (X) QA βA + (X) PA + (X) QA β B − 2 2

1 σ(LˆX W ) = L / X σ − tr(X) πσ 8 % % 1 $(X) 1$ + (4.94) P A + (X) QA ∗ βA + (X) PA + (X) QA ∗ β B 2 2

88

4. COMPARISON

' ( 1 1 (X) (X) ˆAB β B + − β(LˆX W )A = L / X βA − π M − tr π βA 2 8 $ % $ % 3 3 + (X) P A + (X) QA ρ − ϵAB (X) P B + (X) QB σ 4 4 % 1 $(X) (4.95) PB + (X) QB αAB + 4 ( ' 1 (X) (X) ˆ ˆ / X αAB + −2 M − tr π αAB α(LX W )AB = L 8 $(X) $ % % (X) + PA + QA β B + (X) P B + (X) QB β A $ % (4.96) − δAB (X) P + (X) Q · β.

Proof. The proof of this proposition goes by straightforward computation. Now, recalling the definition for the modified Lie derivative (4.13) and the corresponding formula ((4.24), (4.25)), we contract (and project) the tensors to obtain the following. One starts calculating the Lie derivatives LX W4A4B , LX WA434 and LX W3434 : LX W4A4B = L / X αAB − 2(X) M αAB + 2(X) N ρδAB − (X) QA βB − (X) QB βA

LX WA434 LX W3434

− (X) PA βB − (X) PB βA + 2δAB (X) P · β $ % = 2L / X βA + 2 − (X) M + 2(X) M βA − 2(X) N β B $ % − (X) PA + 2(X) QA ρ − 3ϵAB (X) PB σ − (X) P B αAB $ % = 4L / X ρ − 8ρ (X) M + (X) M − 4(X) P A βA + 4(X) PA β A .

We take into account the formulas (X)

[W ]A4B4 = (X) π ˆAC αCB + (X) π ˆBC αCA − (X) π ˆ34 αAB

(X)

[W ]A434

(X)

− 2δAB βC (X) π ˆ4C + ρδAB (X) π ˆ44 + tr(X) παAB ( ' 3 = (X) π ˆB3 αAB + 2 (X) π ˆAB − (X) π ˆ34 δAB βB 2

− (X) π ˆ44 β A − (X) π ˆ4A ρ + 3ϵAB (X) π ˆ4B σ + 2tr(X) πβA

[W ]3434 = 4βA (X) π ˆA3 − 4β A (X) π ˆA4 − 8(X) π ˆ34 ρ + 4tr(X) πρ.

Next, we combine the above two sets of formulas correspondingly, then by signature considerations and identity (4.65) we obtain the expressions (4.91), (4.92) and (4.93) in the proposition. Similarly, everything works for the identities (4.94), (4.95) and (4.96) of the proposition.

4.4. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: DERIVATION

89

Remark: Again, in the formulas (4.91)–(4.96) only the structure of the terms is important, the exact numerical coefficients are not relevant. It is possible to get the said structure of the error terms without really calculating them, but only by applying the principle of conservation of signature and by using the identities from chapter 4.3.2 to show the cancellation of the remaining terms which do not appear in the precise computation. One easily sees that the Lie coefficients (4.90) have the following signatures: s((X) N ) = 2 s((X) N ) = −2

s((X) P, (X) Q) = 1

s((X) N , (X) Q) = −1 s(M, (X) M ) = 0.

Focusing on the right-hand side of the equations (4.91)–(4.96), note that each error term is a product of a null component of the Weyl tensor W and either a component of the Lie coefficients (4.90) or a null component of the deformation tensor of X. Using signature considerations, it turns out that the sum of the signatures of each factor equals the signature of the left-hand side, that is the signature of the null component of LˆX W . To foster the understanding of the structure of these terms, let us do this procedure for the term α(LˆX W ). As 1) α is a traceless, symmetric 2-tensor, and 2) using signature considerations, one writes ˆ/ α + {((X) M, (X) M , tr(X) π), α} α(LˆX W ) = L X + {((X) P, (X) Q), β} + {(X) π ˆ , α}

These are all the terms, that, in principle, could appear in view of 1) and 2). Note that {(X) N, (ρ, σ)} is excluded by 1). The last bracket on the right-hand side, referred to as ‘remaining terms’ above, is excluded by (4.66). One is then left with ˆ/ α + {((X) M, tr(X) π), α} + {((X) P, (X) Q), β}, α(LˆX W ) = L X which corresponds to (4.91). To remind ourselves, the aim of this chapter is to prepare the statement of the Bianchi equations relative to a null frame. In view of this, we still have to assess the structure of these equations and to calculate the error terms. In what follows, we compute the derivatives needed to do so.

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4. COMPARISON

We consider the Bianchi equations (4.9), in particular ((4.18), (4.19)) and ((4.20), (4.21)). From them we have: 1$ ˜ JA4B43 + J˜B4A43 + DB WA434 + DA WB434 2 % + D4 WA3B4 + D4 WB3A4 1$ = J˜A3B34 + J˜B3A34 − DA WB334 − DB WA334 2 % + D3 WB3A4 + D3 WA3B4 .

D3 WA4B4 = (4.97) D4 WA3B3 (4.98)

Next, recalling (4.18):

Dα Wαβγδ = Jβγδ , one has DC WC4A4 + D3 W34A4 + D4 W44A4 = J4A4 , < => ? =0

that is,

1 DC WC4A4 − D4 W34A4 = J4A4 . 2

Thus, it is D4 W34A4 = 2DC WC4A4 − 2J4A4 .

In the same manner, one directly writes from (4.18): (4.99) (4.100) (4.101) (4.102) (4.103) (4.104)

D4 WA434 = 2DC WC4A4 − 2J4A4

D3 WA434 = −2DC WC3A4 + 2J3A4 D4 WA334 = 2DC WA3C4 − 2J4A3

D3 WA334 = −2DC WA3C3 + 2J3A3 D4 W3434 = 2DC WC4A4 − 2J434

D3 W3434 = −2DC WC334 + 2J334 .

In a next step, in view of these formulas, we are going to express the /3 derivatives D3 and D4 of the Weyl tensor in terms of the derivatives D and D /4 of the the Weyl null components and correcting terms which are, in fact, products of a Ricci rotation coefficient ((4.46)–(4.54)) with a Weyl null component. Consider also ((4.55)–(4.63)). We write for D3 WA4B4 : D3 WA4B4 = D /3 WA4B4 − W(D3 A)⊥ 4B4 − WA(D3 4)B4 − WA4(D3 B)⊥ 4 − WA4B(D3 4) .

In view of the first and the third correction terms on the right-hand side, consider (4.62), in view of the second and the fourth, consider (4.58).

4.4. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: DERIVATION

91

Then, with the identity (4.65) we derive for D3 WA4B4 the formula (4.105). Correspondingly, the computations work for the other terms. Thus, we obtain: (4.105)

/3 αAB − 4ΩαAB − 4(ZA βB + ZB βA − Z · βδAB ) D3 WA4B4 = D

(4.106)

/4 αAB − 4ΩαAB + 4(Z A β B + Z B β A − Z · βδAB ) D4 WA3B3 = D

(4.107)

/4 βA + 4ΩβA − 2Z B αAB − 6YA ρ − 6∗ YA σ D4 WA434 = 2D

(4.108)

D3 WA434 = 2D /3 βA − 4ΩβA − 2Y B αAB − 6ZA ρ − 6∗ ZA σ

(4.109)

D4 WA334 = 2D /4 β A − 4Ωβ A + 2YB αAB + 6Z A ρ − 6∗ Z A σ

(4.110)

/3 β A − 4Ωβ A − 2ZB αAB + 6Y A ρ − 6∗ Y A σ D3 WA334 = 2D

(4.111)

D4 W3434 = 4D4 ρ − 8Z A βA + 8YA β A

(4.112)

D3 W3434 = 4D3 ρ + 8ZA β A − 8Y A βA .

Moreover, in the same way, we derive: 1 D3 (WA3B4 + WB3A4 ) = −D3 ρδAB + (Y A βB + Y B βA ) 2 + (∗ Y A ∗ βB + ∗ Y B ∗ βA ) − (ZA β B + ZB β A ) (4.113)

− (∗ ZA ∗ β B + ∗ ZB ∗ β A )

= −D3 ρδAB + (2Y · β − 2Z · β)δAB

1 D4 (WA3B4 + WB3A4 ) = −D4 ρδAB + (Z A βB + Z B βA ) 2 + (∗ Z A ∗ βB + ∗ Z B ∗ βA ) (4.114)

− (YA β B + YB β A ) − (∗ YA ∗ β B + ∗ YB ∗ β A )

= −D4 ρδAB + (−2Y · β + 2Z · β)δAB

Also, following the same idea, we get for DB WA4B4 and DB WA3B3 : (4.115) (4.116)

/ B αAB + 2VB αAB − 2trHβA DB WA4B4 = ∇

DB WA3B3 = ∇ / B αAB − 2VB αAB + 2trHβ A .

92

4. COMPARISON

In addition, one finds the following identities, also applying (4.66). 1 1 (DB WA434 + DA WB434 ) = ∇ /A βB + ∇ /B βA − trHαAB − 3HAB ρ 2 2 1 ˆ AB + (VA βB + VB βA ) ˆ AB σ − (α · H)δ (4.117) − 3∗ H 2 1 1 (DB WA334 + DA WB334 ) = ∇ /A β B + ∇ /B β A + trHαAB + 3H AB ρ 2 2 1 ∗ˆ ˆ AB − (VA β + VB β ) (4.118) − 3 H AB σ + (α · H)δ B A 2 (4.119)

ˆ AB βB DB WA3B4 = −∇ /A ρ + ϵAB ∇ /B σ − trHβ A + 2H

(4.120)

ˆ AB β . /A ρ + ϵAB ∇ /B σ + trHβA − 2H DB WB3A4 = −∇ B

Next, we contract (4.117) and (4.118) to obtain (4.121)

/ β − 3trHρ − α · H + 2V · β DA WA434 = 2div

(4.122)

DA WA334 = 2div / β + 3trHρ + H · α − 2V · β.

Now, having computed the derivatives of the Weyl tensor in this sense, we combine the expressions as follows. Let us do it more explicitly for the case of D4 W3434 . For all the other components, everything works correspondingly. Thus, combining (4.103) with (4.111), we write 1 1 D4 ρ = DC WC434 − J434 + 2Z A βA − 2YA β A . 2 2 Substituting (4.121) for DC WC434 yields:

(4.123)

1ˆ 3 / β − trHρ − H ·α+V ·β D4 ρ = div 2 2 1 + 2(Z · β − Y · β) − J434 . 2

Similarly with the equations (4.104), (4.112) and (4.122) we have:

(4.124)

3 1ˆ ·α+V ·β / β − trHρ − H D3 ρ = −div 2 2 1 + 2(Y · β − Z · β) + J334 . 2

4.4. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: DERIVATION

93

Doing the same with the dual of W , we deduce:

(4.125)

(4.126)

3 1ˆ ∗ / ∗ β − trHσ + H · α − V · ∗β D4 σ = −div 2 2 1 ∗ − 2(Z · ∗ β + Y · ∗ β) − J434 2 3 1ˆ ∗ ∗ / β − trHσ + H · α − V · ∗ β D3 σ = −div 2 2 1 ∗ − 2(Y · ∗ β + Z · ∗ β) + J334 . 2

In the same way, we combine (4.99) and (4.107) with (4.115), then (4.100) and (4.108) with (4.120), then (4.101) and (4.109) with (4.119), as well as we combine (4.102) and (4.106) with (4.115). Proceeding as above, we finally derive: / B αAB − 2trHβA + 2VB αAB + Z B αAB − 2ΩβA D /4 βA = ∇ + 3YA ρ + 3∗ YA σ − J4A4

(4.127)

=∇ / B αAB − 2trHβA − 2ΩβA + (2VB + Z B )αAB + 3(YA ρ + ∗ YA σ) − J4A4

D /3 βA = ∇ /A ρ + ϵAB ∇ /B σ − trHβA ˆ AB β + 2ΩβA + Y B αAB + 3(ZA ρ + ∗ ZA σ) + J3A4 (4.128) + 2H B D /4 β A = −∇ /A ρ + ϵAB ∇ /B σ − trHβ A

(4.129)

ˆ AB βB + 2Ωβ − YB αAB − 3(Z A ρ − ∗ Z A σ) + J3A4 + 2H A

D /3 β A = −∇ / B αAB − 2trHβ A − 2Ωβ A

(4.130)

− (−2VB + ZB )αAB + 3(−Y A ρ + ∗ Y A σ) + J3A3 .

For the remaining two terms D /4 αAB and D /3 αAB , we first combine (4.98) with (4.106) taking into account (4.113) and (4.118) for D /4 αAB , respectively (4.97) with (4.105) taking into account (4.114) and (4.117) for D /3 αAB . Therefore, showing the steps in more details for D /4 αAB , we have: 1 ˆ AB σ D /4 αAB = −∇ /A β B − ∇ /B β A − trHαAB − 3H AB ρ + 3∗ H 2 1 ˆ AB + (VA β + VB β ) − D3 ρδAB − (α · H)δ A A 2 1 + 2Y · βδAB − 2Z · βδAB + (J˜A3B34 + J˜B3A34 ) 2 + 4ΩαAB − 4(Z A β B + Z B β A − Z · β).

Then, substituting equation (4.124) for D3 ρ yields (4.131) for D /4 αAB , /3 αAB . whereas substituting equation (4.123) for D4 ρ yields (4.132) for D

94

4. COMPARISON

This means, we have for D /4 αAB and correspondingly for D /3 αAB the following: 1 /A β B + ∇ /B β A − div / βδAB ) − trHαAB + 4ΩαAB D /4 αAB = −(∇ 2 ∗ˆ ˆ − 3(H AB ρ − H AB σ) + (VA − 4Z A )β B + (VB − 4Z B )β A 1 (4.131) − (V − 4Z) · βδAB + (J˜A3B34 + J˜B3A34 − J334 δAB ) 2 1 /A βB + ∇ /B βA − div / βδAB ) − trHαAB + 4ΩαAB D /3 αAB = (∇ 2 ∗ ˆ ˆ − 3(HAB ρ + HAB σ) + (VA + 4ZA )βB + (VB + 4ZB )βA 1 (4.132) − (V + 4Z) · βδAB + (J˜A4B43 + J˜B4A43 + J434 δAB ). 2 Now, we have derived everything in the form to write down the Bianchi equations relative to a null frame. More precisely, we shall reformulate the equations (4.123)–(4.132) in proposition 8, using the notation of chapter 4.5. This proves proposition 8. To state the homogeneous Weyl equations in the said proposition is subject of chapter 4.5. In the part 4.6, there are given the assumptions for the comparison theorem, which is itself stated and proven in chapter 4.7. 4.5. Bianchi Equations Relative to a Null Frame: Statement Next, we are going to introduce some notation involving the signature, which was given earlier. Recall the signature s to be defined as the difference: the number of contractions with L minus the number of contractions with L. In view of the Bianchi equations that we are going to state relative to a null frame, we define: Definition 20. Let W be an arbitrary Weyl tensor and let ξ be any of its null components. Define the following St,u -tangent tensors: 3−s trχξ 2 3+s /4 ξ + ξ4 = D trχξ 2

(4.133)

ξ3 = D /3 ξ +

(4.134)

/∗1 , D /2 , D /∗2 introduced in a previous Also, we need the operators D /1 , D chapter: (4.135) (4.136) (4.137) (4.138)

D /1 : D /∗1 D /2 D /∗2

ξ 7→ (div / ξ, curl / ξ)

: (ρ, σ) → 7 −∇ /A ρ + ϵAB ∇ / Bσ : ξ → 7 div /ξ % 1@ 1$ ∇ /B ξA + ∇ : ξ 7→ − L /A ξB − div / ξ γAB γξ = − 2 2

4.5. BIANCHI EQUATIONS RELATIVE TO A NULL FRAME: STATEMENT

95

Observe that: D /1 maps any S-tangent 1-form ξ into the pairs of functions (div / ξ, curl / ξ). D /∗1 is the L2 -adjoint of D /1 and takes any pairs of scalars ρ, σ into the S-tangent 1-form in (4.136). D /2 maps any S-tangent, 2-covariant, symmetric, traceless tensors ξ into the S-tangent 1-form div / ξ. D /∗2 is the L2 -adjoint of D /2 and takes any S-tangent 1-form ξ into the 2-form given in (4.138). ˆ for the 2-covariant, symmetric, In addition, we use the notation ξ ⊗η traceless tensor formed out of two covariant S-tangent tensors ξ and η according to the formula (4.139)

ˆ = ξA ηB + ξB ηA − ξ · ηδAB . ξ ⊗η

Now, we state the Bianchi equations in terms of the null components of the curvature, as they will be used in the proof of the comparison theorem below to derive the estimates for the angular derivatives. The Bianchi equations in this form are also deduced in [19] in the chapter about the comparison theorem. We state them as a proposition: Proposition 8. (Bianchi equations) Suppose given the (t, u)foliations introduced above. Then, relative to a null frame compatible with these foliations, the Bianchi equations become: (4.140)

/∗2 β + E3 (α) α3 = −2D ˆ + (V + 4Z) ⊗ ˆ + ∗ Hσ) ˆ β E3 (α) = 4Ωα − 3(Hρ

(4.141)

β3 = D /∗1 (−ρ, σ) + E3 (β) ˆ · β + 2Ωβ + Y · α + 3(Zρ + ∗ Zσ) E3 (β) = 2H

(4.142)

β4 = div / α + E4 (β) E4 (β) = −2Ωβ + (2V + Z) · α + 3(Y ρ + ∗ Y σ)

(4.143)

ρ3 = −div / β + E3 (ρ) 1ˆ E3 (ρ) = − H · α + V · β + 2(Y · β − Z · β) 2

(4.144)

ρ4 = div / β + E4 (ρ) 1ˆ · α + V · β + 2(Z · β − Y · β) E4 (ρ) = − H 2

96

4. COMPARISON

(4.145)

σ3 = −curl / β + E3 (σ) 1ˆ ∗ E3 (σ) = H · α − V · ∗ β − 2(Y · ∗ β + Z · ∗ β) 2

(4.146)

σ4 = −curl / β + E4 (σ) 1ˆ ∗ · α − V · ∗ β − 2(Z · ∗ β + 2Y · ∗ β) E4 (σ) = H 2

(4.147)

/ α + E3 (β) β 3 = −div

E3 (β) = −2Ωβ − (−2V + Z) · α + 3(−Y ρ + ∗ Y σ) (4.148)

/∗1 (ρ, σ) + E4 (β) β4 = D

ˆ · β + 2Ωβ − Y · α − 3(Zρ − ∗ Zσ) E4 (β) = 2H (4.149)

α4 = 2D /∗2 β + E4 (α) ˆ + (V − 4Z) ⊗ ˆ − ∗ Hσ) ˆ β. E4 (α) = 4Ωα − 3(Hρ 4.6. Assumptions for the Comparison Argument

Before we state the assumptions, let us recall, at this point, the definitions of the norms for the null curvature components introduced in the section 3.1.1. Namely, R0 (W ) and R1 (W ) in (3.7)–(3.16) as well as R0 (W ) and R1 (W ) in (3.17)–(3.26). The norms are stated for the interior and exterior regions of the hypersurface Ht . Throughout this work, the interior estimates follow by straightforward calculations, whereas the difficulties appear in the exterior part. Therefore, in order to prove our estimates in the exterior region, we have to proceed in a more subtle way. Let us recall here the definitions of the two regions. The interior region I are all the points in Ht for which r0 (t) . 2 And the exterior region U are all the points in Ht for which r≤

r0 (t) . 2 Here, r0 (t) is the value of r corresponding to the area of St,0 , the surface of intersection between C0 and Ht . And u1 (t) is the value of u corresponding to r0 (t)/2. Next, we are going to state three sets of assumptions on the basic quantities of the (t, u) foliations as well as on the Ricci coefficients with respect r≥

4.6. ASSUMPTIONS FOR THE COMPARISON ARGUMENT

97

to a standard null frame. This all shall be used later in this section to prove the comparison theorem. Assumption 0. At all time t, we assume for r, being a function of u, that: dr 3 2 ≤ ≤ . 3 du 2 Furthermore, we assume that the fundamental constants (defined below) am , aM , hm , hM , km , kM , ζ /a | of each Ht are uniformly bounded for all t. Also, we assume supHt r | a−1 ∇ −1 and Φm = (inf Ht Φ) of each Ht to be uniformly bounded for all t. Assumption 1. Denote by Ut the exterior and by It the interior region. At all points in the exterior region of the spacetime, let the following hold: 3

(4.150)

sup r 2 | (ξ, ζ, ζ, ν, ν) | ≤ ϵ0

(4.151)

sup(| rtrχ − 2 |, | rtrχ − 2 |) ≤ ϵ0

Ut

Ut

3

(4.152)

ˆ | ≤ ϵ0 sup r 2 | χ Ut

1

(4.153)

ˆ | ≤ ϵ0 sup rτ−2 | χ Ut

At all points in the interior region of the spacetime, let (4.154) (4.155)

∥ Φ−1 ∇Φ ∥p,It + ∥ k ∥p,It ≤ ϵ0 /a ∥p,It ≤ ϵ0 . ∥ rθˆ ∥p,It + ∥ ra−1 ∇

Assumption 2. In the interior region, we assume for the Ricci curvature tensor: (4.156)

∥ Ric ∥2,It ≤ ϵ0 .

The fundamental constants of the u-foliation are am = inf am (u);

aM = sup aM (u)

hm = inf hm (u);

hM = sup hM (u)

km = inf kK (u);

kM = sup kK (u)

h = sup h(u);

ζ = sup ζ(u)

u u u

u

κ = sup κ(u) u

u u u u

98

4. COMPARISON

with am (u) = inf a;

aM (u) = sup a

hm (u) = inf rtrθ;

hM (u) = sup rtrθ

Su

Su

Su 2

∥ K∥ − kK (u) = r −

Su

4,Su

h(u) = sup | rtrθ |; Su

ζ(u) = sup r | θˆ | Su

κ(u) = sup | κ | Su

where θˆ denotes the traceless part of θ, and κ = ra−1 (atrθ − atrθ).

(4.157)

4.7. The Comparison Argument and the Controlling Quantities Q 0 and Q 1 in the Comparison Argument Here, we are going to introduce the controlling quantities Q1 and Q0 needed in the comparison argument. The comparison theorem shall then give us estimates on the curvature components up to first derivatives in L2 (Ht ). In view of the quantities Q1 and Q0 , we shall introduce the corresponding Lie derivatives with respect to the scaling vectorfield S and the time vectorfield T . There is no need to work with a Lie derivative with respect to rotational vectorfields. In fact, no rotational vectorfields are needed at all. One might think of introducing them in order to obtain the angular derivatives of the curvature components. But this is not necessary. For, we obtain the angular derivatives of the curvature components more directly by using only Q(W ) and the Lie derivatives of W with respect to T and S together with the Bianchi equations. Moreover, as we only have one derivative of the curvature controlled in Ht , this yields a control in the intersections St,u = Ht ∩ Cu on only the curvature components not on their derivatives. Therefore, the usual argument on the cone would not close. Thus we proceed as follows. First, we have the energy currents (4.158) (4.159)

¯ αT β T γ P0 µ = −Qµ αβγ (W )K ¯ βT γ ¯ α T β T γ − Qµ αβγ (LˆT W )K ¯ αK P1 µ = −Qµ αβγ (LˆS W )K

Then the quantities Q0 and Q1 are defined as the following corresponding energies: # ¯ T, T, T ) (4.160) Q0 (t) = Q(W )(K, Ht # # ˆ ¯ ¯ K, ¯ T, T ). (4.161) Q1 (t) = Q(LS W )(K, T, T, T ) + Q(LˆT W )(K, Ht

Ht

4.7. THE COMPARISON ARGUMENT AND THE CONTROLLING QUANTITIES

99

We recall the norms R0 (W ) and R1 (W ) for the curvature components, and as the exterior norms e R0 and e R1 are going to play an important role right away, we give again the formulas (3.21) and (3.22): # # # # e 2 2 2 2 2 2 2 R0 (W ) = τ− | α | + r |β| + r |ρ| + r2 | σ |2 U U U U # # 2 2 + r |β| + r2 | α |2 U

and e

2

R1 (W ) =

#

U

2

#

4

2

#

4

2

#

|∇ /α | + r |∇ /β | + r |∇ /ρ | + r4 | ∇ /σ |2 U U U # # # # 4 2 4 2 4 2 + r |∇ /β | + r |∇ /α | + τ− | α 3 | + τ−4 | α4 |2 U U #U # # #U 2 2 2 4 2 4 2 + τ− r | β 3 | + r | β4 | + r | ρ3 | + r4 | ρ4 |2 #U # U # U # U 4 2 4 2 4 2 + r | σ3 | + r | σ4 | + r | β3 | + r4 | β4 |2 U U #U #U + r4 | α3 |2 + r4 | α4 |2 . U

τ−2 r2

U

U

From before we see that the Bel-Robinson tensor Q has the following form. The exact coefficients are not important here. Therefore, let us focus on the curvature null components with their weights. ¯ T, T, T ) ∼ τ 2 | α |2 Q(W )(K, − $ % + τ+2 | β |2 + | ρ |2 + | σ |2 + | β |2 + | α |2 (4.162) ¯ K, ¯ T, T ) ∼ τ−4 | α |2 + τ−2 τ+2 | β |2 Q(W )(K, $ % + τ+4 | ρ |2 + | σ |2 + | β |2 + | α |2 (4.163)

Theorem 4. (Comparison Theorem) Assume given a spacetime (M, g) with a (t, u)-foliation as described in the previous subsections. We consider an arbitrary Weyl tensor W satisfying the homogeneous Bianchi equations (4.9). Moreover, we assume that the spacetime fulfills the Assumptions 0 and 1 from the previous subsections, and that the curvature assumption (4.156) is verified in the interior region. Then there exists a constant c such that, if Q0 (W ) and Q1 (W ) are finite, the following holds: $ 1 1% (4.164) R0 (W ) + R1 (W ) ≤ c Q0 (W ) 2 + Q1 (W ) 2 .

Proof. To prove the comparison theorem, we can split inequality (4.164) into two parts. Thus, we obtain (4.165)

1

R0 (W ) ≤ cQ0 (W ) 2

100

4. COMPARISON

and (4.166)

$ 1 1% R1 (W ) ≤ c Q0 (W ) 2 + Q1 (W ) 2 .

Whereas (4.165) is straightforward, the proof of (4.166) needs more attention. Inequality (4.166): In order to show the boundedness of the angular derivatives of the curvature components, let us introduce the following quantity: # / α |2 A(t) := r2 {τ−2 | ∇

(4.167)

Ht + τ+2 (|

∇ / β |2 + | ∇ / α |2 + | ∇ / β |2 + | ∇ / ρ |2 + | ∇ / σ |2 )}.

We are now going to estimate A in (4.167) in terms of Q0 and Q1 (equations (4.160), (4.161)) using the Bianchi equations ((4.140)–(4.149)). To do so, we focus on the Bianchi equations without the error terms. For, the error terms are shown to be controlled appropriately. In the sequel, we denote L = e4 and L = e3 . First, we consider r∇ /α: From the Bianchi equation (4.147) for β 3 we obtain: (4.168) with (4.169)

β 3 = −div / α + l.o.t. div / α = −β 3 , β3 = D /3 β A + 2trχ · β A .

At this point, we recall that S is the scaling field 1 (4.170) S = (uL + uL) 2 while the time translation vectorfiled T is 1 (4.171) T = (L + L). 2 Thus, it is S − uT (4.172) L= 1 2 (u − u) with

(4.173)

u = u + 2r.

Therefore, one writes (4.174)

rL = uT − S.

Now, from the second integral of Q1 in (4.161), we deduce: (4.175)

∥ τ− u D /T β ∥2L2 (Ht ) ≤ cQ1 (t).

4.7. THE COMPARISON ARGUMENT AND THE CONTROLLING QUANTITIES 101

From the first integral of Q1 in (4.161) and from Q0 in (4.160), we obtain: (4.176)

/S β ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)). ∥ τ+ D

In view of (4.174) with L = e3 and we obtain (4.177)

/3 β A + 2trχβ A , β3 = D ∥ τ− rβ 3 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Then, by elliptic theory of (4.168), it follows the estimate for the angular derivative of α: (4.178)

/α ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)). ∥ τ− r∇

The procedure for the estimates of the angular derivatives of the other curvature components is similar. Let us continue with r∇ /β: From the Bianchi equations (4.143) and (4.145) for ρ3 and σ3 , respectively, / β + l.o.t. ρ3 = −div

one obtains (4.179) (4.180) with

σ3 = −curl / β + l.o.t. div / β = −ρ3

curl / β = −σ3

3 ρ3 = D /3 ρ + trχ · ρ 2 3 σ3 = D (4.182) /3 σ + trχ · σ. 2 From the second integral in Q1 , it follows: (4.181)

∥ τ+ uDT ρ ∥2L2 (Ht ) ≤ cQ1 (t)

∥ τ+ uDT σ ∥2L2 (Ht ) ≤ cQ1 (t)

whereas the first integral in Q1 gives:

∥ τ+ DS ρ ∥2L2 (Ht ) ≤ cQ1 (t)

∥ τ+ DS σ ∥2L2 (Ht ) ≤ cQ1 (t). This yields, in view of (4.174), the estimates: (4.183) (4.184)

∥ τ+ rρ3 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t))

∥ τ+ rσ3 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

102

4. COMPARISON

Using the elliptic theory for ((4.179), (4.180)), we obtain the estimate for the angular derivative of β: (4.185)

∥ τ+ r∇ /β ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Next, we prove the analogue for ρ and σ together, that is for r∇ /(ρ, σ): The Bianchi equation (4.141) for β3 β3 = D /∗1 (−ρ, σ) + l.o.t.

without the error terms, reads as: (4.186) where (4.187)

D /∗1 (−ρ, σ) = β3 , β3 = D /3 β + trχ β.

The second integral of Q1 yields: (4.188)

/T β ∥2L2 (Ht ) ≤ cQ1 (t), ∥ τ+ u D

while the first integral of Q1 and Q0 give (4.189)

/S β ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)). ∥ τ+ D

Thus, with (4.174), it follows that (4.190)

∥ τ+ r β3 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

And from the elliptic theory of (4.186), we conclude (4.191)

∥ τ+ r ∇ /(ρ, σ) ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Now, we consider r∇ /β: From the Bianchi equation (4.140) for α3 : we have (4.192)

α3 = −2D /∗2 β + l.o.t., 1 D /∗2 β = − α3 2

with 1 α3 = D /3 α + trχ α. 2 The second integral in Q1 gives

(4.193)

(4.194)

/T α ∥2L2 (Ht ) ≤ cQ1 (t). ∥ τ+ u D

Similarly, the first integral of Q1 and Q0 yield (4.195)

∥ τ+ D /S α ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Therefore, again in view of (4.174), we deduce (4.196)

∥ τ+ rα3 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

4.7. THE COMPARISON ARGUMENT AND THE CONTROLLING QUANTITIES 103

It then follows from the elliptic theory of (4.192) that (4.197)

/β ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)). ∥ τ+ r ∇

As the last component, we consider r∇ /α: The Bianchi equation (4.142) for β4 :

/ α + l.o.t. β4 = div without error terms, is (4.198)

div / α = β4

and (4.199)

β4 = D /4 β + 2trχβ.

Here, let us remind ourselves of the formulas for S and T , that is (4.170) and (4.171). According to them, one has (4.200)

rL = S − uT.

From the second integral in Q1 , we have (4.201)

∥ τ+ u D /T β ∥2L2 (Ht ) ≤ cQ1 (t),

while from the first integral of Q1 and Q0 , it is (4.202)

∥ τ+ D /S β ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

We deduce, in view of (4.200), (4.203)

∥ τ+ r β4 ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Then, the elliptic theory for (4.198) yields: (4.204)

∥ τ+ r ∇ /α ∥2L2 (Ht ) ≤ c(Q1 (t) + Q0 (t)).

Thus, we have shown the angular derivatives of the curvature components to be bounded. That is, we have proven inequality (4.166). Inequality (4.165): This is clear by the definitions of R0 (W ) and Q0 (W ). Therefore we have proven the comparison theorem with inequality (4.164).

CHAPTER 5

Error Estimates This chapter estimates the error terms that are produced while estimating Q0 and Q1 . In the section about the comparison argument, we used Q0 and Q1 to obtain estimates for the curvature components and their derivatives. A major aspect of the proof of the main theorem (boundedness theorem) of the present chapter is the fact, that the error estimates for the most delicate terms are borderline. Thus, any further relaxation of our assumptions would lead to divergence of the critical integrals, that is, the argument would not close anymore. As this is a very crucial point, let us explain it: Contrary to many problems in analysis, where the principal terms, that is the terms containing the highest derivatives, are the most sensitive ones to estimate, whereas the non-principal terms (containing less or no derivatives) are usually easier to handle, throughout this book and especially in the present chapter, the most difficult terms to be estimated are of higher order with respect to asymptotic behaviour (that is they have less decay), but they are non-principal from the point of view of differentiability. On the other hand the expressions, which are principal with respect to derivatives behave better asymptotically, and therefore can be controlled easier. Thus, by ‘borderline’ we always mean borderline in view of decay (asymptotic behaviour). It is an essential difference between the situation investigated by D. Christodoulou and S. Klainerman in [19] and ours that their worst terms still being of lower order in asymptotic behaviour than ours, the borderline case does not appear, whereas in our setting the estimates for the highest order terms from the point of view of asymptotic behaviour are, in fact, borderline. In the proof of the boundedness theorem, we distinguish two types of vectorfields. Namely the multiplier and the commutation vectorfields. That is, there are two kinds of errors that are generated and which we show to ¯ act as multiplier vectorfields, be controlled. While the vectorfields T and K the T and S are the commutation vectorfields. We notice, that T ‘plays both roles’. The two types of vectorfields will be treated in different ways in our estimates. This shall be explained in the corresponding subsection 5.3, before we state and prove the boundedness theorem. In this chapter, we follow the notation of [19]. The decomposition of the deformation tensors and the Weyl current are done in the same way. For completeness, we write down here all these components that are needed in our estimates. As the results in view of the tools required in the main 105

106

5. ERROR ESTIMATES

theorem of this section, are not new, we do not give all the details of the computations, but refer to [19], chapter 8 ‘Error Estimates’. Whenever it fosters a better understanding, we provide the details. As we do not work with rotational vectorfields, the error estimates simplify considerably. However, the borderline terms can only be controlled by subtle estimates. Moreover, the terms to be controlled require to be treated in a corresponding manner, as it is done in this part. At this point, let us remind ourselves of the definitions of Q0 and Q1 given in (4.160) and (4.161): # ¯ T, T, T ) Q0 (t) = Q(W )(K, Ht # # ˆ ¯ ¯ K, ¯ T, T ). Q(LS W )(K, T, T, T ) + Q(LˆT W )(K, Q1 (t) = Ht

Ht

5.1. Setting First, we introduce some notation needed in this section. We define the tensors p and q involving the deformation tensor (X) π for a vectorfield X, and then we write the Weyl current J in terms of p and q. This notation is used throughout this chapter. Following [19], we give the subsequent definitions. Definition 21. Let X be an arbitrary given vectorfield and let its deformation tensor. Denote by (X) π ˆ the traceless part of (X) π. Then we define p and q to be the tensors: (5.1)

(X)

(X) π

be

pγ = (div (X) π ˆ )γ = Dα(X) π ˆαγ

% 1 $(X) pγ gαβ − (X) pβ gαγ . 3 Let W be a Weyl tensor that fulfills the homogeneous Bianchi equations (4.9). We are now going to express Dα (LˆX W )αβγδ in terms of p and q. To do so, we refer to proposition 5. We then have: (5.3) Dα (LˆX W )αβγδ = J(X, W )βγδ (5.2)

(X)

qαβγ = Dβ (X) π ˆγα − Dγ (X) π ˆβα −

with

% 1$ 1 J (X, W ) + J 2 (X, W ) + J 3 (X, W ) , 2 1 2 3 where J , J and J can be written as: (5.4)

J(X, W ) =

ˆ µν Dν Wµβγδ J 1 (X, W )βγδ = (X) π J 2 (X, W )βγδ = (X) pλ W λ βγδ (5.5)

J 3 (X, W )βγδ = (X) qαβλ W αλ γδ + (X) qαγλ W α β λ δ + (X) qαδλ W α βγ λ .

In the chapter about the comparison argument, the quantities Q0 and Q1 are defined to be the integrals on Ht of the Bel-Robinson tensor Q of the Weyl tensor W and of the modified Lie derivatives LˆS and LˆT of W

5.1. SETTING

107

¯ and T as given in the formulas (4.160) and (4.161). contracted with K Equation (4.33) gives for a vectorfield X: div Q(LˆX W )βγδ = (LˆX W )β µ δ ν J(X, W )µγν + (LˆX W )β µ γ ν J(X, W )µδν (5.6) + ∗(LˆXW )β µ δ νJ(X,W )∗µγν + ∗(LˆXW )β µ γ νJ(X,W )∗µδν . Proposition 4 allows us to write the integrals of (4.160) and (4.161) in / the subsequent form. Let Vt denote the spacetime slab t′ ∈[0,t] Ht′ . #

Ht

¯ T, T, T ) = Q(LˆS W )(K,

#

H0

+

#

¯ T, T, T ) Q(LˆS W )(K,

¯ βT γT δ Φ(div Q(LˆS W ))βγδ K

Vt # 1 ¯ + ΦQ(LˆS W )αβγδ (K) π αβ T γ T δ 2 Vt # ¯ γT δ + ΦQ(LˆS W )αβγδ (T )π αβ K

(5.7)

Vt

#

Ht

¯ K, ¯ T, T ) = Q(LˆT W )(K,

#

H0

+ +

#

¯ K, ¯ T, T ) Q(LˆT W )(K,

#Vt Vt

(5.8) #

¯ T, T, T ) = Q(W )(K,

Ht

1 + 2 #

#

¯ ¯ γT δ ΦQ(LˆT W )αβγδ (K) π αβ K

Vt

¯ γK ¯δ ΦQ(LˆT W )αβγδ (T )π αβ K

¯ T, T, T ) Q(W )(K,

H0

+

(5.9)

¯ βK ¯ γT δ Φ(div Q(LˆT W ))βγδ K

#

¯ βT γT δ Φ(div Q(W ))βγδ K

Vt # 1 ¯ + ΦQ(W )αβγδ (K) π αβ T γ T δ 2 Vt # ¯ γT δ. + ΦQ(W )αβγδ (T ) π αβ K Vt

Now, we have (5.10)

Q1 (W, t) ≤ Q1 (W, 0) + E1 (W, t)

and (5.11)

Q0 (W, t) ≤ Q0 (W, 0) + E0 (W, t),

108

5. ERROR ESTIMATES

with E1 (W, t) =

#

Vt

+

(5.12) and

¯ βK ¯ γT δ | Φ | (div Q(LˆT W ))βγδ K

Vt # 1 ¯ Φ | Q(LˆS W )αβγδ (K) π αβ T γ T δ | + 2 # Vt ¯ γT δ | + Φ | Q(LˆS W )αβγδ (T ) π αβ K #Vt ¯ ¯ γT δ | + Φ | Q(LˆT W )αβγδ (K) π αβ K Vt # 1 ¯ γK ¯δ | + Φ | Q(LˆT W )αβγδ (T ) π αβ K 2 Vt

E0 (W, t) =

(5.13)

#

¯ βT γT δ | Φ | (div Q(LˆS W ))βγδ K

#

¯ βT γT δ | Φ | (div Q(W ))βγδ K Vt # 1 ¯ + Φ | Q(W )αβγδ (K) π αβ T γ T δ | 2 # Vt ¯ γT δ |. + Φ | Q(W )αβγδ (T ) π αβ K Vt

The quantity (5.9) has the same structure as (5.7), as in (5.7) the Lie derivative of W is taken with respect to the scaling vectorfield S which comes equipped with the weights u and u, and in which Q is contracted once with ¯ and twice with T . In fact, the error terms for (5.9), namely the vectorfield K E0 (W, t), are easier to handle because they involve the Bel-Robinson tensor Q of the curvature W , and not of its Lie derivative LˆS W . Therefore, if we estimate the error terms for (5.7) in E1 (W, t), that is the first, third and fourth integral on the right-hand side of (5.12), then the estimates for E0 (W, t) are straightforward. Thus, it is clear, that showing inequality (5.10), then also (5.11) follows. Therefore, in the sequel, we shall prove (5.10). To do so, we need some more notation and preliminary computations. They will be presented in the next subsection. In view of the definition (5.15) right below, let us first define the following integral for scalar functions f : ( # # t1 ' # f := aΦf dµγ dt. (5.14) Cu (t0 ,t1 )

t0

St,u

Here, it is 0 ≤ t0 = t0 (u) ≤ t the value of t for which the cone Cu intersects the boundary of the interior region, and Cu (t0 , t) is the region of Cu contained between Ht0 and Ht . In order to prove the boundedness theorem, we shall need, in addition to the quantity Q1 , the following integrals along any null cone Cu (that is,

5.1. SETTING

109

a level hypersurface of the optical function u) in the exterior region r ≥ of the spacetime, using the notation just introduced in (5.14): # ¯ T, T, e4 ) ˜ 1 (W, u, t) := Q(LˆS W )(K, Q Cu (t0 ,t) # ¯ K, ¯ T, e4 ). + (5.15) Q(LˆT W )(K,

r0 2

Cu (t0 ,t)

Finally, we introduce the quantity Q∗1 for a given fixed value t∗ of t as follows: . ˜ 1 (W, u, t) , (5.16) Q∗1 := max sup Q1 (W, t), sup sup Q t∈[0,t∗ ]

t∈[0,t∗ ] u∗ ≥u0 (t)

with, for a given t, the term u0 (t) is the value of u corresponding to the boundary of the interior region r ≤ r20 . This quantity Q∗1 can be estimated as in the subsequent proposition. Proposition 9. With the notation introduced above, it is $ % (5.17) Q∗1 ≤ 2 Q1 (0) + E1 (t∗ )

To prove this proposition, we need the following lemma (divergence lemma), which is an analogue of proposition 4 of the chapter about the comparison argument. Lemma 5. Let D(u, t0 , t1 ) be a domain of the spacetime bounded by a null hypersurface Cu , which is given by a level hypersurface of u, and the two spacelike hypersurfaces Ht0 and Ht1. Let P µ be a vector satisfying the divergence equation Dµ Pµ = F on a domain D(u, t0 , t1 ). Then it is # # + , P, T dµg −

Cu (t0 ,t1 )

Ht1

with

#

+

Cu (t0 ,t1 )

and

#

,

P, e4 =

D

F =

#

+

, P, e4 =

t1

t0

#

t1

t0

dt

dt

0#

#

Ht0

St,u

0#

Ht ∩D

+

+

, P, T dµg − ,

P, e4 aΦdµγ

#

F

D

1

1

ΦF dµg .

Proof of Lemma 5: This is a well-known result. For the proof of this lemma, we refer to [19], chapter 8, Lemma 8.1.1, p. 209–211. Proof of Proposition 9: The proof relies on inequality (5.10) as well as on proposition 4 and lemma 5.

110

5. ERROR ESTIMATES

To proceed, we recall from proposition 4 that denoting for arbitrary vectorfields X, Y, Z, Pα (W, X, Y, Z) = Q(W )αβγδ X β Y γ Z δ , then, it is 1 divP = X β Y γ Z δ (divQ)βγδ + Qαβγδ ((X) π αβ Y γ Z δ 2 + (Y ) π αβ Z γ X δ + (Z) π αβ X γ Y δ ). Now, we apply lemma 5 twice. First, we apply it to W replaced by LˆS W ¯ Y = T, Z = T , then to W replaced by LˆT W and X = K, ¯ Y = and X = K, ¯ Z = T in any domain D(u, t1 , t2 ) bounded by a null hypersurface Cu(t ,t ) K, 1 2 contained in the exterior region. Then, we use the quantity Q1 (t). That is, we estimate the integrals on the spacelike hypersurfaces by the quantities Q1 (t) and the volume integral on the domain D by supt∈[0,t∗ ] | E1 (t) |. As we have in view of (5.10), that Q1 (t) ≤ Q1 (0) + E1 (t∗ ),

the inequality (5.17) follows, and proposition 9 is proved. At this point, we also write down the following coarea formula for domains bounded by null hypersurfaces, as it can be found in [19], as we shall need this result in the proof of the main theorem. Lemma 6. Let D(u1 , u2 , t0 , t1 ) be a domain of the spacetime bounded by the null hypersurfaces u = u1 , u = u2 and the spacelike hypersurfaces Ht0 , Ht1 . Then, given any scalar function f on D, it is ( # # u2 ' # f dµg = du f D

with

#

Cu (t0 ,t1 )

u1

Cu (t0 ,t1 )

f=

#

t1

t0

dt

'#

(

aΦf dµγ ,

St,u

where Cu(t0 ,t1 ) is the region of the cone Cu between t0 and t1 . 5.2. Weyl Current and Its Null Decomposition The Weyl current Jβγδ is given above. In general, we have the following definition. Definition 22. A tensorfield Jβγδ is called a Weyl current, if it satisfies the equations: Jβγδ + Jγδβ + Jδβγ = 0 Jβγδ + Jβδγ = 0 trJδ = g βγ Jβγδ = 0.

5.2. WEYL CURRENT AND ITS NULL DECOMPOSITION

111

Now, let us introduce the following null decomposition for any given Weyl current J. We give the notation as in [19].

(5.18)

1 Λ(J) = J434 ; 4 1 K(J) = ϵAB J4AB ; 4 1 Ξ(J)A = J44A ; 2 1 I(J)A = J34A ; 2 2Θ(J)AB =

One sees that

1 Λ(J) = J343 4 1 K(J) = ϵAB J3AB 4 1 Ξ(J)A = J33A 2 1 I(J)A = J43A 2 JA4B + JB4A − (δ DC JC4D )δAB

2Θ(J)AB = JA3B + JB3A − (δ DC JC3D )δAB . JA4B = Θ(J)AB − ΛδAB + KϵAB

JA3B = Θ(J)AB − ΛδAB + KϵAB

JABC = ϵBC (∗ I(J)A + ∗ I(J)A ). For the duals, one finds:

(5.19)

Λ(J ∗ ) = K(J) K(J ∗ ) = − Λ(J) Ξ(J ∗ ) = −∗ Ξ(J) I(J ∗ ) = −∗ I(J) Θ(J ∗ ) = −∗ Θ(J)

; ; ; ; ;

Λ(J ∗ ) = −K(J) K(J ∗ ) = Λ(J) Ξ(J ∗ ) = ∗ Ξ(J) I(J ∗ ) = ∗ I(J) Θ(J ∗ ) = ∗ Θ(J).

With this notation, we can now write the divergence of Q(LˆS W ) as well as the divergence of Q(LˆT W ) as the sum of products of a curvature component with a null component of the corresponding current J. This follows directly from equation (5.6), and is formulated in the next proposition. We recall again the fact, that the precise numerical coefficients are not important. Proposition 10. Let W be any Weyl tensor. Let S and T be the scalar and the time vectorfields, respectively. Denote by D(S, W ) and D(T, W ) the divergences: D(S, W ) = div Q(LˆS W ) and D(T, W ) = div Q(LˆT W ). Then the following formulas hold everywhere in the exterior region: $ % $ % ¯ K, ¯ T ) ≈ τ+4 D444 + D344 + τ−2 τ+2 D344 + D334 D(T, W )(K, $ % + τ−4 D334 + D333 (5.20) $ % ¯ T, T ) ≈ τ 2 D444 + D344 + D334 D(S, W )(K, + $ % + τ−2 D344 + D334 + D333 (5.21)

112

5. ERROR ESTIMATES

with the following components of D(X, W ), where X denotes S and T , respectively: D(X, W )444 ≈ α(LˆX W ) · Θ(X, W ) − β(LˆX W ) · Ξ(X, W ) D(X, W )443 ≈ ρ(LˆX W )Λ(X, W ) + σ(LˆX W )K(X, W ) − β(LˆX W )I(X, W )

D(X, W )334 ≈ ρ(LˆX W )Λ(X, W ) − σ(LˆX W )K(X, W ) − β(LˆX W )I(X, W )

D(X, W )333 ≈ α(LˆX W ) · Θ(X, W ) + β(LˆX W ) · Ξ(X, W ),

(5.22)

and Λ(X, W ), Λ(X, W ), K(X, W ), K(X, W ), I(X, W ), I(X, W ), Ξ(X, W ), Ξ (X, W ), Θ(X, W ), Θ(X, W ) are the null components of the Weyl current J(X, W ) given above. Proof: This proposition is directly derived from signature considerations, formula (5.6) for the divergence of Q(LˆX W ) and considering the duality relative to the components 3 and 4. Also note that D334 = D433 . Going back to equation (5.3), the Weyl current J(X, W ) on the righthand side is decomposed into J 1 (X, W ), J 2 (X, W ), J 3 (X, W ), which are themselves expressed in terms of (X) p and (X) q introduced in definition 21. We remark that (X) q satisfies all the properties of a Weyl current. Therefore, it can be split into the null components Λ((X)q), Λ((X)q), K((X)q), K((X)q), . . . In a next step, in view of (5.3), we want to write down the null decomposition of the Weyl current J(X, W ) relative to the null decomposition of LˆX W and that of (X) π ˆ , (X) p, (X) q. To do this, we first give the null decomposition of (X) π ˆ and (X) p. Definition 23. Let (X) π be the deformation tensor of an arbitrary vectorfield X. Then the null decomposition of its traceless part (X) π ˆ is defined to be as follows: (X)

iAB = (X) π ˆAB

(X)

mA = (X) π ˆ4A

(X)

n = (X) π ˆ44 j = (X) π ˆ34

(X) (X)

mA = (X) π ˆ3A

(X)

(5.23) Given the vector

(X) p

as in definition 21. Then we decompose (X)

(X) p

p3 , (X) p4

/pA = (X) pA ,

(X)

(5.24) where

n = (X) π ˆ33 .

(X) p

3

and

(X) p

4

are scalars and

(X)/ p

A

is an S-tangent vector.

into

5.2. WEYL CURRENT AND ITS NULL DECOMPOSITION

113

In the next proposition, we state the null decompositions of J 1 , J 2 and They have been derived by D. Christodoulou and S. Klainerman in [19] and can be found in their chapter 8, proposition 8.1.4, p. 214–218. They also give the proof of these formulas (p. 219–222). As we need them to do the error estimates, we consider it as necessary to write them down at this place. We remark that the exact numerical coefficients in these terms are not important in anything that follows. Thus, we give the leading terms without considering the coefficients. Let us have a closer look at these leading terms. They are quadratic terms, that is, a product of a null component of the curvature or its derivative with a null component of the deformation tensor. To simplify the notation, we introduce an expression as follows. We denote by

J 3.

{, } any quadratic form with coefficients depending only on the induced metric and the area form of St,u . For instance, {(X) i, ∇ / α} denotes a quadratic (X) (X) form between the null component i of π ˆ and the derivative ∇ / of the curvature null component α. Moreover, we write l.o.t. for the lower order terms. Here, they are cubic with respect to (X) π, W and ˆ χ, ˆ and linear with regard to each of them the Ricci coefficients ξ, ζ, ζ, ν, ν, χ, separately. They are lower order than all other terms both with regard to their asymptotic behaviour in the wave zone as well as the order of differentiability relative to W . This can be checked by direct computations. Thus, when estimating our quantities in the proof of the boundedness theorem, the lower order terms are dominated by the quadratic ones, they (l.o.t.) decay considerably faster than these dominating terms and therefore can be neglected. Using this notation, we state the following proposition. Proposition 11. Let X be an arbitrary vectorfield, and let (X) π ˆ , (X) p, 1 2 3 be defined as above. Let J = J(X, W ) = J + J + J be the Weyl current defined by (5.3). Then the following holds: (i) The null decomposition of J 1 is given by the formulas: (X) q

Ξ(J 1 ) = {(X) i, ∇ / α} + {(X) m, α3 } + {(X) m, α4 }

/ β} + {(X) j, β 3 } + {(X) n, β 4 } + {(X) m, ∇ % $ + trχ {(X) m, α} + {((X) i, (X) j), β} + {(X) m, (ρ, σ)} $ % + trχ {(X) m, α} + {(X) n, β} + l.o.t.

114

5. ERROR ESTIMATES

Θ(J 1 ) = {(X) m, ∇ / α} + {(X) n, α3 } + {(X) j, α4 }

+ {(X) i, ∇ / β} + {(X) m, β 3 } + {(X) m, β 4 }

/ (ρ, σ)} + {(X) j, (ρ3 , σ3 )} + {(X) n, (ρ4 , σ4 )} + {(X) m, ∇ % $ + trχ {(X) n, α} + {(X) m, β} + {((X) i, (X) j), (ρ, σ)} + {(X) m, β} $ % + trχ {(X) j, α} + {(X) m, β} + {(X) n, ∇ / (ρ, σ)} + l.o.t.

Λ(J 1 ) = {(X) i, ∇ / β} + {(X) m, β 3 } + {(X) m, β 4 }

/ (ρ, σ)} + {(X) j, (ρ3 , σ3 )} + {(X) n, (ρ4 , σ4 )} + {(X) m, ∇ % $ + trχ {(X) m, β} + {(X) j, (ρ, σ)} + {(X) m, β} $ % + trχ {(X) i, α} + {(X) m, β} + {(X) n, (ρ, σ)} + l.o.t.

/ β} + {(X) m, β 3 } + {(X) m, β 4 } K(J 1 ) = {(X) i, ∇

/ (ρ, σ)} + {(X) j, (ρ3 , σ3 )} + {(X) n, (ρ4 , σ4 )} + {(X) m, ∇ $ % + trχ {(X) m, β} + {(X) j, (ρ, σ)} + {(X) m, β} $ % + trχ {(X) i, α} + {(X) m, β} + {(X) n, (ρ, σ)} + l.o.t.

I(J 1 ) = {(X) m, ∇ / β} + {(X) n, β 3 } + {(X) j, β 4 }

/ (ρ, σ)} + {(X) m, (ρ3 , σ3 )} + {(X) m, (ρ4 , σ4 )} + {(X) i, ∇ % $ + trχ {(X) n, β} + {(X) m, (ρ, σ)} + {(X) i, β} $ % + trχ {(X) m, α} + {((X) i, (X) j), β} + {(X) m, (ρ, σ)} + l.o.t.

/ α} + {(X) m, α4 } + {(X) m, α3 } Ξ(J 1 ) = {(X) i, ∇

+ {(X) m, ∇ / β} + {(X) j, β4 } + {(X) n, β3 } $ % + trχ {(X) m, α} + {((X) i, (X) j), β} + {(X) m, (ρ, σ)} % $ + trχ {(X) m, α} + {(X) n, β} + l.o.t.

Θ(J 1 ) = {(X) m, ∇ / α} + {(X) n, α4 } + {(X) j, α3 }

+ {(X) i, ∇ / β} + {(X) m, β4 } + {(X) m, β3 }

+ {(X) m, ∇ / (ρ, σ)} + {(X) j, (ρ4 , σ4 )} + {(X) n, (ρ3 , σ3 )} $ (X) % + trχ { n, α} + {(X) m, β} + {((X) i,(X) j), (ρ, σ)} + {(X) m, β} $ % + trχ {(X) j, α} + {(X) m, β} + {(X) n, (ρ, σ)} + l.o.t.

/ β} + {(X) m, β4 } + {(X) m, β3 } Λ(J 1 ) = {(X) i, ∇

+ {(X) m, ∇ / (ρ, σ)} + {(X) j, (ρ4 , σ4 )} + {(X) n, (ρ3 , σ3 )} $ % + trχ {(X) m, β} + {(X) j, (ρ, σ)} + {(X) m, β} % $ + trχ {(X) i, α} + {(X) m, β} + {(X) n, (ρ, σ)} + l.o.t.

5.2. WEYL CURRENT AND ITS NULL DECOMPOSITION

115

K(J 1 ) = {(X) i, ∇ / β} + {(X) m, β4 } + {(X) m, β3 }

+ {(X) m, ∇ / (ρ, σ)} + {(X) j, (ρ4 , σ4 )} + {(X) n, (ρ3 , σ3 )} $ % + trχ {(X) m, β} + {(X) j, (ρ, σ)} + {(X) m, β} % $ + trχ {(X) i, α} + {(X) m, β} + {(X) n, (ρ, σ)} + l.o.t.

/ β} + {(X) n, β4 } + {(X) j, β3 } I(J 1 ) = {(X) m, ∇

+ {(X) i, ∇ / (ρ, σ)} + {(X) m, (ρ4 , σ4 )} + {(X) m, (ρ3 , σ3 )} $ % + trχ {(X) n, β} + {(X) m, (ρ, σ)} + {(X) i, β} % $ + trχ {(X) m, α} + {((X) i, (X) j), β} + {(X) m, (ρ, σ)} + l.o.t.

By signature considerations only, the term {(X) i, α}

could appear in the expressions for Θ(J 1 ) and for Θ(J 1 ), but in fact, it vanishes. (ii) The null decomposition of J 2 is given by the following formulas: Ξ(J 2 ) = {(X)/p , α} + {(X) p3 , β}

Θ(J 2 ) = {(X) p4 , α} + {(X)/p , β} + {(X) p3 , (ρ, σ)} Λ(J 2 ) = {(X)/p , β} + {(X) p3 , (ρ, σ)}

K(J 2 ) = {(X)/p , β} + {(X) p3 , (ρ, σ)}

I(J 2 ) = {(X) p4 , β} + {(X)/p , (ρ, σ)} I(J 2 ) = {(X) p3 , β} + {(X)/p , (ρ, σ)}

K(J 2 ) = {(X)/p , β} + {(X) p4 , (ρ, σ)} Λ(J 2 ) = {(X)/p , β} + {(X) p4 , (ρ, σ)}

Θ(J 2 ) = {(X) p3 , α} + {(X)/p , β} + {(X) p4 , (ρ, σ)} Ξ(J 2 ) = {(X)/p , α} + {(X) p4 , β}.

(iii) The null decomposition of J 3 is given by the following formulas: Ξ(J 3 ) = {α, (I, I)((X) q)} + {β, (K, Λ, Θ)((X) q)} + {(ρ, σ), Ξ((X) q)}

Θ(J 3 ) = {α, K((X) q)} + {α, Λ((X) q)} + {β, (I, I)((X) q)} + {(ρ, σ), Θ((X) q)} Λ(J 3 ) = {α, Θ((X) q)} + {(ρ, σ), (K, Λ)((X) q)} + {β, Ξ((X) q)}

K(J 3 ) = {α, Θ((X) q)} + {(ρ, σ), (K, Λ)((X) q)} + {β, Ξ((X) q)}

116

5. ERROR ESTIMATES

I(J 3 ) = {α, Ξ((X) q)} + {β, (K, Λ, Θ)((X) q)} + {(ρ, σ), (I, I)((X) q)} + {β, (K, Λ, Θ)((X) q)}

I(J 3 ) = {β, (K, Λ, Θ)((X) q)} + {(ρ, σ), (I, I)((X) q)} + {β, (K, Λ, Θ)((X) q)} + {α, Ξ((X) q)}

K(J 3 ) = {α, Θ((X) q)} + {(ρ, σ), (K, Λ)((X) q)} + {β, Ξ((X) q)} Λ(J 3 ) = {α, Θ((X) q)} + {(ρ, σ), (K, Λ)((X) q)} + {β, Ξ((X) q)}

Θ(J 3 ) = {α, K((X) q)} + {α, Λ((X) q)} + {β, (I, I)((X) q)} + {(ρ, σ), Θ((X) q)} Ξ(J 3 ) = {α, (I, I)((X) q)} + {β, (K, Λ, Θ)((X) q)} + {(ρ, σ), Ξ((X) q)}.

By signature considerations only, the following terms could appear in the corresponding formulas, but in fact, they vanish: {α, Θ((X) q)}, {β, Ξ((X) q)}

in Θ(J 3 )

{β, (I, I)((X) q)}

in K(J 3 )

{β, (I, I)((X) q)}

in Λ(J 3 )

{α, Ξ((X) q)}

in I(J 3 )

{α, Ξ((X) q)}

in I(J 3 )

{β, (I, I)((X) q)}

in Λ(J 3 )

{β, (I, I)((X) q)}

in K(J 3 )

{α, Θ((X) q)}, {β, Ξ((X) q)}

in Θ(J 3 ).

Remark: In view of these last terms which do not appear, but from the point of view of signature considerations in fact could, the {α, Θ((X) q)} as well as the {α, Θ((X) q)} cannot be part of Θ(J 3 ) respectively Θ(J 3 ) because these are symmetric, trace-free tensors. Whereas the fact that the term {β, Ξ((X) q)} is not present in Θ(J 3 ), is obtained by calculations, as are also the remaining ones. Moreover, the corresponding results directly follow for the duals. Proof: The proof of this proposition can be found in [19], chapter 8, p. 219–222. We only remark two points here. First, the asymptotic behaviour of the Ricci coefficients and of the null components of W are given in the chapter about the comparison theorem. And second, the null components of the Weyl current have the following signatures: s(Ξ) = +2 s(K, Λ, Θ) = +1 s(I, I) = 0

5.2. WEYL CURRENT AND ITS NULL DECOMPOSITION

117

s(K, Λ, Θ) = −1

s(Ξ) = −2.

(X) p

and (X) q: For (X) p and (X) q with respect to a vectorfield X, one can derive the following formulas. They will be needed to prove the boundedness theorem. Steps to calculate them can be found in [19] in the proof of their proposition 8.2.1, p. 229–232. (X) p

3

= div /

(X) m

1$ D4 2

−

(X) n

+ D3

(X) j

% 1 $ 1 − trχ (X) j + tr (X) i − trχ 2 2 (X) (X) j)} + {(ϵ, ζ, ζ), + {χ, ˆ ( i,

(5.25) (X) p

4

= div /

(X) m

1$ D3 2

−

(X) n

+ D4

(5.26) (X)/ p

A

(5.27)

(X) n

+ {χ, ˆ

(X) m}

(X) j

% 1 $ 1 − trχ (X) j + tr (X) i − trχ 2 2 + {χ, ˆ ((X) i, (X) j)} + {(ϵ, ζ, ζ),

%

%

(X) n

(X) n}

+ {ξ,

+ {χ, ˆ

(X) m}

(X) n}

(X) m}

% 1$ D4 (X) m + D3 (X) m 2 3 3 − trχ (X) m − trχ (X) m + {χ, ˆ (X) m} + {χ, ˆ (X) m} 4 4 + {ν, (X) m} + {ν, (X) m} + {(ζ, ζ), ((X) i, (X) j)}

= div /

(X) i

+ {ξ,

−

(X) n}.

And, ΞA ((X) q) = (5.28) (5.29)

1$ D / 2 3 + {χ, ˆ

Λ((X) q) = D4 (5.30)

1 + trχ (X) mA 2 (X) m} + {ν, (X) m} + {(ζ, ϵ), (X) n} + {ξ, (

(X) m A

(X) n

−∇ /A

− D3

(X) j

(X) n

+

2 3

%

(X) p

3

+ {(ζ, ζ),

(X) m}

(X) i, (X) j)}

+ {ν,

(X) j}

1 / (X) m + {χ, ˆ (X) n} + {χ, ˆ ( (X) i, (X) j)} + {ϵ, (X) m} K((X) q) = curl 2 $ % / 3 (X) iAB − ∇ / A (X) mB + ∇ / B (X) mA − div / (X) mδAB Θ((X) q) = 2D $ % 1 + trχ (X) iAB − tr (X) iδAB + {ξ, (X) m} + {(ζ, ϵ), (X) m} 2 (X) + {χ, ˆ (5.31) n} + {χ, ˆ ( (X) i, (X) j)}

118

5. ERROR ESTIMATES

I A ((X) q) =

'

% 1 1$ D / 3 (X) mA −∇ / A (X) j + trχ (X) mA +trχ (X) mA 2 2 ( 2 ˆ (X) m}+{ˆ χ, (X) m} +{ν, (X) m} + (X)/pA + {χ, 3

+ {ν, (X) m} + {ξ, (X) n} + {ζ, ((X) i, (X) j)} ' % 1 1$ (X) / A (X) j + trχ (X) mA + trχ (X) mA D / 4 (X) mA − ∇ IA ( q) = 2 2 ( 2 χ, (X) m} +{χ, ˆ (X) m} +{ν, (X) m} + (X)/pA +{ˆ 3

(5.32)

(5.33)

+ {ν,

(X) m}

+ {ζ, ( (X) i, (X) j)} $ % / 4 (X) iAB − ∇ / A (X) mB + ∇ / B (X) mA − div / (X) mδAB Θ((X) q) = 2D ' ( 1 (X) (X) + trχ iAB − tr iδAB + {(ζ, ϵ), (X) m} 2

(5.34)

+ {χ, ˆ

(X) n}

(5.35)

+ {χ, ˆ ((X) i,

1 K((X) q) = curl / (X) m + {χ, ˆ 2 (5.36) Λ((X) q) = D3 ΞA ((X) q) = (5.37)

(X) n

− D4

(X) n}

(X) j

+

2 3

(X) j)}

+ {χ, ˆ ((X) i, (X) p

4

(X) j)}

+ {(ζ, ζ),

+ {ϵ,

(X) m}

(X) m}

+ {ν,

(X) j}

% 1 1$ / A (X) n + trχ (X) mA D / 4 (X) mA − ∇ 2 2 (X) (X) + {χ, ˆ m} + {ν, m} + {(ζ, ϵ), (X) n}.

5.3. Boundedness Theorem: Statement and Proof Goal of this chapter is to state and to prove the boundedness theorem for the controlling quantities Q0 (t) and Q1 (t). In order to control the corresponding error terms, one has to make the subsequent assumptions on the ¯ traceless parts of the deformation tensors for the vectorfields T, S, K. The chapter splits into the subsections: • Assumptions • Consequences • Statement of the Theorem • Proof of the Theorem. In the next subsection, we state the assumptions on the components of the deformation tensors. We split them into an interior and an exterior part. In the exterior region, L∞ -norms in Ht are required in the assumptions ((5.48)–(5.54)). In the proof of the boundedness theorem, the deformation tensors appear up to first derivatives. Inequalities ((5.55)–(5.67)) control the

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

119

first derivatives of the deformation tensors in the L2 -norm in Ht . However, we have to make the assumptions ((5.68)–(5.80)) on the second derivatives in addition, in order to obtain the first derivatives bounded in the surfaces St,u , as we need this in the proof. In fact, from controlling the second derivatives of the deformation tensors in L2 (Ht ) we obtain by Sobolev inequalities that the deformation tensors are pointwise bounded and by the trace lemma that they lie in L4 (St,u ) up to first derivatives. In the subsection ‘Consequences’, we then present inequalities for the corresponding norms on the surfaces St,u as well as for the L2 (Ht )-norms of (T ) p, (T ) q, (S) p, (S) q. Estimating the error terms, we have to show that we control the corresponding integrals on Vte∗ . To achieve this, we use two ways of integration, applying the coarea formulas for these situations. One way is to integrate first on Hte and then with respect to t, the other one is to integrate first on Cu and then with respect to u. This shall be done explicitly in the proof of the main theorem later in this chapter. We recall from the introduction of this chapter, that there are two kinds of errors that are generated and which we show to be controlled in the proof of the boundedness theorem. Therefore, we distinguish two types of vectorfields: the multiplier and the commutation vectorfields. While the ¯ act as multiplier vectorfields, the T and S are the vectorfields T and K commutation vectorfields, T showing up as either one. The two types of vectorfields will be treated in different ways in our estimates. Denote by X the multiplier and by Y the commutation vectorfields. The error terms involve products of the form (5.38)

(X) π ˆ

· W (LˆT R),

(Y ) π ˆ

· W (LˆT R),

as well as (5.39)

∇

and

(Y ) π ˆ

(5.40)

· ∇W (LˆT R),

(X) π ˆ

∇

· W (LˆS R)

(Y ) π ˆ

(Y ) π ˆ

· W (LˆS R)

· ∇W (LˆS R).

In order to estimate the corresponding integrals, for the type (5.38), that is the terms involving the multiplier vectorfields, we use that (X) π ˆ

∈ L∞ (Ht ),

and W (LˆT R), W (LˆS R) ∈ L2 (Ht ).

In view of the commutation vectorfields, we distinguish between the kinds (5.39) and (5.40). Estimating the integrals involving (5.40), we use that (Y ) π ˆ

∈ L∞ (Ht ),

and ∇W (LˆT R), ∇W (LˆS R) ∈ L2 (Ht ).

The error terms of type (5.40) appear in the first component J 1 of the Weyl current and are integrated first on Hte and then with respect to t, yielding the required bound for the integral on Vte∗ . Contrary to that, the

120

5. ERROR ESTIMATES

error terms involving (5.39), which appear in the second and third components J 2 and J 3 of the Weyl current, have to be treated in a different way. First, we integrate them on the surfaces St,u using the fact that ∇

(Y ) π ˆ

as well as W (LˆT R), W (LˆS R) ∈ L4 (St,u ).

∈ L4 (St,u ),

Then there are two choices for (5.39), namely, the integration on Hte or on Cu as explained above. 5.3.1. Assumptions. The assumptions are stated using the interior ∥ · ∥p,i and the exterior ∥ · ∥p,e norms with p = 2, ∞, introduced earlier. Interior Assumptions

∥ (T ) π ˆ ∥∞,i ≤ ϵ0 (1 + t)−1

(5.41)

ˆ ∥∞,i ≤ ϵ0 ∥ (S) π

(5.42)

¯

∥ (K) π ˆ ∥∞,i ≤ ϵ0 (1 + t)1

(5.43) (5.44)

∥D

(5.45)

∥D

∥ D2

(5.46)

(T ) π ˆ

1

(S) π ˆ

∥2,i ≤ ϵ0 (1 + t) 2

(T ) π ˆ

∥ D2

(5.47)

1

∥2,i ≤ ϵ0 (1 + t)− 2 3

∥2,i ≤ ϵ0 (1 + t)− 2 1

(S) π ˆ

∥2,i ≤ ϵ0 (1 + t)− 2

Exterior Assumptions 1

(5.48)

∥ rτ−2

(5.49) (5.50) (5.51)

∥r

(5.54) (5.55) (5.56) (5.57) (5.58) (5.59)

$(T )

∥r

(5.52) (5.53)

3 2

∥r

1 2

∥r

$ ¯ − 1 (K) 2

i,

(S) j, (S) m, (S) n 3 2

$ −1 (S)

∥ r τ−

i,

(T ) j

(T ) m, (T ) n, (T ) n

m,

$(S)

3 2

(T ) i

m,

(S) n

¯ ¯ ¯ (K) j, (K) m, (K) n 3

$ ¯ −2 (K)

∥ r + 2 τ−

∥r

− 12

m,

¯ (K) n

% %

%

%

%

∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0 ∥∞,e ≤ ϵ0

+1 τ− 2 ∇ / (T ) i

3

3

∥2,e ≤ ϵ0 (1 + t)−1

/ 3 (T ) i ∥2,e ∥ r− 2 τ−2 D 1 ∥D / 4 (T ) i + trχ (T ) i ∥2,e 2 $(T ) (T ) % (T ) (T ) ∥∇ / j, m, m, n, (T ) n ∥2,e $ % ∥D / 4 (T ) j, (T ) m, (T ) m, (T ) n (T ) n ∥2,e

≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

(5.60) (5.61) (5.62) (5.63) (5.64) (5.65) (5.66) (5.67)

∥ r−1 τ− D /3

$(T )

/ ∥ r−1 ∇

j,

$(S)

(T ) m, (T ) m, (T ) n

i,

/3 ∥ r−2 τ−1 D

/4 ∥ τ−−1 D

$(S)

i,

(5.70) (5.71) (5.72) (5.73) (5.74) (5.75) (5.76) (5.77) (5.78) (5.79) (5.80)

(T ) n

(S) j, (S) m, (S) n

/ ∥ τ−−1 ∇ $ −1 (S) (S) /4 i, j, ∥r D

$(S)

m,

(S) n

(S) m, (S) n

$(S)

m,

(S) n

(S) j, (S) m, (S) n

/3 ∥ r−1 D

$(S)

1

m,

(S) n

+1

∥ r 2 τ− 2 ∇ /2

(5.68) (5.69)

/3 ∥ r−1 τ− D

1

% %

%

%

%

%

%

(T ) i

3

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

/D / 3 (T ) i ∥2,e ∥ r− 2 τ−2 ∇ $ % 1 ∥ r∇ / D / 4 (T ) i + trχ (T ) i ∥2,e 2 % 2 $(T ) (T ) (T ) (T j, m, m, ) n, (T ) n ∥2,e ∥ r∇ / $ % ∥ r∇ /D / 4 (T ) j, (T ) m, (T ) m, (T ) n (T ) n ∥2,e $ % /D / 3 (T ) j, (T ) m, (T ) m, (T ) n ∥2,e ∥ τ− ∇ ∥∇ /

2 $(S)

/D /3 ∥ τ− ∇

i,

(T ) n

(S) j, (S) m, (S) n

/2 ∥ r1 τ−−1 ∇ $ ∥∇ /D / 4 (S) i, (S) j,

/D /4 ∥ rτ−−1 ∇ $ /D / 3 (S) i, (S) j, ∥ r−1 τ−1 ∇ ∥∇ /D /3

$(S)

m,

(S) n

(S) m, (S) n

$(S)

m,

(S) n

(S) m, (S) n

$(S)

m,

(S) n

%

%

%

%

%

%

121

≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1 ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1 ∥2,e ≤ ϵ0 (1 + t)−1

∥2,e ≤ ϵ0 (1 + t)−1

These assumptions imply estimates on other quantities that we shall use to prove the main theorem in this section. To state these implications, is the purpose of the next subsection.

5.3.2. Consequences. We are now going to state in the next proposition the consequences of the assumptions (5.41)–(5.80), that are required in what follows. In addition to the norms used in the ‘Assumptions’, here, we give the inequalities on St,u in L4 -norms of St,u . Proposition 12. Let (T ) p, (S) p and torfields T and S be defined as before.

(T ) q, (S) q

with respect to the vec-

122

5. ERROR ESTIMATES

Then the Interior and Exterior Assumptions (5.41)–(5.80) imply: 1. Interior Estimates for

(T ) p, (T ) q 1

∥ (T ) p, (T ) q ∥2,i ≤ c(1 + t)− 2 % $ 3 ∥ D (T ) p, (T ) q ∥2,i ≤ c(1 + t)− 2

(5.81) (5.82)

2. Exterior Estimates for ∥ r−1 τ−

(5.83)

∥ r−1 τ−

(5.84)

∥ r−1 τ−

(5.85)

(T ) p, (T ) q

(T ) p

4

(T ) p

3

(T )/ p

∥2,e ≤ cϵ0 (1 + t)−1 ∥2,e ≤ cϵ0 (1 + t)−1 ∥2,e ≤ cϵ0 (1 + t)−1

∥ r−1 τ− Λ((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.86)

∥ r−1 τ− Λ((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.87)

∥ K((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.88) (5.89)

∥ r−1 τ− K((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.90)

∥ r− 2 τ−2 Θ((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.91)

∥ r− 2 τ−2 Θ((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.92) (5.94) (5.95) L4 -norms in St,u :

(5.97) (5.98) (5.99) (5.100) (5.101) (5.102) (5.103) (5.104)

1

3

3

∥ r−1 τ− I(

(T ) q)

∥ Ξ(

(T ) q)

∥2,e ≤ cϵ0 (1 + t)−1

∥ r−1 τ− I((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.93)

(5.96)

1

∥2,e ≤ cϵ0 (1 + t)−1

∥ r−1 τ− Ξ((T ) q) ∥2,e ≤ cϵ0 (1 + t)−1 1

∥ τ−2

(T ) i

1

1

∥4,St,u ≤ cϵ0 (1 + t)− 2 1

/ (T ) i ∥4,St,u ≤ cϵ0 (1 + t)− 2 ∥ τ−2 r∇ 1 $ % 1 1 ∥ τ−2 r D / 4 (T ) i + trχ (T ) i ∥4,St,u ≤ cϵ0 (1 + t)− 2 2 3

∥ τ−2 D /3

(T ) i

1

∥4,St,u ≤ cϵ0 (1 + t)− 2

∥ (T ) j, (T ) m, (T ) m, (T ) n, (T ) n ∥4,St,u % $(T ) (T ) j, m, (T ) m, (T ) n, (T ) n ∥4,St,u ∥ r∇ / % $ ∥ rD / 4 (T ) j, (T ) m, (T ) m, (T ) n, (T ) n ∥4,St,u $ % / 3 (T ) j, (T ) m, (T ) m, (T ) n ∥4,St,u ∥ τ− D /3 ∥ τ− D

(T ) n

≤ cϵ0 (1 + t)−1 ≤ cϵ0 (1 + t)−1 ≤ cϵ0 (1 + t)−1 ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1

Denote by an index H the components of (T ) q estimated on Ht and then integrated with respect to t and denote by an index C the remaining ones,

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

123

which are estimated on Cu and then integrated with respect to u. This will be introduced and used in the proof of the boundedness theorem. (5.105) (5.106)

1

∥ r2 τ−2 (KH , ΛH , ΘH )((T ) q) ∥∞,e ≤ cϵ0 5

∥ r 2 ΞH ((T ) q) ∥∞,e ≤ cϵ0 5

∥ r 2 ΞH ((T ) q) ∥∞,e ≤ cϵ0

(5.107) (5.108) (5.109)

1

∥ r2 τ−2 (K H , ΛH , ΘH )((T ) q) ∥∞,e ≤ cϵ0 5

∥ r 2 (IH , I H )((T ) q) ∥∞,e ≤ cϵ0 1

1

1

3

(5.110)

∥ r 2 τ−2 ΘC ((T ) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

(5.111)

∥ r− 2 τ−2 ΘC ((T ) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

(5.112) (5.113) (5.114)

∥ τ− (ΛC ((T ) q), KC ((T ) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ τ− (ΛC ((T ) q), K C ((T ) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1 ∥ rΞC ((T ) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ τ− ΞC ((T ) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

(5.115) (5.116) (5.117)

∥ τ− (IC ((T ) q), I C ((T ) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1 ∥ τ−

(5.118)

∥ τ−

(5.119)

∥ τ−

3. Interior Estimates for (5.120) (5.121)

∥ ∥D

$(S) $(S)

(5.122)

(S) q

p,

(S) q

(5.123) (5.124) (5.125) (5.126) (5.127) (5.128)

∥ r−1

∥ r−1

∥ r−2 τ−

(T ) p

3

(T ) p

4

∥4,St,u ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1 ∥4,St,u ≤ cϵ0 (1 + t)−1

(S) p, (S) q

p,

4. Exterior Estimates for

(T )/ p

% %

1

∥2,i ≤ cϵ0 (1 + t) 2

1

∥2,i ≤ cϵ0 (1 + t)− 2

(S) p, (S) q

(S)/ p (S) p

4

(S) p

3

∥2,e ≤ cϵ0 (1 + t)−1

∥2,e ≤ cϵ0 (1 + t)−1 ∥2,e ≤ cϵ0 (1 + t)−1

∥ r−1 Λ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

∥ r−2 τ− Λ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1 ∥ r−1 K((S) q) ∥2,e ≤ cϵ0 (1 + t)−1 ∥ r−1 K((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

124

5. ERROR ESTIMATES

∥ r−1 Θ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.129)

∥ r−2 τ− Θ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.130)

∥ r−1 I((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.131)

∥ r−2 τ− I((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.132)

∥ r−1 Ξ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.133)

∥ r−2 τ− Ξ((S) q) ∥2,e ≤ cϵ0 (1 + t)−1

(5.134) L4 -norms in St,u : (5.135) (5.136) (5.137) (5.138) (5.139) (5.140) (5.141) (5.142)

∥ r−1 ∥∇ / ∥D /4

$(S) $(S)

i,

(S)j, (S) m, (S) n

∥ τ−−1

i, i,

m,

(S) n

(S)j, (S) m, (S) n

/ ∥ τ−−1 r∇

$(S)

$(S)

$(S)

m,

(S) n

(S)j, (S) m, (S) n

/4 ∥ τ−−1 rD $ / 3 (S) i, (S)j, ∥ τ− r−1 D ∥D /3

$(S)

m,

(S) n

(S) m, (S) n

$(S)

m,

(S) n

%

%

%

%

%

%

%

%

∥4,St,u ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1 ∥4,St,u ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1 ∥4,St,u ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1 ∥4,St,u ≤ cϵ0 (1 + t)−1

∥4,St,u ≤ cϵ0 (1 + t)−1

Similarly as above and in view of the proof of the boundedness theorem, we denote by an index H the components of (S) q estimated on Ht and then integrated with respect to t and we denote by an index C the remaining ones, which are estimated on Cu and then integrated with respect to u. (5.143) (5.144) (5.145) (5.146) (5.147) (5.148) (5.149) (5.150) (5.151) (5.152)

3

∥ r 2 (K H, ΛH, ΘH )((S) q) ∥∞,e ≤ cϵ0 3

∥ r 2 ΞH ((S) q) ∥∞,e ≤ cϵ0

3

∥ r 2 (KH, ΛH, ΘH )((S) q) ∥∞,e ≤ cϵ0 3

∥ r 2 (IH , I H )((S) q) ∥∞,e ≤ cϵ0 3

∥ r 2 ΞH ((S) q) ∥∞,e ≤ cϵ0

∥ Θ((S) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ r−1 τ− ΘC ((S) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ r−1 τ− (ΛC ((S) q), K C ((S) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1 % $ ∥ ΛC ((S) q), KC ((S) q) ∥4,St,u ≤ cϵ0 (1 + t)−1 % $ ∥ IC ((S) q), I C ((S) q) ∥4,St,u ≤ cϵ0 (1 + t)−1

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

(5.153) (5.154) (5.155) (5.156) (5.157)

125

∥ r−1 τ− ΞC ((S) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1 ∥ rτ−−1 Ξ((S) q)) ∥4,St,u ≤ cϵ0 (1 + t)−1 ∥ (S)/p ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ (S) p4 ∥4,St,u ≤ cϵ0 (1 + t)−1

∥ τ− r−1

(S) p

3

∥4,St,u ≤ cϵ0 (1 + t)−1

Proof: The proof is straightforward computation. The calculations of the most important formulas can be found in [19]. Let us remark here, that one should consider the formulas ((5.25)–(5.37)). ((5.83)–(5.95)), respectively ((5.122)–(5.134)), follow directly from the assumptions ((5.48)–(5.52)), ((5.55)–(5.67)). Also the inequalities expressed for the L∞ -norms in Ht follow immediately. Moreover, the additional assumptions ((5.68)–(5.80)) on the second derivatives of the components of the deformation tensors for the vectorfields T and S yield by the trace lemma the estimates ((5.96)–(5.104)), ((5.110)–(5.119)), ((5.135)–(5.142)), ((5.148)–(5.157)). That is: Let X denote T or S. As the components of the deformation ˆ are in L2 (Ht ) up to second derivatives, the Sobolev inequalities tensors (X) π ∞ ˆ . And in view of L4 (St,u ), it is by the trace yield L -estimates for (X) π lemma: (X) π ˆ ∈ W 2,2 (Ht ) ⇒ (X) π ˆ ∈ W 1,4 (St,u ). 5.3.3. Statement of the Boundedness Theorem. Having done all the computations and decompositions above, we now have the required tools to state and to prove the main theorem of this chapter. Theorem 5. Assume a given spacetime (M, g) together with a (t, u)– foliation as it was described previously. Consider an arbitrary Weyl tensor W satisfying the homogeneous Bianchi equations (4.9). Furthermore, let the assumptions of the comparison theorem hold in the spacetime slab 4 Ht , Vt∗ = t∈[0,t∗ ]

and hence, as a consequence, there exists a constant c such that

(5.158)

R[1] ≤ cQ1

in the spacetime slab Vt∗. Moreover, we require that the vectorfields T and S satisfy the assumptions ((5.41)–(5.80)). Then there exists a constant c1 such that (5.159)

Q∗1 ≤ Q1 (0) + c1 ϵ0 Q∗1 .

Thus, if ϵ0 is chosen sufficiently small, then it follows that (5.160)

Q∗1 ≤ 2Q1 (0).

126

5. ERROR ESTIMATES

5.3.4. Proof of the Boundedness Theorem. To prove the boundedness theorem 5, we first recall the formulas (5.7) and (5.8) as well as the inequality (5.10). Let us write the latter again for a time t∗ : (5.161)

Q1 (W, t∗ ) ≤ Q1 (W, 0) + E1 (W, t∗ ),

recalling also for E1 (W, t∗ ) the expression: E1 (W, t∗ ) =

#

¯ βT γT δ | Φ | divQ(LˆS W )βγδ K

Vt∗

+

#

V

¯ βK ¯ γT δ | Φ | (divQ(LˆT W ))βγδ K

#t∗ 1 ¯ + Φ | Q(LˆS W )αβγδ (K)π αβ T γ T δ | 2 Vt∗ # ¯ γT δ | + Φ | Q(LˆS W )αβγδ (T ) π αβ K Vt∗ # ¯ ¯ γT δ | + Φ | Q(LˆT W )αβγδ (K) π αβ K Vt∗ # 1 ¯ γK ¯ δ |. + Φ | Q(LˆT W )αβγδ (T )π αβ K 2 Vt∗

(5.162)

Thus, we have to estimate the error term E1 (W, t∗ ). The six integrals in the definition of E1 (W, t∗ ) shall be estimated in the sequel. We can split them into the interior region Vti , where r ≤ r20 , and the exterior region Vte , where r ≥ r20 . For the interior, the calculations are easier and the integrals for Vti are estimated in a straightforward way. To do so, one uses the interior assumptions ((5.41)–(5.47)) as well as the comparison theorem. We stress here the fact, that in the interior, all the components of the tensors W , DW , LˆT W , LˆS W and all the components ¯ behave in the same of the deformation tensors of the vectorfields T, S, K manner. In the exterior, things are different, as the components of the said tensors behave differently. Therefore, the exterior exstimates are more complicated, as they depend on the structure of the nonlinear terms. In the sequel, we shall enumerate the integrals in (5.162) according to their order of appearance by Int1 . . . Int6 . Let us estimate the six integrals separately. Estimates for the Integral Int2 :

Int2 =

#

Vt∗

¯ βK ¯ γT δ | Φ | (divQ(LˆT W ))βγδ K

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

127

Now, formula (5.20) in proposition 10 yields # ¯ βK ¯ γT δ | Φ | (divQ(LˆT W ))βγδ K Int2 = Vt∗ # -# Φτ+4 | D(T, W )344 | ≤ c′ Φτ+4 | D(T, W )444 | + V V # t∗ #t∗ + Φτ+2 τ−2 | D(T, W )344 | + Φτ+2 τ−2 | D(T, W )334 | V Vt∗ # t∗ # . + (5.163) Φτ−4 | D(T, W )334 | + Φτ−4 | D(T, W )333 | Vt∗

Vt∗

As the interior estimates are straightforward, we concentrate on the exterior part of this integral. The most difficult term to estimate is # (5.164) Φτ+2 τ−2 | D(T, W )334 |, Vte∗

as D(T, W )334 , involving parts without optimal decay properties, is multiplied by τ+2 τ−2 . These parts require more subtle treatment. The part # (5.165) Φτ+4 | D(T, W )344 |, Vte∗

where | D(T, W )344 | is multiplied by the highest weight τ+4 , is seen to be controlled in a straightforward way. It only contains terms with better decay properties than the worst ones in (5.164). The integral involving the most terms of better decay is # (5.166) Φτ+4 | D(T, W )444 | . Vte∗

In view of what we said at the beginning of this section, we shall, besides estimating the most delicate terms, which are of higher order with respect to asymptotic behaviour, but non-principal from the point of view of differentiability, also discuss the situation for the principal terms from the point of view of differentiability. We recall, here, that the higher order terms with respect to asymptotic behaviour (less decay) are the most difficult ones to control. These terms are non-principal from the point of view of differentiability. On the other hand, the principal terms with respect to differentiability behave better asymptotically, indeed, they decay faster. We shall see in what follows, that in the estimates for the principal terms from the point of view of differentiability, there is room left. Whereas the critical case will appear in the estimates for Int1 , involving Q(LˆS W ), where the most sensitive terms will be borderline.

128

5. ERROR ESTIMATES

Now, in view of (5.164), proposition 10, expression (5.22) yields: D(T, W )334 ≈ ρ(LˆT W )Λ(T, W ) − σ(LˆT W )K(T, W ) − β(LˆT W )I(T, W ).

(5.167)

The most sensitive terms to estimate, appear in the first two products on the right-hand side. In view of formula (5.4) and proposition 11, one can write % 1$ Λ(T, W ) = Λ1 (T, W ) + Λ2 (T, W ) + Λ3 (T, W ) 2 % 1$ K(T, W ) = K 1 (T, W ) + K 2 (T, W ) + K 3 (T, W ) 2 Applying proposition (11) to the vectorfield T , one has

/ β} + {(T ) m, β 3 } + {(T ) m, β 4 } Λ1 (T, W ) = K 1 (T, W ) = {(T ) i, ∇

/ (ρ,σ)} + {(T ) j,(ρ3 ,σ3 )} + {(T ) n, (ρ4 , σ4 )} + {(T ) m,∇ % $ (T ) + trχ { m, β} + {(T ) j, (ρ, σ)} + {(T ) m, β} $ % + trχ {(T ) i, α} + {(T ) m, β}+{(T ) n, (ρ, σ)} +l.o.t.

(5.168) (5.169)

Λ2 (T, W ) = K 2 (T, W ) = {(T )/p , β} + {(T ) p3 , (ρ, σ)}

Λ3 (T, W ) = K 3 (T, W ) = {α, Θ((T ) q)}+{(ρ, σ), (K, Λ)((T ) q)} + {β, Ξ((T ) q)}

(5.170)

Analogously, one finds I(T, W ) = with

% 1$ 1 I (T, W ) + I 2 (T, W ) + I 3 (T, W ) 2

I 1 (T, W ) = {(T ) m, ∇ / β} + {(T ) n, β 3 } + {(T ) j, β 4 }

(5.171) (5.172)

/ (ρ, σ)} + {(T ) m, (ρ3 , σ3 )} + {(T ) m, (ρ4 , σ4 )} + {(T ) i, ∇ % $ (T ) + trχ { n, β} + {(T ) m, (ρ, σ)} + {(T ) i, β} $ % + trχ {(T ) m, α} + {((T ) i,(T ) j), β} + {(T ) m, (ρ, σ)} + l.o.t.

I 2 (T, W ) = {(T ) p4 , β} + {(T )/p , (ρ, σ)}

I 3 (T, W ) = {α, Ξ((T ) q)} + {β, (K, Λ, Θ)((T ) q)} + {(ρ, σ), (I, I)((T ) q)}

(5.173)

+ {β, (K, Λ, Θ)((T ) q)}

Now, the term of lowest order in decay appears in Λ1 (T, W ), K 1 (T, W ), namely: (5.174)

trχ

(T )ˆ i·

α.

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

In Λ3 (T, W ), K 3 (T, W ), we find: ' ' 1 (T ) (5.175) αAB 2D /4 iAB + trχ (T ) iAB − tr 2

(T ) iδ AB

((

129

.

In view of (5.174), the corresponding principal term from the point of view of differentiability is:

(5.176)

D /4

(T )ˆ i·

α.

It has the same decay as (5.174), but (5.174) is non-principal with respect to differentiation. In order to show this precisely, we shall first give the calculations for (5.174), and then also for (5.176). Thus, in view of (5.174) we now estimate # τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α | . (5.177) Vte∗

&t & & We can write in view of the coarea formula V e = 0 ∗ dt H e Φdµg : t∗ t # τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α | Vte∗

(5.178)

=

#

t∗

0

#

Hte

τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α | dt.

Then, we calculate for the integral on Hte : # (5.179) τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α | Hte

(5.180) (5.181)

≤ c sup {τ+ | trχ |} Hte

− 32

′

≤ c (1 + t)

− 32

′

≤ c (1 + t) (5.182)

#

×

'#

Hte

τ+2 τ−3

#

Hte

'#

Hte 3

τ+3 τ−2 | (ρ, σ)(LˆT W ) || (T )ˆi || α |

Hte

|

τ+ τ−2 | (ρ, σ)(LˆT W ) || (T )ˆi || α |

τ+4

(T )ˆ i |2 |

| (ρ, σ)(LˆT W ) |2 2

α|

(1

(1 2

2

The latter integral can be estimated as follows: (1 '# '# 1 2 2 3 (T )ˆ 2 2 (T ) 2 τ+ τ − | i | | α | ≤ sup {τ+ τ− | ˆi |} Hte

onHte

Hte

(5.183) (5.184)

≤ ϵ0

'#

Hte

τ−2

(1 2 |α| . 2

τ−2

2

|α|

(1 2

130

5. ERROR ESTIMATES

Thus, (5.179) is bounded by: # (5.185) τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α | Hte

3 ≤ c(1 + t)− 2 ∥ r2 (ρ, σ)(LˆT W ) ∥2,e 1

(5.186)

(T )ˆ i∥

× ∥ rtrχ ∥∞,e ∥ rτ−2 − 23

(5.187)

≤ cϵ0 (1 + t)

∞,e ∥ τ− α

∥2,e

Q1 ,

which yields for (5.177): # 1 τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || trχ || (T )ˆi || α |≤ cϵ0 (1 + t)− 2 Q∗1 . (5.188) Vte∗

Next, considering the term (5.176), let us estimate the following integral, to which we first apply the coarea formula of lemma 6: # (5.189) τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || D / 4 (T )ˆi || α | Vte∗

#

=

(5.190)

u∗

#

du

−∞

Cu

τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || D /4

(T )ˆ i ||

α |.

Then, focusing on the integral on Cu , we obtain: #

τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || D / 4 (T )ˆi || α | # 3 − 32 ≤ cτ− τ+2 τ−2 τ−2 | (ρ, σ)(LˆT W ) || D /4

Cu

Cu

(T )ˆ i ||

α|

(5.191) ≤

−3 cτ− 2

(5.192)

'#

Cu

τ+2 τ−2

| (ρ, σ)(LˆT W ) |2

(1 ' # 2

Cu

τ+2 τ−5

|D /4

In view of the latter integral, we estimate '#

Cu

τ+2 τ−5

'# ≤c

|D /4

t∗

0

'# ≤c

0

dt

(T )ˆ i |2 |

'#

St,u

t∗

τ+2 τ−5

2

α|

τ+2 τ−5

∥| D /4

(1 2

|D /4

(T )ˆ i ||

(T )ˆ i |2 |

α

2

α|

| ∥2L2 (St,u )

(( 1

dt

2

(1 2

(T )ˆ i |2 |

2

α|

(1 2

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

'# ≤c

t∗

0

τ+2 τ−5

∥

D / 4 (T )ˆi ∥2L4 (St,u ) ∥

'# ≤ c sup {r τ− ∥ α ∥L4 (St,u ) } 1 2

3 2

1 2

3 2

α

t∗

τ+ τ−2

0

t,onCu

∥2L4 (St,u )

1

= c sup {r τ− ∥ α ∥L4 (St,u ) } ∥ r 2 τ− D /4 (5.193)

≤

t,onCu cϵ0 Q∗1

This yields for (5.192): # τ+2 τ−2 Φ | (ρ, σ)(LˆT W ) || D /4 Cu

(T )ˆ i ||

∥

dt

(1

131

2

(1

D / 4 (T )ˆi ∥2L4 (St,u )

(T )ˆ i∥

2

L2 ([0,t∗ ],L4 (St,u ))

α|

−3

3

1

≤ cτ− 2 ∥ rτ− (ρ, σ)(LˆT W ) ∥2,Cu sup{r 2 τ−2 ∥ α ∥L4 (St,u ) } t,Cu

(5.194)

3

1 2

− / 4ˆi ∥L2 ([0,t∗ ],L4 (St,u )) ≤ cϵ0 τ− 2 Q∗1 , × ∥ r τ− D

which gives for (5.189): # (5.195) Φτ+2 τ−2 | (ρ, σ)(LˆT W ) || D /4 Vte∗

(T )ˆ i ||

α | ≤ cϵ0 Q∗1 .

We remark that, in view of the last integral in formula (5.193), the corresponding situation is studied later in the discussion of Int1 in (5.266) ff. The calculations for (5.189) can also be carried out first on Hte and 1 then with integration in t, which would yield a room of decay of (1 + t)− 2 , similarly as in the case (5.178). All the other terms in (5.164) # Φτ+2 τ−2 | D(T, W )334 | Vte∗

behave better and can be calculated in a straightforward manner. That is, taking into account all the above estimates, and proceeding with all the other terms in D(T, W )334 analogously, we first give the results for the terms estimated on Hte and then integrated with respect to t (as in (5.178)), and second we give the results for the quantities estimated on Cu and then integrated with respect to u (as in (5.190)). The two parts together will finally yield the estimate (5.219) below. In the following estimates, the terms in (5.196)

Λ((T ) q), Λ((T ) q), K((T ) q), K((T ) q), . . . ,

which are integrated on the cone Cu , we denote by (5.197)

ΛC ((T ) q), ΛC ((T ) q), KC ((T ) q), K C ((T ) q), . . .

and the ones integrated on the hypersurface Ht , we denote by (5.198)

ΛH ((T ) q), ΛH ((T ) q), KH ((T ) q), K H ((T ) q), . . .

132

5. ERROR ESTIMATES

They are involved in the third components of (5.199)

Λ(T, W ), Λ(T, W ), K(T, W ), K(T, W ), . . . .

In fact, while the first components of (5.199) are integrated only on Ht , and the second components only on Cu , the third ones are split into terms integrated on Ht and terms integrated on Cu . Namely, in view of (5.200) and (5.212) it is Λ3H (T, W ) = K 3H (T, W ) = {α, ΘH ((T ) q)} + {(ρ, σ), (K H , ΛH )((T ) q)} + {β, ΞH ((T ) q)}

I 3H (T, W ) = {α, ΞH ((T ) q)} + {β, (KH , ΛH , ΘH )((T ) q)} + {(ρ, σ), (IH , I H )((T ) q)}

+ {β, (K H , ΛH , ΘH )((T ) q)} and correspondingly for Λ3C (T, W ), K 3C (T, W ), I 3C (T, W ). The terms (5.197) integrated on Cu are the ones in (5.196) which are derivatives of the components of the deformation tensor, whereas (5.198), to be integrated on Ht , involve all the other terms in (5.196), that is, the ones not involving any derivatives of the deformation tensor. We refer to the formulas given above in ((5.25)–(5.37)). Now, we estimate (5.164). On Hte , it is: # Φτ+2 τ−2 | D(T, W )334 | Hte

(5.200)

A 3 3 ≤ c(1 + t)− 2 ∥ r2 (ρ, σ)(LˆT W ) ∥2,e · ∥ r 2 τ−2 (Λ1 , K 1 ) ∥2,e B 3 3 + ∥ r 2 τ−2 (Λ3H , K 3H ) ∥2,e + c(1 + t)− 2 ∥ rτ− β(LˆT W ) ∥2,e A B 5 5 · ∥ r 2 τ− I 1 ∥2,e + ∥ r 2 τ− I 3H ∥2,e .

And we estimate the components as follows. 3

∥ r 2 τ−2 (Λ1 , K 1 ) ∥2,e A 1 ≤ c ∥ τ− α ∥2,e ∥ rtrχ ∥∞,e ∥ rτ−2 (T )ˆi ∥∞,e 3 3 + ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 (T ) m ∥∞,e + ∥ r(ρ, σ) ∥2,e ∥ r 2 3 + ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 (T ) m ∥∞,e . 3 3 + ∥ r(ρ, σ) ∥2,e ∥ r 2 (T )j ∥∞,e + ∥ rβ ∥2,e ∥ r 2 (T ) m ∥∞,e 1

/ β ∥2,e ∥ rτ−2 + ∥ r2 ∇

(T ) i

3

∥∞,e + ∥ rτ− β 3 ∥2,e ∥ r 2

(T ) m

∥∞,e

(T ) n

. ∥∞,e

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF 3

+ ∥ r2 β 4 ∥2,e ∥ r 2

3

(T ) m 3

+ ∥ r2 (ρ3 , σ3 ) ∥2,e ∥ r 2

∥∞,e + ∥ r2 ∇ / (ρ, σ) ∥2,e ∥ r 2

(T ) j

(5.201)

(T ) m 3

∥∞,e + ∥ r2 (ρ4 , σ4 ) ∥2,e ∥ r 2

133

∥∞,e

(T ) n

B ∥∞,e

3

∥ r 2 τ−2 (Λ3H , K 3H ) ∥2,e A 1 ≤ c ∥ τ− α ∥2,e ∥ r2 τ−2 ΘH ((T ) q) ∥∞,e + ∥ r(ρ, σ) ∥2,e 2

1 2

)((T ) q)

× ∥ r τ− (ΛH , K H

(5.202)

5 2

∥∞,e + ∥ rβ ∥2,e ∥ r ΞH

B

((T ) q)

∥∞,e

5

∥ r 2 τ− I 1 ∥2,e A 3 ≤ c ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 . 1 + ∥ rβ ∥2,e ∥ rτ−2 (T ) i ∥∞,e 3 + ∥ rtrχ ∥∞,e ∥ τ− α ∥2,e ∥ r 2 3

+ ∥ rβ ∥2,e ∥ r 2

(T ) j 3

+ ∥ r2 ∇ / β ∥2,e ∥ r 2 + ∥ r2 β 4 ∥2,e ∥ r

3 2

∥∞,e

(T ) m

(T ) j

(T ) n

3

∥∞,e + ∥ r(ρ, σ) ∥2,e ∥ r 2 1

(T ) m

∥∞,e + ∥ rβ ∥2,e ∥ rτ−2 . 3 + ∥ r(ρ, σ) ∥2,e ∥ r 2 (T ) m ∥∞,e 3

∥∞,e + ∥ rτ− β 3 ∥2,e ∥ r 2

(T ) n

(T ) m

(T ) i

∥∞,e

∥∞,e

∥∞,e 1

+ ∥ r2 ∇ / (ρ, σ) ∥2,e ∥ rτ−2

3

∥∞,e + ∥ r2 (ρ3 , σ3 ) ∥2,e ∥ r 2 B 3 + ∥ r2 (ρ4 , σ4 ) ∥2,e ∥ r 2 (T ) m ∥∞,e (T ) i

(T ) m

∥∞,e

(5.203)

A 5 5 ∥ r 2 τ− I 3H ∥2,e ≤ c ∥ τ− α ∥2,e ∥ r 2 ΞH ((T ) q) ∥∞,e 1

+ ∥ rβ ∥2,e ∥ r2 τ−2 (KH , ΛH , ΘH )((T ) q) ∥∞,e 5

+ ∥ r(ρ, σ) ∥2,e ∥ r 2 (IH , I H )((T ) q) ∥∞,e

B 1 + ∥ rβ ∥2,e∥ r2 τ−2 (K H , ΛH , ΘH )((T ) q) ∥∞,e

(5.204)

In view of (5.202) and (5.204) it is (5.205) 1

∥ r2 τ−2 (K H , ΛH , ΘH )((T ) q) ∥∞,e A 3 3 3 3 ≤ c ∥ r 2 (ζ, ζ) ∥∞,e ∥ r 2 (T ) m ∥∞,e + ∥ r 2 ν ∥∞,e ∥ r 2

(T ) j

∥∞,e

∥∞,e

134

5. ERROR ESTIMATES 3

3

+ ∥ r2χ ˆ ∥∞,e ∥ r 2 1 2

(T ) n 3

(T ) j

+ ∥ rτ− χ ˆ ∥∞,e ∥ r 2

1 2

1

3

(5.206)

3

∥∞,e + ∥ r 2 ϵ ∥∞,e ∥ r 2

(T )ˆ i∥

+ ∥ rtrχ ∥∞,e ∥ rτ−

1

∥∞,e + ∥ rτ−2 χ ˆ ∥∞,e ∥ rτ−2 3 2

+ ∥ r ξ ∥∞,e ∥ r

∞,e

3 2

(T ) i

(T ) m (T ) m

∥∞,e

∥∞,e

B

∥∞,e

5

∥ r 2 ΞH ((T ) q) ∥∞,e A 3 ≤ c · ∥ rtrχ ∥∞,e ∥ r 2 3

3

+ ∥ r 2 ν ∥∞,e ∥ r 2 3 2

(T ) m

(T ) m

1 2

+ ∥ r ξ ∥∞,e ∥ rτ−

(T ) i

(5.207)

1

3

(T ) m

∥∞,e + ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 3

3

∥∞,e + ∥ r 2 (ζ, ϵ) ∥∞,e ∥ r 2 3 2

∥∞,e + ∥ r ξ ∥∞,e ∥ r

3 2

(T ) j

(T ) n

∥∞,e

∥∞,e B

∥∞,e

5

∥ r 2 ΞH ((T ) q) ∥∞,e A 3 ≤ c · ∥ rtrχ ∥∞,e ∥ r 2

3

(T ) m

3

(T ) m

∥∞,e + ∥ r 2 (χ, ˆ ν) ∥∞,e ∥ r 2 B 3 ∥ r 2 (T ) n ∥∞,e

3 2

+ ∥ r (ζ, ϵ) ∥∞,e

(5.208)

1

∥∞,e

∥ r2 τ−2 (KH , ΛH , ΘH )((T ) q) ∥∞,e A 1 3 3 ≤ c ∥ rtrχ ∥∞,e ∥ rτ−2 (T )ˆi ∥∞,e + ∥ r 2 (ζ, ζ, ϵ) ∥∞,e ∥ r 2 (T ) m ∥∞,e 1 1 $ 3 3 + ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 (T ) n ∥∞,e + ∥ r 2 χ ˆ ∥∞,e ∥ rτ−2 (T ) i ∥∞,e B % 3 3 3 + ∥ r 2 (T ) j ∥∞,e + ∥ r 2 ν ∥∞,e ∥ r 2 (T ) j ∥∞,e

(5.209) 5

∥ r 2 (IH , I H )((T ) q) ∥∞,e A 3 ≤ c ∥ rtrχ ∥∞,e ∥ r 2

(T ) m

3 2

3 2

3 2

3 2

(T ) m

3

3

(T ) n

+∥r χ ˆ ∥∞,e ∥ r + ∥ r ν ∥∞,e ∥ r

+ ∥ r 2 ξ ∥∞,e ∥ r 2 3 2

+ ∥ r (ζ, ζ) ∥∞,e (5.210)

(T ) m

3

∥∞,e + ∥ rtrχ ∥∞,e ∥ r 2 1 2

∥∞,e + ∥ rτ− χ ˆ ∥∞,e ∥ r 3 2

∥∞,e + ∥ r ν ∥∞,e ∥ r 3

(T ) m

(T ) m 1

∥∞,e + ∥ r 2 (ζ, ζ) ∥∞,e ∥ rτ−2 B 3 ∥ r 2 (T ) j ∥∞,e

Then, we have (5.211)

3 2

3 2

3

(5.200) ≤ cϵ0 (1 + t)− 2 Q∗1 .

(T ) m

∥∞,e

∥∞,e

∥∞,e

(T ) i

∥∞,e

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

135

On Cu , it is: # Φτ+2 τ−2 | D(T, W )334 | Cu

3

− ≤ cτ− 2 ∥ rτ− (ρ, σ)(LˆT W ) ∥2,Cu A B 3 3 · ∥ r2 τ−2 (Λ2 , K 2 ) ∥2,Cu + ∥ r 2 τ−2 (Λ3C , K 3C ) ∥2,Cu A B 3 3 −3 + cτ− 2 ∥ τ−2 β(LˆT W ) ∥2,Cu · ∥ r2 τ−2 I 2 ∥2,Cu + ∥ r2 τ−2 I 3C ∥2,Cu .

(5.212)

The components are estimated as follows. 3

∥ r2 τ−2 (Λ2 , K 2 ) ∥2,Cu A 3 1 1 ≤ c sup{r 2 τ−2 ∥ β ∥4,St,u } ∥ r 2 τ−

(T )/ p

∥L2 ([0,t∗ ],L4 (St,u ))

(T ) p

3∥L2 ([0,t∗ ],L4 (St,u ))

t,Cu

1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r 2 τ− t,Cu

(5.213)

B

3

∥ r 2 τ−2 (Λ3C , K 3C ) ∥2,Cu A 1 1 3 ≤ c sup{r 2 τ−2 ∥ α ∥4,St,u } ∥ rτ−2 ΘC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

1

+ sup{r2 ∥(ρ, σ) ∥4,St,u } ∥ r 2 τ− (ΛC ((T ) q), K C ((T ) q)) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

B 1 + sup{r2 ∥ β ∥4,St,u } ∥ r 2 τ− ΞC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

(5.214)

3

∥ r2 τ−2 I 2 ∥2,Cu A 3 1 1 ≤ c sup{r 2 τ−2 ∥ β ∥4,St,u } ∥ r 2 τ− t,Cu

1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r 2 τ− t,Cu

(5.215)

(T ) p

4

(T )/ p

∥L2 ([0,t∗ ],L4 (St,u ))

B ∥L2 ([0,t∗ ],L4 (St,u ))

3

∥ r2 τ−2 I 3C ∥2,Cu A 1 3 3 ≤ c sup{r 2 τ−2 ∥ α ∥4,St,u } ∥ r 2 ΞC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

3

1

+ sup{r 2 τ−2 ∥ β ∥4,St,u } t,Cu

% $ 1 × ∥ r 2 τ− ΛC ((T ) q), KC ((T ) q), ΘC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u ))

136

5. ERROR ESTIMATES 1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r 2 τ− (IC ((T ) q), I C ((T ) q)) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

+ sup{r2 ∥ β ∥4,St,u } t,Cu

B 3$ % × ∥ τ−2 ΛC ((T ) q), K C ((T ) q), ΘC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u ))

(5.216)

In view of (5.214) and (5.216) we have A 1 1 /4 ∥ rτ−2 ΘC ((T ) q) ∥L2 ([0,t∗ ],L4 (St,u )) ≤ c ∥ rτ−2 D 3

(5.217)

+ ∥ r2∇ /

(T ) i

(T ) m

∥L2 ([0,t∗ ],L4 (St,u ))

∥L2 ([0,t∗ ],L4 (St,u ))

All the other terms in ΛC ((T ) q), KC ((T ) q), . . . of the inequalities (5.214) and (5.216) are estimated correspondingly. We refer to ((5.25)–(5.37)). Then, we have −3

(5.212) ≤ cϵ0 τ− 2 Q∗1 .

(5.218)

Now, integrating (5.200) with respect to t as in (5.178), and integrating (5.212) with respect to u as in (5.190), proves # Φτ+2 τ−2 | D(T, W )334 |≤ cϵ0 Q∗1 . (5.219) Vte∗

Next, in view of (5.166), proposition 10, expression (5.22) yields: (5.220)

D(T, W )444 ≈ α(LˆT W ) Θ(T, W ) − β(LˆT W )Ξ(T, W ).

Proceeding as above, we obtain from (5.4) and proposition 11: % 1$ 1 Θ (T, W ) + Θ2 (T, W ) + Θ3 (T, W ) 2 % 1$ 1 Ξ(T, W ) = Ξ (T, W ) + Ξ2 (T, W ) + Ξ3 (T, W ) 2

Θ(T, W ) =

We first give the estimate for a principal term with respect to differentiability, but with better decay. We shall see, that there is room left in these estimates. Let us estimate # Φτ+4 | α(LˆT W ) || (T ) m || ∇ / α |, (5.221) Vte∗

which appears in the first product on the right-hand side of (5.220), has good decay properties and is principal in view of differentiability.

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

137

The coarea formula yields for (5.221): # Φτ+4 | α(LˆT W ) || (T ) m || ∇ /α| Vte∗

(5.222)

#

=

t∗

0

#

Hte

Φτ+4 | α(LˆT W ) || (T ) m || ∇ / α | dt.

Then, we calculate for the integral on Hte : # Φτ+4 | α(LˆT W ) || (T ) m || ∇ /α| Hte

3

≤ c(1 + t)− 2 − 32

′

≤ c (1 + t) (5.223)

#

7

Hte

Φτ+2 τ+2 | α(LˆT W ) || (T ) m || ∇ /α|

'#

Hte

(1 ' # 2 | α(LˆT W ) | ·

τ+4

2

Hte

For the latter integral, we have: (1 C 3 '# 2 7 (T ) 2 2 τ+ | m | | ∇ /α| ≤ sup τ+2 | onHte

Hte

(T ) m

which yields for (5.223): # /α| Φτ+4 | α(LˆT W ) || (T ) m || ∇

τ+7

|

(T ) m |2 |

D'# |

Hte

τ+4

2

(1

2

(1

∇ /α|

|∇ /α|

2

2

,

Hte

(5.224) (5.225)

3 3 ≤ c(1 + t)− 2 ∥ r2 α(LˆT W ) ∥2,e ∥ r 2 3

≤ cϵ0 (1 + t)− 2 Q∗1 .

(T ) m

∥∞,e ∥ r2 ∇ / α ∥2,e

Thus, we deduce for (5.221): # 1 Φτ+4 | α(LˆT W ) || (T ) m || ∇ / α |≤ cϵ0 (1 + t)− 2 Q∗1 . (5.226) Vte∗

Next, let us present the situation for the following term in D(T, W )444 , which appears in the first product on the right-hand side of (5.220). # (5.227) Φτ+4 | α(LˆT W ) || trχ || (T ) n || α | . Vte∗

Note that the integrand is non-principal with respect to differentiability. Similarly as before, it is # Φτ+4 | α(LˆT W ) || trχ || (T ) n || α | Vte∗

(5.228)

=

#

0

t∗

#

Hte

Φτ+4 | α(LˆT W ) || trχ || (T ) n | | α | dt

138

5. ERROR ESTIMATES

On Hte , we have #

Hte

Φτ+4 | α(LˆT W ) || trχ || (T ) n || α |

≤ c sup {r | trχ |} onHte

− 32

′

≤ c (1 + t) ≤ c (1 + t) (5.229)

− 32

≤ c(1 + t)

− 32

≤ c(1 + t) (5.230) That is, #

Hte

Hte

τ+3 | α(LˆT W ) || (T ) n || α |

Hte

5

τ+2 τ+2 | α(LˆT W ) || (T ) n || α |

'#

τ+4

| α(LˆT W ) |2

(1 ' # 2 ·

'#

τ+4

| α(LˆT W ) |

(1

τ+4

(1 ' # 2 | α(LˆT W ) | ·

− 32

′

#

#

Hte

Hte

'#

Hte

2

Hte

2

τ+5

|

-# 3 sup {r 2 | (T ) n |}

onHte

2

Hte

τ+2

2

|α|

(1

2

α|

Hte

(1 2

τ+2 | α |2

.1 2

2

Φτ+4 | α(LˆT W ) || trχ || (T ) n || α |

3 3 ≤ c(1 + t)− 2 ∥ r2 α(LˆT W ) ∥2,e ∥ rtrχ ∥∞,e ∥ r 2

− 32

(5.231)

(T ) n |2 |

≤ cϵ0 (1 + t)

(T ) n

Q∗1 .

∥∞,e ∥ rα ∥2,e

And we conclude for (5.227) that # 1 (5.232) Φτ+4 | α(LˆT W ) || trχ || (T ) n || α |≤ cϵ0 (1 + t)− 2 Q∗1 . Vte∗

Now, the highest order terms with respect to asymptotic behaviour (less decay) in D(T, W )444 are α(LˆT W )trχ (T ) i(ρ, σ) : where ˆ (T ) iα : where α(LˆT W )χ α(LˆT W ) (T ) p3 α : where β(LˆT W )trχ

(T ) iβ

: where

trχ

(T ) i(ρ, σ)

χ ˆ

(T ) iα

(T ) p

trχ

3α

(T ) iβ

∈ Θ1 (T, W )

∈ Θ3 (T, W )

∈ Θ3 (T, W )

∈ Ξ1 (T, W ).

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

139

They are all treated in the same manner. We are now going to present the details for # Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) | . (5.233) Vte∗

We write

#

Vte∗

(5.234) #

On

Hte

= Hte ,

Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) | #

t∗

0

#

Hte

we estimate

Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) | dt.

Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) | − 32

≤ c(1 + t)

− 32

≤ c(1 + t) (5.235)

sup{r | trχ |} Hte

-#

Hte

#

3 r3 r 2 | α(LˆT W ) || (T ) i || (ρ, σ) |

Hte

.1 ' # 2 r | α(LˆT W ) | · 4

2

Hte

3 ≤ c(1 + t)− 2 ∥ r2 α(LˆT W ) ∥2,e '# 2 · sup{r ∥ (ρ, σ) ∥L4 (St,u ) }

Hte

− 32

≤ c(1 + t)

2 3

∥ r2 α(LˆT W ) ∥2,e

u∗

−∞ 1

· sup{r2 ∥ (ρ, σ) ∥L4 (St,u ) } ∥ r 2 Hte

r∥

(T ) i

(T ) i

r r |

∥2L4 (St,u )

(T ) i |2 |

du

2

(ρ, σ) |

(1 2

∥L2 ((−∞,u∗ ],L4 (St,u ))

(5.236) That is,

#

Hte

Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) |

3 ≤ c(1 + t)− 2 ∥ rtrχ ∥∞,e ∥ r2 α(LˆT W ) ∥2,e 1

· sup{r2 ∥ (ρ, σ) ∥L4 (St,u ) } ∥ r 2 Hte

(5.237)

(T ) i

∥L2 ((−∞,u∗ ],L4 (St,u ))

3

≤ cϵ0 (1 + t)− 2 Q∗1 ,

from which it directly follows that (5.233) is bounded: # 1 Φτ+4 | α(LˆT W ) || trχ || (T ) i || (ρ, σ) |≤ cϵ0 (1 + t)− 2 Q∗1 . (5.238) Vte∗

(1 2

140

5. ERROR ESTIMATES

Estimating all the other terms in D(T, W )444 in the same manner and proceeding as above for D(T, W )334 , we obtain # | Φτ+4 D(T, W )444 |≤ cϵ0 Q∗1 . (5.239) Vte∗

Finally, proceeding in the same way with all the other terms in (5.163), we deduce Int2 ≤ cϵ0 Q∗1 .

(5.240)

Estimates for the Integral Int1 : Int1 =

#

Vt∗

¯ βT γT δ | Φ | divQ(LˆS W )βγδ K

Applying formula (5.21) in proposition 10 yields # ¯ βT γT δ | Int1 = Φ | (divQ(LˆS W ))βγδ K Vt∗

′

≤c

+

'# #

Vt∗

Vt∗

+

#

Vt∗

(5.241)

+

#

Vt∗

Φ|

τ+2 D(S, W )444

#

|+

Φ | τ+2 D(S, W )334 | Φ|

τ−2 D(S, W )344

|+

Φ|

τ−2 D(S, W )333

|

(

Vt∗

#

Vt∗

Φ | τ+2 D(S, W )344 |

Φ | τ−2 D(S, W )334 |

Also here, the estimates in the interior are straightforward and we focus on the far more subtle exterior part of the integral. Now, the most delicate term to estimate is # Φτ+2 | D(S, W )334 |, (5.242) Vte∗

as D(S, W )334 , being multiplied by τ+2 , involves parts with the worst decay properties. These parts require more subtle estimates, as we encounter, here, the borderline case. In view of the formula (5.22) in proposition 10, we write D(S, W )334 ≈ ρ(LˆS W )Λ(S, W ) − σ(LˆS W )K(S, W ) − β(LˆS W )I(S, W ).

(5.243)

As before, the most sensitive terms to estimate, appear in the first two products on the right-hand side. In view of formula (5.4) and proposition 11,

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

141

one can write % 1$ 1 Λ (S, W ) + Λ2 (S, W ) + Λ3 (S, W ) 2 % 1$ K(S, W ) = K 1 (S, W ) + K 2 (S, W ) + K 3 (S, W ) 2 One applies proposition (11) to the vectorfield S to find Λ(S, W ) =

Λ1 (S, W ) = K 1 (S, W ) = {(S) i, ∇ / β} + {(S) m, β 3 } + {(S) m, β 4 }

/ (ρ,σ)}+{(S) j,(ρ3 ,σ3 )}+{(S) n, (ρ4 , σ4 )} + {(S) m,∇ % $ + trχ {(S) m, β} + {(S) j, (ρ, σ)} + {(S) m, β} $ % + trχ {(S) i, α} + {(S) m, β} + {(S) n, (ρ, σ)} +l.o.t.

(5.244) (5.245)

Λ2 (S, W ) = K 2 (S, W ) = {(S)/p , β} + {(S) p3 , (ρ, σ)}

Λ3 (S, W ) = K 3 (S, W ) = {α, Θ((S) q)}+{(ρ, σ), (K, Λ)((S) q)} + {β, Ξ((S) q)}

(5.246)

Analogously, one has I(S, W ) = with

% 1$ 1 I (S, W ) + I 2 (S, W ) + I 3 (S, W ) 2

/ β} + {(S) n, β 3 } + {(S) j, β 4 } I 1 (S, W ) = {(S) m, ∇

(5.247)

/ (ρ, σ)} + {(S) m, (ρ3 , σ3 )} + {(S) m, (ρ4 , σ4 )} + {(S) i, ∇ % $ (S) + trχ { n, β} + {(S) m, (ρ, σ)} + {(S) i, β} $ % + trχ {(S) m, α} + {((S) i,(S) j), β} + {(S) m, (ρ, σ)} + l.o.t.

I 2 (S, W ) = {(S) p4 , β} + {(S)/p , (ρ, σ)}

(5.248)

I 3 (S, W ) = {α, Ξ((S) q)} + {β, (K, Λ, Θ)((S) q)} + {(ρ, σ), (I, I)((S) q)} (5.249)

+ {β, (K, Λ, Θ)((S) q)}

The terms of higher order with respect to asymptotic behaviour (less decay) appear in Λ1 (S, W ), K 1 (S, W ), I 1 (S, W ) and Λ3 (S, W ), K 3 (S, W ) I 3 (S, W ). Namely, the terms with worst decay in these six quantities are: (5.250)

trχ

(5.251)

trχ

(S)ˆ i·

(S) m

α · α.

The first one (5.250) appears in Λ1 (S, W ), K 1 (S, W ), Λ3 (S, W ), K 3 (S, W ), whereas the second one (5.251) shows up in I 1 (S, W ), I 3 (S, W ). The terms (5.250) and (5.251), involving the factor α, are the most delicate ones to

142

5. ERROR ESTIMATES

estimate, and we shall take special care of them. Actually, (5.250) is the worst term, that is, (5.251) behaves slightly better. The other higher order terms with respect to asymptotic behaviour in these six quantities are the following, remarking that contrary to above, where the worst terms involve α, here, they involve β, β or (ρ, σ), which behave better. A B trχ (S) m · β + (S) n · (ρ, σ) + (S) i · β + (S) j · β + (S) m · (ρ, σ) A B (S) (S) (S) (S) (S) m · β + i · (ρ, σ) + j · (ρ, σ) + m · β + i · β trχ (5.252)

Moreover, the terms corresponding to (5.250) and (5.251), but which are principal from the point of view of differentiability appear in Λ3 (S, W ), K 3 (S, W ). They are D / 4 (S)ˆi · α $ / B (S) mA − div / ∇ / A (S) mB + ∇ % $ (S) (S) mA − ∇ /A n α D /A

(5.253) (5.254) (5.255)

(S) mδ

AB

%

α

We also refer to our comments in the subsection for the Integral Int2 above. Considering (5.250), we now estimate #

(5.256)

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | .

Vte∗

In view of the coarea formula (lemma 6), we write #

Vte∗

(5.257)

=

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | #

u∗

du

−∞

#

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α |

We calculate on Cu : # (5.258) τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | Cu # ′ ≤ c sup {τ+ | trχ |} τ+ | (ρ, σ)(LˆS W ) || (S)ˆi || α | Cu

(5.259)

'# ≤c

Cu

Cu

τ+2

| (ρ, σ)(LˆS W ) |2

(1 ' # 2 ·

Cu

|

(S)ˆ i |2 |

2

α|

(1 2

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

Let us study the latter integral: '# ( 1 ' # t∗ A # 2 2 2 (S) ˆ | i| |α| = 0

Cu

= (5.260)

≤

'#

St,u

t∗

∥|

0

'#

t∗

∥

0

|

(S)ˆ i |2 |

(S)ˆ i ||

α

B ( 12 α | dt 2

| ∥2L2 (St,u )

(S)ˆ i ∥2L4 (St,u ) ∥

143

α

dt

(1 2

∥2L4 (St,u )

dt

(1 2

Heuristically, we can say that, writing in the dimensionless L4 -norm on St,u , the component α is of order − ∥ α− ∥

And we have (5.261)

−3

1

L4 (St,u )

sup

t,onCu

= r− 2 ∥ α ∥L4 (St,u ) = O(t−1 τ− 2 ).

2

1

r 2 ∥ α ∥L4 (St,u )

3

−3

≤ cτ− 2 .

Then, (5.260) becomes ' # t∗ (1 2 2 2 (S) ˆ ∥ i ∥L4 (St,u ) ∥ α ∥L4 (St,u ) dt 0

(5.262)

≤

−3 cτ− 2

'#

0

t∗

−1

(1 + t)

∥

(S)ˆ i ∥2L4 (St,u )

dt

(1 2

One could also write the following. Though we work with (5.262). ' # t∗ (1 (1 ' # t∗ 2 2 − 32 2 2 2 (S) (S) ˆ ˆ ∥ i ∥L4 (St,u ) ∥ α ∥L4 (St,u ) dt ≤ cτ− ∥ i ∥L∞ (St,u ) dt . 0

0

(5.263)

We encounter here in (5.262) the borderline case. In fact, the integral (5.262) is critical. Thus, the estimates for (5.256) are really borderline. We can bound the integral (5.262), and therefore also (5.256), using the fact ˆ the corresponding norms of which are bounded. that (S)ˆi involves χ, Now, we are going to give the rigorous estimates. We proceed as follows. In view of the definition of (S)ˆi and writing out the Lie derivative 1 1 LˆS g = uLˆL g + uLˆL g, 2 2 (S) ˆ the i involves the terms rχ ˆ and uˆ χ. Whereas the latter has order of decay 1

1

O(r−1 τ−2 ), the first one is only of order o(r− 2 ). Therefore, we concentrate on estimating rχ. ˆ In previous chapters, we computed from the Codazzi equations 1 div /χ ˆ − /d trχ = β + f 2

144

5. ERROR ESTIMATES

and the assumption of β ∈ L4 (St,u ) that ∇ /χ ˆ ∈ L4 (St,u ),

(5.264)

where in the estimates we achieved control in terms of β. More precisely, by elliptic estimates we obtained /χ ˆ ∥L4 (St,u ) ≤ c ∥ r2 div /χ ˆ ∥L4 (St,u ) ∥ rχ ˆ ∥L4 (St,u ) + ∥ r2 ∇

5

≤ c ∥ r2 β ∥L4 (St,u ) = cr 2− ∥ β− ∥

(5.265)

L4 (St,u ) .

Next, in view of the term (1 + t)−1 ∥ (S)ˆi ∥2L4 (St,u ) in (5.262) we estimate #

t∗

0

3 2

∥r β

∥2L4 (St,u )

dt ≤ c

(5.266)

#

t∗

∥ rβ ∥L2 (St,u ) · ∥ rβ ∥L2 (St,u ) dt

0

+c

#

t∗

0

'# ≤c

t∗

∥ rβ

0

'# +c ×

t∗

t∗

∥2L2 (St,u )

2

2

Cu

-# +c

2

r |β|

Cu

(1 ' # 2

(1

Cu

∥ rβ

0

dt

(1

Cu

dt

(1 2

2

2

r |β|

2

∥2L2 (St,u )

2

2

.1 - #

r2 | β |2

t∗

2

∥2L2 (St,u )

∥r ∇ /β

0

dt

( 1 '#

∥ rβ ∥2L2 (St,u ) dt

0

'#

'# =c (5.267)

∥ rβ ∥L2 (St,u ) · ∥ r2 ∇ / β ∥L2 (St,u ) dt

(1 2

r4 | ∇ / β |2

.1 2

And (5.267) is bounded. To show that the inequality (5.266), in fact, holds, we apply the isoperimetric inequality on St,u for a function f and isoperimetric constant I(u) #

St,u

$

-# %2 ¯ f − f ≤ I(u)

St,u

|∇ / f | dµγ

with f¯ =

1 A(St,u )

#

St,u

f dµγ

.2

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

145

3

to | β |2 yielding an estimate for ∥ r 2 β ∥2L4 (St,u ) . Thus, we obtain by first applying the isoperimetric and then the H¨ older inequality # .2 -# $ %2 2 2 ≤ I(u) |∇ / | β |2 | dµγ | β | −| β | St,u St,u -# .2 ≤ cI(u) |β·∇ / β | dµγ St,u -# .- # . 2 ≤ cI(u) | β | dµγ |∇ / β |2 dµγ (5.268) St,u

For a function f it is # # $ %2 f − f¯ = St,u

St,u

f2 −

St,u

#

f f¯ =

St,u

#

St,u

f2 −

#

f¯2 .

2

(

St,u

Then, writing

#

1 Area(St,u )

| β |2 =

St,u

| β |2 ,

we obtain from (5.268) the following: #

-# |β| ≤c 4

St,u

| β |2 dµγ

St,u

+ cr ≤ cr (5.269)

+

−2

−2

'#

| β | dµγ

St,u

( C' # | β | dµγ ·

2

St,u

(' #

.

| β | dµγ

2

St,u

'#

|∇ / β |2 dµγ

2

St,u

'#

.-

2

r |∇ / β | dµγ

2

St,u

(D

| β | dµγ

(

We multiply (5.269) by r6 to obtain #

St,u

'# r |β| ≤c 6

(5.270)

4

2

St,u

·

C' #

2

r | β | dµγ

St,u

2

2

(

r | β | dµγ

(

+

'#

St,u

4

2

r |∇ / β | dµγ

(D

That is 3

∥ r 2 β ∥2L4 (St,u ) ≤ c ∥ rβ ∥L2 (St,u ) ∥ rβ ∥L2 (St,u ) + c ∥ rβ ∥L2 (St,u ) ∥ r2 ∇ / β ∥L2 (St,u ) .

(5.271)

146

5. ERROR ESTIMATES

Next, we multiply (5.271) by aM (t)ΦM (t) and integrate with respect to t, which yields inequality (5.266): #

0

t∗

3 2

∥r β

∥2L4 (St,u )

dt ≤ c

#

t∗

0

+c

#

∥ rβ ∥L2 (St,u ) · ∥ rβ ∥L2 (St,u ) dt t∗

/ β ∥L2 (St,u ) dt ∥ rβ ∥L2 (St,u ) · ∥ r2 ∇

0

Then, applying the Cauchy-Schwarz inequality yields (5.267). Therefore, the part rχ ˆ of (S)ˆi, which has the worst decay behaviour, is controlled in terms of β. Going back to (5.262), we have thus shown that the integral on the right-hand side of (5.262) is bounded by (5.267): ∥

1 r− 2 (S)ˆi ∥

L2 ([0,t∗ ],L4 (St,u ))

'# ≤c

2

Cu

'# +c

2

r |β|

Cu

2

(1 ' # 2

2

Cu

2

r |β|

≤ cϵ0 Q∗1 .

2

r |β|

(1 ' # 2

Cu

(1

4

2

2

r |∇ /β|

(1 2

Therefore, we have #

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | 3

− ≤ cτ− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,Cu ∥ rtrχ ∥∞,Cu 2 1 3 3 1 × sup r 2 τ−2 ∥ α ∥L4 (St,u ) ∥ r− 2 (S)ˆi ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu −3

≤ cτ− 2 ϵ0 Q∗1

Thus, we have now proven that the integral (5.256) (i.e. (5.257)) is bounded: # (5.272) τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S)ˆi || α | ≤ cϵ0 Q∗1 Vte∗

Remark: Note that these estimates, being borderline, do not allow any further relaxation. We see here, similarly as in other parts of this work, that our assumptions lead to borderline estimates. Let us now discuss the corresponding situation for (5.253): (5.273)

#

Vte∗

τ+2 Φ | (ρ, σ)(LˆS W ) || D /4

(S)ˆ i ||

α |.

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

147

First, we write by the coarea formula (lemma 6) # τ+2 Φ | (ρ, σ)(LˆS W ) || D / 4 (S)ˆi || α | Vte∗

=

(5.274)

#

u∗

du

−∞

#

#

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || D /4

(S)ˆ i ||

α|

By similar argumentations as above in (5.258)–(5.262), we obtain on Cu

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || D /4

(5.275) ≤

−3 cτ− 2

(5.276)

'#

Cu

τ+2

(S)ˆ i ||

α|

| (ρ, σ)(LˆS W ) |2

(1 ' # 2 ·

Cu

Then, for the latter integral, it is '# (1 2 2 3 2 2 (S) τ + τ− | D / 4 ˆi | | α | Cu

= ≤

'#

t∗

0

'#

C#

τ+2 τ−3

τ+2 τ−3

D / 4 (S)ˆi ∥2L4 (St,u ) ∥

St,u

t∗

0

2

1 2

∥

3 2

|D /4

(S)ˆ i |2 |

≤ c sup r τ− ∥ α ∥L4 (St,u ) t,Cu

(5.277)

3

·

τ+2 τ−3

D (1 2 α | dt

|D /4

(S)ˆ i |2 |

2

α|

(1 2

2

α

'#

0

∥2L4 (St,u )

t∗

τ+ ∥

dt

(1 2

D / 4 (S)ˆi ∥2L4 (St,u )

dt

(1 2

As the the worst term in rD / 4 (S)ˆi is r∇ / χ, ˆ it follows directly from the elliptic estimates (5.265) that (1 ' # t∗ 2 2 (S) ˆ r∥D /4 ≤ cϵ0 Q∗1 . i ∥L4 (St,u ) dt 0

Hence, we have for (5.275): # τ+2 Φ | (ρ, σ)(LˆS W ) || D /4 Cu

(S)ˆ i ||

α|

−3

≤ cτ− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,Cu 2 1 3 3 1 × sup r 2 τ−2 ∥ α ∥L4 (St,u ) ∥ r 2 D /4 t,Cu −3

≤ cτ− 2 ϵ0 Q∗1 .

(S)ˆ i∥

L2 ([0,t∗ ],L4 (St,u ))

148

5. ERROR ESTIMATES

Therefore, (5.273) is estimated by # (5.278) τ+2 Φ | (ρ, σ)(LˆS W ) || D /4 Vte∗

(S)ˆ i ||

α | ≤ cϵ0 Q∗1 .

Similarly as above, we now estimate the integral for (5.251). # (5.279) τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || α | . Vte∗

The coarea formula (lemma 6) gives # τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m | | α | Vte∗

=

(5.280)

#

u∗

du

−∞

#

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || α |

One estimates on Cu : (5.281) #

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || α | Cu # ′ ≤ c sup {τ+ | trχ |} τ+ | (ρ, σ)(LˆS W ) || (S) m || α | Cu

≤

−3 cτ− 2

Cu

0#

Cu

τ+2

11 0 # 2 | (ρ, σ)(LˆS W ) | · 2

Cu

τ−3

|

(S) m |2 |

2

α|

11 2

2 1 3 3 ≤ cτ− ∥ r(ρ, σ)(LˆS W ) ∥2,Cu ∥ rtrχ ∥∞,Cu sup r 2 τ−2 ∥ α ∥L4 (St,u ) − 32

t,Cu

×∥r

−3

1 2

τ−−1 (S) m

∥L2 ([0,t∗ ],L4 (St,u ))

≤ cτ− 2 ϵ0 Q∗1

(5.282)

This yields for (5.279) # (5.283) τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || α | ≤ cϵ0 Q∗1 . Vte∗

Next, in view of (5.252) we estimate # τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || β | . (5.284) Vte∗

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

149

The coarea formula yields # τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || β | Vte∗

(5.285) #

=

#

#

t∗

0

We compute on Hte

Hte

τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || β | dt.

Hte

Φτ+2 | (ρ, σ)(LˆS W ) || trχ || (S) m || β | #

≤ c′ ∥ rtrχ ∥∞,e 3

≤ c(1 + t)− 2 − 32

≤ c(1 + t)

#

τ+ | (ρ, σ)(LˆS W ) || (S) m || β |

Hte 3

Hte

rr 2 | (ρ, σ)(LˆS W ) || (S) m || β |

'#

r | (ρ, σ)(LˆS W ) | 2

Hte

2

(1 2

·

'#
'

1 ≤c∥r 2 (S) m∥∞,e

(5.286) That is, we obtain for (5.286) # Φτ+2 | (ρ, σ)(LˆS W ) || trχ || (S) m || β |

!

Hte

2

β|

(1

r 2 |β|2

2

? ( 12

Hte

3 1 ≤ c(1 + t)− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ rtrχ ∥∞,e ∥ r 2

− 32

≤ c(1 + t)

(S) m

ϵ0 Q∗1 .

∥∞,e ∥ rβ ∥2,e

(5.287)

Then, one has for (5.284) # 1 τ+2 Φ | (ρ, σ)(LˆS W ) || trχ || (S) m || β |≤ c(1 + t)− 2 ϵ0 Q∗1 . (5.288) Vte∗

Next, from Λ1 (S, W ) we estimate # (5.289) τ+2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ /β|. Vte∗

By the coarea formula it is # τ+2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ /β| Vte∗

(5.290)

=

#

0

t∗

#

Hte

τ+2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ / β | dt.

150

#

5. ERROR ESTIMATES

Thus, on Hte , we calculate

Hte

τ+2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ /β| 3

≤ c(1 + t)− 2 − 32

≤ c(1 + t)

#

3

Hte

r2 r 2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ /β|

'#

(1 ' # 2 r | (ρ, σ)(LˆS W ) | · 2

Hte

2

2 3

Hte


'

Hte

2

? ( 12

Hte

3

1

≤ c(1 + t)− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ r 2

(5.292)

(S) i

3

≤ c(1 + t)− 2 ϵ0 Q∗1

∥∞,e ∥ r2 ∇ / β ∥2,e

Then, (5.289) is # 1 τ+2 Φ | (ρ, σ)(LˆS W ) || (S) i || ∇ / β | ≤ c(1 + t)− 2 ϵ0 Q∗1 . (5.293) Vte∗

From Λ2 (S, W ) = K 2 (S, W ) we estimate # (5.294) τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β | . Vte∗

Similarly as above, the coarea formula yields # τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β | Vte∗

=

(5.295)

#

u∗

−∞

du

#

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β |

We deduce for the integral on Cu : # τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β | Cu

(5.296)

≤ (5.297)

−3 cτ− 2

'#

Cu

τ+2

| (ρ, σ)(LˆS W ) |2

(1 ' # 2 ·

Cu

τ+2 τ−3

|

(S)/ p

2

2

| |β|

(1 2

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

151

The latter term in (5.297) satisfies the following inequality: '#

τ+2 τ−3

Cu

|

(S)/ p

2

2

| |β|

(1 2

C D '# 3 1 2 ≤ c sup r 2 τ− ∥ β ∥L4 (St,u ) ·

(5.298)

0

t,Cu

t∗

τ− ∥

(S)/ p

∥24,St,u

(1 2

.

Then, inequality (5.296) becomes #

τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β |

Cu

−3

≤ cτ− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,Cu 2 3 1 3 1 × sup r 2 τ−2 ∥ β ∥L4 (St,u ) ∥ r 2

(S)/ p

t,Cu

∥L2 ([0,t∗ ],L4 (St,u ))

−3

≤ cτ− 2 ϵ0 Q∗1

(5.299)

Thus, in view of (5.294) it is #

(5.300)

Vte∗

τ+2 Φ | (ρ, σ)(LˆS W ) || (S)/p || β | ≤ cϵ0 Q∗1 .

The remaining term in Λ2 (S, W ) = K 2 (S, W ) is estimated as follows. First, the coarea formula yields #

(5.301)

Vte∗

=

(5.302)

τ+2 Φ | (ρ, σ)(LˆS W ) || (S) p3 || (ρ, σ) | #

u∗

−∞

du

#

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || (S) p3 || (ρ, σ) |

Then, we compute on Cu : #

Cu

τ+2 Φ | (ρ, σ)(LˆS W ) || (S) p3 || (ρ, σ) |

(5.303) ≤

−3 cτ− 2

(5.304)

'#

Cu

τ+2

| (ρ, σ)(LˆS W ) |2

(1 ' # 2 ·

Cu

τ+2 τ−3

|

(S) p

3

2

2

| | (ρ, σ) |

(1 2

152

5. ERROR ESTIMATES

In view of the latter integral, it is '# (1 2 2 3 (S) 2 2 τ+ τ− | p3 | | (ρ, σ) | Cu

2

3

2

≤ c sup r ∥ (ρ, σ) ∥L4 (St,u ) · t,Cu

(5.305)

'#

t∗

0

r−1 τ−2

∥

(S) p

2 3 ∥4,St,u

(1 2

Then, (5.303) becomes # τ+2 Φ | (ρ, σ)(LˆS W ) || (S) p3 || (ρ, σ) | Cu

3

− ≤ cτ− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,Cu 2 3 1 × sup r2 ∥ (ρ, σ) ∥L4 (St,u ) · ∥ r− 2 τ− t,Cu

(S) p

3

∥L2 ([0,t∗ ],L4 (St,u ))

−3

(5.306)

≤ cτ− 2 ϵ0 Q∗1

This yields for (5.294) # τ+2 Φ | (ρ, σ)(LˆS W ) || (S) p3 || (ρ, σ) |≤ cϵ0 Q∗1 . (5.307) Vte∗

All the other terms in (5.242) # Φτ+2 | D(S, W )334 | Vte∗

behave as good as the ones just estimated or better and can be calculated in a straightforward way. This means, that we proceed analogously as above. Some of the terms are estimated on Ht and then integrated with respect to t (as in (5.284)) and the remaining ones are estimated on Cu and then integrated with respect to u (as in (5.257)). Which terms are estimated on Ht respectively on Cu , we specify below. These sets of estimates shall yield the estimate (5.335). Analogously to the procedure for estimating Int2 above, here, we denote the terms to be integrated on Ht by a lower index H and the ones to be integrated on Cu by a lower index C. The terms integrated on Cu are: All the second components of the corresponding null components of the Weyl current, namely, Λ2 (S, W ), K 2 (S, W ), I 2 (S, W ). From the first components, only the terms involving α are integrated on Cu and we denote them by Λ1C (S, W ), K 1C (S, W ), I 1C (S, W ). From the third components, we integrate on Cu all the terms of the form (derivative of π · curvature), except from I 3 (S, W ), which is multiplied by β(LˆS W ), the part (β · (Λ, K, Θ)((S) q)). Moreover, from the remaining only the ones involving α. They are denoted by Λ3C (S, W ), K 3C (S, W ), I 3C (S, W ).

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

153

The terms integrated on Ht are: From the first components of the corresponding null components of the Weyl current, we integrate on Ht all the terms except the ones involving α. They are denoted by Λ1H (S, W ), K 1H (S, W ), I 1H (S, W ). And from the third components, from I 3 (S, W ) the part (β · (Λ, K, Θ)((S) q)). Moreover, the terms not involving any derivatives, nor involving α, are estimated on Ht . We refer to them as Λ3H (S, W ), K 3H (S, W ), I 3H (S, W ). We obtain for (5.242) the following. On Hte , it is: #

Hte

3 Φτ+2 | D(S, W )334 | ≤ c(1 + t)− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,e

·

A

B 5 5 ∥ r 2 (Λ1H , K 1H ) ∥2,e + ∥ r 2 (Λ3H , K 3H ) ∥2,e

3 + c(1 + t)− 2 ∥ rβ(LˆS W ) ∥2,e A B 5 5 · ∥ r 2 I 1H ∥2,e + ∥ r 2 I 3H ∥2,e .

(5.308)

The components are estimated as follows. 5

∥ r 2 (Λ1H , K 1H ) ∥2,e A 1 1 ≤ c ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 (S) m ∥∞,e + ∥ r(ρ, σ) ∥2,e ∥ r 2 3 + ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 τ−−1 (S) m ∥∞,e . 1 1 + ∥ r(ρ, σ) ∥2,e ∥ r 2 (S) j ∥∞,e + ∥ rβ ∥2,e ∥ r 2 (S) m ∥∞,e 1

/ β ∥2,e ∥ r 2 + ∥ r2 ∇

+ ∥ r2 β 4 ∥2,e ∥ r

1 2

(S) i

(S) m 1

+ ∥ r2 (ρ3 , σ3 ) ∥2,e ∥ r 2 (5.309)

3

∥∞,e + ∥ rτ− β 3 ∥2,e ∥ r 2 τ−−1 ∥∞,e + ∥ r2 ∇ / (ρ, σ) ∥2,e ∥ r

(S) j

1 2

(S) m

(S) m 1

∥∞,e + ∥ r2 (ρ4 , σ4 ) ∥2,e ∥ r 2

(S) n

∥∞,e

.

∥∞,e ∥∞,e

(S) n

5

B ∥∞,e

∥ r 2 (Λ3H , K 3H ) ∥2,e A B 3 3 ≤ c ∥ r(ρ, σ) ∥2,e ∥ r 2 (ΛH , K H )((S) q) ∥∞,e + ∥ rβ ∥2,e ∥ r 2 ΞH ((S) q) ∥∞,e (5.310) 5

∥ r 2 I 1H ∥2,e A 3 ≤ c ∥ rtrχ ∥∞,e ∥ rβ ∥2,e ∥ r 2 τ−−1 3

+ ∥ r(ρ, σ) ∥2,e ∥ r 2 τ−−1

(S) m

(S) n

∥∞,e

1

∥∞,e + ∥ rβ ∥2,e ∥ r 2

(S) i

. ∥∞,e

154

5. ERROR ESTIMATES

-

1

∥ rβ ∥2,e ∥ r 2 ((S) i, (S) j) ∥∞,e . 1 3 + ∥ r(ρ, σ) ∥2,e ∥ r 2 (S) m ∥∞,e + ∥ r2 ∇ / β ∥2,e ∥ r 2 τ−−1 + ∥ rtrχ ∥∞,e

3

+ ∥ rτ− β 3 ∥2,e ∥ r 2 τ−−1 + ∥ r2 ∇ / (ρ, σ) ∥2,e ∥ r

1 2

(S) n

1

∥∞,e + ∥ r2 β 4 ∥2,e ∥ r 2

(S) m

(S) j 3 2

∥∞,e

∥∞,e + ∥ r2 (ρ3 , σ3 ) ∥2,e ∥ r τ−−1 B 1 + ∥ r2 (ρ4 , σ4 ) ∥2,e ∥ r 2 (S) m ∥∞,e (S) i

∥∞,e (S) m

∥∞,e

(5.311) 5

∥ r 2 I 3H ∥2,e A 3 ≤ c ∥ rβ ∥2,e ∥ r 2 (KH , ΛH , ΘH )((S) q) ∥∞,e 3

+ ∥ r(ρ, σ) ∥2,e ∥ r 2 (IH , I H )((S) q) ∥∞,e B 2 3 1 + sup r2 ∥ β ∥4,St,u ∥ r 2 (K, Λ, Θ)((S) q) ∥L2 ((−∞,u∗ ],L4 (St,u )) Hte

(5.312)

In view of (5.310) and (5.312) it is (5.313) 3

∥ r 2 (K H , ΛH , ΘH )( (S) q) ∥∞,e A 3 1 3 1 ≤ c ∥ r 2 (ζ, ζ) ∥∞,e ∥ r 2 (S) m ∥∞,e + ∥ r 2 ν ∥∞,e ∥ r 2 3

1

(S) n

+ ∥ r2χ ˆ ∥∞,e ∥ r 2 3 2

1 2

3 2

3 2

+ ∥ r ϵ ∥∞,e ∥ r

(S) m

+ ∥ r ξ ∥∞,e ∥ r τ−−1

(5.314)

1

1

∥∞,e + ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 (

∥∞,e + ∥ rtrχ ∥∞,e ∥ r B (S) m ∥ ∞,e

1 2

(S)ˆ i∥

1

∥ r 2 (K, Λ, Θ)( (S) q) ∥L2 ((−∞,u∗ ],L4 (St,u )) A 3 1 ≤ c ∥ r 2 (ζ, ζ) ∥∞,e ∥ r− 2 (S) m ∥L2 ((−∞,u∗ ],L4 (St,u )) 1

3

1

+ ∥ r 2 ν ∥∞,e ∥ r− 2 (S) j ∥L2 ((−∞,u∗ ],L4 (St,u ))

+ ∥ r2χ ˆ ∥∞,e ∥ r− 2 (S) n ∥L2 ((−∞,u∗ ],L4 (St,u )) 1

1

+ ∥ rτ−2 χ ˆ ∥∞,e ∥ r− 2 ((S) i, 3 2

+ ∥ r ϵ ∥∞,e ∥ r

− 12

(S) m

1

+ ∥ rtrχ ∥∞,e ∥ r− 2 3 2

+ ∥ r ξ ∥∞,e ∥

∥L2 ((−∞,u∗ ],L4 (St,u ))

∥L2 ((−∞,u∗ ],L4 (St,u ))

(S)ˆ i∥

1 r− 2 (S) m

(S) j)

L2 ((−∞,u∗ ],L4 (St,u ))

∥L2 ((−∞,u∗ ],L4 (St,u ))

∥∞,e

(S) i, (S) j)

(5.315)

3

(S) j

∞,e

∥∞,e

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF 1

1

1

1

155

+ ∥ r2D / 3 j ∥L2 ((−∞,u∗ ],L4 (St,u )) + ∥ r 2 p3 ∥L2 ((−∞,u∗ ],L4 (St,u ))

+ ∥ r2D / 3 i ∥L2 ((−∞,u∗ ],L4 (St,u )) + ∥ r 2 D / 4 n ∥L2 ((−∞,u∗ ],L4 (St,u )) B 1 + ∥ r2∇ / m ∥L2 ((−∞,u∗ ],L4 (St,u ))

(5.316) 3

∥ r 2 ΞH ((S) q) ∥∞,e A 1 1 1 ≤ c ∥ rtrχ ∥∞,e ∥ r 2 (S) m ∥∞,e + ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 (S) m ∥∞,e 3

1

3

1

+ ∥ r 2 ν ∥∞,e ∥ r 2 (S) m ∥∞,e + ∥ r 2 (ζ, ϵ) ∥∞,e ∥ r 2 (S) n ∥∞,e B 3 1 + ∥ r 2 ξ ∥∞,e ∥ r 2 ((S) i, (S) j) ∥∞,e

(5.317) 3

∥ r 2 (KH , ΛH , ΘH )((S) q) ∥∞,e A 1 3 3 ≤ c ∥ rtrχ ∥∞,e ∥ r 2 (S)ˆi ∥∞,e + ∥ r 2 (ζ, ζ, ϵ) ∥∞,e ∥ r 2 τ−−1 1

3

3 2

+ ∥ r ν ∥∞,e ∥ r

1 2

(S) j

(5.318)

3

(S) n

+ ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 τ−−1

1

∥∞,e + ∥ r 2 χ ˆ ∥∞,e ∥ r 2 ((S) i, B

(S) m

(S) j)

∥∞,e

∥∞,e

∥∞,e

3

∥ r 2 (IH , I H )( (S) q) ∥∞,e A 3 ≤ c ∥ rtrχ ∥∞,e ∥ r 2 τ−−1 3

1

3 2

1 2 (S)

3 2

3 2

+ ∥ r2χ ˆ ∥∞,e ∥ r 2 + ∥ r ν ∥∞,e ∥ r

(S) m

1

∥∞,e + ∥ rtrχ ∥∞,e ∥ r 2 (S) m ∥∞,e 1

3

∥∞,e + ∥ rτ−2 χ ˆ ∥∞,e ∥ r 2 τ−−1 3 2

3 2

m ∥∞,e + ∥ r ν ∥∞,e ∥ r τ−−1

+ ∥ r ξ ∥∞,e ∥ r τ−−1

(5.319)

(S) m

(S) n

3 2

(S) m

(S) m 1 2

∥∞,e

∥∞,e

∥∞,e + ∥ r (ζ, ζ) ∥∞,e ∥ r ((S) i,

(S) j)

B ∥∞,e

Thus, it is 3

(5.308) ≤ cϵ(1 + t)− 2 Q∗1 .

(5.320) On Cu , it is: #

Cu

Φτ+2 | D(S, W )334 | −3

≤ cτ− 2 ∥ r(ρ, σ)(LˆS W ) ∥2,Cu A 3 B 3 3 · ∥ r 2 τ− (Λ2 , K 2 ) ∥2,Cu + ∥ rτ−2 (Λ1C , K 1C ) ∥2,Cu + ∥ rτ−2 (Λ3C , K 3C ) ∥2,Cu

156

5. ERROR ESTIMATES 3

− + cτ− 2 ∥ τ− β(LˆS W ) ∥2,Cu A B 1 1 1 · ∥ r2 τ−2 I 2 ∥2,Cu + ∥ r2 τ−2 I 1C ∥2,Cu + ∥ r2 τ−2 I 3C ∥2,Cu .

(5.321)

The components are estimated as follows. 3

∥ r 2 τ− (Λ2 , K 2 ) ∥2,Cu A 1 3 1 ≤ c sup{r 2 τ−2 ∥ β ∥4,St,u } ∥ r 2

(S)/ p

∥L2 ([0,t∗ ],L4 (St,u ))

t,Cu

1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r− 2 τ−

(S) p

t,Cu

3

(5.322)

B ∥L2 ([0,t∗ ],L4 (St,u ))

3

∥ rτ−2 (Λ1C , K 1C ) ∥2,Cu 1

3

1

≤ c · sup{r 2 τ−2 ∥ α ∥4,St,u } · sup{| rtrχ |}· ∥ r− 2 t,Cu

t,Cu

(S)ˆ i∥

L2 ([0,t∗ ],L4 (St,u ))

(5.323) 3

∥ rτ−2 (Λ3C , K 3C ) ∥2,Cu A 1 3 1 ≤ c sup{r 2 τ−2 ∥ α ∥4,St,u } ∥ r 2 Θ((S) q) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r− 2 τ− (ΛC ((S) q), K C ((S) q)) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

2

+ sup{r ∥ β ∥4,St,u } ∥ r

− 12

t,Cu

τ− ΞC

((S) q)

B

∥L2 ([0,t∗ ],L4 (St,u ))

(5.324) 1

∥ r2 τ−2 I 2 ∥2,Cu A 3 1 1 ≤ c sup{r 2 τ−2 ∥ β ∥4,St,u } ∥ r 2

(S) p

t,Cu

1

+ sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r 2

(S)/ p

t,Cu

(5.325)

4

∥L2 ([0,t∗ ],L4 (St,u ))

B ∥L2 ([0,t∗ ],L4 (St,u ))

1

∥ r2 τ−2 I 1C ∥2,Cu 1

3

1

≤ c · sup{r 2 τ−2 ∥ α ∥4,St,u } · sup{r | trχ |}· ∥ r 2 τ−−1 t,Cu

(5.326)

t,Cu

(S) m

∥L2 ([0,t∗ ],L4 (St,u ))

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

157

1

∥ r2 τ−2 I 3C ∥2,Cu A 1 3 3 ≤ c sup{r 2 τ−2 ∥ α ∥4,St,u } ∥ r 2 τ−−1 Ξ((S) q) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

3

1

+ sup{r 2 τ−2 ∥ β ∥4,St,u } t,Cu

% 1$ × ∥ r 2 ΛC ((S) q), KC ((S) q), ΘC ((S) q) ∥L2 ([0,t∗ ],L4 (St,u ))

B 1 + sup{r2 ∥ (ρ, σ) ∥4,St,u } ∥ r 2 (IC ((S) q), I C ((S) q)) ∥L2 ([0,t∗ ],L4 (St,u )) t,Cu

(5.327)

In view of (5.324) and (5.327) we have 1

∥ r 2 Θ((S) q) ∥L2 ([0,t∗ ],L4 (St,u )) A 1 ≤ c ∥ r2D / 4 (S) i ∥L2 ([0,t∗ ],L4 (St,u )) 3

+ ∥ r 2 τ−−1 ∇ /

(S) m

∥L2 ([0,t∗ ],L4 (St,u )) 1

+ sup{r | trχ |} ∥ r− 2 t,Cu

3

(S)ˆ i∥

L2 ([0,t∗ ],L4 (St,u ))

1

+ sup{r 2 (| ζ |, | ϵ |)} ∥ r 2 τ−−1

(S) m

t,Cu

1

1

+ sup{rτ−2 | χ ˆ |} ∥ r 2 τ−−1

(S) n

t,Cu

3

1

+ sup{r 2 | χ ˆ |} ∥ r− 2 ((S) i, t,Cu

(5.328)

∥L2 ([0,t∗ ],L4 (St,u ))

∥L2 ([0,t∗ ],L4 (St,u ))

(S) j)

B ∥L2 ([0,t∗ ],L4 (St,u ))

1

∥ r− 2 τ− (ΛC ((S) q), K C ((S) q)) ∥L2 ([0,t∗ ],L4 (St,u )) A 1 1 ≤ c ∥ r 2 D4 (S) n ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r− 2 τ− D3 1

1

+ ∥ r− 2 τ− p3 ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 ∇ /

(5.329)

(S) m

(S) j

∥L2 ([0,t∗ ],L4 (St,u )) B

∥L2 ([0,t∗ ],L4 (St,u ))

1

∥ r− 2 τ− ΞC ((S) q)) ∥L2 ([0,t∗ ],L4 (St,u )) A 1 1 ≤ c ∥ r − 2 τ− D / 3 (S) m ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 ∇ /

(S) n

(5.330)

B ∥L2 ([0,t∗ ],L4 (St,u ))

3

∥ r 2 τ−−1 Ξ((S) q)) ∥L2 ([0,t∗ ],L4 (St,u )) A 3 3 ≤ c ∥ r 2 τ−−1 D / 4 (S) m ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 τ−−1 ∇ /

(S) n

∥L2 ([0,t∗ ],L4 (St,u ))

158

5. ERROR ESTIMATES 1

+ sup{r | trχ |} ∥ r 2 τ−−1

(S) m

t,Cu

3

1

+ sup{r 2 (| χ ˆ |, | ν |)} ∥ r 2 τ−−1

∥L2 ([0,t∗ ],L4 (St,u )) (S) m

t,Cu

3

1

+ sup{r 2 (| ζ |, | ϵ |)} ∥ r 2 τ−−1

(S) n

t,Cu

(5.331)

∥L2 ([0,t∗ ],L4 (St,u ))

B ∥L2 ([0,t∗ ],L4 (St,u ))

% 1$ ∥ r 2 ΛC ((S) q), KC ((S) q) ∥L2 ([0,t∗ ],L4 (St,u )) A 1 1 ≤ c ∥ r 2 D3 (S) n ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 D4 1

+ ∥ r2

(S) p

(5.332)

3

4

∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 τ−−1 ∇ /

% 1$ ∥ r 2 IC ((S) q), I C ((S) q) ∥L2 ([0,t∗ ],L4 (St,u )) A 1 1 ≤ c ∥ r2D / 4 (S) m ∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 ∇ / 1

+ ∥ r2

(S)/ p

(5.333)

1

∥L2 ([0,t∗ ],L4 (St,u )) + ∥ r 2 D /3

(S) j

∥L2 ([0,t∗ ],L4 (St,u ))

(S) m

(S) j

(S) m

B ∥L2 ([0,t∗ ],L4 (St,u ))

∥L2 ([0,t∗ ],L4 (St,u )) B

∥L2 ([0,t∗ ],L4 (St,u ))

We obtain −3

(5.321) ≤ cϵτ− 2 Q∗1 .

(5.334)

Then, integrating (5.308) with respect to t and integrating (5.321) with respect to u, proves # (5.335) Φτ+2 | D(S, W )334 |≤ cϵQ∗1 . Vte∗

Proceeding in the same manner with all the other terms in (5.241), we derive Int1 ≤ cϵ0 Q∗1 .

(5.336)

Estimates for the Integral Int5 : #

Vte∗

¯ ¯ γT δ | . Φ | Q(LˆT W )αβγδ (K) π αβ K

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

159

Here, we use the decomposition of Q into null components as it is done in the chapter about the comparison theorem. We do this with the integrand. It then follows, that the most delicate terms to estimate are τ+2 | Q(LˆT W )αβ34

¯ αβ (K) π ˆ

| ≤ cτ+2 | Q4443 + | Q3334

¯ (K) π ˆ

+ | QA334

(5.337)

¯ (K) π ˆ 44

¯ (K) π ˆ

33

| + | Q4334

| + | QA434

A4

¯ (K) π ˆ

¯ (K) π ˆ

| + | QAB34

A3

¯ (K) π ˆ

34

|

AB

|

| .

Here, we integrate certain terms first on Hte and then with respect to t, and we denote them by an index H. We will see that there is one term that has to be treated in a separate way, because it is borderline. By the assumptions ((5.53)–(5.54)), we obtain for the right hand side of (5.337) the first five terms to be controlled as in (5.338), also from the last quantity in (5.337) we integrate the term not involving β in the same manner. The remaining term (involving β) is borderline and requires more subtle treatment. We denote it by an index X. The same subtle estimates have to ¯ ˆAB involving α. The two be done in Int3 for the critical term in QAB33 (K) π problems are of the same quality. We shall carry out the computations in details after the presentation of the estimates for Int3 below. Now, let us state the results for the terms of Int5 integrated on Hte : #

Hte

¯ Φτ+2 | Q(LˆT W )αβ34 (K) π αβ |H 1

≤ c ∥ r− 2

3

¯ (K) n 1

+ c ∥ r− 2 3

− ∥∞,e ∥ r2 β(LˆT W ) ∥22,e · r0 2 % $ −3 ¯ (K) j ∥∞,e ∥ r2 ρ(LˆT W ), σ(LˆT W ) ∥22,e ·r0 2

1

+ c ∥ r− 2

3

¯ (K) n

− ∥∞,e ∥ rτ− β(LˆT W ) ∥22,e ·r0 2 % $ ¯ (K) m ∥∞,e ∥ r2 ρ(LˆT W ), σ(LˆT W ) ∥2,e

+ c ∥ r 2 τ−−2

3

− · ∥ r2 β(LˆT W ) ∥2,e · r0 2 3

+ c ∥ r 2 τ−−2

(5.338)

¯ (K) m

$ % ∥∞,e ∥ r2 ρ(LˆT W ), σ(LˆT W ) ∥2,e −3

· ∥ rτ− β(LˆT W ) ∥2,e · r0 2 % $ 1 −3 ¯ + c ∥ r− 2 (K) i ∥∞,e ∥ r2 ρ(LˆT W ), σ(LˆT W ) ∥22,e · r0 2 −3

≤ cϵ0 r0 2 Q∗1 .

160

5. ERROR ESTIMATES

For the remaining term in QAB44 #

Vte∗

(T ) π ˆ

AB

we obtain:

¯ Φτ+2 | Q(LˆT W )AB34 (K) π AB |X

≤c

#

0

t∗

2 3 ¯ sup r−1 ∥ (K) i ∥L∞ (St,u ) ∥ r2 β(LˆT W ) ∥2,e ∥ rτ− β(LˆT W ) ∥2,e dt u

≤ cϵ0 Q∗1 .

(5.339)

This last part (5.339) is proven after the subsection for 3 Int3 below. In fact, 2 −1 ¯ ( K) we shall give an estimate for supu r ∥ i ∥L∞ (St,u ) to show that it is integrable in t. Now, we deduce #

¯

Vte∗

Φτ+2 | Q(LˆT W )αβ34 (K) π αβ |≤ cϵ0 Q∗1 .

All the other terms in Int5 are estimated straightforward. Thus, it is Int5 ≤ cϵ0 Q∗1 .

(5.340)

Estimates for the Integral Int6 : #

Vte∗

¯ γK ¯δ | . Φ | Q(LˆT W )αβγδ (T ) π αβ K

As in the estimates just derived, we present the treatment of the terms that are most sensitive. Decomposing the integrand similarly as in (5.337), one finds them to be ˆ αβ | ≤ cτ+4 | Q4444 (T ) π ˆ33 | + | Q3444 (T ) π ˆ34 | τ+4 | Q(LˆT W )αβ44 (T ) π + | Q3344

(5.341)

+ | QA344

(T ) π ˆ

44

| + | QA444

(T ) π ˆA4

(T ) π ˆ

| + | QAB44

A3

(T ) π ˆ

|

AB

|

.

Similarly as in the previous calculations for Int1 and Int2 we integrate here certain terms on Hte and the remaining on Cu . The first five terms on the right-hand side of (5.341) are integrated on Hte and we denote the corresponding quantity by an index H below, whereas the last term in (5.341) is integrated on Cu and we denote it by an index C as below.

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

161

By the assumptions ((5.48)–(5.50)), we obtain for the integral over Hte # Φτ+4 | Q(LˆT W )αβ44 (T ) π ˆ αβ |H Hte

3

(T ) n

≤ c ∥ r2

− 32

∥∞,e ∥ r2 α(LˆT W ) ∥22,e · r0

+ c ∥ r2

3

(T ) j

+ c ∥ r2

3

(T ) n

+ c ∥ r2

3

(T ) m

+ c ∥ r2

3

(T ) m

3

− ∥∞,e ∥ r2 (ρ, σ)(LˆT W ) ∥22,e · r0 2

−3

(5.342)

3

− ∥∞,e ∥ r2 β(LˆT W ) ∥22,e · r0 2

− 32

∥∞,e ∥ r2 α(LˆT W ) ∥2,e ∥ r2 β(LˆT W ) ∥2,e · r0

3

− ∥∞,e ∥ r2 (ρ, σ)(LˆT W ) ∥2,e ∥ r2 β(LˆT W ) ∥2,e · r0 2

≤ cϵ0 r0 2 Q∗1 .

Correspondingly, we estimate the integral over Cu # Φτ+4 | Q(LˆT W )αβ44 (T ) π ˆ αβ |C Cu

1

≤ c ∥ rτ−2

(T )ˆ i∥

∞,Cu

-

∥ r2 β(LˆT W ) ∥22,Cu

+ ∥ rτ− (ρ, σ)(LˆT W ) ∥2,Cu ∥ r2 α(LˆT W ) ∥2,Cu (5.343) That is,

−3

≤ cϵ0 τ− 2 Q∗1 . #

Vte∗

Φτ+4 | Q(LˆT W )αβ44

(T ) π αβ

.

−3

· τ− 2

|≤ cϵ0 Q∗1 .

Similarly as above, all the other terms in Int6 are estimated in a straightforward manner, which yields: Int6 ≤ cϵ0 Q∗1 .

(5.344)

Estimates for the Integral Int3 : # ¯ Φ | Q(LˆS W )αβγδ (K) π αβ T γ T δ | . Vte∗

The most difficult terms are estimated below. First, decomposing the integrand in a corresponding way as in (5.337), it is ¯ αβ ¯ ¯ ˆ | ≤ c | Q4433 (K) π ˆ33 | + | Q4333 (K) π ˆ34 | | Q(LˆS W )αβ33 (K) π + | Q3333

(5.345)

+ | QA333

¯ (K) π ˆ

44

¯ (K) π ˆ

| + | QA433

A4

¯ (K) π ˆ

| + | QAB33

A3

¯ (K) π ˆ

|

AB

|

.

162

5. ERROR ESTIMATES

Similarly as in Int5 , we encounter here the borderline case which requires a more subtle treatment. On Hte we integrate the first five terms on the right-hand side of (5.345) and from the last quantity the term not involving α. We indicate these terms by an index H. The remaining term (involving α) is critical and we denote it by an index X. The detailed estimates for this borderline term shall be given after this subsection. Now, assumptions ((5.53)–(5.54)) yield # ¯ αβ Φ | Q(LˆS W )αβ33 (K) π ˆ |H Hte

1

≤ c ∥ r− 2

¯ (K) n

¯ (K) j

1

+ c ∥ r− 2 3

+ c ∥ r 2 τ−−2 3

+ c ∥ r 2 τ−−2

¯ (K) n

Vte∗

3

∞,e ∥

− 32

rβ(LˆS W ) ∥22,e ·r0 (T ) π ˆ

AB

−3

≤ cϵ0 r0 2 Q∗1 .

we obtain:

¯ AB Φ | Q(LˆS W )AB33 (K) π ˆ |X

≤c ≤

3

− ∥∞,e ∥ τ− α(LˆS W ) ∥2,e ∥ rβ(LˆS W ) ∥2,e · r0 2

For the remaining term in QAB33 #

− 32

∥∞,e ∥ τ− α(LˆS W ) ∥22,e · r0

− ∥∞,e ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ rβ(LˆS W ) ∥2,e · r0 2

¯ (K) m

¯ ˆ (K) i∥

1

+ c ∥ r− 2

3

− ∥∞,e ∥ rβ(LˆS W ) ∥22,e · r0 2

¯ (K) m

1

+ c ∥ r− 2

(5.346)

3

− ∥∞,e ∥ r(ρ, σ)(LˆS W ) ∥22,e · r0 2

#

0

t∗

2 3 ¯ sup r−1 ∥ (K) i ∥L∞ (St,u ) ∥ τ− α(LˆS W ) ∥2,e ∥ r(ρ, σ)(LˆS W ) ∥2,e dt u

cϵ0 Q∗1 .

(5.347) This gives #

Vte∗

Φ | Q(LˆS W )αβ33

¯ αβ (K) π ˆ

| ≤ cϵ0 Q∗1 .

Estimating the other terms in Int3 correspondingly, we derive (5.348)

Int3 ≤ cϵ0 Q∗1 .

2 3 ¯ Estimates for supu r−1 ∥ (K) i ∥L∞ (St,u ) in (5.339) and (5.347): We present here the estimates for (5.347). As the situation in (5.339) is in principle the same, the estimates follow directly.

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

163

¯

The integral of the term | QAB33 (K) π ˆAB |X reads # ¯ Φ | (K) i || (ρ, σ)(LˆS W ) || α(LˆS W ) |

(5.349)

Vte∗

=c (5.350) =c ≤c

# #

dt

0

× (5.351)

t∗

'#

Hte

t∗

0

#

du

#

dt

0

#

u∗

−∞

t∗

#

#

dt

0

× ≤c

t∗

dt

St,u

St,u

Φ2 r−1 rτ−−1 τ− | (K) i || (ρ, σ)(LˆS W ) || α(LˆS W ) |

C' #

Hte

r−2 τ−−2 r2

| α(LˆS W ) |2

C' #

u∗

−∞

r

−2

¯ Φ2 | (K) i || (ρ, σ)(LˆS W ) || α(LˆS W ) |

¯

Hte

τ−2

#

|

|

¯ (K) i |2 |

(ρ, σ)(LˆS W ) |2

(1 B

(1 2

2

duτ−−2

¯ (K) i |2

r | (ρ, σ)(LˆS W ) |2 2

¯

(1 2

∥ τ− α ∥2,e

D

1

2 . In order to estimate (5.351), We recall that the term (K) i grows 2 −1like(rK) 3 ¯ i ∥L∞ (St,u ) out of the integral we have to take the supremum supu r ∥ over u, and to estimate this supremum in terms of β. Moreover, we transform the integral over u into an integral on a finite interval by the following substitution: d˜ u 2 = 2τ−−2 , d˜ u = 2τ−−2 du. u ˜ = 2 arctan u, = du 1 + u2 Then it is # #

u∗

−∞

2τ−−2 du =

u ˜∗

d˜ u.

−π

Denote the new interval by I = (−π, u ˜∗ ]. Therefore, the said integral in (5.351) becomes # # u∗ ¯ −2 duτ− r−2 | (K) i |2 r2 | (ρ, σ)(LˆS W ) |2 −∞ S # # t,u ¯ (5.352) = d˜ u r−2 | (K) i |2 r2 | (ρ, σ)(LˆS W ) |2 I

St,˜ u

2 3 ¯ Let us now estimate supu r−1 ∥ (K) i ∥L∞ (St,u ) . First, we remind ourselves of the computations in the subsection for Int1 in order to estimate

164

5. ERROR ESTIMATES

(5.262) involving (S) i. (See (5.262)–(5.272).) Similarly as in that situation ¯ for (S) i we write now in view of the definition of (K) i: 1 1 LˆK¯ g = τ+2 LˆL g + τ−2 LˆL g. 2 2 ¯ ˆ and τ−2 χ ˆ . Whereas the latter has order of The (K)ˆi involves the terms r2 χ 1

1

growth O(τ−2 ), the first one is of order o(r 2 ). Thus, we proceed with r2 χ. ˆ In (5.265) we recalled the formulas how χ ˆ is estimated in terms of β. From our chapter about general inequalities and 3d-results and from [19], chapter 3, we apply the results to estimate the supremum of the L4 -norm on S as well as the L6 -norm on H in terms of the L2 -norm up to first derivative on H. In the notation of the said chapter, F denoting an arbitrary tensor on H, tangent to S at every point, it is: -# .1 .1 4 2 sup (5.353) r4 | F |4 ≤ cS | F |2 +r2 | ∇F |2 u∈H

(5.354)

S

-#

S

r6 | F |6

.1 6

.1 2 ≤ cS | F |2 +r2 | ∇F |2 .

As we have transformed our integral (5.352) into one over a finite interval for u ˜, we can integrate on St,˜u0 for a fixed value u0 and compute the difference from the supremum of the integral on a surface St,˜u . We drop the tilde for the subsequent calculations. Thus by straightforward computations we obtain # B # A 6 4 | sup r |β| r6 | β |4 | − u

Su

Su 0

1 ≤ c ∥ r 2 β ∥2L2 (Hu,u

3

0)

3 2

+∥r ∇ / β ∥2L2 (Hu,u

(5.355)

· ∥ r2∇ / N β ∥2L2 (Hu,u

0)

0)

3 2

·∥r ∇ / N β ∥2L2 (Hu,u

0)

.

with Hu,u0 denoting the domain in Ht between the values u and u0 . One sees that also the square root of (5.355) is integrable in time: # t∗ 1 3 dt ∥ r 2 β ∥2L2 (Hu,u ) · ∥ r 2 ∇ / N β ∥2L2 (Hu,u ) 0

0

+ ∥ r2∇ / β ∥2L2 (Hu,u

· ∥ r2∇ / N β ∥2L2 (Hu,u

0

3

0)

(5.356)

≤ cϵ0 Q∗1 .

3

0)

.1 2

We have shown that the supremum satisfies the inequality # 1 3 r6 | β |4 ≤ c′ ∥ r 2 β ∥2L2 (Hu,u ) · ∥ r 2 ∇ / N β ∥2L2 (Hu,u ) sup 0 0 u Su . 3 3 + ∥ r2∇ / β ∥2L2 (Hu,u ) · ∥ r 2 ∇ / N β ∥2L2 (Hu,u ) 0

(5.357)

0

5.3. BOUNDEDNESS THEOREM: STATEMENT AND PROOF

165

Now, we go back to (5.351). Observing that 2 3 3 ¯ sup r−1 ∥ (K) i ∥L∞ (St,u ) ≤ c sup ∥ r 2 β ∥L∞ (St,u ) u u 1 3 ≤ c ∥ r 2 β ∥2L2 (Hu,u ) · ∥ r 2 ∇ / N β ∥2L2 (Hu,u

0)

0

3

3

+ ∥ r2∇ / β ∥2L2 (Hu,u

(5.358)

· ∥ r2∇ / N β ∥2L2 (Hu,u

0)

.1 2

0)

We estimate, taking into account (5.356) and (5.358): (5.351) # t∗ C' # 2 32 ¯ dt d˜ u sup r−1 ∥ (K) i ∥L∞ (St,˜u ) ≤c 0

×

≤c

#

×

=c

r | (ρ, σ)(LˆS W ) | 2

0

≤ cϵ0 Q∗1

2

(1 2

∥ τ− α ∥2,e

A 2 3 ¯ dt sup r−1 ∥ (K) i ∥L∞ (St,˜u )

'#

t∗

u ˜

I

St,˜ u

t∗

0

#

#

u ˜

d˜ u

I

A

#

St,˜ u

(1 2

r | (ρ, σ)(LˆS W ) | 2

2

D

∥ τ− α ∥2,e

B

B 2 3 ¯ dt sup r−1 ∥ (K) i ∥L∞ (St,˜u ) ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ τ− α ∥2,e u ˜

(5.359)

This ends the proof for (5.339) and (5.347). Estimates for the Integral Int4 : # Φ | Q(LˆS W )αβγδ

(T ) π αβ K ¯ γT δ

Vt∗

|.

Let us estimate the most delicate terms for this situation. To do so, we decompose the integrand similarly as in (5.337) and obtain 2 αβ 2 (T ) ˆ π ˆ | ≤ cτ+ | Q4443 (T ) π ˆ33 | + | Q4334 (T ) π ˆ34 | τ+ | Q(LS W )αβ34 + | Q3334

(5.360)

+ | QA334

(T ) π ˆ

44

(T ) π ˆ

| + | QA434

A4

(T ) π ˆ

| + | QAB34

A3

(T ) π ˆ

|

AB

. |

In a similar way as for Int6 above, we calculate the first five terms on the right-hand side of (5.360) on Hte and denote them by an index H, the last term in (5.360) is integrated on Cu and the corresponding quantity is denoted by an index C.

166

5. ERROR ESTIMATES

In view of assumptions ((5.48)–(5.50)), one estimates on Hte # Φτ+2 | Q(LˆS W )αβ34 (T ) π ˆ αβ |H Hte

3

≤ c ∥ r2

(T ) n

3

− ∥∞,e ∥ rβ(LˆS W ) ∥22,e · r0 2

+ c ∥ r2

3

(T ) j

+ c ∥ r2

3

(T ) n

+ c ∥ r2

3

(T ) m

+ c ∥ r2

3

(T ) m

− 32

∥∞,e ∥ rβ(LˆS W ) ∥22,e · r0

−3

(5.361)

3

− ∥∞,e ∥ r(ρ, σ)(LˆS W ) ∥22,e · r0 2 3

− ∥∞,e ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ rβ(LˆS W ) ∥2,e · r0 2 3

− ∥∞,e ∥ r(ρ, σ)(LˆS W ) ∥2,e ∥ rβ(LˆS W ) ∥2,e · r0 2

≤ cϵ0 r0 2 Q∗1 .

On Cu , we have #

Φτ+2 | Q(LˆS W )αβ34 (T ) π ˆ αβ |C # = Φτ+2 | Q(LˆS W )AB34 (T ) π ˆ AB |

Cu

Cu

1

(5.362)

-

∥ r(ρ, σ)(LˆS W ) ∥22,Cu . −3 + ∥ rβ(LˆS W ) ∥2,Cu ∥ τ− β(LˆS W ) ∥2,Cu · τ− 2

≤ c ∥ rτ−2

(T )ˆ i∥

∞,Cu

−3

≤ cϵ0 τ− 2 Q∗1 .

Therefore, we obtain # Φ | Q(LˆS W )αβ34 Vte∗

(T ) π ˆ αβ

| ≤ cϵ0 Q∗1 .

All the other terms in Int4 are estimated correspondingly, yielding

(5.363)

Int4 ≤ cϵ0 Q∗1 .

Now, we have estimated all the terms of E1 (W, t∗ ) in (5.162). Thus, we have shown (5.364)

E1 (W, t∗ ) ≤ cϵ0 Q∗1 .

Therefore, we have proven the boundedness theorem.

CHAPTER 6

Second Fundamental Form k: Estimates for the Components of k Here, we estimate the components δ, ϵ, ηˆ of the second fundamental form k of the time foliation. 6.1. Decomposing the Equations for k Relative to the Radial Foliation In this section, we shall decompose the elliptic system for the second fundamental form k relative to the radial foliation. First, recall that for k we have the following elliptic system on each slice Ht . (6.1)

trk = 0

(6.2)

curl k = H

(6.3)

div k = 0.

Also, recall that H is the magnetic part of the spacetime curvature relative to the time foliation. It is also true that (6.4)

Rij = kia k a j + Eij ,

where E denotes the electric part of the spacetime curvature relative to the time foliation. The equations (6.1–6.3) can be decomposed relative to the radial foliation as follows. 3 div / ϵ = −∇N δ − trθ · δ + ηˆ · θˆ − 2(a−1 ∇ (6.5) / a) · ϵ 2 (6.6) curl / ϵ = σ + θˆ ∧ ηˆ 1 ∇ /N ϵ + trθ · ϵ = (β − β) + ∇ (6.7) / δ − 2θˆ · ϵ 2 3 (6.8) / a) · δ − ηˆ · (a−1 ∇ / a) + (a−1 ∇ 2 1 1 1 div / ηˆ = (β − β) − ∇ (6.9) / δ + θˆ · ϵ − trθ · ϵ 2 2 2 1 1 1 3 ˆ ˆ ϵ+ δ·θ ∇ /N ηˆ + trθ · ηˆ = (α − α) + ∇ (6.10) /⊗ 2 4 2 2 −1 ˆ + (a ∇ (6.11) / a) ⊗ ϵ. 167

168

6. SECOND FUNDAMENTAL FORM k

We will see later on, that the nonlinear terms in these equations have better decay properties than the worst linear ones. So, if we control the linear terms, we also control the rest. In a ‘linear’ theory, these equations reduce to: (6.12) (6.13)

3 div / ϵ + trθ · δ + ∇N δ = 0 2 curl / ϵ=σ 1 / δ + (β − β) ∇ /N ϵ + trθ · ϵ = ∇ 2 1 1 1 / δ − trθ · ϵ + (β − β) div / ηˆ = − ∇ 2 2 2 1 1 1 ˆ ϵ + (α − α). ∇ /N ηˆ + trθ · ηˆ = ∇ /⊗ 2 2 4

(6.14) (6.15) (6.16)

ˆ of the Second 6.2. Estimating the Components δ, ϵ, η Fundamental Form Next, let us estimate the components δ, ϵ, ηˆ of the second fundamental form. We will first discuss the setting and the structure of the problem, then we shall formulate the idea of the proof, and finally we will give the proof in details. We will see that, in our case, it is possible to get most of the estimates for δ and ϵ together. That is, we combine them into the 1-form iZ k as it is carried out below. First, we shall estimate δ and ϵ. To do so, recalling the position vectorfield Z to be Z = rN, let us introduce the following 1-form: (6.17)

(iZ k)i = kij Z j = r(ϵi + δNi ).

One can compute (6.18) (6.19)

curl(iZ k) = F + G 1 ij div(iZ k) = π ˆ kij , 2

where (6.20) (6.21)

Fi = Hij Z j 1 j Gi = ϵmn Nm knj 2 i

and N reads as (6.22)

j j =π ˆm − ra−1 (Nm ∇ /j a − N j ∇ /m a). Nm

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

169

Here, π = (Z) π denotes the deformation tensor of Z. And π ˆij are the components of its traceless part. The deformation tensor is given as: (6.23) (6.24) (6.25)

1 /j a + Nj ∇ /i a) π ˆij = 2rθˆij + κ(gij − 3Ni Nj ) − ra−1 (Ni ∇ 3 trπ = ra−1 (2atrθ + atrθ) 1 πij = π ˆij + trπgij , 3

and κ = ra−1 (atrθ − atrθ).

(6.26)

In equation (6.19), we write for π ˆ on the right-hand side 1 π ˆ ij = θˆij? + κ(g ij − 3N i N j ) − ra−1 (N i ∇ / ja + Nj∇ / i a) 3 =2r ηˆij

(6.27)

−1

= O(τ− 2 ).

Therefore, we have for the right-hand side of equation (6.19):

(6.28) (6.29) (6.30)

1 ij π ˆ kij = rθˆij kij + l.o.t. 2 = rθˆij ηij + l.o.t. 1 = rθˆij (ˆ ηij + trη γij ) + l.o.t. 3 ij ˆ = rθ ηˆij + l.o.t. = r | ηˆ |2 + l.o.t.

= O(r−1 τ−−1 ).

Thus, we see that (6.31)

1 ij ˆ kij ∈ L2 (H). div(iZ k) = π 2

Here, one can work with the ‘linear’ theory as in view of (6.19) and (6.30), the right-hand side of the equation (6.19) has better decay in the wave zone than the components of its left-hand side (see (6.39)). That is, if we control the corresponding components in the ‘linear’ theory, we control the general nonlinear problem. But before we will proceed, we shall give an explanation of what we have just introduced.

170

6. SECOND FUNDAMENTAL FORM k

In view of (6.30) recall that 1 ij 1 1 / ja + Nj∇ / i a)kij π ˆ kij = r | ηˆ |2 + κ(g ij − 3N i N j )kij − ra−1 (N i ∇ 2 6 2 1 1 = r | ηˆ |2 + κtrk − κδ − ra−1 (∇ / a)ϵ ? 2 =0

1 (6.32) / a)ϵ. = r | ηˆ | − κδ − ra−1 (∇ 2 So, let us compute explicitly the right-hand sides of the equations (6.18) and (6.19). First, in view of (6.18), and recalling that ϵBC ∇B kjC = curlkj , we write 2

curl(iZ k) = ϵBC ∇B (iZ k)C = ϵBC ∇B (kjC Z j ) = ϵBC (kjC ∇B Z j + (∇B kjC )Z j )

(6.33)

= ϵBC kjC ∇B Z j + curlkj Z j

(6.34)

= G + F.

The last equation holds because of the fact that curlkij = Hij as F is given by Fi = Hij Z j ; and as Gi = 12 ϵi mn Nm j knj in the following caldr = 2r atrθ: culation, with λ = du 1 1 j ∇B Z j = NBj + r(trθ − λa−1 )gB 2 3 1 BC 1 j BC j ϵ kjC ∇B Z = ϵ kjC NBj + r(trθ − λa−1 ) ϵBC kjC gB 2 3 < => ?

1 = ϵBC kjC NBj = G, 2 as k is symmetric. For (6.19) we write




=0

?

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

171

j Recalling equation (6.22) and ∇B Z j = 12 NBj + 13 r(trθ − λa−1 )gB , we then write 1 1 g li kij ∇l Z j = kij g li Nli + (rtrθ + λa−1 ) kij g li glj 2 3 < => ? =trk=0

1 = kij N ij 2 1 1 = kij π ˆ ij − ra−1 (kij N i ∇ / j a − kij N j ∇ / i a) 2 2 < => ? =0

1 = kij π ˆ ij . 2 From now on, we shall work with the linear terms. For, as explained above, if we control the corresponding components in the ‘linear’ theory, we control them in the general nonlinear theory. In the ‘linear’ case, note the following fact. As the lapse function u is linear too, a =| ∇u |−1 is constant. (Also, consider the constraint equation R = | k |2 .) Then, in the ‘linear’ theory, equations (6.18) and (6.19) read as (6.37)

curl(iZ k) = iZ H

(6.38)

div(iZ k) = 0.

So, one writes (iZ H)i = Hij Z j = Fi , that is curl(iZ k) = iZ H = F. From the calculation for curl(iZ k) above, we see that curl(iZ k) = G+F , and G is of order k 2 . Thus, in the linear theory, it only remains curl(iZ k) = F . And in view of equation (6.38), the divergence of iZ k is also of order k 2 , which means that, in fact, in the linear theory div(iZ k) = 0. One computes for the equation (6.38) (6.39)

3 div / ϵ + ∇N δ + trθ · δ = 0. 2

ˆ – Preliminary Discussion. In the 6.2.1. Estimates for δ, ϵ and η next chapter, it will be shown that the L2 -norms of the components δ, ϵ and ηˆ of the second fundamental form k are bounded. And their derivatives with respective weights will be investigated. Purpose of the present short chapter is to give a preliminary discussion of the problem, in order to see where the main difficulties arrive, which later on will be solved. In particular, in the next chapter it shall bee seen that # r2 | ϵ |2 is bounded (6.40) H

172

6. SECOND FUNDAMENTAL FORM k

and that

#

(6.41)

H

r2 | δ |2 is bounded.

One can leave δ and ϵ together to obtain their estimates and then derive the ones for ηˆ from the equations (6.15), (6.16). In a next step, we will estimate derivatives of the components δ, ϵ, η. Here, we discuss the expected results and their behaviour in the wave zone. To do so, we first recapitulate the orders of these terms. 3

(6.42)

δ = o(r− 2 )

(6.43)

ϵ = o(r− 2 )

(6.44)

η = O(r−1 τ− 2 ).

3

−1

First, notice that η yields the borderline case, which we can bound and therefore obtain that η ∈ L2 (H). In (6.31), we saw that div(iZ k) ∈ L2 (H). And iZ k only involves δ and ϵ. So, angular and normal derivatives of iZ k behave nicely, i.e. decay fast enough. Recall that (see (6.29), (6.30) and (6.31)) 1 ij ˆ kij = r | ηˆ |2 + l.o.t. = O(r−1 τ−−1 ). div(iZ k) = π 2 From this it follows that ∇iZ k = O(r−1 τ−−1 ). At first sight, the term η could cause troubles in the wave zone, as it only −1

decays like r−1 τ− 2 . That is, if we take the second derivative of iZ k in the normal direction, that is ∇ /N ∇iZ k,

that will yield the integral # # r2 | ∇ /N ∇iZ k |2 = c τ−−4 + l.o.t. H H # 0# ∞

=c

Su

u0

′

(6.45)

′

≥cA

#

∞

u0

aτ−−4 dµγ

r2 du. τ−4

1

du + l.o.t.

And this integral has a bad behaviour in the wave zone. Recall here that r > τ− in the wave zone, and τ− > r far enough outside this region. Nevertheless, we can derivate ∇iZ k tangentially to Su to obtain: # # r2 | ∇ /∇iZ k |2 ≤ c r−2 τ−−2 < ∞. (6.46) H

H

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

173

Remark: Note that ∇ /N ∇iZ k leads to the following degenerate integral, which is finite: # τ−2 | ∇ /N ∇iZ k |2 < ∞. (6.47) H

Also, we have # # # 2 −6 2 2 r |∇ /N iZ k | = c τ− + l.o.t. = c H

H

#

≥ c′ A′

(6.48)

∞

u0

∞

u0

r2 du. τ−6

0#

Su

1

aτ−−6 dµγ du + l.o.t.

And we see again, that this is bad in the wave zone. Now, we consider the derivatives for η. Here, we encounter a problem, /N η |2 . That is, when taking the normal derivative of η and integrating r2 | ∇ the said integral diverges in the wave zone. Nevertheless, we can derivate η tangentially to Su . Let us first give the orders for the normal and angular derivatives of η. We obtain for the angular derivative −1

(6.49)

∇ /η = O(r−2 τ− 2 )

(6.50)

r∇ /η = O(r−1 τ− 2 ),

−1

whereas for the normal derivative, one only has −3

∇ /N η = O(r−1 τ− 2 )

(6.51)

−3

(6.52)

r∇ /N η = O(τ− 2 ).

Therefore, we calculate with the help of (6.52): # # # 0# 2

2

H

r |∇ /N η | = c

H

(6.53)

′

′

≥cA

τ−−3

+ l.o.t. = c

#

r2 du. τ−3

∞

u0

∞

u0

Su

aτ−−3 dµγ

1

du + l.o.t.

So, we see that this integral has a bad behaviour in the wave zone. On the other hand, we infer from (6.50) that # (6.54) r2 | ∇ /η |2 dµγ < ∞. St,u

/ η |2 decays fast In other words, everything is fine in the wave zone; r2 | ∇ enough for the integral over St,u to be bounded. Integrating the term over the hypersurface H, we find the borderline case. This is good enough, for,

174

6. SECOND FUNDAMENTAL FORM k

we can control it by giving corresponding bounds. Now, we have 1 # # # ∞0# 2 2 −2 −1 −2 −1 r |∇ /η | ≤ c r τ− ≤ c ar τ− dµγ du H

H

≤ c′ A

(6.55)

Su

u0

#

∞

u0

τ−−1 du.

The borderline case (6.55) yields a term, which can be controlled by adequate bounds. It remains to show how the second derivatives of η behave. So, let us first take the angular derivative of ∇ / η and calculate: 1 # # # ∞0# 2 r4 | ∇ / η |2 ≤ c r−2 τ−−1 ≤ c ar−2 τ−−1 dµγ du H

H

≤ c′ A

(6.56)

Su

u0

#

∞

u0

τ−−1 du.

This is the borderline case, which we have encountered above in (6.55). Similarly, taking the normal derivative of ∇ / η yields the borderline case: # r2 τ−2 | ∇ /∇ /N η |2 . (6.57) H

Finally, for ∇ /N2 η, we see that, again, the integral # # ∞0# # 2 4 2 2 −5 r |∇ /N η | = c r τ− + l.o.t. = c H

(6.58)

H

′

≥cA

′

#

u0

∞

u0

Su

ar2 τ−−5 dµγ

1

du

r4 du. τ−−5

has a bad behaviour in the wave zone.

ˆ – Proof. Here, we will carry out 6.2.2. Estimates for δ, ϵ and η the proof in details. Estimates for δ and ϵ: As announced above, we will obtain most of the estimates for these quantities together by studying the 1-from iZ k. Recall equation (6.17): (iZ k)i = kij Z j = r(ϵi + δNi ). Also recall the Hodge system (6.37, 6.38) for the linear case: curl iZ k = iZ H = H · Z = rN · H div iZ k = 0. Now, if the right-hand side of (6.37) is bounded, then we can use proposition 7 to estimate the norms of iZ k and ∇iZ k.

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

175

To do so, we first have to check that, indeed, ∥ H · Z ∥L2 is finite. Therefore, we write # (6.59) ∥ H · Z ∥2L2 = r2 | N · H |2 . H

Comparing this with our norms for the curvature tensor (3.82) and recalling (3.7)–(3.26) as well as the components of H ((4.73)–(4.75)), we approve that, ∥ H · Z ∥L2 lies within our assumptions on the components of the curvature tensor. Thus, in fact, ∥ H · Z ∥L2 is finite. Next, we shall apply proposition 7 to the system (6.37, 6.38). That is, considering equation (52) and the fact that div iZ k = 0, we have # # # 2 2 (6.60) | ∇iZ k | + R | iZ k | = | curl iZ k |2 . H

H

So, we obtain (6.61) (6.62)

#

#

H

H

2

H

r

| ∇iZ k | ≤ c1

−2

2

| iZ k | ≤ c2

#

H

| curl iZ k |2

H

| curl iZ k |2

#

with constants c1 and c2 . Here, the second result follows directly by the first one and by the Poincar´e inequality. Thus, we have shown that ∥ r−1 iZ k ∥L2 and ∥ ∇iZ k ∥L2 are bounded. As we have iZ k = rϵ + rδN (see (6.17)), and from the above (equation (6.62)), it follows that (6.63) (6.64)

∥ ϵ ∥L2 ≤ cR[1]

∥ δ ∥L2 ≤ cR[1] .

Similarly, in view of equation (6.61), we deduce (6.65) (6.66)

∥ r∇ϵ ∥L2 ≤ cR[1]

∥ r∇δ ∥L2 ≤ cR[1] .

In a next step, we will estimate second derivatives of iZ k. We shall see that the corresponding norms of ∇ / 2 iZ k and ∇ /∇ /N iZ k can be bounded in the proper way, i.e. the estimates are non-degenerate, whereas ∇ /N2 iZ k leads to degenerate estimates. First, let us consider the angular and the mixed derivatives, that is /∇ /N iZ k, respectively. We are going to estimate them together. ∇ / 2 iZ k and ∇ At a certain point in this procedure, we shall need a decomposition of iZ k to handle the terms of lower decay of type r · Riem · ξ, which will show up in the computations. In fact, the said term involving the component of the Riemannian curvature tensor with worst decay, does not decay fast enough. Thus, by introducing a suitable decomposition, the terms of order higher than r−2 will cancel out. And we shall prove the terms ∇ / 2 iZ k and ∇ /∇ /N iZ k to have the desired decay.

176

6. SECOND FUNDAMENTAL FORM k

Before we start with the detailed estimates, we shall recall here some facts about 1-forms and 2-forms and also prove a statement that will be used later on. So, let ξ be a 1-form on H and consider the div − curl system div ξ = ∇k ξk = ρ

(6.67)

(curl ξ)i = ϵijk ∇j ξ k = σi ,

where ρ is a given function and σ is a given 1-form. To obtain estimates for the second derivatives of a 1-form, we shall take the covariant derivative of such a system. That is, we obtain: (6.68)

∇m div ξ = ∇k ∇m ξk − Rm k ξk = ∇m ρ

∇m (curl ξ)i = ϵijk ∇j ∇m ξ k + ϵijk Rm jk l ξ l = ∇m σi .

To carry on, we shall work with the 2-form A instead of the 1-form ξ. So, set Aij := ∇i ξj .

(6.69)

In fact, we use the 2-form Aij with one index frozen. Therefore, in view of the above system (6.68), it is: (6.70)

∇m div ξ = ∇k Amk − Rm k ξk = ∇m ρ

∇m (curl ξ)i = ϵijk ∇j Am k + ϵijk Rm jk l ξ l = ∇m σi .

We put the curvature terms on the right-hand side in (6.70) to obtain the following system for A: div Ai = ∇j Aij = ρi

(6.71)

(curl Ai )m = ϵmjk ∇j Ai k = σim

with ρi = ∇i ρ + Ri k ξk

σim = ∇i σm − ϵmjk Ri jk l ξ l . Now, we state the following proposition. Proposition 13. Let A be a smooth, compactly supported 2-tensor on H. Then the following identity holds: (6.72) # # 2 | ∇A | = H

H

2

|ρ| +

#

H

2

|σ| −

#

H

R

i

mjk Ai

k

mj

A

−

#

Rmk Ai k Aim

H

with Ri mjk denoting the components of the Riemannian curvature tensor and Rmk denoting the components of the Ricci tensor of H.

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

177

Proof. For the 2-tensor Aij we compute curl A as follows. | curlA |2 = (curlAi )m (curlAi )m

= ϵmjk ϵm ab ∇j Ai k ∇a Aib

= (gja gkb − gjb gka )∇j Ai k ∇a Aib

= (∇j Ai k )(∇j Ai k ) − (∇j Ai k )(∇k Ai j )

=| ∇A |2 −(∇j Ai k )(∇k Ai j ).

(6.73)

Integrating on H, we obtain for the last term, using integration by parts and commuting derivatives: # # (∇j Ai k )(∇k Ai j ) = − Ai k ∇j ∇k Ai j H #H =− Ai k {∇k ∇j Ai j +Ri mjk Amj + Rj mjk Aim } < => ? H =−

=ρi

#

k

H

i

Ai ∇k ρ −

#

R

i

mjk Ai

k

A

mj

H

−

#

Rmk Ai k Aim .

H

Integrating by parts again the first term on the right-hand side yields: # # # (6.74) − Ai k ∇k ρi = + ∇ k Ai k · ρ i = + | ρ |2 . H

H

We obtain # # k j i (∇ Ai )(∇k A j ) = H

2

H

|ρ| −

and

(6.75) # # 2 | curlA | = H

H

2

| ∇A | −

This ends the proof.

#

H

H

#

R

i

mjk Ai

k

mj

A

H

2

|ρ| +

#

R

i

mjk Ai

H

k

−

A

#

mj

Rmk Ai k Aim ,

H

+

#

Rmk Ai k Aim .

H

Let us go back to the systems (6.67), (6.68), (6.71). Taking the covariant derivative of the system (6.67) requires the multiplication of the system by r. From now on, let us write ξi for (iZ k)i , that is, set (6.76)

ξi := (iZ k)i = rϵi + rδNi .

And recall our div–curl system (6.37, 6.38). Also Πm n denotes the following projection: (6.77) Now, let (6.78)

Πm n = δm n − Nm N n . ζmi = rΠm n ∇n ξi .

178

6. SECOND FUNDAMENTAL FORM k

Taking the derivative of ζmi yields: ∇k ζmi = ∇k (rΠm n ∇n ξi ) (6.79)

= rΠm n ∇k ∇n ξi + (∇k (rΠm n ))(∇n ξi )

= rΠm n (∇n ∇k ξi ) + rΠm n Rkni j ξj + (∇k (rΠm n ))(∇n ξi ).

The last term in (6.79) behaves well, as we have estimated ∇n ξi to be bounded above, and as (6.80)

| ∇(rΠ) |≤ C,

for a constant C. But the curvature term needs a special treatment, as it is not decaying fast enough in the following sense. Considering the divergence ∇i ζmi or the curl ϵl ki ∇k ζmi , the resulting curvature term (6.81)

rRiemξ

1

− 32

is of order O(r− 2 τ− ), which is in L2 (H), but the L2 -norm grows like 3

− 12

√

t.

− O(r−1 τ− 2 ).

(The worst compoThis can be seen as ξ = O(r ), Riem = nent, α, comes in.) We can get rid of the bad term (6.81) by choosing a suitable decomposition of ξ. That is, we decompose ξ into a component /ξ tangential to St,u and a component normal to St,u : ξi = /ξ i + Ni ξN

(6.82) with

/ξ i = Πi j ξj

and

ξN = N j ξj .

Then, we introduce (6.83)

ηmi = r(∇ /m/ξ i + Ni ∇ /m ξN ).

By a straightforward calculation, we obtain: (6.84)

ηmi = ζmi + r(Ni θm j − θmi N j )ξj .

In order to prove (6.84), first let us remind ourselves of the following facts: ∇k r = N k

θk m = ∇k N m − Nk ∇N N m = ∇k N m + a−1 Nk ∇ / ma

∇k (Πm n ) = −(∇k Nm )N n − Nm (∇k N n ).

And in the linear case, the second equation reduces to In view of (6.83), we calculate (6.85)

θk m = ∇k N m .

/m (Πi j ξj ) = (∇ /m Πi j )ξj + Πi j ∇ /m ξj ∇ /m/ξ i = ∇ = (Πm n ∇n Πi j )ξj + Πi j Πm n ∇n ξj . < => ? < => ? =:A

=:B

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

179

Considering A, we compute Πm n ∇n Πi j = (δm n − Nm N n )[−N j ∇n Ni − Ni ∇n N j ] = −N j ∇m Ni − Ni ∇m N j

which gives for A

+ N m N n N j ∇n N i + N m N n N i ∇ n N j ,

(6.86) A = −N j ξj ∇m Ni − Ni ξj ∇m N j + Nm ξj N j ∇N Ni + Nm Ni ξj ∇N N j .

And for B, it is (6.87) (6.88)

B = Πi j Πm n ∇n ξj = Πi j [∇m ξj − Nm N n ∇n ξj ]

= ∇m ξi − Nm ∇N ξi −Ni N j ∇m ξj + Ni N j Nm ∇N ξj . < => ? =r−1 ζmi

Again in view of (6.83), we calculate

Ni ∇ /m ξN = Ni Πm n ∇n ξN = Ni Πm n [N j ∇n ξj + ξj ∇n N j ]

= Ni [N j ∇m ξj + ξj ∇m N j − N j Nm N n ∇n ξj − Nm N n ξj ∇n N j ]

(6.89)

= Ni N j ∇m ξj + Ni ξj ∇m N j

− Ni N j Nm ∇N ξj − Ni Nm ξj ∇N N j .

Eventually, bringing everything together in view of equation (6.83), the corresponding terms cancel, and we obtain: /m/ξ i + rNi ∇ /m ξN ηmi = r∇

= ζmi + rNi ξj ∇m N j − rN j ξj ∇m Ni = ζmi + rNi ξj θm j − rN j ξj θmi .

And this proves equation (6.84). Continuing with ηmi and the equation (6.84), we calculate the derivative of (6.84). Also taking into account equation (6.79) for ∇k ζmi , this yields: (6.90)

∇k ηmi = ∇k ζmi + ∇k (rNi θm j ξj ) − ∇k (rθmi N j ξj )

= rΠm n (∇n ∇k ξi ) + rΠm n Rkni j ξj + (∇k (rΠm n ))(∇n ξi )

(6.91)

+ (∇k (rNi ))θm j ξj + rNi ξj ∇k θm j + rNi θm j ∇k ξj

− (∇k (rN j ))θmi ξj − rN j ξj ∇k θmi − rN j θmi ∇k ξj

= rΠm n (∇n ∇k ξi ) + rΠm n Ri j kn ξj + (∇k (rΠm n ))(∇n ξi ) (6.92) (6.93) with (6.94)

+ r(Ni ∇k θm j − N j ∇k θmi )ξj

+ [(∇k (rNi ))θm j − θmi (∇k (rN j ))]ξj

= rΠm n (∇n ∇k ξi ) + rPi j km ξj + Lkmi

Pi j km = Πm n Ri j kn + Ni ∇k θm j − N j ∇k θmi

180

6. SECOND FUNDAMENTAL FORM k

and Lkmi = (∇k (rΠm n ))(∇n ξi ) + [(∇k (rNi ))θm j − θmi (∇k (rN j ))]ξj .

(6.95)

Note that the Lkmi are the lower order terms, and they behave well. The terms that cause difficulties are in Pi j km , which are O(r−1 ). So, we have to focus on these terms. Compute Πm i Πn j Rij as follows: (6.96) which gives:

/m ∇ /n a − ∇ /N θmn − θm i θin − Kγmn Πm i Πn j Rij = −a−1 ∇ /N θmn = O(r−2 ). Πm i Πn j Rij + ∇

(6.97) Further, it is

(6.98)

Rijkn = gik Rjn + gjn Rik − gjk Rin − gin Rjk 1 − (gik gjn − gjk gin )R. 2

And we have: R = | k |2 = O(r−2 ).

(6.99) Now, set

Smn = Πm i Πn j Rij

(6.100)

for the tangential-tangential component of Rij . Only these Smn are O(r−1 ). Thus, we can write (6.101) And by (6.99), we obtain:

Rij ≃ Sij . trS ≃ R = O(r−2 ).

(6.102)

Next, we substitute (6.99) and (6.101) in (6.98) and take into account (6.102), which yields: (6.103)

Rijkn ≃ gik Sjn + gjn Sik − gjk Sin − gin Sjk .

Then, we obtain: (6.104) with (6.105)

Πm n Rijkn ≃ gik Sjm + γjm Sik − gjk Sim − γim Sjk γij = gik Πk j = gik (δ k j − N k Nj ) = gij − Ni Nj

being the induced metric on St,u . To prove (6.104), just calculate for the first term on the right-hand side, using (6.103) and remembering that the

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

181

Sij are the tangential-tangential components of Rmn : Πm n gik Sjn = (δm n − Nm N n )gik Sjn

= gik Sjm − gik Nm N n Sjn = gik Sjm .

For the second term on the right-hand side, it is: Πm n gjn Sik = (δm n − Nm N n )gjn Sik

= gjm Sik − Nm N n gjn Sik < => ? =Nj

= (gjm − Nj Nm )Sik = γjm Sik .

Similarly, the third and the fourth term on the right-hand side follow. Now, with (6.105), we write (6.104) as follows: (6.106) with (6.107)

Πm n Rijkn = Qijkm + Ni Nk Sjm − Nj Nk Sim Qijkm = γik Sjm + γjm Sik − γjk Sim − γim Sjk .

Note that, Qijkm is antisymmetric in the first two indices, also antisymmetric in the last two indices, and belongs to St,u , as it only has tangential components. Thus, one can write: (6.108)

Qijkm = qϵij ϵkm

for some function q, where ϵij is the area 2-form on St,u . Observe that, g ik g jm Qijkm = g ik g jm ϵij ϵkm q (6.109)

= 2q = O(trS).

And with (6.102) the last term is O(r−2 ) and so it is: (6.110)

Qijkm = O(r−2 ).

Therefore, together with (6.106), this yields: (6.111)

Πm n Rijkn = Ni Nk Sjm − Nj Nk Sim + O(r−2 ).

Now, consider (6.97) and (6.100): (6.112)

/N θmn + O(r−2 ). Smn = Πm i Πn j Rij = −∇

Substituting (6.112) in (6.111) yields: (6.113)

Πm n Rijkn = −Ni Nk ∇ /N θjm + Nj Nk ∇ /N θim + O(r−2 ).

182

6. SECOND FUNDAMENTAL FORM k

Let us now go back to (6.94). Since only the normal derivatives of θij are O(r−1 ), consider the following: (6.114)

∇k θij = Πk n ∇n θij + Nk ∇N θij = Nk ∇ /N θij + O(r−2 ).

We substitute (6.114) in (6.94) to obtain: (6.115)

/N θmj − Nj Nk ∇ /N θmi + O(r−2 ). Pijkm = Πm n Rijkn + Ni Nk ∇

Thus, with the result of (6.113) the terms of order higher than (r−2 ) cancel, and it directly follows that: Pijkm = O(r−2 ).

(6.116)

Therefore, in view of (6.93), it is: ∇k ηmi = O(r−2 ).

(6.117)

Now, with the system (6.71), we have the following div − curl system for the 2-form η: (6.118) with

div ηi = ∇j ηij = ρi

(curl ηi )m = ϵmjk ∇j ηi k = σim ρi = rΠi n (∇n ∇j ξj ) + rPj sj i ξs + Lj ij

That is: (6.119)

σim = ϵmjk rΠi n (∇n ∇j ξ k ) + ϵmjk rP ksj i ξs + ϵmjk Lj i k . div ηi = rΠi n ∇n div ξ + rPj sj i ξs + Lj ij

(curl ηi )m = rΠi n ∇n (curl ξ)m + ϵmjk rP ksj i ξs + ϵmjk Lj i k .

From above we know that the terms involving P and L on the right-hand side are in L2 (H). It remains to check, that the derivative terms for the divergence of ξ as well as for the curl of ξ also are in L2 (H). Therefore, recalling the Hodge system (6.37, 6.38): curl ξ = iZ H = H · Z = rN · H div ξ = 0, we are now going to verify that rΠi n ∇n (rN · H) ∈ L2 (H). In order to do so, let us remind ourselves of the components of H (see (4.73–4.75)). They are: 1 ∗ 1 1 αAB + ∗ αAB − σδAB 4 4 2 1 ∗ 1 ∗ = β − βA 2 A 2 = σ.

HAB = − HAN HN N

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

183

As we want to estimate the corresponding L2 -norms involving HAN and HN N , we have to focus on the β, β and σ terms. We have to prove that they have enough decay and lie within our assumptions on the components of the curvature tensor. As their corresponding norms have the same weights, it suffices to show it only for β. So, we have: rΠi n ∇n (rN · H) = r2 Πi n ∇n (N · H) + r Πi n N · H ∇n r = r2 Πi n ∇n (N · H). ?


=Nn

=0

?

Πi n ∇n β m = δm a Πi n ∇n β a

= Πm a Πi n ∇n β a + N a Nm Πi n ∇n β a =∇ /i β m − N m β a θ i a .

The last equation holds because of the identity N a Πi n ∇n β a = −β a Πi n ∇n N a ,

as N and β are orthogonal. We conclude that

r 2 Πi n ∇ n β m = r 2 ∇ /i β m − r 2 N m β a θ i a

lies in L2 (H), as both terms on the right-hand side correspond with our assumptions on β and its derivatives, and noting that θk m = r−1 Πk m . Therefore, as indeed rΠi n ∇n (rN · H) ∈ L2 (H), all the terms on the right-hand side of the system (6.119) belong to L2 (H). And so we can now apply proposition 13 to the div–curl system (6.119) for η, respectively the system (6.118). We obtain (6.120) # # # # # H

| ∇η |2 =

H

| ρ |2 +

H

| σ |2 −

H

Ri mjk ηi k η mj −

Rmk ηi k η im

H

So, in view of the previous results, the definition of η and of the assumptions on the components of the curvature tensor, we see that the right-hand side of (6.120) is in L2 (H). Thus, we have:

(6.121)

∥ ∇η ∥L2 (H)

is bounded.

Then, recalling again equation (6.93), r∇ /m ∇k ξi = rΠm n (∇n ∇k ξi ) = ∇k ηmi − rPi j km ξj − Lkmi ,

it directly follows by the previous results: (6.122) Now, recall (6.17),

∥ r∇ / ∇ξ ∥L2 (H)

is bounded.

ξ = iZ k = rϵ + rδN,

184

6. SECOND FUNDAMENTAL FORM k

to conclude for the norms of the corresponding derivatives of ϵ and δ: / 2 ϵ ∥L2 ≤ cR[1] ∥ r2 ∇

(6.123)

/ 2 δ ∥L2 ≤ cR[1] ∥ r2 ∇

(6.124)

∥ r2 ∇ /∇ /N ϵ ∥L2 ≤ cR[1]

(6.125)

∥ r2 ∇ /∇ /N δ ∥L2 ≤ cR[1] .

(6.126)

Summarizing, we have now shown that the second angular derivatives as well as the mixed second derivatives (i.e. one normal and one angular derivative) of δ and ϵ are bounded by non-degenerate estimates. This is in accordance with our bootstrap assumptions. What remains as a last step of this paragraph is to study the second normal derivatives of δ and ϵ. In the sequel, we shall show how they lead to degenerate estimates. Consider again the Hodge system (6.37, 6.38) on H: curl iZ k = iZ H = H · Z = rN · H div iZ k = 0. Next, one takes the normal derivative on both sides of equation (6.37). (6.127)

/N (H · Z) = ∇ /N (rN · H). ∇ /N (curl iZ k) = ∇

Focus the right-hand side and compute: /N r + r∇ /N (N H). ∇ /N (rN H) = N H∇ Immediately, one sees that the term r∇ /N (N H) is of the same order in r as rN H. Thus, ∥ r2 ∇ /N (N H) ∥L2 (H) does not lie within our bootstrap assumptions. So, we obtain the following degenerate estimate: (6.128) ∥ τ− ∇ /N (HZ) ∥2L2 (H) ≤ c ∥ τ− r∇ /N (N H) ∥2L2 (H) = c Then, we conclude that (6.129)

#

H

τ−2 r2 | ∇ /N (N H) |2 .

/N2 iZ k ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ). ∥ τ− r∇

And this yields (6.130) (6.131)

/N2 ϵ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ τ− r∇

/N2 δ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ). ∥ τ− r∇

Summarizing, what we have shown, it is (6.132)

K[2] (δ) ≤ c(R[1] + ϵ0 K[2] )

and (6.133)

K[2] (ϵ) ≤ c(R[1] + ϵ0 K[2] ).

6.2. ESTIMATING THE COMPONENTS δ, ϵ, ηˆ

185

Estimates for η ˆ: Finally, to estimate ηˆ, we consider equations (6.15) and (6.16). 1 1 1 / δ − trθ · ϵ + (β − β) div / ηˆ = − ∇ 2 2 2 1 1 1 ˆ ϵ + (α − α) ∇ /N ηˆ + trθ · ηˆ = ∇ /⊗ 2 2 4 1 1 /j ϵi − γij div / ϵ) + (α − α). = (∇ /i ϵj + ∇ 4 2 Then, similarly as above, we first compute 1 1 1 1 ∥ div / ηˆ ∥ ≤ ∥ ∇ / δ ∥ + ∥ trθ · ϵ ∥ + ∥ β ∥ + ∥ β ∥ 2 2 2 2 ≤ c(R[1] + ϵ0 K[2] ) 1 1 1 1 ˆϵ∥+ ∥α∥+ ∥α∥ ∥∇ /N ηˆ ∥ ≤ ∥ trθ · ηˆ ∥ + ∥ ∇ /⊗ 2 2 4 4 ≤ c(R[1] + ϵ0 K[2] ),

In equation (6.16), the term α appears, which will make the estimates for ∇ /N ηˆ degenerate, as r∇ /N α does not lie in L2 (H), i.e. does not lie within our bootstrap assumptions. Therefore, we conclude that ∥ rdiv / ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] )

/N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ). ∥ τ− ∇

As ηˆ is a symmetric, traceless 2-tensor on S, the proposition 5 applies. And as also ∥ rdiv / ηˆ ∥L2 (H) is bounded, it follows that ∥ ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] )

∥ r∇ / ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ).

For the second derivatives, we calculate from equations (6.15) and (6.16). 1 1 1 2 1 / δ− ∇ / (trθ · ϵ) + ∇ /β − ∇ /β ∇ /div / ηˆ = − ∇ 2 2 2 2 1 1 1 1 ˆ ϵ) + ∇ ∇ /∇ /N ηˆ = − ∇ / (trθ · ηˆ) + ∇ /(∇ /⊗ /α − ∇ / α. 2 2 4 4 Further, it is 1 1 1 1 ∇ /N div / ηˆ = − ∇ /δ − ∇ /N ∇ /N (trθ · ϵ) + ∇ /N β − ∇ / β 2 2 2 2 N 1 1 1 1 ˆ ϵ) + ∇ ∇ /N2 ηˆ = − ∇ /⊗ / (trθ · ηˆ) + ∇ / (∇ / α− ∇ / α. 2 N 2 N 4 N 4 N This yields ∥ r2 ∇ / div / ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] )

/∇ /N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) ∥ rτ− ∇

/N2 ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ). ∥ τ−2 ∇

186

6. SECOND FUNDAMENTAL FORM k

Thus, we derive directly by the Hodge type estimates for H2 (see propositions 5 and 6) that (6.134) (6.135) (6.136) (6.137) (6.138) (6.139)

∥ ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] )

(from above)

/N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ τ− ∇

(from above)

∥ r∇ / ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) 2

(from above)

2

/ ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) ∥r ∇

/∇ /N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) ∥ τ− r∇

/N2 ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ). ∥ τ−2 ∇

That is, (6.140)

K[2] (ˆ η ) ≤ c(R[1] + ϵ0 K[2] ).

Let us summarize, here, the results in the following theorem. Theorem 6. Let the bootstrap assumptions hold. Then there exists a constant c such that the components of the second fundamental form k, that is δ, ϵ, η, verify the estimates (6.141) (6.142) (6.143) (6.144) (6.145) (6.146) (6.147) (6.148) (6.149) (6.150) (6.151) (6.152) (6.153) (6.154)

∥ ϵ ∥L2 (H) ≤ cR[1]

∥ δ ∥L2 (H) ≤ cR[1]

∥ r∇ / δ ∥L2 (H) ≤ cR[1] ∥ r∇ / ϵ ∥L2 (H) ≤ cR[1]

/ 2 ϵ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ r2 ∇

/ 2 δ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ r2 ∇

∥ r2 ∇ /∇ /N δ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) ∥ r2 ∇ /∇ /N ϵ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) /N2 δ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ τ− r∇ /N2 ϵ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ τ− r∇ ∥ ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] )

∥ r∇ / ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] )

/N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[1] ) ∥ τ− ∇ / 2 ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ) ∥ r2 ∇

6.3. ESTIMATES AND RECOVERING THE BOOTSTRAP ASSUMPTIONS

(6.155) (6.156)

187

∥ τ− r∇ /∇ /N ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] )

/N2 ηˆ ∥L2 (H) ≤ c(R[1] + ϵ0 K[2] ). ∥ τ−2 ∇

These estimates show, that: (6.157)

δ, ϵ, η ∈ W 2,2 (H).

Thus, the Sobolev imbeddings yield (6.158)

δ, ϵ, η ∈ L∞ (H).

For Sobolev imbeddings see the corresponding chapter. Thus we have shown the following proposition: Proposition 14. Let the bootstrap assumptions hold. Then there exists a numerical constant c such that eK [2]

≤ c(R[1] + ϵ0 K[2] ).

6.3. Estimates and Recovering the Bootstrap Assumptions The interior estimates are straightforward. Combining the exterior and interior estimates for the second fundamental form k, we deduce that (6.159)

K[2] ≤ c(R[1] + ϵ0 K[2] ).

This yields the main theorem of this section:

Theorem 7. Let the bootstrap assumptions be valid.Then, if ϵ0 is chosen sufficiently small, the following inequality holds, with a numerical constant c, (6.160)

K[2] ≤ cR[1] .

By the Sobolev imbeddings, we obtain

K0∞ ≤ cR[1] ≤ cϵ0 .

CHAPTER 7

Second Fundamental Form χ: Estimating χ and ζ The main ideas of this chapter are due to Demetrios Christodoulou, and it also appears in [16]. Notation: In the following, if nothing else is specified, ∇ will denote differentiation on M and ∇ / differentiation on S. Correspondingly, div / is the divergence on S. The goal of the present chapter is to prove theorem 8. The method we use is the treatment of systems of ordinary differential equations along the generators of a null hypersurface C, coupled to elliptic equations on the surfaces of a foliation of C by sections. It was introduced by D. Christodoulou and S. Klainerman in [19]. However, our estimates differ fundamentally from [19], as we do not have any L∞ bounds on the curvature, but we only control it in H 1 (H). Therefore, by the trace lemma, the Gauss curvature K is in L4 (S). This leads to borderline estimates for χ. 7.1. Statement of the Main Theorem for χ and ζ – Sketch of the Proof Theorem 8. Suppose that there is a constant B independent of t such that for all s ∈ [0, t] the function Ψ(s) associated to the two-manifold (Ss , γ(s)) (which is diffeomorphic to S 2 ) by the uniformization theorem satisfies (7.1)

ΨM (s) ≤ B,

Ψm (s) ≥ B −1 ,

E p (s) ≤ B, K

E p are defined as in (7.23), (7.30), (7.34) and where Ψ(s), ΨM , Ψm and K (7.43) respectively. Let # t −1 (7.2) r0 κ(s)ds 0

with κ = a + b and a, b given in (7.188), (7.189) respectively; and

(7.3)

1

sup κ 2 (s) s∈[0,t]

be bounded by suitably small constants depending only on B. (For r, r0 , see (7.66).) Also, let ν (7.4) z1 r0−1 < ¯ , 2C 189

190

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

where C¯ is a constant depending only on B, and ν is a fixed constant with 2 r0 . (7.5) ν< 3 y0 ¯ , one could require z0 r−1 to be likewise bounded instead of (As z1 = z0 + CM 0 (7.4).) Assume that y0 > 0. Further, let yr00 be bounded by a fixed numerical constant. Let also the following bootstrap assumptions hold at s = 0 with strict inequalities: ¯ ∥ ζ ∥L∞ ≤ 1 (7.6) Cr 2 r (7.7) ∥ Lp ≤ C − ∥ trχ − 2 (7.8) ∥ Lp ≤ δ r− ∥ χ ˆ− r (7.9) ∥ trχ ∥L∞ ≤ C 2 with C a numerical constant, and δ a numerical constant which we choose such that C¯ 2 δ ≤ 12 . Then equations (7.6) to (7.9) hold for all s and with z = P + Q the following inequality holds: (for P , Q, see (7.187), (7.62) respectively) (7.10)

z ≤ 2z1 .

Sketch of the Proof: Consider (7.12)–(7.16). We derive 1. Estimates in terms of f in (7.16), assuming estimates for the curvature component β in f (see (7.15)), (content of subsection 7.2), 2. Estimates including lower order terms involving ζ, (content of subsection 7.3). This is done by a bootstrap argument with bootstrap assumptions on the curvature and on the lower order terms. The core method of the whole proof is the treatment of systems of ordinary differential equations along the generators of a null hypersurface C, coupled to elliptic equations on the surfaces of a foliation of C by sections, as cited above. Thus, we obtain estimates for the shear χ, ˆ the traceless part of χ, from a Hodge system on S. In order to do so, we use the uniformization theorem with K ∈ L4 (S), which is proven in the next chapter. We also derive the Hodge system for the torsion ζ (see (7.119)–(7.120)): 1 ˆ·χ ˆ div / ζ = −µ − ρ + χ 2 1 curl / ζ =σ− χ ˆ∧χ ˆ 2 with µ denoting the mass aspect function (7.116): 1 µ = K + trχ trχ − div / ζ. 4

7.2. PROOF OF THE MAIN THEOREM: ESTIMATES FOR χ

191

Thus, we have a Hodge system for ζ on each section S, coupled to a propagation equation for µ along each generator of C. The propagation equation for µ is derived to be (7.128):

(7.11)

1 1 dµ 3 ˆ |2 + trχ | ζ |2 + trχµ = − trχ | χ ds 2 4 2 ˆ + 2div / χ·ζ +χ ˆ·∇ / ⊗ζ,

ˆ ζ)ab = ∇ with (∇ / ⊗ / a ζb + ∇ / b ζa − γab div / ζ. The proof of the main theorem 8 for χ and ζ is carried out in the remaining part of this chapter (sections 7.2–7.3). 7.2. Proof of the Main Theorem: Estimates for χ Now, recall the equation ∂trχ 1 + (trχ)2 + | χ ˆ |2 = 0. ∂s 2 And consider the following equations in terms of Jacobi field frames:

(7.12)

∂d / a trχ + trχd / a trχ + 2χ ˆbc ∇ / aχ ˆbc = 0 ∂s 1 1 (7.14) ˆa b ζb − trχζa = −βa . div /χ ˆa − /da trχ + χ 2 2 Equation (7.13) is an ordinary differential equation along the generators of C. We call such equations propagation equations. (7.13) is obtained by differentiation of (7.12) tangentially to Ss . And equation (7.14) is an elliptic system on each section Ss of C. The following is the method of treatment of such coupled systems of equations. Consider the Codazzi equations. Setting (7.13)

1 fa = −βa + trχζa − χ ˆa b ζb , 2 the Codazzi equations read as follows: (7.15)

1 div /χ ˆa = /da trχ + fa . 2 What we are going to do is, assuming estimates for the spacetime curvature to obtain estimates for the quantities controlling the geometry of C as described by its foliation {Ss }. Here, we want to derive estimates for χab . We have for χ ˆ on each section Ss an elliptic system of the form (7.16)

(7.17)

div θa = fa

for a trace-free, symmetric, 2-covariant tensorfield θab on a two-dimensional, compact, Riemannian manifold (M, gab ) (without boundary). Moreover, M in the case of interest is diffeomorphic to S 2 . We want to derive an estimate of the form (7.18)

∥ ∇θ ∥Lp (M,g) ≤ C ∥ f ∥Lp (M,g) ,

192

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

for any 1 < p < ∞. (In fact, in our application we need an estimate of θ in L∞ , thus in view of the Sobolev inequality on (M, g) an estimate of ∇θ in Lp (M, g) for some p > 2.) By the uniformization theorem there exists a function Φ on M such that g˜ab = e2Φ gab ˜ = 1 so that (M, g˜) is isometric to the is a metric of Gauss curvature K standard unit sphere. Moreover, the equation (7.17)

(7.19)

div θa = fa is conformally covariant; that is, we have: F θa = f˜a ; div

(7.20)

f˜a = e−2Φ fa .

For this system the standard Calderon-Zygmund Lp estimates hold: E ∥p p (7.21) ∥ θ ∥p p + ∥ ∇θ ≤ C(p)p ∥ f˜ ∥p p g) L (M,˜

g) L (M,˜

g) L (M,˜

for any 1 < p < ∞. From this we want to derive estimates for ∥ θ ∥Lp (M,g)

and ∥ ∇θ ∥Lp (M,g)

in terms of ∥ f ∥Lp (M,g) . To do this more neatly, we first make a (constant) rescaling to a metric g ′ of area 4π by setting (7.22) where r =

G

′ gab = r−2 gab , Area(M,g) . 4π

We then make the conformal change

′ g˜ab = e2Ψ gab ˜ = 1. So, Φ = Ψ − log r and to a metric g˜ of Gauss curvature K # # # 2Ψ e dµg′ = dµg′ , 4π = dµg˜ =

(7.23)

M

M

M

i.e. the mean value of e2Ψ on M relative to g ′ is equal to 1. Under the (constant) rescaling we have that: # p | θ |pg′ dµg′ , (7.24) ∥ θ ∥Lp (M,g′ ) = M

and

| θ |2g′ = (g ′−1 )ac (g ′−1 )bd θab θcd = r2 (g −1 )ac r2 (g −1 )bd θab θcd = r4 | θ |2g ,

that is

| θ |g′ = r2 | θ |g .

(7.25) Also, it is

dµg′ = r−2 dµg . And we have # # | θ |pg′ dµg′ = M

M

r2p−2 | θ |pg drg = r2p−2 ∥ θ ∥pLp (M,g) .

7.2. PROOF OF THE MAIN THEOREM: ESTIMATES FOR χ

193

We have thus obtained that 2p−2 p

∥ θ ∥Lp (M,g′ ) = r

(7.26)

∥ θ ∥Lp (M,g) .

Next, the connection coefficients of and gab coincide, hence ∇′ θ = ∇θ. (∇′ denotes the covariant differentiation with respect to g ′ .) Also, one sees that ′ gab

1

| ∇′ θ |g′ = | ∇θ |g′ = ((g ′−1 )ad (g ′−1 )be (g ′−1 )cf ∇a θbc ∇d θef ) 2

1

= (r2 (g −1 )ad r2 (g −1 )be r2 (g −1 )cf ∇a θbc ∇d θef ) 2 = r3 | ∇θ |g .

Thus we have:

∥ ∇′ θ ∥pLp (M,g′ ) =

#

M

| ∇′ θ |pg′ dµg′ =

#

= r3p−2 ∥ ∇θ ∥pLp (M,g) ,

that is,

∥ ∇′ θ ∥Lp (M,g′ ) = r

(7.27)

3p−2 p

M

r3p | ∇θ |pg r−2 dµg

∥ ∇θ ∥Lp (M,g) .

Next, we consider the conformal transformation ′ g˜ab = e2Ψ gab .

(7.28) Then it is

1

(7.29)

1

| θ |g˜ = ((˜ g −1 )ac (˜ g −1 )bd θab θcd ) 2 = e−2Ψ ((g ′−1 )ac (g ′−1 )bd θab θcd ) 2

and

= e−2Ψ | θ |g′

dµg˜ = e2Ψ dµg′ ,

hence ∥θ

∥pLp (M,˜g)

=

#

M

|θ

|pg˜

dµg˜ = #

≥ e−(2p−2)ΨM

(7.30)

M

#

M

e−(2p−2)Ψ | θ |pg′ dµg′

| θ |pg′ dµg′ = e−(2p−2)ΨM ∥ θ ∥pLp (M,g′ )

with ΨM = supM Ψ. Combining this with the previous result yields: ∥ θ ∥Lp (M,g) ≤ (r−1 eΨM )

(7.31)

≤ (r−1 eΨM )

(7.32)

On the other hand, f˜a = e−2Φ fa So, we have 1

(7.33)

2p−2 p 2p−2 p

∥ θ ∥Lp (M,˜g)

C(p) ∥ f˜ ∥Lp (M,˜g) .

(Φ = Ψ − log r). 1

| f˜ |g˜ = ((˜ g −1 )ab f˜a f˜b ) 2 = (e−2Φ (g −1 )ab e−2Φ fa e−2Φ fb ) 2 = e−3Φ | f |g .

194

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

It follows that (7.34) ∥ f˜ ∥pLp (M,˜g) =

#

M

e−(3p−2)Φ | f |pg dµg ≤ e−(3p−2)Φm ∥ f ∥pLp (M,g)

with Φm = Ψm − log r and Φm = inf M Φ, Ψm = inf M Ψ. That is, ∥ f˜ ∥Lp (M,˜g) ≤ (re−Ψm )

(7.35)

3p−2 p

∥ f ∥Lp (M,g) .

Substituting above in the estimate (7.32) for ∥ θ ∥Lp (M,g) we obtain: (7.36)

∥ θ ∥Lp (M,g) ≤ re

(2p−2)ΨM −(3p−2)Ψm p

C(p) ∥ f ∥Lp (M,g) .

To obtain the estimate for ∇θ in Lp (M, g), we write:

E a θbc + △d θdc + △dac θbd , ∇′a θbc = ∇ ab

(7.37)

where

E c − Γ′c △cab = Γ ab ab

(7.38)

= δac ∂b Ψ + δbc ∂a Ψ − (˜ g −1 )cd g˜ab ∂d Ψ.

(7.39) It follows that:

E |g˜ + C | dΨ |g˜| θ |g˜, | ∇′ θ |g˜ ≤ | ∇θ

(7.40)

where C is a numerical constant. Hence, it is E ∥Lp (M,˜g) + C ∥| dΨ |g˜| θ |g˜∥Lp (M,˜g) (7.41) ∥| ∇′ θ |g˜∥Lp (M,˜g) ≤ ∥ ∇θ and

E p sup | θ |g˜, ∥| dΨ |g˜| θ |g˜∥Lp (M,˜g) ≤ K

(7.42)

M

where

E p := ∥ dΨ ∥Lp (M,˜g) . K

(7.43)

Taking now p > 2, the Sobolev inequality on the standard 2-sphere gives: E ∥Lp (M,˜g) + ∥ θ ∥Lp (M,˜g) ), (7.44) sup | θ |g˜ ≤ C(p)(∥ ∇θ M

where C(p) → ∞ as p → 2. Substituting the Calderon-Zygmund estimate for the right-hand side of (7.44) gives: (7.45) sup | θ |g˜ ≤ C(p) ∥ f˜ ∥Lp (M,˜g) M

and substituting again above in (7.42), (7.41), yields: (7.46)

E p ) ∥ f˜ ∥Lp (M,˜g) . ∥| ∇′ θ |g˜∥Lp (M,˜g) ≤ C(p)(1 + K

Finally, using the fact that

| ∇′ θ |g˜ = e−3Ψ | ∇′ θ |g′

and the relations between the norms derived above, we obtain: (7.47)

E p )e ∥ ∇θ ∥Lp (M,g) ≤ C(p)(1 + K

(ΨM −Ψm )(3p−2) p

∥ f ∥Lp (M,g) ,

7.2. PROOF OF THE MAIN THEOREM: ESTIMATES FOR χ

195

Ep, where the constant C(p) actually depends on the following terms: C(p, K ΨM − Ψm ). From (7.45) and (7.34) we see that sup | θ |g˜ ≤ C(p) ∥ f˜ ∥Lp (M,˜g) ≤ C(p)(re−Ψm )

3p−2 p

M

∥ f ∥Lp (M,g) .

Also, we have that (7.48) So,

g −1 )ac (˜ g −1 )bd θab θcd = (r2 e−2Ψ )2 (g −1 )ac (g −1 )bd θab θcd | θ |2g˜ = (˜ = (r2 e−2Ψ )2 | θ |2g .

| θ |g = r−2 e2Ψ | θ |g˜

(7.49) and

sup | θ |g ≤ r−2 e2ΨM | θ |g˜ .

(7.50)

M

Substituting, we obtain (7.51)

sup | θ |g ≤ r

p−2 p

− 3p−2 Ψm p

e2ΨM e

M

∥ f ∥Lp (M,g) .

We apply the above estimates to the null Codazzi equations (see (7.14), (7.15) and(7.16)). 1 div /χ ˆa = /da trχ + fa , 2 where 1 fa = −βa + trχζa − χ ˆa b ζb . 2 We are going to assume an Lp estimate for β on each section Ss . We shall first present a preliminary simplified treatment which ignores the lower order terms involving ζ. We shall later give the actual treatment which derives an Lp estimate for ∇ / ζ on each section Ss , from which an L∞ estimate for ζ follows. Let us introduce now the following dimensionless norms: For any tensorfield t on Ss we write (1 ' # p 1 p (7.52) − ∥ t(s) − ∥ Lp (S) = | t |γ dµγ Area(Ss ) Ss (7.53)

− p1

= (4πr2 )

∥ t ∥Lp (Ss ,γ) .

(Here, γ(s) is the induced metric on Ss .) We apply the estimates derived (for div θ = f ), obtaining: (7.54)

− /d trχ(s) − ∥ Lp (S) + − ∥ f− ∥ Lp (S) } − ∥ ∇ / χ(s) ˆ − ∥ Lp (S) ≤ C(s){∥

E p , ΨM − Ψm , p). with C(s) = C(K

196

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

By virtue of (7.51) we can also bound in the same way r−1 ∥ χ(s) ˆ ∥L∞ (S) : (7.55) E P ){ − r−1 ∥ χ(s) ˆ ∥L∞ (S) ≤ C(ΨM , Ψm , K ∥ /d trχ(s) − ∥ Lp (S) + − ∥ f− ∥ Lp (S) }.

E P ) ≤ C¯ is independent of s. This holds if there Assume that C(ΨM , Ψm , K is a uniform (i.e. independent of s) upper bound for ΨM , a uniform lower EP . bound for Ψm and a uniform upper bound for K The idea is to derive an ordinary differential inequality for − ∥ /d trχ(s)∥ − Lp (S) . We have: (writing A(s) for Area(Ss )) ' (1 # p 1 d d − ∥ /d trχ(s) − ∥ Lp (S) = | /d trχ(s) |pγ(s) dµγ(s) ds ds A(s) S C # 1 dA 1 1−p ∥ /d trχ(s)− ∥ Lp (S) − 2 = − | /d trχ(s) |pγ(s) dµγ(s) p A ds S # ' ∂ 1 + | /d trχ(s) |pγ(s) A S ∂s D ( p + | /d trχ(s) |γ(s) trχ dµγ(s) . Now, it is

# 1 1 dA trχdµγ = trχ, = A ds A A where trχ denotes the mean value of trχ on Ss . Also, p ∂ ∂ | /d trχ(s) |pγ(s) = {((γ −1 )ab ∂a trχ ∂b trχ) 2 } ∂s ∂s p = | /d trχ(s) |p−2 γ(s) 2C D ∂ ab −1 ab × −2χ ∂a trχ ∂b trχ + 2(γ ) ∂b trχ (∂a trχ) . ∂s

So, it is

C 1 d 1−p ∥ /d trχ(s) − ∥ pLp (S) − ∥ /d trχ(s) − ∥ Lp (S) = − ∥ /d trχ(s) − ∥ Lp (S) − trχ − ds p # p − trχ | /d trχ(s) |pγ(s) dµγ(s) 2A S # p − | /d trχ(s) |p−2 ˆab/da trχ /db trχ dµγ(s) γ(s) χ A S # p ∂ + | /d trχ(s) |p−2 (γ −1 )ab ∂b trχ (∂a trχ)dµγ(s) γ(s) A S ∂s D # 1 + | /d trχ(s) |pγ(s) trχ dµγ(s) . A S

We write

trχ = trχ + (trχ − trχ)

7.2. PROOF OF THE MAIN THEOREM: ESTIMATES FOR χ

197

to obtain: H Hd 1 H − ∥ Lp (S) + trχ − ∥ /d trχ(s) − ∥ Lp (S) H ds ∥ /d trχ(s) − 2 # H Hp−2 H 1 ∂ H H H −1 ab (γ ) ∂ trχ trχ)dµ (∂ / d trχ(s) −− ∥ /d trχ(s) − ∥ 1−p H H a b γ(s) H Lp (S) A ∂s γ(s) C' S ( # 1 p 1 1−p ≤ − | trχ − trχ || /d trχ |pγ(s) dµγ(s) ∥ /d trχ − ∥ Lp (S) −1 p 2 A S D # p p + |χ ˆ |γ(s) | /d trχ |γ(s) dµγ(s) . A S We now appeal to the equation: ∂ / a trχ + 2χ ˆbc ∇ / aχ ˆbc = 0. /d trχ + trχd ∂s a Thus, (γ −1 )ab ∂b trχ

∂ ∂a trχ = −trχ | /d trχ |2 − 2d / a trχ · χ ˆbc · ∇ / aχ ˆbc ∂s

and 1 − A

#

∂ | /d trχ |p−2 (γ −1 )ab ∂b trχ (∂a trχ)dµγ(s) ∂s S # # 1 2 = trχ | /d trχ |p dµγ(s) + | /d trχ |p−2 /d a trχ · χ ˆbc · ∇ / aχ ˆbc dµγ(s) . A S A S

Again substitute trχ = trχ + (trχ − trχ) and the result in the inequality above. We obtain: |

d 3 − ∥ /d trχ(s) − ∥ Lp (S) + trχ − ∥ /d trχ(s) − ∥ Lp (S) | ds 2 # C 1 ≤C− ∥ /d trχ(s) − ∥ 1−p (| trχ − trχ | + | χ ˆ |γ(s) ) | /d trχ |p dµγ(s) Lp (S) A S D # 1 p−1 + | /d trχ | | χ ˆ || ∇ /χ ˆ | dµγ(s) . A S

Now, by H¨ older’s inequality, # 1 | /d trχ |p−1 | ∇ /χ ˆ | dµγ(s) A S '# ( p−1 ' # (1 p p 1 p p ≤ | /d trχ | dµγ(s) · |∇ /χ ˆ | dµγ(s) A S S = − ∥ /d trχ − ∥ p−1 ∥ ∇ /χ ˆ− ∥ Lp (S) . Lp (S) −

198

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

We thus obtain the following differential inequality for − ∥ /d trχ(s) − ∥ Lp (S) : | (7.56)

d − ∥ /d trχ(s) − ∥ Lp (S) + ds ≤ C{(∥ trχ − trχ ∥L∞

3 trχ − ∥ /d trχ(s) − ∥ Lp (S) | 2 +∥χ ˆ ∥L∞ ) − ∥ /d trχ(s) − ∥ Lp (S)

+∥χ ˆ ∥L∞ − ∥ ∇ /χ ˆ− ∥ Lp (S) }.

Now we make use of the elliptic estimate on each Ss for − ∥ ∇ /χ ˆ− ∥ Lp (S) and ˆ ∥L∞ in terms of − ∥ /d trχ − ∥ Lp (S) and − ∥ f− ∥ Lp (S) . We have r−1 ∥ χ $ %2 ¯ ∥ f− ∥ Lp (S) }. (7.57) − ∥ ∇ /χ ˆ− ∥ Lp (S) , r−1 ∥ χ ˆ ∥L∞ ≤ C{r − ∥ /d trχ − ∥ Lp (S) + − Also, we have:

∥ /d trχ − ∥ Lp (S) . r−1 ∥ trχ − trχ ∥L∞ ≤ C¯ −

We then obtain the ordinary differential inequality d 3 | ∥ /d trχ(s) − ∥ Lp (S) | − ∥ /d trχ(s) − ∥ Lp (S) + trχ − ds 2 ¯ − (7.58) ∥ /d trχ(s) − ∥ Lp (S) − ∥ f− ∥ Lp (S) }. ≤ C{r ∥ /d trχ − ∥ 2Lp (S) + r −

Now, suppose that we have an ordinary differential equation of the form: dα + λtrχα = β, (7.59) ds where α, β are non-negative functions of s and λ is a constant. We have: 1 dA = trχ, A = Area of Ss . A ds The equation (7.59) is thus equivalent to d λ (A α) = Aλ β. (7.60) ds Integrating on [s0 , s1 ], we obtain # s1 λ λ (Aλ β)(s)ds. (7.61) (A α)(s1 ) − (A α)(s0 ) = s0

Analogously with an inequality in place of the equation. In the case we are considering, it is λ = 32 . Thus, setting

(7.62)

Q = r3 − ∥ /d trχ − ∥ Lp ,

we obtain the following ordinary differential inequality for Q: dQ ¯ −2 Q2 + b), | ≤ C(r (7.63) | ds ∥ f− ∥ 2Lp . For, according to the inequality (7.58) just where we set b = r4 − derived, we have 3 d ∥ /d trχ(s) − ∥ Lp (S) | − ∥ /d trχ(s) − ∥ Lp (S) + trχ − | 2 ds ¯ − (7.64) ∥ f− ∥ 2Lp (S) ). ≤ Cr( ∥ /d trχ(s) − ∥ 2Lp (S) + −

7.2. PROOF OF THE MAIN THEOREM: ESTIMATES FOR χ

199

We need the upper bound for dQ ds if we are given initial data, the lower bound if we are given final data. To proceed, we make the assumption: # s b(s′ )ds′ ≤ M : independent of s. (7.65) 0

We must also assume that:

r ≥ r0 + νs

(7.66)

for some positive constant ν, to be chosen later. Set (7.67)

'

r0 = r |s=0 =

G

(

Area(S0 ) . 4π

¯ Q1 = Q0 + CM

(Q0 = Q |s=0 ). Under the above two assumptions (7.65) and (7.66) we can show: Proposition 15. We have (7.68)

Q ≤ 2Q1 ,

provided that Q1 is suitably small.

Proof: The inequality holds for small positive s. Let s∗ be the maximum value of s for which the inequality holds in [0, s∗ ]. Then we have, integrating the upper bound for dQ ds from s = 0. C# s∗ D # s∗ 4Q21 ds ¯ Q(s∗ ) ≤ Q0 + C + bds (r0 + νs)2 0 0 ' 2 ( ¯ 2 4Q1 4CQ 1 + M = Q1 + . ≤ Q0 + C¯ νr0 νr0

Thus, if

that is, (7.69)

¯ 2 4CQ 1 < Q1 , νr0 Q1
r0 + νs∗ , 3y0

if we require, in place of (7.80) the strict inequality: 2r0 ; (7.84) ν< 3y0

202

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

r0 contradicting the maximality of s∗ . (We may fix ν = 2y .) Then also the 0 lower bound on r continues to hold. Therefore, all the above inequalities continue to hold as long as the assumption (7.65), i.e., # s b(s′ )ds′ ≤ M 0

holds.

7.3. Proof of the Main Theorem: The Torsion (ζ) System In this section we consider the torsion (ζ) system, a coupled elliptic system on each section of the null hypersurface with a propagation equation along each (null geodesic) generator of the hypersurface. We start from the equation ∇L L = −2Z,

(7.85)

where Z is the Ss -tangential vectorfield on C which corresponds to the torsion-1-form ζ. (7.86)

ζ(ΠX) = g(Z, X)

for all X in Tp M , where Π denotes the projection to Tp Ss and p ∈ Ss . Let now X, Y be vectorfields defined on and tangential to S0 . We extend them to C by: (7.87)

[L, X] = [L, Y ] = 0.

Then X, Y are Ss -tangential for all s. Then the second fundamental form χ of Ss relative to the (C-transversal) null normal L is defined by: (7.88)

χ(X, Y ) = g(∇X L, Y ).

This is symmetric (like χ). We consider the propagation of χ along the generators of C. We have: (7.89)

L(χ(X, Y )) = g(∇L ∇X L, Y ) + g(∇X L, ∇L Y ).

Now, look at the first term on the right-hand side of equation (7.89): (7.90) Hence,

∇L ∇X L = ∇X ∇L L + R(L, X)L = −2∇X Z + R(L, X)L.

g (∇L ∇X L, Y ) = −2(∇X ζ)(Y ) − R(Y, L, X, L). On the other hand, for the second term in the same equation: " (7.91) ∇X L = ζ(X)L + χ(EA , X)EA , A

where (EA ; A = 1, 2) is a local orthonormal frame for Ss . Hence, it is " χ(X, EA )χ(EA , Y ). g(∇X L, ∇L Y ) = g(∇X L, ∇Y L) = 2ζ(X)ζ(Y ) + A

Substituting and noting that (7.92)

/ X ζ)(Y ) : as Y is Ss -tangential, like χ, (∇X ζ)(Y ) = (∇

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

203

we obtain: (7.93)

L(χ(X, Y )) = −2(∇ / X ζ)(Y ) − R(Y, L, X, L) " + 2ζ(X)ζ(Y ) + χ(X, EA )χ(EA , Y ). A

This is an equation along each generator of C. We can take X = Ea and Y = Eb , where Ea , Eb are Jacobi fields along the generator. With the notation (7.94)

χ(Ea , Eb ) = χab ,

the equation (7.93) takes the form: (7.95)

∂χab

= −2∇ / a ζb − R(Eb , L, Ea , L) + 2ζa ζb + χa c χcb . ∂s Let (EA ; A = 1, 2) be a local orthonormal frame field for Ss ; we complete it with E3 = L and E4 = L to a null frame for M along C. We want to express RA3B4 . We decompose: 1 1 (7.96) RA3B4 = (RA3B4 + RB3A4 ) + (RA3B4 − RB3A4 ). 2 2 For the symmetric part we use: " 1 RACBC − (RA3B4 + RB3A4 ) = RicAB = 0 < => ? 2 C

=RA4B3

by the vacuum equations. Now, there exists a function ρ such that (7.97) Hence, it is (7.98)

RABCD = −ρϵAB ϵCD . " C

RABCD = −ρ

" B

ϵAB ϵCD = −ρδAC .

We conclude that the symmetric part of RA3B4 is given by: 1 (7.99) (RA3B4 + RB3A4 ) = −ρδAB . 2 Another expression for ρ is obtained by considering " 1 1 RA4A3 = Ric43 = 0. − R3443 − R4433 + 2 2 < => ? =0

A

Hence, from above we see that: " 1 R3434 = − (7.100) RA3A4 = 2ρ. 2 A

So, ρ can equivalently be defined as: 1 (7.101) ρ = R3434 . 4

204

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

Next, we consider the antisymmetric part of RA3B4 : 1 1 (RA3B4 − RB3A4 ) = (RA3B4 + RA43B ) 2 2 1 1 (7.102) = − (RAB43 ) = RAB34 = σϵAB . 2 2 The last equality defines the function σ. We conclude from the above that RA3B4 = −ρδAB + σϵAB ,

(7.103)

or with respect to an arbitrary frame for Ss ,

R(Ea , L, Eb , L) = −ργab + σϵab .

(7.104)

∂χ

Substituting this expression in the equation (7.95) for ∂sab , the equation takes the form: ∂χab (7.105) = −2∇ / a ζb + ργab + σϵab + 2ζa ζb + χa c χcb . ∂s The antisymmetric part of (7.105) is the equation: 1 ˆ∧χ ˆ. (7.106) curl / ζ =σ− χ 2 Here, 1 curl / ζ = ϵab (∇ / a ζb − ∇ / b ζa ) 2 and 1 χ ˆ∧χ ˆ = ϵab (χ ˆa c χ ˆ cb − χ ˆb c χ ˆ ca ). 2 The symmetric part of (7.105) is the following propagation equation for χ: 1 = −∇ / a ζb − ∇ / b ζa + ργab + 2ζa ζb + (χa c χcb + χb c χca ) ∂s 2 The trace of (7.107) is the following propagation equation for trχ:

(7.107)

∂χab

∂trχ = −2div / ζ + 2 | ζ |2 − χ · χ + 2ρ. ∂s

(7.108)

∂χ

∂ ∂ (Recall that ∂s trχ = ∂s ((γ −1 )ab χab ) = −2χab χab + (γ −1 )ab ∂sab .) The equation (7.108) will lose one derivative tangentially to the sections Ss . The trace-free part of (7.107) is the propagation equation for χ ˆ:

(Note that (7.109)

∂ˆ ∂s

denotes the trace-free part of

∂ ∂s .)

∂ˆχ ˆ ab

1 1 ˆ ab + 2ζa ζb − | ζ |2 γab , ˆab − (∇ / ⊗ζ) − trχˆ χab = trχχ ∂s 2 2

where: (7.110)

ˆ ab = ∇ / a ζb + ∇ / b ζa − γab div / ζ, (∇ / ⊗ζ)

the trace-free part of the symmetrized covariant derivative. Note the absence of a curvature term on the right-hand side of (7.109).

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

205

We must also consider the Gauss equation of the embedding of a surface (section of C) Ss in spacetime. Using the fact that the projection operator Π to Ss is given by: 1 1 (7.111) ΠX = X + g(L, X)L + g(L, X)L 2 2 for all X ∈ Tp M and p ∈ Ss ⊂ C. We derive the equation, in a local / ABCD the curvature tensor of orthonormal frame for Ss . First, denote by R Ss and by RABCD the Ss -tangential components of the curvature of M . So, we then have (7.112) Now, we have: (7.113)

R / ABCD + χAC χBD − χAD χBC = RABCD . R / ABCD = K(δAC δBD − δAD δBC ),

where K is the Gauss curvature of Ss . Recall the equation (7.97): Moreover,

RABCD = −ρϵAB ϵCD .

ϵAB ϵCD = δAC δBD − δAD δBC . Thus, the Gauss equation is equivalent to: 1 1 (7.114) K + trχtrχ − χ · χ = −ρ 2 2 or, 1 1 (7.115) K + trχtrχ − χ ˆ·χ ˆ = −ρ. 4 2 We now define the mass aspect function µ by: 1 /ζ (7.116) µ = K + trχtrχ − div 4 and its conjugate µ by: 1 / ζ. µ = K + trχtrχ + div 4 These are functions associated to a closed spacelike surface in spacetime (in particular one diffeomorphic to S 2 ). In terms of µ the propagation equation for trχ reads: (7.117)

∂ 1 trχ + trχtrχ = −2µ + 2 | ζ |2 . ∂s 2 If µ is given, the equation defining µ supplements the curl equation for ζ to a Hodge (div − curl) system: (7.118)

(7.119) (7.120)

1 ˆ∧χ ˆ curl / ζ =σ− χ 2 1 div / ζ = −µ − ρ + χ ˆ·χ ˆ 2

206

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

(substituting for K from the Gauss equation). This is an elliptic system on each section which is coupled to a propagation equation for µ along each generator. Propagation Equation for µ: The propagation of K is according to: ∂K + trχK = div / div / χ−△ / trχ. ∂s

(7.121)

This is derived as follows: / ab 2K = (γ −1 )ab R

(7.122)

(the scalar curvature of Ss )

and ∂Γc ∂Γc ∂R / ab =∇ / c ( ab ) − ∇ / a ( cb ) ∂s ∂s ∂s

(7.123) while (7.124)

∂Γcab / b χc a − ∇ / c χab =∇ / a χc b + ∇ ∂s

(7.125)

∂Γccb =∇ / b trχ. ∂s

We then obtain: 2

∂R / ab ∂(γ −1 )ab ∂K = R / ab + (γ −1 )ab ∂s ∂s ∂s

and the first term is −2χab R / ab = −2χab γab K = −2trχK while the second term is 2div / div / χ − 2△ / trχ. From all this we conclude (7.121), the propagation equation for K. Now, we want to find also the propagation of div / ζ. ∂ ∂ / a ζb ) div / ζ = ((γ −1 )ab ∇ ∂s ∂s C D ∂Γcab ∂ ∂ζb −1 ab = (γ ) ∇ / a( / a ζb )− ζc + (γ −1 )ab ∇ ∂s ∂s ∂s ∂ζ = div / ( ) − 2div / a ζb . / χζ + /d trχζ − 2χab ∇ ∂s We now recall the propagation equation for ζ, namely (7.126)

∂ζa + trχζa = div / χa − /da trχ ∂s

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

207

which results from eliminating the curvature β using the null Codazzi equation. Substituting, we obtain the following propagation equation for div / ζ: ∂ ˆ + div div / ζ = −2trχdiv / ζ − 2div / χ·ζ −χ ˆ·∇ / ⊗ζ / div / χ−△ / trχ. ∂s (7.127) ∂ 1 ( 4 trχtrχ) from the propagation equations of trχ and We also calculate ∂s trχ. Combining with the above results (7.121) and (7.127) we obtain in this way the propagation equation for µ:

1 1 ∂µ 3 ˆ + trχµ = − trχ | χ ˆ |2 + trχ | ζ |2 +2div / χ·ζ +χ ˆ·∇ / ⊗ζ. 2 ∂s 2 4 (7.128) 7.3.1. The div-curl System for 1-Forms on a 2-Dimensional, Compact, Riemannian Manifold. Let (M, gab ) be a 2-dimensional, compact, Riemannian manifold. And let ξa be a 1-form on M . We now consider the system (7.129) (7.130)

div ξ = ρ curl ξ = ω.

Proposition 17. The system (equations (7.129) and (7.130)) is conformally covariant. Proof. Setting g˜ab = e2Φ gab , we have F = (˜ E a ξb divξ g −1 )ab ∇

= e−2Φ (g −1 )ab (∇a ξb − △cab ξc ) A B = e−2Φ (g −1 )ab ∇a ξb − (δac ∂b Φ + δbc ∂a Φ − (g −1 )cd gab ∂d Φ)ξc

that is,

= e−2Φ {div ξ − (2dΦ · ξ − 2dΦ · ξ)}

F ξ = e−2Φ div ξ. div

(7.131) And, (7.132)

I ξ = ϵ˜ab ∇ E a ξb = ϵ˜ab (dξ)ab = e−2Φ ϵab (dξ)ab = e−2Φ curl ξ. curl

Thus, with

ρ˜ = e−2Φ ρ,

ω ˜ = e−2Φ ω

we have that (7.133) (7.134) This proves the proposition.

F ξ = ρ˜ div I ξ=ω curl ˜.

208

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

The L2 -Theory: We have: ∇a ξb − ∇b ξa = curl ξϵab . < => ?

(7.135)

=(dξ)ab

Hence,

1 1 (curl ξ)2 = curl ξϵab · curl ξϵab = (∇a ξb − ∇b ξa )(∇a ξ b − ∇b ξ a ) 2 2 (7.136) = | ∇ξ |2 −∇a ξ b ∇b ξ a . On the other hand,

∇a ξ b ∇b ξ a = ∇b (ξ a ∇a ξ b ) − ξ a ∇b ∇a ξ b

and

∇b ∇a ξ b − ∇a ∇b ξ b = Rb cba ξ c = Rca ξ c = Kξa ,

that is,

∇b ∇a ξ b = ∇a (div ξ) + Kξa .

Thus, we have

∇a ξ b ∇b ξ a = −K | ξ |2 +∇b (ξ a ∇a ξ b ) − ξ a ∇a (div ξ),

(7.137) or,

∇a ξ b ∇b ξ a = −K | ξ |2 +(div ξ)2 + ∇a (ξ b ∇b ξ a − ξ a div ξ).

(7.138)

Integrating (7.136) and (7.137) on M and adding yields: # # 2 2 (7.139) {| ∇ξ | +K | ξ | }dµg = {(div ξ)2 + (curl ξ)2 }dµg . M

M

The conformal invariance of the div–curl system together with the uniformization theorem shows that this system is injective when M is diffeomorphic to S 2 . The L2 -adjoint of (div, curl) is the operator A acting on pairs of functions (u, v). A(u, v) is a 1-form ζ such that: # # (7.140) (u div ξ + v curl ξ)dµg = ξ a ζa dµg M

M

for every 1-form ζ. Now, # # u div ξ dµg = − (7.141) M

and

(7.142)

#

M

Hence, (7.143) or (7.144)

v curl ξ dµg =

ξ a ∂a u dµg

M

#

M

P.I.

vϵ a ∇b ξ dµg = − b

a

ζa = −∂a u − ∂b v · ϵb a ζ = −du −∗ dv.

#

M

ξ a ϵb a ∂b v dµg .

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

209

The operator A is also conformally covariant. Thus its Kernel depends only on the conformal class of (M, g). Taking M diffeomorphic to S 2 , the Kernel of A is that on the standard sphere. Setting (7.145)

w = u + iv

the equations expressing the condition that w lies in the Kernel are the Cauchy-Riemann equations ∂w =0 (7.146) ∂ z¯ on the complex plane (representing the standard sphere through the stereographic projection). The condition that w is bounded at infinity (north pole) implies then that w is a complex constant (Liouville’s theorem). Then u and v are constants and the div − curl system is integrable if and only if the source functions ρ and ω are L2 -orthogonal to the constants, that is, they have vanishing mean. 7.3.2. Treatment of the Propagation Equation for µ. This equation is of the form ∂µ 3 + trχµ = g, (7.147) ∂s 2 where g denotes the right-hand side of equation (7.128), namely, 1 1 ˆ g = − trχ | χ ˆ |2 + trχ | ζ |2 + 2div / χ·ζ +χ ˆ·∇ / ⊗ζ. 4 2 We shall derive the following inequality: 3 d ∥ µ− ∥ Lp (S) | − ∥ µ− ∥ Lp (S) + trχ − | ds 2 3 (7.148) ∥ µ− ∥ Lp (S) + − ∥ g− ∥ Lp (S) . ≤ ∥ trχ − trχ ∥L∞ (S) · − 2 We have: (in the following we write Lp for Lp (S)) ' # (1 p d 1 d p | µ | dµγ − ∥ µ− ∥ Lp = ds ds A S C # 1 dA 1 − = − | µ |p dµγ ∥ µ− ∥ 1−p Lp p A2 ds S D # 1 ∂µ p−1 p + (p | µ | sgn µ + | µ | trχ)dµγ A S ∂s C 1 = − − trχ − ∥ µ− ∥ pLp ∥ µ− ∥ 1−p Lp p D ( ' # 1 3 p−1 p + [p | µ | sgn µ − trχµ + g + | µ | trχ]dµγ . A S 2 We rewrite the last term as follows: # # 1 1 | µ |p trχdµγ = trχ − ∥ µ− ∥ pLp + | µ |p (trχ − trχ)dµγ . A S A S

210

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

Also, for the preceding term, # 1 3 p | µ |p−1 sgn µ(− trχµ)dµγ A S 2 # 3p =− | µ |p trχ dµγ 2A S C D # 3p 1 p p =− trχ − ∥ µ− ∥ Lp + | µ | (trχ − trχ)dµγ . 2 A S Substituting, we obtain:

d 3 ∥ µ− ∥ Lp − ∥ µ− ∥ Lp + trχ − ds 2( ' # 3 1 1 − − ∥ µ− ∥ 1−p =− · | µ |p (trχ − trχ)dµγ Lp 2 p A S # 1−p 1 +− ∥ µ− ∥ Lp · | µ |p−1 sgn µg dµγ . A S

(7.149)

And we estimate: # 1 | µ |p | trχ − trχ | dµγ ≤∥ trχ − trχ ∥L∞ − ∥ µ− ∥ pLp (7.150) A S as well as (7.151) (7.152)

1 A

#

S

p−1

|µ|

1 | g | dµγ ≤ A =− ∥

'#

p

| µ | dµγ

S µ− ∥ 1−p ∥ Lp −

( p−1 '# p

S

p

| g | dµγ

g− ∥ Lp ,

(1

p

where (7.151) is obtained by H¨ older’s inequality. The ordinary differential inequality for − ∥ µ− ∥ Lp then follows. We now estimate − ∥ g− ∥ Lp . We begin with the principal terms: (7.153)

− ∥ div / χ·ζ− ∥ Lp ≤ C ∥ ζ ∥L∞ (∥ −∇ /χ ˆ− ∥ Lp + − ∥ /d trχ − ∥ Lp ), ˆ − ∥ χ ˆ·∇ / ⊗ζ ∥ Lp ≤ C ∥ χ (7.154) − ˆ ∥L∞ − ∥ ∇ / ζ− ∥ Lp , where C is a numerical constant. Then, let us estimate the lower order terms: (7.155) (7.156)

ˆ |2 − ∥ Lp ≤ ∥ χ ˆ ∥2L∞ − ∥ trχ − ∥ Lp , − ∥ trχ | χ − ∥ trχ | ζ |2 − ∥ Lp ≤ ∥ trχ ∥L∞ ∥ ζ ∥2L∞ .

Thus, we obtain: −∇ /χ ˆ− ∥ Lp + − ∥ /d trχ − ∥ Lp ) − ∥ g− ∥ Lp ≤ C{∥ ζ ∥L∞ (∥ (7.157)

+∥χ ˆ ∥L∞ − ∥ ∇ / ζ− ∥ Lp + ∥ χ ˆ ∥2L∞ − ∥ trχ − ∥ Lp + ∥ trχ ∥L∞ ∥ ζ ∥2L∞ }.

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

211

Elliptic Estimates. We now go back to the null Codazzi equation: (recall (7.16) and (7.15)) 1 (7.158) div /χ ˆ = /d trχ + f 2 and include in f the lower order terms involving ζ: 1 (7.159) f = −β + trχ ζ − χ ˆ · ζ. 2 We had derived the estimate: (see equation (7.54)) ¯ − /d trχ − ∥ p+− ∥ f− ∥ p }, (7.160) − ∥ ∇ /χ ˆ− ∥ p ≤ C{∥ L

L

L

where C¯ depends on an upper bound for ΨM , a lower bound for Ψm and an ˜ p. upper bound for K At this point let us make the following convention: From now on we ∥ χ ˆ− ∥ Lp in the definition of − ∥ ∇ /χ ˆ− ∥ Lp , i.e. instead of − ∥ ∇ /χ ˆ− ∥ Lp + include r−1 − r−1 − ∥ χ ˆ− ∥ Lp we will just write − ∥ ∇ /χ ˆ− ∥ Lp . Similarly, we include r−1 − ∥ ζ− ∥ Lp in the definition of − ∥ ∇ / ζ− ∥ Lp . Writing trχ = trχ + (trχ − trχ), we estimate: 1 ∥ β− ∥ Lp + trχ ∥ ζ ∥L∞ − ∥ f− ∥ Lp ≤ − 2 ( ' 1 (7.161) + ˆ ∥L∞ ∥ ζ ∥L∞ , ∥ trχ − trχ ∥L∞ + ∥ χ 2 provided that (7.162)

trχ ≥ 0.

Recall from equation (7.75) that we had set 2 , (7.163) y= trχ and we had obtained the ordinary differential equation (7.76) for y, i.e. dy 1 = 1 − y 2 h, ds 2 where

1 ˆ |2 h = (trχ − trχ)2 − | χ 2 as we see in equation (7.73). Recall also proposition 16, which shows that 3 (7.164) y ≤ y0 + s 2 and (7.165)

r ≥ r0 + νs,

where ν is a fixed constant satisfying the inequality (7.84): 2r0 . ν< 3y0

212

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

(We are assuming that y0 > 0.) The above inequalities were proved under the smallness condition 4Q21 ν 2 y0 +M < ¯ . νr0 2C

(7.166)

In fact, the same condition yields the lower bound: 1 y ≥ y0 + s 2 (which in particular shows that y stays positive.) The lower bound follows because the upper bound on y and the lower bound on r, which has been established, imply that

(7.167)

1 y ≤ , r ν

(7.168) hence,

1 2 1 C¯ y | h | ≤ 2 r2 | h | ≤ 2 (r−2 Q2 + b) 2 2ν ν

(7.169) and

#

0

s

C# s D # s 1 2 C¯ 4Q21 ′ ′ ′ ds + b(s )ds y | h | ds ≤ 2 ′ 2 2 ν 0 (r0 + νs ) 0 ' ( C¯ 4Q21 +M . ≤ 2 ν νr0

So, integrating the equation

1 dy = 1 − y 2 h, ds 2 or the inequality

from s = 0 we obtain:

H H H H H H H H dy H − 1 H ≤ 1 y 2 H hH H 2 H H H ds

' ( C¯ 4Q21 1 + M < y0 | y − y0 − s |≤ 2 ν νr0 2

by the smallness condition. Thus a continuity argument shows that (7.170) or

1 | y − y0 − s | ≤ y0 2

3 1 y0 + s ≤ y ≤ y0 + s. 2 2 We then integrate (from s = 0) the equation

(7.171)

(7.172)

1 d log r = ds y

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

to obtain (7.173) or (7.174) In particular, (7.175) and (7.176)

0

log

1 2 y0 + 1 2 y0

r0

'

s

1

1 0 ' ( 3 y + s r 0 ≥ log ≥ log 2 3 r0 2 y0

2s 1+ y0

(

≥ r ≥ r0

'

( 2s 1+ . 3y0

r ≥ r0 + νs 3 y0 + s y ≤ max ≤ 2 r r0 + νs

Also, (7.177)

213

r ≤ r0 +

C

3y0 1 , 2r0 ν

D

=

1 . ν

2r0 s y0

and (7.178)

0 r0 + 2r r 2r0 y0 s = . ≤ 1 y y0 2 y0 + s

Remark that

2r0 is initial data, y0 so, we can assume that it is bounded by a fixed numerical constant (as it is dimensionless). We now substitute the bound (7.161) for − ∥ f− ∥ Lp in the elliptic estimate (7.54) after using the Sobolev inequality to bound r−1 ∥ trχ − trχ ∥L∞ in terms of − ∥ /d trχ − ∥ Lp and r−1 ∥ χ ˆ ∥L∞ in terms of − ∥ ∇ /χ ˆ− ∥ Lp , i.e. we have used the Sobolev inequality on Ss (p > 2) to estimate ¯ − ¯ − ∥ trχ − trχ ∥L∞ ≤ Cr ∥ /d trχ − ∥ p, ∥ χ ˆ ∥L∞ ≤ Cr ∥ ∇ /χ ˆ− ∥ p. L

L

This yields:

¯ − /d trχ − ∥ Lp + − ∥ β− ∥ Lp − ∥ ∇ /χ ˆ− ∥ Lp ≤ C{∥

(7.179)

+ r−1 ∥ ζ ∥L∞ + r( − ∥ /d trχ − ∥ Lp + − ∥ ∇ /χ ˆ− ∥ Lp ) ∥ ζ ∥L∞ }

Here, we have made use of the inequality r ≤C (7.180) y to estimate 1 trχ by Cr−1 . 2 We now introduce the bootstrap hypothesis ¯ ∥ ζ ∥L∞ ≤ 1 . (7.181) Cr 2

214

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

This can be assumed to hold with a strict inequality at s = 0, that is for the initial data. Under the assumption (7.181) we can eliminate the last term on the right-hand side of (7.179) obtaining: ¯ − /d trχ − ∥ p+− ∥ β− ∥ p + r−1 ∥ ζ ∥L∞ }. (7.182) − ∥ ∇ /χ ˆ− ∥ p ≤ C{∥ L

L

L

Next, we consider the (elliptic) (div / , curl / ) system for ζ. We obtain the following estimate: ¯ − − ∥ ∇ / ζ− ∥ Lp ≤ C{ (7.183) ∥ div / ζ− ∥ Lp + − ∥ curl / ζ− ∥ Lp } ¯ − ∥ ρ− ∥ p+− ∥ σ− ∥ p ≤ C{ ∥ µ− ∥ p+− L

(7.184)

L

L

+∥χ ˆ ∥L∞ · ∥ χ ˆ ∥Lp }.

∥ Lp (which To proceed, we must have estimates for the quantities − ∥ trχ − enters the estimate for − ∥ g− ∥ Lp , the right-hand side of the propagation ∥ Lp (which enters the above estimate equation for µ, see (7.157)) and − ∥ χ ˆ− for − ∥ ∇ /ζ− ∥ Lp , see (7.184)). We thus introduce the following bootstrap assumptions: r (7.185) ∥ Lp ≤ C, − ∥ trχ − 2 where C is a numerical constant (that can be large); and (7.186)

r− ∥ χ ˆ− ∥ Lp ≤ δ,

where δ is a numerical constant which shall be chosen suitably small in the following. We introduce the notation (7.187) in addition to (7.62), that is Also, (7.188) in addition to (7.189)

∥ µ− ∥ Lp P = r3 −

∥ /d trχ − ∥ Lp . Q = r3 − a = r4 ( − ∥ ρ− ∥ 2Lp + − ∥ σ− ∥ 2Lp ) ∥ β− ∥ 2Lp . b = r4 −

In terms of this notation the elliptic estimates above take the form: ¯ −3 Q + r−2 b 12 + − ∥ ∇ / ζ− ∥ Lp }, − ∥ ∇ /χ ˆ− ∥ Lp ≤ C{r ¯ − ∥ ∇ / ζ− ∥ Lp . where we have substituted ∥ ζ ∥L∞ ≤ Cr (7.190)

¯ −3 P + r−2 a 12 + δ − − ∥ ∇ / ζ− ∥ Lp ≤ C{r ∥ ∇ /χ ˆ− ∥ Lp }, ¯ where we have substituted ∥ χ ˆ ∥L∞ ≤ Cr − ∥ ∇ /χ ˆ− ∥ Lp . We choose δ such that

(7.191)

1 C¯ 2 δ ≤ . 2

Then with z = P + Q,

κ=a+b

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

215

the above inequalities imply: ¯ −3 z + r−2 κ 12 }. − ∥ ∇ /χ ˆ− ∥ Lp , − ∥ ∇ / ζ− ∥ Lp ≤ C{r

(7.192)

We now substitute the above in the estimate of − ∥ g− ∥ Lp obtained previously, see (7.157). To handle the cubic term ∥ trχ ∥L∞ ∥ ζ ∥2L∞ ,

(7.193)

we also make, for uniformity of treatment, the basic bootstrap assumption: r ∥ trχ ∥L∞ ≤ C, (7.194) 2 where C is a numerical constant (which can be large). We then obtain ¯ −5 z 2 + r−3 κ} (7.195) − ∥ g− ∥ p ≤ C{r L ∞ L -norms of

(if we estimate the χ ˆ and ζ in terms of the Lp -norms of ∇ /χ ˆ and ∇ / ζ). Multiplying the ordinary differential inequality for − ∥ µ− ∥ Lp by r3 (see (7.148)) and substituting the estimate for − ∥ g− ∥ Lp , i.e. (7.195), as well as the estimate: ¯ − ∥ trχ − trχ ∥L∞ ≤ Cr (7.196) ∥ /d trχ − ∥ p ¯ −2 Q = Cr ¯ −2 z, ≤ Cr

(7.197) (7.198)

we obtain:

L

dP ¯ −2 z 2 + κ}. ≤ C{r ds Also, we go back to the ordinary differential inequality for − ∥ /d trχ − ∥ Lp . 3 We multiply this by r . (See (7.56) and (7.62).) In view of (7.199)

ˆ ∥L∞ ) − ∥ /d trχ − ∥ Lp + ∥ χ ˆ ∥L∞ − ∥ ∇ /χ ˆ− ∥ Lp (∥ trχ − trχ ∥L∞ + ∥ χ 2 2 ¯ − /d trχ − ≤ Cr{∥ ∥ p+− ∥ ∇ /χ ˆ− ∥ p} L

L

¯ −5 z 2 + r−3 κ}, ≤ C{r

we obtain:

dQ ¯ −2 z 2 + κ}. ≤ C{r ds Adding the two ordinary differential inequalities (7.199) and (7.200), we obtain the following ordinary differential inequality for z: dz ¯ −2 z 2 + κ}. (7.201) ≤ C{r ds It is formally identical to the one obtained earlier for Q, with κ in the role of b. Therefore, by the same argument, under the curvature hypothesis: # s κ(s′ )ds′ ≤ M, (7.202) (7.200)

0

where M is a constant independent of s; as well as the bootstrap assumption:

(7.203)

r ≥ r0 + νs,

216

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

we can show that: (7.204)

z ≤ 2z1 ,

¯ ; provided that the following smallness condition holds: where z1 = z0 + CM νr0 (7.205) z1 < ¯ 2C (on the initial data as well as the ambient spacetime curvature). To close the argument, we must recover the bootstrap assumptions. That is, we are employing the method of continuity: the bootstrap assumptions hold with strict inequalities at s = 0. We consider the maximal value s∗ of s such that these assumptions hold on [0, s∗ ). If then s∗ is finite, the inequalities must be saturated at s∗ . We proceed to show that this cannot happen. This yields the conclusion that s∗ is infinite or as long as the spacetime curvature hypothesis holds. We first revisit the argument of proposition 16. Recall (7.73): 1 ˆ |2 . h = (trχ − trχ)2 − | χ 2 We have that, on [0, s∗ ]: 1 ˆ ∥2L∞ | h | ≤ ∥ trχ − trχ ∥2L∞ + ∥ χ 2 ¯ 2( − ≤ Cr ∥ /d trχ − ∥ 2Lp + − ∥ ∇ /χ ˆ− ∥ 2Lp ) ¯ −4 z 2 + r−2 κ), ≤ C(r (7.206) and we have the ordinary differential inequality: dy 1 − 1 ≤ | h | y2 ds 2 ¯ y )2 (r−2 z 2 + κ). (7.207) ≤ C( r Thus, choosing ν such that 2r0 , (7.208) ν≤ 3y0 we have, on [0, s∗ ]: D C 3 1 y 3y0 1 2 y0 + s = . (7.209) , ≤ ≤ max r r0 + νs 2r0 ν ν Hence, since z ≤ 2z1 on [0, s∗ ], (7.210)

' ( 4z12 C¯ dy +κ − 1 |≤ 2 | ds ν (r0 + νs)2

holds on [0, s∗ ]. Therefore, integrating from s = 0, we obtain C¯ 4z 2 | y − y0 − s | ≤ 2 ( 1 + M ) (7.211) ν νr0 1 < y0 : holds for all s ∈ [0, s∗ ], (7.212) 2

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

217

in particular at s = s∗ , showing that the inequalities concerning y are not saturated at s = s∗ . Since 3 1 y0 + s ≤ y ≤ y0 + s ∀s ∈ [0, s∗ ] 2 2 integrating 1 d log r = ds y yields ' ( 2s 2s ∀s ∈ [0, s∗ ]. ) ≤ r ≤ r0 1 + (7.213) r0 (1 + 3y0 y0

So, if we fix ν to be any positive, real number strictly less than conclude that the inequality (7.203):

2r0 3y0 ,

we

r ≥ r0 + νs

cannot be saturated at s = s∗ . Next, we consider the bootstrap assumptions (7.194) and (7.181). For (7.194) we write 2 ∥ trχ ∥L∞ = + | trχ − trχ |L∞ . y So, it is r r 2r0 C¯ 2 r + r − ∥ /d trχ − ∥ Lp ∥ trχ ∥L∞ = + ∥ trχ − trχ ∥L∞ ≤ 2 y 2 y0 2 z1 2r0 (7.214) + C¯ ≤ y0 r0 (7.215) < C, preassigned (numerical) constant. Thus, under this smallness condition, the inequality of (7.194) cannot be saturated at s = s∗ . For (7.181), ¯ 2− (7.216) ∥ ∇ / ζ− ∥ Lp elliptic estimate r ∥ ζ ∥L∞ ≤ Cr ¯ −1 z + κ 12 } ≤ C{r

(7.217)

1

−1 ¯ ≤ C{2z 1 r0 + κ 2 }.

To show that the inequality (7.181), namely, ¯ ∥ ζ ∥L∞ ≤ 1 Cr 2 is not saturated at s = s∗ , we assume that 1 2z1 1 (7.218) + sup κ 2 < ¯ 2 , r0 2C s so that in addition to the hypothesis (7.202), that is, # s κ(s′ )ds′ ≤ M, 0

218

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

(suitably small), we must also make the hypothesis that 1

(7.219)

sup κ 2 s

is suitably small.

These are the hypotheses on the ambient spacetime curvature. Finally, we recover the bootstrap assumptions (7.185) and (7.186). First, consider (7.185): Recall the propagation equation for trχ (see (7.118)): 1 ∂ trχ + trχtrχ = −2µ + 2 | ζ |2 . ∂s 2 By the Gauss equation (see (7.114) and (7.115)) it is: 1 µ + µ = 2K + trχtrχ = −2ρ + χ ˆ · χ. ˆ 2

(7.220)

We can thus express µ in terms of µ. So, setting l := µ + 2ρ − χ ˆ·χ ˆ + | ζ |2

(7.221) we have

−µ+ | ζ |2 = l

and we obtain the ordinary differential inequality H H H Hd 1 H H − ∥ trχ − ∥ + trχ − ∥ trχ − ∥ p p L L H H ds 2 1 ∥ trχ − trχ − ≤ − (7.222) ∥ L∞ · − ∥ trχ − ∥ Lp + 2 − ∥ l− ∥ Lp . 2 We then set (7.223)

X=

r ∥ Lp − ∥ trχ − 2

and we obtain (7.224)

1 dX ≤ ∥ trχ − trχ ∥L∞ X + r − ∥ l− ∥ Lp . ds 2

Again, let the assumptions (7.185) and (7.186) hold at s = 0 with C and δ replaced by C2 and 2δ , also let s∗ be the maximal value of s such that the assumptions (7.185) and (7.186) hold on [0, s∗ ). Using what we know in the interval [0, s∗ ), we deduce: (7.225)

1

and: (7.226)

1

¯ −2 (r−1 z + κ 2 ) ≤ Cr ¯ −2 (r−1 2z1 + κ 2 ) − ∥ l− ∥ Lp ≤ Cr ¯ −2 z ≤ Cr ¯ −2 2z1 . ∥ trχ − trχ ∥L∞ ≤ Cr

7.3. PROOF OF THE MAIN THEOREM: THE TORSION (ζ) SYSTEM

219

Integrating the ordinary differential inequality (7.224) for X on [0, s∗ ] then yields: # s∗ 1 −1 ¯ ¯ X(s∗ ) ≤ X(0) + Cr0 z1 + C r−1 κ 2 ds 0

≤ X(0) +

¯ −1 z1 Cr 0

+ C¯

'#

0

s∗

r

−2

ds

( 1 '# 2

s∗

0

κ ds

¯ −1 z1 + C¯ (ν −1 r−1 ) 12 M 12 ≤ X(0) + Cr 0 0 < => ? ≤(ν −1 r0−1

(1 2

z1 1 ¯ )2 C

¯ = z1 ). (recalling that z0 + CM C Since X(0) ≤ 2 , we then obtain: (7.227) (7.228)

C ¯ −1 z1 ) 12 + C(r 0 2 < C : if r0−1 z1 is suitably small,

X(s∗ ) ≤

contradicting the maximality of s∗ in regard to the assumption (7.185). Next, we handle (7.186): Recall the propagation equation for χˆ (see (7.109)): (7.229)

∂ˆχ ˆab

1 − trχχ ˆab = mab , ∂s 2

where (7.230) It follows that

(7.231)

1 ˆ + 2ζ ⊗ ζ − γ | ζ |2 . ˆ−∇ / ⊗ζ m = trχχ 2 H H H H∂ 1 H H − − ∥ + trχ − ∥ χ ˆ − ∥ ∥ χ ˆ p p L L H H ∂s 2 ≤ C(∥ trχ − trχ ∥L∞ − ∥ χ ˆ− ∥ Lp + − ∥ m− ∥ Lp )

with C a numerical constant. We set (7.232)

∥ Lp . Y = r− ∥ χ ˆ−

We then obtain the ordinary differential inequality: (7.233)

dY ≤ C(∥ trχ − trχ ∥L∞ Y + r − ∥ m− ∥ Lp ). ds

Using what we know in the interval [0, s∗ ), we can estimate: (7.234) (7.235)

¯ −2 (r−1 z + κ 12 ) − ∥ m− ∥ Lp ≤ Cr ¯ −2 (r−1 2z1 + κ 12 ). ≤ Cr

220

7. SECOND FUNDAMENTAL FORM χ: ESTIMATING χ AND ζ

Integrating the ordinary differential inequality (7.233) for Y on [0, s∗ ], then yields: (7.236) (7.237)

¯ −1 z1 δ + C(r ¯ −1 z1 ) 12 Y (s∗ ) ≤ Y (0) +Cr 0 0 < => ? ≤ 2δ

? =−K

Setting

µ = r2 λ,

this becomes

∂µ µ2 + 2 = −r2 K. ∂r r Since µ → 0 as r → 0, we obtain along each ray the integral equation ( # r' µ(r′ , θ)2 ′2 ′ + r K(r , θ) dr′ . (8.30) µ(r, θ) = − r′2 0 Analyzing this equation, we conclude that (8.31)

λ(r, θ) = O(r)

and λ 1 → − KP , as r → 0. r 3 Thus we see that △g w0 is bounded and

(8.32)

2 △g w0 → KP 3 as we approach P . Next, we set w = w0 + w1 . Then w1 is to satisfy the equation (8.33)

△g w1 = △g w − △g w0 = K − △g w0 on M \P.

Let f be the function on M given by: (8.34)

f = K − △g w0 .

We have shown that f extends to a continuous function on M . There is a solution w1 of (8.35)

△g w1 = f

unique up to an additive constant, provided that # f dµg = 0. (8.36) M

To show this, we integrate f on M \Bδ (P ) with 0 < δ ≤ ϵ. We have # # (8.37) − △g w0 dµg = ∇N w0 ds, M \Bδ (P )

∂Bδ (P )

226

8. UNIFORMIZATION THEOREM

¯δ (P ) we have, in polar (ds is the element of arc length of ∂Bδ (P )). In B coordinates: (8.38)

w0 = −2 log r

and (8.39)

∇N =

so

∂ , ∂r

2 ∇N w0 = − . r

(8.40) Moreover, it is (8.41)

ds = R dθ.

So, we have # (8.42)

2 ∇N w0 ds = − δ ∂Bδ (P )

On the other hand, (8.43)

lim

#

δ→0 M \Bδ (P )

#

0

2π

R(δ, θ)dθ

K dµg =

#

→ −4π

as δ → 0.

K dµg = 4π

M

(by Gauss-Bonnet). We conclude that indeed # (8.44) f dµg = 0. M

So, the equation is solvable for w1 . In fact, we can show that w1 is bounded on M (in fact continuous). (For, f = △g w1 being bounded, in particular f ∈ L2 (M ) implies w1 ∈ H2 (M ), hence w1 is bounded.) Step 4: Now, it is u=w+v w = w0 + w1 , where w1 is bounded on M , while w0 = −2η log dP

and dP is the g-distance from P . On the other hand 1 , (8.45) ev = d˜2 1 + 4O where d˜O is the g˜-distance from O. So, it is (8.46) v = −2 log d˜O + O(1).

It follows that u is bounded on M if and only if in Bϵ (P ) (relative to g) dP · d˜O is bounded above and below by positive constants. a) Upper bound for dP d˜O : Consider an arbitrary point Q ∈ Bϵ (P ).

8.1. STANDARD SITUATION

227

Let γ˜ be the unique g˜-geodesic joining Q to O. Let Q1 be the first point of intersection of γ˜ with the geodesic circle ∂Bϵ (P ). Consider the g-ray γ from P to Q, extended up to ∂Bϵ (P ). Let Q2 be the point where γ meets ∂Bϵ (P ). Then we write ˜ O) = d(Q ˜ 1 , O) + d(Q, ˜ Q1 ) d˜O (Q) = d(Q, ˜ Q1 ). = d˜O (Q1 ) + d(Q, Now, d˜O (Q1 ) is bounded, while ˜ Q2 ) + d(Q ˜ 1 , Q2 ) ˜ Q1 ) ≤ d(Q, d(Q,

and

˜ Q2 ) ≤ g˜ − length of the segment γ(Q, Q2 ) of γ =: L ˜ 1, d(Q,

also

˜ 1 , Q2 ) ≤ g˜ − length of the arc A(Q1 , Q2 ) of the circle ∂Bϵ (P ) d(Q ˜ 2. between Q1 and Q2 =: L In polar normal coordinates centered at P , we have Q1 = (ϵ, θ1 ),

Q2 = (ϵ, θ2 )

Q = (r, θ2 ),

r = dP (Q).

and Thus, it is ˜1 = L

#

ϵ

dP (Q)

≤ e(w1 )M

ew(r,θ2 ) dr = #

ϵ

dP (Q)

#

ϵ

dP (Q)

e(w0 +w1 )(r,θ2 ) dr

r−2 dr ≤ e(w1 )M

1 , dP (Q)

˜ 2 we write where (w1 )M = supM w1 , for L # θ2 # 2π w(ϵ,θ) ˜ e R(ϵ, θ)dθ ≤ ew(ϵ,θ) R(ϵ, θ)dθ =: C(ϵ). L2 = θ1

0

We conclude that (8.47) (8.48)

˜ Q1 ) ≤ L ˜1 + L ˜2 d(Q, ≤ e(w1 )M

1 + C(ϵ). dP (Q)

This yields the upper bound. b) Lower bound for dP d˜O : First, notice that ˜ Q1 ) = g˜ − length of the g˜ − minimal geodesic from Q to Q1 , d˜O (Q) ≥ d(Q, ˜ 3. that is of γ˜ (Q, Q1 ) =: L

228

8. UNIFORMIZATION THEOREM

We can express γ˜ (Q, Q1 ) in parametric form: r = r(λ) θ = θ(λ), r(0) = dP (Q), r(1) = ϵ,

λ ∈ [0, 1]

θ(0) = θ2

θ(1) = θ1 .

Thus, we have ˜3 = L ≥

#

1

0

#

0

1

ew(r(λ),θ(λ))

dr dλ

(2

+ R2

ew1 (r(λ),θ(λ)) e ?|

≥ e(w1 )m (w1 )m

=e

J'

#

0

=(r(λ))−2

1

(r(λ))−2

'

dθ dλ

(2 K 12

dλ

dr | dλ dλ

dr dλ dλ

(r(0)−1 − r(1)−1 ),

where (w1 )m = inf M w1 . We conclude that ˜ Q1 ) ≥ e(w1 )m (dP (Q)−1 − ϵ−1 ), (8.49) d(Q,

which yields the lower bound. So, we have derived the upper and lower bounds for dP d˜O . This concludes the proof of the uniformization theorem. 8.2. Uniformization with K ∈ L4 (S)

In this section we shall derive the uniformization theorem for the case where K ∈ L4 (S). Thus, we relax the conditions on the curvature. The fact that the curvature is not pointwise bounded in our situation, but we only have K ∈ L4 (S), requires more work. We also need the Gauss-Bonnet (G.B.) theorem to hold with the relaxed conditions. Therefore, we will first formulate Gauss-Bonnet as theorem 10 and then show that it applies to our more general situation. Afterwards, we shall prove the uniformization 11. Let us recall that in the work [19], it is K ∈ L∞ (S). Therefore, the authors can directly use the uniformization theorem 9 from section 8.1. We saw in that previous section how the Gauss-Bonnet theorem (8.1) enters the proof of the uniformization 9. In particular, if γ = 0, that is χ = 2, the G.B. theorem reads: # Kdµg = 4π. M

So, M is then diffeomorphic (topologically) to a sphere. In the standard situation, discussed in section 8.1, the uniformization theorem 9 says that g is conformal to a metric of constant Gauss curvature. Throughout this chapter, we are considering a compact, 2-dimensional, Riemannian manifold (M, g) (without boundary), sometimes denoted by

8.2. UNIFORMIZATION WITH K ∈ L4 (S)

229

(S, γ). Now, we are assuming an L4 bound on the curvature R, that is on the Gauss curvature K. To prove the uniformization theorem 11 for this more general situation, we shall use the Gauss-Bonnet theorem 10. The conditions for the latter to hold are K ∈ L4 (S) and the isoperimetric constant of S to be bounded. Thus, we still have to show that the isoperimetric constant indeed is bounded, in order to apply Gauss-Bonnet 10 to our situation. In what follows, we shall make use of the following imbeddings: W 1,2 (S) 4→ L4 (S), W 2,2 (S) 4→ L∞ (S), W 1,4 (S) 4→ L∞ (S). 8.2.1. Gauss-Bonnet Theorem for K ∈ L4 (S). The isoperimetric constant I(S) was introduced as: ' ( Area U I(S) = sup , (Perimeter ∂U )2 U

with U being the smaller of the two domains bounding the closed curve Γ = ∂U . The supremum is over curves Γ which are homologous to 0. For a function Φ on S the isoperimetric inequality reads (2 '# # 2 2 ¯ (Φ − Φ )dµγ ≤ I(S) |∇ / Φ | dµγ . (8.50) S

S

We cite the G.B. theorem as follows:

Theorem 10. Let (S, γ) be a 2-dimensional, compact, Riemannian manifold with Gauss curvature K and isoperimetric constant I(S), satisfying: ∥ K ∥4,S < ∞, I(S) < ∞.

Then, assuming that S is diffeomorphic to the 2-dimensional sphere, the Gauss-Bonnet theorem holds. That is # Kdµγ = 4π. S

Remark: In fact, the Gauss-Bonnet theorem is valid with Lp constraints on the curvature, where p > 1, if an upper bound on the isoperimetric constant is assumed. What remains to be shown in our situation is, that the isoperimetric constant I(S) is bounded. That is, there is an upper and a lower bound. Then, we can apply the G.B. theorem 10. We are going to do this in the following exposure. Proof of the boundedness of I(S): The lower bound on the isoperimetric constant follows naturally.

230

8. UNIFORMIZATION THEOREM

To show that we have an upper bound on the isoperimetric constant, let us proceed as follows: First, we consider (8.51)

A(s) γ(0) = γ(s)

to estimate the smallest and the largest eigenvalue of A(s), where λm (s) denotes the smallest and λM (s) the largest eigenvalue. Then we estimate the isoperimetric constant I(S) for each S(s′ ) with s′ ∈ [0, s]. We have 1 det γ(s) =: 2 (8.52) λm λM = det γ(0) a and (8.53)

λm + λM = trγ(0) γ(s) = (γ(0)−1 )ab (γ(s))ab .

(8.54) Write a =

√ 1 , λm λM

then one can easily see that da = −a trχ, ds

from which it follows a = e−

!s 0

trχ(s′ )ds′

.

To be precise, writing a(s), there is the additive constant 1 on the righthand side, but one can neglect it in view of what we are going to show. In terms of a Jacobi field frame we have: ∂γab (8.55) = 2χab . ∂s Here, we have an elliptic symmetric operator. We are going to show that there is an upper bound for the largest eigenvalue λM as well as a lower bound for the smallest eigenvalue λm , of γab (s) relative to γab (0). For this we have to integrate (8.55) along each generator. # s # s ∂γab ′ (8.56) ds = 2 χab ds′ . ′ ∂s 0 0 This yields (8.57)

γab (s) − γab (0) = M ≤ | χ |L1 ([0,s],L∞ (S)) ≤ C | χ |L1 ([0,s],W14 (S)) .

Here, we mean by the pointwise norm of χ the norm with respect to γ(0), i.e. | χ | := | χ |γ0 . Then, (8.57) becomes

(8.58) (8.59) (8.60)

γab (s) = γab (0) + M ≤ γab (0) + | χ |L1 ([0,s],L∞ (S))

≤ γab (0) + C | χ |L1 ([0,s],W14 (S)) < ∞.

8.2. UNIFORMIZATION WITH K ∈ L4 (S)

231

The integration of (8.55) along each generator needs χ ∈ L1 ([0, s], L∞ (S)). Inequalities (8.57) respectively (8.60) hold because of the imbedding W1p (S) 4→ L∞ (S)

(8.61) for p > 2. That means, it is

∥ χ ∥L∞ (S) ≤ C ∥ χ ∥W1p (S) . As we have χ ∈ W14 (S), all the corresponding inequalities above hold. For the upper bound of λM we consider the fact that χ ∈ W14 (S). This yields directly an upper bound for the eigenvalue λM . That is, we have the following: It is λM (s) = sup {γab (s)ξ a ξ b }. |ξ|γ0 =1

Then we calculate

#

s

χab (s′ )ξ a ξ b ds′ γab (s)ξ ξ = γab (0)ξ ξ + 2 0 # s =1+2 χab (s′ )ξ a ξ b ds′ . a b

a b

0

Therefore, (8.62)

sup γab (s)ξ ξ = 1 + 2 sup a b

|ξ|γ0 =1 0

|ξ|γ0 =1

that is, (8.63)

#

λM (s) ≤ 1 + 2

#

0

s

s

χab (s′ )ξ a ξ b ds′ ,

| χ(s′ ) |γ0 ds′ < ∞.

To make sure that the right-hand side is really bounded, we are going to estimate | χ(s′ ) |γ0 by | χ(s′ ) |γ(s′ ) , which we control. That is, we first split χ into its trace and traceless part and then do the estimates. To proceed as announced, we now compute | χ(s′ ) |γ(s′ ) . Note that with respect to γ(0), in an orthonormal basis relative to γ(0), it is | χ(s′ ) |2γ(0) = χ211 + 2χ212 + χ222 . In the sequel, we write γ0 for γ(0). For the components of γ(s′ ), if we pick a basis of eigenvectors relative to γ(s′ ), denote γ12 = γ21 = 0, γ11 = λ1 , γ22 = λ2 . Therefore, we obtain (8.64) (8.65) (8.66) (8.67)

| χ(s′ ) |2γ(s′ ) = (γ(s′ )−1 )ac (γ(s′ )−1 )bd χab χcd

−2 2 −1 −1 2 2 = λ−2 1 χ11 + λ2 χ22 + 2λ1 λ2 χ12 2 2 2 ≥ λ−2 M (χ11 + χ22 + 2χ12 )

′ 2 ≥ λ−2 M | χ(s ) |γ0 .

232

8. UNIFORMIZATION THEOREM

That is | χ(s′ ) |γ0 ≤ λM (s′ ) | χ(s′ ) |γ(s′ ) .

(8.68)

Now, we consider the integral on the right-hand side of (8.62). Substituting 1 ˆab (s′ ) + trχ(s′ ) γab (s′ ), χab (s′ ) = χ 2 it becomes # s # s $ % 1 χab (s′ )ξ a ξ b ds′ = (8.69) χ ˆab (s′ )ξ a ξ b + trχ(s′ )γab (s′ )ξ a ξ b ds′ . 2 0 0

Subtracting from trχ and adding again its mean value on the surface S(s), namely, # 1 trχ = trχ dµγ , A(S(s)) Ss we obtain for (8.69) in view of the definition of λM : # s # s L 1 χab (s′ )ξ a ξ b ds′ = χ ˆab (s′ )ξ a ξ b + (trχ(s′ ) − trχ(s′ ))γab (s′ )ξ a ξ b 2 0 0 M 1 + trχ(s′ )γab (s′ )ξ a ξ b ds′ # 2s L ≤ | χ(s ˆ ′ ) |γ0 + | trχ(s′ ) − trχ(s′ ) | λM (s′ ) 0 M + | trχ(s′ ) | λM (s′ ) ds′ # s L ≤ ˆ ′ ) |γ(s′ ) + λM (s′ ) | trχ(s′ ) − trχ(s′ ) | λM (s′ ) | χ(s 0 M + λM (s′ ) | trχ(s′ ) | ds′ . (8.70) In the last inequality we used the fact that we can estimate | χ(s ˆ ′ ) |2γ(0) = χ ˆ211 + 2χ ˆ212 + χ ˆ222

as follows, in view of (8.64)–(8.68):

−1 2 | χ(s ˆ ′ ) |2γ(s′ ) = λ−2 ˆ211 + λ−2 ˆ222 + 2λ−1 ˆ12 1 χ 2 χ 1 λ2 χ

Thus, we have: (8.71)

≥ λ−2 ˆ ′ ) |2γ0 . M | χ(s

| χ(s ˆ ′ ) |γ0 ≤ λM (s′ ) | χ(s ˆ ′ ) |γ(s′ ) ,

where we control |χ(s ˆ ′ )|γ(s′ ) on the right-hand side. Substituting the estimate (8.70) in (8.62), yields # s $ L λM (s) ≤ 1 + 2 ˆ ′ ) |γ(s′ ) λM (s′ ) · | χ(s 0 %M (8.72) + | trχ(s′ ) − trχ(s′ ) | + | trχ(s′ ) | ds′ .

8.2. UNIFORMIZATION WITH K ∈ L4 (S)

233

Applying the Gronwall lemma gives: (8.73) ( ' # s ′ ′ ′ ′ ′ λM (s) ≤ exp 2 (| χ(s ˆ ) |γ(s′ ) + | trχ(s ) − trχ(s ) | + | trχ(s ) |)ds . 0

The first two terms in the integral in the exponent decay fast enough according to our results in the corresponding chapter. The only thing remaining to be shown, is that the integral of | trχ(s′ ) | is bounded. This term having a decay behaviour as sc , which was also shown in the same chapter. Therefore, we compute H# H # s # s H H 1 H H | trχ(s′ ) | ds′ = trχ(s′ )dµγ(s′ ) H ds′ H H 0 0 A(Ss′ ) H Ss′ H H2 ( 1 ( 1 ' # s H# '# s H 2 2 H H −2 ′ ′ ′ ≤ A(Ss ) ds trχ(s )dµγ(s′ ) H ds′ H H 0 0 H Ss ′ < => ? ≤

'#

≤ C.

0

s

−2

A(Ss′ )

ds

′

(1 ' # 2

0

=A(Ss′ )2 (trχ)2

s

2

2

A(Ss′ ) (trχ) ds

′

(1 2

Thus, the right-hand side of (8.73) and therefore λM is shown to be bounded from above. To obtain a lower bound for λm , we recall (8.52) and write for λm : 1 λm = . λM a2 Together with the result (8.73), this yields: ' # s 1 1 λm (s) = ≥ exp − 2 (| χ(s ˆ ′ ) |γ(s′ ) λM (s)a2 a2 0 ( ′ ′ ′ ′ (8.74) + | trχ(s ) − trχ(s ) | + | trχ(s ) |)ds

(8.75)

> 0.

That is, λm , in fact, is bounded from below. Summarizing, we have shown that (8.76)

0 < λm ≤ λM < ∞.

Now, the isoperimetric constant I(S) of S is bounded in terms of these eigenvalues λM and λm . This can be seen as follows: Let I(S0 ) be the isoperimetric constant of S0 . Recall that this is finite. Remember the isoperimetric inequality (8.50) and the isoperimetric constant: ' ( Area U , (8.77) I(S) = sup (Perimeter ∂U )2 U

234

8. UNIFORMIZATION THEOREM

where U is the smaller of the two domains bounding the closed curve Γ = ∂U , and the supremum is over curves Γ which are homologous to 0. So, we have A0 ≤ I(S0 )P02 ,

(8.78)

where A0 and P0 denote the area and the perimeter of a corresponding domain U0 in S0 , respectively. Consider S0 × [0, s]. Now, as λM is the largest eigenvalue of γab (s) relative to γab (0), it is for A denoting the area and P the perimeter with respect to s (as explained above for s = 0): (8.79)

A ≤ λ M A0

by(8.78)

≤

λM I(S0 )P02 .

And on the other hand, as λm is the smallest eigenvalue, one writes: ! (8.80) P ≥ λ m P0 .

We obtain:

2 −1 2 A ≤ λM I(S0 )P02 ≤ λM I(S0 )λ−1 m P = I(S0 )λM λm P ,

(8.81)

which yields

I(S) ≤ I(S0 )λM λ−1 m < ∞.

(8.82)

So, the isoperimetric constant I(S) of each surface S(s) is bounded. In view of A ≤ I(S)P 2 ,

(8.83)

the proof of the Gauss-Bonnet theorem works in the well-known way. Considering the left-hand side in the conclusion of theorem 10, we observe: # # 1 3 K dµγ ≤ | K | dµγ ≤ ∥ K ∥2,S vol(S) 2 ≤∥ K ∥4,S vol(S) 4 . S

S

That is, for a bounded domain Ω in S, we have: # 1 3 Kdµγ ≤ ∥ K ∥2,Ω vol(Ω) 2 ≤ ∥ K ∥4,Ω vol(Ω) 4 . Ω

Summarizing, we have shown, that the assumptions in theorem 10 hold in our case, in particular the assumption on the isoperimetric constant, and that then the usual proof of the Gauss-Bonnet theorem applies. 8.2.2. Uniformization Theorem for K ∈ L4 (S). We are going to prove the uniformization theorem under these new conditions. That is, that the metric g of our manifold (M, g) is conformal to a metric of constant Gauss curvature. That is, we are going to show that we can find a smooth function Ω on M such that the metric g˜˜ = Ω2 g has constant Gauss curvature ˜˜ to be 1. (See section 8.1 for ˜ ˜ And by constant rescaling, one can take K K. transformations of curvature etc.) The uniformization theorem is a central point in the proof for the estimates of χ and ζ in chapter 7, as explained there and in the ‘Introduction’.

8.2. UNIFORMIZATION WITH K ∈ L4 (S)

235

In particular, from (8.89) together with the standard uniformization in section 8.1 we obtain the desired bounds for Ψm and ΨM in theorem 8. (See (7.23), (7.30), (7.34).) At this point, we recall that the former proof involved derivatives of the distance function. This would not work here. As K does not lie in L∞ (M ), that means there is no supremum for the curvature, the distance function between two points p and q in M cannot be defined using a supremum as in the former case. Nevertheless, one can define the distance function with an infimum instead. But still, one cannot handle the derivative of the distance. Therefore, one has to find another way. Here, we will first show that the metric g is conformal to a metric g˜ ˜ Then we know from the former case with pointwise bounded curvature K. (section 8.1), that g˜ itself is conformal to a metric g˜˜ with constant Gauss curvature. This idea for the proof is due to Demetrios Christodoulou and it was communicated personally. Next, we want to show the uniformization theorem. Consider (M, g) a compact, 2-dimensional, Riemannian manifold (without boundary) with Gauss curvature K ∈ Lp (M ) for p > 1. Assume that M is diffeomorphic to the 2-dimensional sphere. Theorem 11. (Uniformization) g is conformal to a metric g˜˜ with constant Gauss curvature 1. If there is a metric g˜ with g˜ = e2u g, where (M, g) is our manifold with ˜ being curvature K only in L4 and the curvature related to g˜, namely K, pointwise bounded, then the arguments of the former case will apply. So, we state the following theorem: Theorem 12. Let (M, g) be a compact, 2-dimensional, Riemannian manifold (without boundary) with Gauss curvature K ∈ Lp (M ) for p > 1. Assume that M is diffeomorphic to the 2-dimensional sphere. Then g is ˜ bounded in L∞ , that is: conformal to a metric g˜ with Gauss curvature K ˜ ∈ L∞ (M ). (8.84) g˜ = e2u g and K

Assume this theorem to hold, then we consider the manifold (M, g˜). We have shown before, that g˜ is conformal to a metric g˜˜ with constant Gauss ˜ ˜ i.e.: g˜ curvature K, ˜ = e2v g˜. Now, we are going to prove theorem 12. Proof. First, let us assume K ∈ L2 (M ). Then we will show the theorem to hold for p > 1, (so, also for p = 4). Denote by u ¯ the mean value of u on M and # 1 ¯ = K. (8.85) K Area(M ) M Take u such that ¯ ∈ L2 (M ). (8.86) △g u = K − K

236

8. UNIFORMIZATION THEOREM

Consider the metric g˜ with g˜ = e2u g.

(8.87)

˜ Then we obtain for the curvature K: ˜ = e−2u (K − △g u). (8.88) K

We have to show first that the function u fulfilling equation (8.86) obeys: | u |≤ C ′ .

(8.89)

˜ is shown to be pointwise bounded as well. Then with this result, K The following estimates are valid for any p > 1 (especially for p = 2), as shall be pointed out below. For the moment, let us take K ∈ L4 (M ) and ¯ = 4π ∈ L2 (M ), that is in particular K A ¯ ∈ L2 (M ). K −K

(8.90)

¯ Then equation (8.86) reads Now, write f := K − K. △g u = f.

(8.91)

Our aim is to derive an H2 estimate for u. Then by Sobolev inequalities we will get an L∞ estimate for u, as desired. For this purpose, let us first take the derivative of equation (8.91) as follows: ∇a △u = ∇a f.

(8.92) Then we obtain:

∇a △u = ∇a (∇b ∇b u) = ∇b (∇a ∇b u) + Rb c a b ∇c u

= (∇b ∇b )(∇a u) − Rc a ∇c u = △(∇a u) − K∇a u.

(8.93) That is

△(∇a u) − K∇a u = ∇a f.

(8.94)

Now, multiply this equation (8.94) by −∇a u and integrate on M . For the first term in (8.94) we obtain: # # # P.I. −∇a u · △(∇a u) = −∇a u · ∇b ∇b (∇a u) = ∇b ∇a u∇b ∇a u M M #M = (8.95) | ∇2 u |2 . M

For the second term in (8.94) we have: # # # a a (8.96) −∇ u(K∇a u) = − K∇ u∇a u = − M

M

M

And the third term in (8.94) gives: # # # P.I. (8.97) −∇a u∇a f = ∇a ∇a u · f = M

M

M

△u · f =

K | ∇u |2 . #

M

f 2.

8.2. UNIFORMIZATION WITH K ∈ L4 (S)

Summarizing, we have: # # (8.98) | ∇2 u |2 + M

M

K | ∇u |2 =

#

M

237

f 2 < ∞.

¯ ∈ L2 (M ). And this is bounded because f = K − K Writing equation (8.98) as # # # # 2 2 2 2 ¯ ¯ K | ∇u | + |∇ u| + (K − K) | ∇u | = (8.99) M

it follows (8.100)

#

M

M

2

2

|∇ u| +

#

M

M

¯ | ∇u |2 = K

#

f 2,

M

2

f −

M 4π A(S) .

#

M

¯ | ∇u |2 . (K − K)

¯ = ¯ is positive, it follows The Gauss-Bonnet theorem gives K As K from equation (8.100) that # # # 2 2 2 ¯ | ∇u |2 |∇ u| ≤ f − (K − K) (8.101) M

as well as

#

(8.102)

M

M

¯ | ∇u |2 ≤ K

#

M

M

2

f −

#

M

¯ | ∇u |2 . (K − K)

In view of (8.102), it is also # # 2 K | ∇u | ≤ (8.103) M

Then, we write # # 2 (8.104) K | ∇u | = M

M

f 2.

M

¯ | ∇u |2 + K

#

M

¯ | ∇u |2 ≤ (K − K)

So, the first term on the right-hand side of (8.104) is # # # 2 2 ¯ ¯ K | ∇u | = K | ∇u | ≤ f 2. (8.105) M

M

#

M

M

This shows

∇u ∈ L2 (M ).

(8.106)

From (8.101), and (8.98) one finds # | ∇2 u |2 ≤ ∥ f ∥2L2 (M ) . (8.107) M

That is

∇2 u ∈ L2 (M ).

(8.108)

Finally, the second term on the right-hand side of (8.104) reads (1 ' # (1 '# # 2 2 2 2 4 ¯ | ∇u | ≤ ¯ (K − K) (K − K) · | ∇u | M

(8.109)

M

M

¯ ∥L2 (M ) · ∥ ∇u ∥2 4 =∥K −K L (M ) .

f 2.

238

8. UNIFORMIZATION THEOREM

¯ ∥L2 (M ) < ∞. So, it remains to We know from above (8.90) that ∥ K − K 2 show that ∥ ∇u ∥L4 (M ) is finite. This follows directly from the Sobolev imbedding theorem. Theorem 13. Let (M, g) be our two-dimensional, compact, Riemannian manifold. Then the following imbedding holds: W 1,2 (M ) 4→ L4 (M ).

(8.110)

More generally, the theorem holds for mp = n and p ≤ q < ∞, where n denotes the dimension of the manifold: W m,p (M ) 4→ Lq (M ).

(8.111)

Thus from (8.110) and ∇u, ∇2 u ∈ L2 (M ), it follows:

∥ ∇u ∥L4 (M ) ≤ C ∥ ∇u ∥W 1,2 (M ) = C(∥ ∇u ∥L2 (M ) + ∥ ∇2 u ∥L2 (M ) ).

(8.112)

This holds by Moser iteration. Let us stress, here, the fact that C depends on the isoperimetric constant I(M ), and that we showed the latter to be bounded in the previous section. Therefore, the right-hand side of this inequality is bounded. Now, one has to show that u ∈ L2 (M ). Applying the Poincar´e inequality on M to u and by the result that ∇u ∈ L2 (M ) from above (8.105), we obtain that u ∈ L2 (M ).

(8.113) Thus it is

u ∈ H 2 (M ).

(8.114)

p Now, with the imbedding Wm 4→ L∞ where mp > n = dim M , that is: 2 ∞ W2 4→ L , it follows that

u ∈ L∞ (M ).

(8.115)

This proves (8.89). We have seen above that # # 2 2 2 ¯ 2 1. Then it is: H# H H 1 HH −2u H ˜ sup | K | = sup H (8.121) e KH M M A M ' ( 1 p ∥ e−2u ∥ p−1 (8.122) ≤ ∥ K ∥Lp (M ) < ∞. L (M ) A Especially, for K ∈ L4 (M ) it is ˜ |≤ 1 (∥ e−2u ∥ 4 (8.123) sup | K ∥ K ∥L4 (M ) ) < ∞. L 3 (M ) A M

At this point, we discuss equation (8.116). We can take K in Lp (M ) for any p > 1. This will yield the same result. For, the estimates hold in the ¯ lies in L2 (M ). Thus, we obtain again u ∈ H 2 (M ), same way, also K − K ∞ that is u ∈ L . And ∥ e−2u ∥ p is bounded for any p > 1 as u ∈ L∞ . p−1 ˜ ∈ L∞ (M ), we are back in the former Now, after having shown that K case. Recall that we used Gauss-Bonnet in the former proof. As GaussBonnet holds also here, the same proof applies. So, the result from above ˜˜ says that g˜ is conformal to a metric g˜˜ with constant Gauss curvature K. p That is, we have proven the uniformization theorem for K in L (M ).

CHAPTER 9

χ on the Surfaces S – Changes in r and s 9.1. Setting To recapitulate the notation we use here, we refer to the chapter about the comparison as well as to [16] and [19], Chapter 9 and beginning of Chapter 13, where everything is introduced and the detailed derivations can be found. The Hessian of the optical function u decomposes as follows: χAB : null second fundamental form ζA : 1-form ω : function. Along the generators of each level set Cu of u, they fulfill the following equations: (9.1) (9.2) (9.3)

d χAB = −χAC χCB − αAB ds d ζA = −χAB ζB + χAB ζ B − βA ds d ω = 2ζ · ζ− | ζ |2 −ρ. ds

The null Codazzi equation reads as (9.4)

(div / χ) ˆA=

% 1$ ∇ /A trχ + ζA trχ − χ ˆAB ζB − βA 2

whereas the conjugate null Codazzi equation is (9.5)

(div / χ) ˆA=

% 1$ ∇ /A trχ − ζA trχ + χ ˆ AB ζB + β A 2

Moreover, the Gauss equation for the Gauss curvature K on St,u is (9.6)

1 1 ˆ·χ ˆ − ρ. K = − trχtrχ + χ 4 2

For the torsion ζ, one derives the propagation equation: (9.7)

d ζA = −χAB ζB + χAB ζ B − βA . ds 241

242

9. χ ON THE SURFACES S – CHANGES IN r AND s

In addition, we also find the div − curl-system as follows: 1 ˆ∧χ ˆ (9.8) curl / ζ =σ− χ 2 1 ˆ · χ. ˆ div / ζ = −µ − ρ + χ (9.9) 2 Here, µ denotes the mass aspect function defined by 1 / ζ. (9.10) µ = K + trχ trχ − div 4 Let us define its conjugate µ as follows: 1 / ζ. µ = K + trχ trχ + div 4 In the chapter, where we estimate χ, we also use the propagation equation of µ: 3 1 1 d ˆ µ + trχ µ = − trχ | χ ˆ |2 + trχ | ζ |2 + 2div / χ·ζ +χ ˆ·∇ /⊗ζ. (9.12) ds 2 4 2 For the definition of the latter expression, see the chapter about the comparison. Further equations are derived in details in our chapter ‘Second Fundamental Form χ: Estimating χ and ζ’. Let us now state some facts, which we shall use later on. For further information and detailed proofs, we refer to our Chapter ‘The Last Slice’ and to [19], Chapter 13. (9.11)

Lemma 7. Let TA1 ···Ak as well as FA1 ···Ak be k-covariant tensorfields tangential to the surfaces St,u satisfying: dTA1 ···Ak + λ0 trχ TA1 ···Ak = FA1 ···Ak ds with λ0 being a nonnegative real number. Denote λ1 = 2λ0 − p2 . Also, we define λ2 = λ1 + p2 . Then, the following holds. Let c be a constant independent of u. ' ( # t∗ λ1 λ1 λ1 | r F |p,S (t, u) dt . | r T |p,S (t∗ , u) ≤ c | r0 T |p,S (t0 , u) + t0

Equivalently, we have

r −T ∥ − ∥ p,S (t∗ , u) ≤ c λ2

'

r0λ2 −T ∥ − ∥ p,S (t0 , u)

+

#

t∗

t0

(

r −F ∥ − ∥ p,S (t, u)dt . λ2

Lemma 8. Let TA = TA1 ···Ak be a k-covariant tensorfield in spacetime, being tangent to the surfaces St,u everywhere. Then, we have the equations: $ % D /4 ∇ /B T A − ∇ /B D /4 TA = −χBC ∇ /C TA + ζB + ζ B D /4 TA "$ % χAj B ζ C − χBC ζ A + ϵAj C ∗ βB TA1 ···C···Ak + j

j

9.2. SECOND FUNDAMENTAL FORM χ ON THE SURFACES S

243

$ % D /4 D /3 TA − D /3 D /4 TA = 2 ζ B − ζB ∇ /B TA + 2ωD /4 TA "$ % +2 ζAj ζ B − ζ A ζB − σϵAj B × TA1 ···B···Ak j

j

D /3 ∇ /B T A − ∇ /B D /3 TA = −χBC ∇ /C T A + ξ B D /4 TA "$ + χAj B ξ C − χBC ξ A + χA j

j

jB

ζC − χBC ζAj

% − ϵAj C ∗ β B TA1 ···C···Ak

As straightforward consequences, we derive the following two corollaries. Corollary 1. Let T be a scalar which obeys d T = F. ds Then, the following holds: % $ d ∇ /A T + χAB ∇ /B T = ∇ /A F + ζA + ζ A F ds $ % d D3 T = D3 F + 2ωF + 2 ζ A − ζA ∇ /A T. ds Corollary 2. Let TA be a 1-form in spacetime which is tangent to the surfaces St,u everywhere and satisfies the equation d TA = F A . ds Then, the following holds: ( ' $ % 1 d div / T + χAB ∇ trχζ A − χ / F + ζ +ζ F + /A TB = div ˆAB ζ B + βA TA . ds 2 9.2. Second Fundamental Form χ on the Surfaces S – Changes in r and s We always work with the surfaces Ss where s is a parameter along the cone C, and with the surfaces St,u with t being the time and u the optical function. There is a difference, if we consider the surfaces Ss along the cone C or the surfaces St,u . In order to express the quantities that are defined or estimated with respect to one of these surfaces in terms of the other one, we have to take this fact into account. Thus, we have (9.13) With (9.14) it then is (9.15)

1

| /d s |= o(r− 2 ). dtrχ = o(r−2 ) ds H H H dtrχ H −5 H H H ds H | /d s |= o(r 2 ).

244

9. χ ON THE SURFACES S – CHANGES IN r AND s

That is, the decay rates change, switching from Ss to St,u or vice versa, which means that the decay in s differs from the decay in r by the above formulas. In the chapter, where we estimated χ, we worked with the surfaces Ss . There we obtained our estimates by the method of treating elliptic systems on the surfaces Ss together with evolution equations along the cones C. In the following, we shall deduce the corresponding estimates for the surfaces St,u . 9.2.1. Surfaces S s and S t,u . In general, we have to deal with the following fact. Let Tp M denote the tangent space of M at the point p. Consider the tangent space Tp Cu of Cu at a point p. As in previous chapters, L is the generating vectorfield of the cone Cu . Moreover, let X be a vectorfield in Tp St,u (that is, also in Tp Cu ). Then, we define Y ∈ Tp Cu as (9.16)

Y = X + λL

for λ ∈ R such that

(9.17)

Y (s) = 0.

Thus, it is (9.18) Now, we replace

Y ∈ T p Ss .

Xtrχ by Y trχ. Therefore, one has, recalling that L(s) = 1: (9.19) (9.20) One then writes

0 = Y (s) = X(s) + λ λ = −X(s).

(9.21)

Y (trχ) = Xtrχ + λLtrχ dtrχ (9.22) . = Xtrχ − (Xs) ds From now on, we denote by /χ the following: dtrχ /d s. (9.23) /χ = /d trχ − ds 9.2.2. Estimates for χ in S t,u . We have dtrχ ∇ /A s ( 'ds 1 2 2 =∇ /A trχ + (trχ) + | χ /A s. (9.25) ˆ| ∇ 2 At this point, we recall the evolution equations for trχ and χ: ˆ 1 dtrχ = − (trχ)2 − | χ (9.26) ˆ |2 ds 2 dχ ˆAB = −trχχ ˆAB − αAB . (9.27) ds (9.24)

/A trχ − /χA = ∇

9.2. SECOND FUNDAMENTAL FORM χ ON THE SURFACES S

The evolution equation for (9.24) reads ' ( % d$ d dtrχ d /χ = ∇ / trχ − ∇ / s . (9.28) ds A ds A ds ds A

In view of the second term in (9.28) we compute ' ( 1 d d2 trχ 2 2 − (trχ) − | χ (9.29) = ˆ| ds2 ds 2 dtrχ = −trχ + 2χ ˆ · α + 2trχ | χ ˆ |2 (9.30) ds 1 = (trχ)3 + 3trχ | χ (9.31) ˆ |2 + 2χ ˆ · α. 2 Also, we obtain from corollary 1 the following equation for ∇ /A s. (9.32)

d ∇ / s = −χAB ∇ /B s + ζA + ζ A . ds A

Then it is ' (( ' . d dtrχ 1 ∇ /A s = − (trχ)2 + | χ −χAB ∇ ˆ |2 /B s + ζA + ζ A ds ds 2 ' ( 1 3 2 + (trχ) + 3trχ | χ (9.33) ˆ | + 2χ ˆ·α ∇ /A s 2 This together with corollary 1 applied to trχ, yields for (9.24): 3 d /χA = − trχ∇ /A trχ − χ ˆAB ∇ /B trχ ds 2

(9.34)

(9.35)

. -1 − 2χ ˆBC ∇ /A χ ˆBC − (ζA + ζ A ) (trχ)2 + | χ ˆ |2 2 ' (. 1 + (trχ)2 + | χ − χAB ∇ ˆ |2 /B s + ζA + ζ A 2 ' ( 1 − (trχ)3 + 3trχ | χ ˆ |2 + 2χ ˆ·α ∇ /A s 2 3 = − trχ∇ /A trχ − χ ˆAB ∇ /B trχ 2 ( ' 1 − 2χ ˆBC ∇ (trχ)2 + | χ /A χ ˆBC − χAB ∇ /B s ˆ |2 2 ' ( 1 3 2 − (trχ) + 3trχ | χ ˆ | + 2χ ˆ·α ∇ /A s. 2

Let us study the terms proportional to ∇ /A s. ( ' (' ( ' 1 1 1 2 2 3 2 (trχ) + | χ (trχ) + 3trχ | χ − trχ − (9.36) ˆ| ˆ | + 2χ ˆ·α 2 2 2 3 1 7 = − trχ (trχ)2 − trχ | χ ˆ |2 −2χ (9.37) ˆ·α 2 2 2 ' ( 3 1 = − trχ (trχ)2 + | χ ˆ |2 −2χ (9.38) ˆ |2 − 2trχ | χ ˆ · α. 2 2

245

246

9. χ ON THE SURFACES S – CHANGES IN r AND s

Finally, we deduce for (9.24) the following equation. Here, we make use of (9.38). d 3 /χA = − trχχ /A − χ ˆAB /χB ds 2 ' ( . 1 2 2 (trχ) + | χ /B s + − 2trχ | χ +χ ˆAB ˆ| ∇ ˆ |2 − 2χ ˆ·α ∇ /A s 2 ( ' 1 2 2 − 2χ ˆBC ∇ (trχ) + | χ ˆAB ∇ /A χ ˆBC − ˆ| χ /B s 2 . 3 /A − χ (9.39) = − trχχ ˆAB /χB + −2trχ | χ ˆ |2 −2χ ˆ·α ∇ / A s − 2χ ˆBC ∇ /A χ ˆBC . 2 Thus, we have the propagation equation . 3 d /χA + trχχ /A = −χ ˆAB /χB − 2trχ | χ ˆ |2 + 2χ ˆ·α ∇ /A s − 2χ ˆBC ∇ /A χ ˆBC . ds 2 (9.40) Let us denote by F the right-hand side of (9.40). Directly from lemma (7) we then derive the subsequent inequalities. Here, we refer to the chapter about χ, where we estimated χ on the surfaces Ss on Cu . Also, similar techniques but requiring more subtle estimates in the different setting of the last slice Ht∗ are applied in the chapter ‘The Last Slice’. Moreover, we refer to [19], Chapter 13, ‘Derivatives of the Optical Function’, where the corresponding situation is studied. Now, we obtain: H# t∗ H( ' H H 3 3 3 r −χ (9.41) ∥ /A− ∥ 4,S (t∗ ) ≤ c r0 −χ ∥ /A− ∥ 4,S0 (t0 ) + HH r −F ∥ − ∥ 4,S (t)dtHH t0

1 2

Dividing by r both sides of this inequality gives ' 5 ( 5 1 r 2 −χ ∥ /A− ∥ 4,S ≤ c r02 −χ ∥ /A− ∥ 4,S0 + c′ r− 2

(9.42)

Now, we have:

≤ cϵ0 . 3

(9.43)

∥ χ∥ ˆ−4,S ≤ cϵ0 r2 −

5

(9.44)

∥ /χ∥ ˆ−4,S ≤ cϵ0 r 2 −∇

(9.45) (9.46) Also, one obtains: (9.47) (9.48) (9.49) (9.50)

5 2

r −trχ ∥ − ∥ 4,S ≤ cϵ0

∥ / trχ∥ −4,S ≤ cϵ0 r −∇ 3

r 2 −ζ∥ ∥ −4,S ≤ cϵ0

5

∥ /ζ∥ −4,S ≤ cϵ0 r 2 −∇ 1

∥ −4,S ≤ cϵ0 r 2 −ω∥

3

∥ /ω∥ −4,S ≤ cϵ0 r 2 −∇

CHAPTER 10

The Last Slice This is the chapter about the last slice Ht∗ of the spacetime. Here, we construct the function u on Ht∗ . Starting from the given surface St∗ ,0 = Ht∗ ∩ C0 , where C0 is the standard cone, we define the function u∗ to be the solution of the inverse lapse problem, taking the value 0 on St∗ ,0 (see below), and we shall show the global existence of solutions. 10.1. Inverse Lapse Problem – Equation of Motion of Surfaces Constructing u as a solution of the inverse lapse problem, that is solving an equation of motion of surfaces, was first done by D. Christodoulou and S. Klainerman in [19]. One might first think of applying other, easier methods in order to construct u. But, as we are going to explain now, they would not match our requirements. What we refer to as the ‘last slice’ is the maximal hypersurface Ht∗ , which bounds in the future the spacetime slab that we are constructing in the continuity argument. The obvious choice of u on Ht∗ , namely minus the signed distance function from St∗ ,0 , is inappropriate because this distance function is only as smooth as the induced metric g¯t∗ . It is not one order better, which would be the maximal possible. Thus, there would be a loss of one order of differentiability. But the problem does not allow to lose derivatives. That is, with this loss of one order of differentiability, the estimates would fail to close. To overcome this difficulty, we define u on Ht∗ in a different way, namely by solving an equation of motion of surfaces on Ht∗ . The initial surface being St∗ ,0 . St∗ ,0 = Ht∗ ∩ C0 , where C0 is the standard cone, we define the function u∗ to be the solution of the inverse lapse problem, taking the value 0 on St∗ ,0 . We shall show the global existence of solutions. That is, we define u∗ to be the solution of the inverse lapse problem below. Why would the use of other methods like the inverse mean curvature flow (IMCF) not work here? Using IMCF, the problem could be solved in the outward direction only. Whereas the equation of motion of surfaces can be solved in both directions, which is what we need. The equation, we use here, has to have the smoothing property described above as well as it has to be solvable in both directions. This excludes the IMCF and similar methods. It turns out that the equation of motion of surfaces yields exactly what we need. 247

248

10. THE LAST SLICE

In solving the inverse lapse problem, we first show that if the first and second fundamental forms of Ht∗ satisfy certain estimates, then the problem (10.2), (10.3) has a solution which is global in the exterior of St∗ ,0 and extends in the interior up to a distance that is a given fraction of r0 . Given a function u on such a manifold, a function whose level sets define locally a foliation, we have the associated lapse function (10.1)

1

a = (¯ g ij ∂i u ∂j u)− 2 ,

which measures the normal separation of the leaves. We can think of a as the normal velocity of a surface, leaf of the foliation. An equation of motion of surfaces (10.2) is then a rule which assigns a positive function a to a given surface. And the rule which defines the equation of motion is given in (10.3). The main part in constructing the function u∗ in the last slice is again a bootstrap argument. From the propagation equation (10.8) for trχ in Ht∗ we obtain the propagation equation (10.12) for trχ. We couple this evolution equation to the elliptic system ((10.6), (10.7)) on each St∗ ,u . The right-hand sides of these equations are controlled by assumption within the bootstrap argument. This yields estimates on the quantities on the left-hand sides. Using the evolution equation to obtain the desired estimates in the bootstrap argument finally yields the global results. 10.2. Setting As mentioned above, in this chapter about the last slice Ht∗ of the spacetime, we construct the function u on Ht∗ . Starting from the given surface St∗ ,0 = Ht∗ ∩ C0 , where C0 is the standard cone, we define the function u∗ to be the solution of the inverse lapse problem, taking the value 0 on St∗ ,0 (see below), and we shall show the global existence of solutions. That is, we define u∗ to be the solution of the following: (10.2)

| ∇u∗ |−1 = a,

u∗ |St∗ ,0 = 0

with a satisfying the following equation on each level surface St∗ ,u∗ of u∗ : (10.3) △ / log a = f − f¯ − div / ϵ, log a = 0,

where

1 f = K − (trχ)2 . 4 The goal is to prove that there exist global solutions to the inverse lapse problem (10.2), (10.3). For this purpose, we shall have to assume estimates for the spacetime curvature on the last slice Ht∗ . However, we cannot assume estimates for the components of the curvature with respect to a null frame adapted to the surfaces St∗ ,u∗ , since these surfaces, have not been constructed yet, with the exception of St∗ ,0 . Instead, what we can do is, to assume estimates for the curvature components relative to a null frame defined by the level surfaces of the function u′ on Ht∗ of our background foliation. This function u′ has been constructed in the previous chapters,

(10.4)

10.2. SETTING

249

and the curvature components with respect to its foliation have been shown to be small. So, we have constructed the level surfaces St,u′ . We can extend this function u′ onto the last slice Ht∗ . This gives us the foliation St∗ ,u′∗ of Ht∗ . However, it is not clear if this function fulfills the equations of the inverse lapse problem (10.2), (10.3). Therefore, we will use the estimates on the background foliation to control the curvature and geometric components of the foliation by u∗ . Later on, for convenience of notation, we shall write u instead of u∗ . In the sequel, we will work with the normalized null normals (10.5)

e′3 = T − N = a−1 e3 ,

e′4 = T + N = ae4 ,

where N is the outward unit normal to St∗ ,u∗ relative to Ht∗ . In this chapter, we drop the primes. We will show that if the first and second fundamental forms of Ht∗ satisfy certain estimates, then the problem (10.2), (10.3) has a solution which is global in the exterior of St∗ ,0 and extends in the interior up to a distance that is a given fraction of r0 . Let us recall some equations, derived in the previous sections, involving the second fundamental forms. Then, we can write equation (10.3) in the form (10.6) div / ζ = f − f¯,

and we remind ourselves of the corresponding curl / -equation: (10.7)

curl / ζ = σ − ηˆ ∧ χ. ˆ

(10.6) and (10.7) form a system for ζ. Again considering results from before, we have the following equation for ∇N trχ on each Ht , including Ht∗ : 1 1 ˆ |2 − ηˆ · χ ˆ − | ζ |2 − div / ζ − ρ. (10.8) ∇N trχ = − (trχ)2 − δtrχ − | χ 2 2 For the Gauss curvature, we have: 1 1 1 ˆ − δtrχ − ρ. (10.9) K = (trχ)2 − | χ ˆ |2 − ηˆ · χ 4 2 2 We obtain the following propagation equation for trχ on Ht∗ : 1 (10.10) ∇N trχ + (trχ)2 = −F0 − f¯ 2 with 1 (10.11) F0 = | χ ˆ |2 + | ζ |2 . 2 / ], Applying the commutation formula for [∇ /N , ∇ / ] trχ)j = a−1 ∇ /j a∇ /N trχ − θjm ∇ / m trχ, ([∇ /N , ∇

this yields the propagation equation for ∇ / trχ: 3 / trχ) + trθ ∇ (10.12) ∇ /N (∇ / trχ = −F1 2

250

10. THE LAST SLICE

with (10.13) F1 = (χ ˆ + ηˆ) · ∇ / trχ + δ∇ / trχ + ∇ / F0 +

'

( 1 (trχ)2 + F0 + f¯ (ζ − ϵ). 2

Also, we will use the normalized null Codazzi equation 1 1 (10.14) div /χ ˆ− ∇ / trχ + ϵ · χ ˆ − ϵtrχ = −β. 2 2 10.3. Comparing Null Frames

As mentioned above, we shall make assumptions on the curvature components with respect to a null frame defined by the level surfaces of a function u′ , to derive the desired estimates with respect to a null frame given by the foliation by the function u of the inverse lapse problem. To do so, we have to switch from one null frame into the other and to compare the curvature components and geometrical quantities of the two frames. In view of this, we are now going to prove the following lemma: Lemma 9. Let H be a 3-dimensional Riemannian manifold. And let S0 be a surface in H diffeomorphic to S 2 . Let u′ and u be smooth functions without critical points defined in tubular neighbourhoods U ′ and U with U ′ ⊃ U of S0 and vanishing on S0 . So, we take 4 4 Su′ ′ , U = Su (10.15) U′ = u′ ∈I ′

u∈I

with and I being open real intervals including 0, whereas Su′ ′ and Su are the level surfaces of u′ and u respectively. With (1 (1 ' ' Area(Su′ ′ ) 2 Area(Su ) 2 ′ , r= , (10.16) r = 4π 4π I′

let

(10.17) (10.18)

diam(Su′ ′ ) ≤ 2πr′ , 1 ≤ a′ ≤ 2, 2

diam(Su ) ≤ 2πr,

1 ≤ a ≤ 2, 2

where Also, let (10.19) (10.20)

a′ = | ∇u′ |−1 ,

a = | ∇u |−1 .

( ' ′ ( r′ ′ 1 r trθ ≥ , sup trθ′ ≤ 2, 2 2 2 Su ′ Su′ ′ -r . 1 -r . inf trθ ≥ , sup trθ ≤ 2 Su 2 2 2 Su inf ′

'

hold for all u′ in I ′ and all u in I with θ′ and θ being the second fundamental forms of Su′ ′ and Su respectively.

10.3. COMPARING NULL FRAMES

251

Then there is a numerical constant c such that in U , it is: c−1 u ≤ u′ ≤ cu,

(10.21) Assume that

c−1 r′ ≤ r ≤ cr′ . 3

sup | trθ′ − trθ′ | ≤ cr′− 2

(10.22)

Su′ ′

and 1

′− sup | θˆ′ | ≤ cr′−1 τ− 2

(10.23)

Su′ ′

1

hold for all u′ in I ′ and with τ−′ = (1 + u′2 ) 2 . Denote 3

(10.24)

/ ′ log a′ |, A′ = sup sup r′ 2 | ∇

(10.25)

/ log a |, A = sup sup r 2 | ∇

u′ ∈I ′ S ′ ′ u

3

u∈I Su

and ∇ / denote the derivatives tangential to Su′ ′ and Su respectively. where Consider the unit normal vectorfields N ′ and N of the foliations {Su′ ′ } and {Su }, also, consider the angle ϕ between N and N ′ . Then, if A and A′ are sufficiently small, ϕ < π2 in U , and there exists a numerical constant c such that ∇ /′

3

sin ϕ ≤ c(A + A′ ) | u′ | r′− 2 .

(10.26)

Further, denoting by

1

B ′ = sup sup r′ 2 | ∇N ′ log a′ |,

(10.27)

u′ ∈I ′

Su′ ′

and if A, A′ , B ′ ≤ 1, then we have for all u in I

1

oscSu r′ ≤ c(A + A′ + B ′ ) | u | r− 2

(10.29) (10.30)

1

oscSu u′ ≤ c(A + A′ + B ′ ) | u | r− 2

(10.28)

1

sup | r − r′ | ≤ c(A + A′ + B ′ ) | u | r− 2 . Su

Proof. As on S0 = S0′ it is ϕ = 0, we have ϕ < π2 in a neighbourhood of 0 by continuity. Denote by Iˆ the maximal subinterval of I where ϕ < π2 . We will show below that Iˆ = I. Now, we are going to compare the functions u and u′ in 4 ˆ= U Su . u∈Iˆ

ˆ on Su and consider the integral curve Γ of N Consider a point p in U joining p to S0 . By s we denote the arc length of Γ. Then we have ds =a (10.31) du

252

10. THE LAST SLICE

and therefore, it is: u′ (p) =

(10.32) as |

1 a′

#

Γ

∇N u′ ds ≤

#

Γ

| ∇u′ | ds =

#

u

0

a |Γ du, a′

|= and ds = adu. ˆ . We use Let Γ′ be the integral curve of N ′ joining p to S0 . Then Γ′ ⊂ U integral curves of N to construct a diffeomorphism of U → I × S0 . Then the metric in U can be expressed in the form ∇u′

(10.33)

gij dxi dxj = a2 du2 + γAB dω A dω B

with (ω 1 , ω 2 ) being local coordinates on S0 . Now, Γ′ is given by an equation of the form ω A = ω A (u). Let us calculate u′ (p) as an integral along Γ′ instead of Γ. So, denoting the arc length on Γ′ by s′ and as ds′2 = gij (u)dxi (u)dxj (u) = a2 du2 + γAB (u)dω A (u)dω B (u), we obtain: H # # 1 HH ′ ′ ′ ∇N ′ u ds = ds′ u (p) = ′H Γ′ Γ′ a Γ′ (1 H # u ' # u H 1 dω A dω B 2 HH a HH 2 a (10.34) = + γ du ≥ du. AB H ′ ′ ′H du du 0 a 0 a Γ′ Γ That is, we have found that # u # u a a ′ ′ (10.35) | du ≤ u (p) ≤ | du ′ Γ ′ Γ a a 0 0 and therefore it is # u # u -a. -a. (10.36) inf ′ du ≤ u′ ≤ sup ′ du. a 0 Su a 0 Su It follows

′

(10.37)

oscSu u ≤

#

u

oscSu

-a.

du. a′ Then with the assumptions of the lemma, one obtains

(10.38) Also, recalling that (10.39) As it is

dr du

c−1 u ≤ u′ ≤ cu.

= 2r atrθ and considering the assumptions, it is

1 1 dr′ ≤ ′ = a′ r′ trθ′ ≤ 4, 4 du 2

inequality (10.38) implies (10.40)

0

1 dr 1 ≤ = artrθ ≤ 4. 4 du 2

r′ |u′ =0 = r |u=0 = r0 , c−1 r′ ≤ r ≤ cr′ .

Next, we want to obtain a more precise comparison of Su and Su′ ′ . For this, we construct a diffeomorphism of U ′ → I ′ × S0 , using the integral curves of N ′ . The metric in U ′ can be expressed as (10.41)

′ dω A dω B . gij dxi dxj = a′2 du′2 + γAB

10.3. COMPARING NULL FRAMES

253

Then the surface Su is given by an equation of the form u′ = f (u, ω), and the metric induced on Su writes as ∂f ∂f . ∂ω A ∂ω B It is easy to see then that the determinant of γ can be expressed as follows: ' ( ′ ′2 ′AB ∂f ∂f det γ = det γ 1 + a γ . ∂ω A ∂ω B This implies that ′ γAB = γAB + a′2

(10.42) Area(Su ) =

#

S0

Also, it is

$

det γ

%1 2

2

d ω≥

#

S0

$

%1 det γ ′ (f (u, ω), ω) 2 d2 ω = Area(Su′ ′ ).

∂det γ ′ = a′ trθ′ detγ ′ > 0. ∂u

(10.43) Set

u′m (u) = inf u′ , Su

u′M (u) = sup u′ . Su

Therefore, it is det γ ′ (f (u, ω), ω) ≥ det γ ′ (u′m (u), ω)

and consequently, we have

Area(Su ) ≥ Area(Su′ ′m (u) ).

Similarly, interchanging the roles of u′ and u, denoting u, um (u′ ) = inf ′ Su ′

uM (u′ ) = sup u, Su′ ′

and using the fact that trθ > 0, it directly follows that Area(Su′ ′ ) ≥ Area(Sum (u′ ) ).

Next, we shall show that (10.44)

um (c′ ) = c ⇒ u′M (c) = c′ .

So, let p ∈ Sc′ ′ achieve inf S ′ ′ u = c. Then the following holds: c

∀q ∈ Sc : u′ (q) ≤ c′ .

This is seen by the following arguments leading to a contradiction: If there exists a q ∈ Sc such that u′ (q) > c′ , then q belongs to the exterior of Sc′ ′ . Note that ∇N ′ u = cos ϕ∇N u =

du cos ϕ = a−1 cos ϕ > 0. ds

254

10. THE LAST SLICE

Thus, following the integral curve of N ′ through q inward, we obtain a point q ′ ∈ Sc′ ′ with u(q ′ ) < c. But this contradicts the definition of c. That is, (10.44) is shown. This yields (10.45)

Area (Su′ ′

M (u)

) ≥ Area (Su ) ≥ Area (Su′ ′m (u) ),

that is, r′ (u′M (u)) ≥ r(u) ≥ r′ (u′m (u)).

(10.46)

Therefore, we also have

sup | r − r′ | ≤ oscSu r′ .

(10.47)

Su

We shall have to switch between the vectorfields N and N ′ . In view of this fact, we express N ′ with respect to N and a vectorfield Y tangential to Su , and respectively for N . That is, we have N ′ = cos ϕ N + Y,

(10.48)

N = cos ϕ N ′ + Y ′

with Y and Y ′ being vectorfields tangential to Su and Su′ ′ respectively. And it is | Y | = | Y ′ | = sin ϕ.

(10.49) Recall, here,

/j a + ∇ i N j = N i ∇ /j log a + ∇i Nj θij = ∇i Nj − Ni ∇N Nj = a−1 Ni ∇

in view of the fact that

∇ /j log a = a−1 ∇ /j a.

So, with respect to the vectorfields N and N ′ it is: /j log a, ∇i Nj = θij − Ni ∇

′ ∇i Nj′ = θij − Ni′ ∇ / j′ log a′ .

Therefore, differentiating N ′ in (10.48) yields:

(10.50)

′ − Ni′ ∇ / j′ log a′ = ∇i (Nj cos ϕ) + ∇i Yj ∇i Nj′ = θij = ∇i Yj + cos ϕ∇i Nj + Nj ∇i cos ϕ /j log a). = ∇i Yj + Nj ∇i cos ϕ + cos ϕ(θij − Ni ∇

Note that, by (10.48), the following identities hold:

θ′ (N, N ) = θ′ (Y ′ , Y ′ ) (N, ∇ / ′ log a′ ) = ∇Y ′ log a′ (∇N Y, N ) = −(Y, ∇N N ) = ∇Y log a.

In view of this, we contract equation (10.50) with N i N j and obtain θ′ (N, N ) − cos ϕ∇Y ′ log a′ = ∇N cos ϕ + ∇Y log a (10.51) ∇N cos ϕ = θ′ (Y ′ , Y ′ ) − ∇Y log a − cos ϕ∇Y ′ log a′ .

10.3. COMPARING NULL FRAMES

Considering (10.49) and letting Yˆ = (10.51) as (10.52) sin2 ϕ θ′ (Yˆ′ , Yˆ′ ) < => ?

= 12 trθ′ +θˆ′ (Yˆ′ ,Yˆ′ )

that is, (10.53)

∇N ϕ +

'

Y |Y |

as well as Yˆ′ =

255 Y′ |Y ′ | ,

we write

= − sin ϕ ∇N ϕ + sin ϕ ∇Yˆ log a + sin ϕ cos ϕ ∇Yˆ′ log a′ ,

( 1 ′ ˆ′ ˆ′ ˆ′ trθ + θ (Y , Y ) sin ϕ = cos ϕ∇Yˆ′ log a′ + ∇Yˆ log a. 2

Next, we denote λ′ = ∇N ′ r′ and we remind ourselves of the fact r′ λ′ = ∇N ′ r′ = ′ a′ trθ′ . 2a Then, we have ∇N r′ = λ′ cos ϕ. Now, we set γ = r′ sin ϕ and we use r′ as a parameter along the integral curves of N . One concludes ' ( 1 ′ ′ ′ dγ (a trθ − a′ trθ′ ) + θˆ′ (Yˆ′ , Yˆ′ ) γ |≤ r′ (| ∇ / ′ log a′ | + | ∇ / log a |). |λ ′+ dr 2a′

Integrating this, we obtain the following, considering that γ(r0 ) = 0 as well as the assumptions and (10.40): D ( C # r′ ' 1 1 ′ ′ ′ ′ ′ ′ ˆ | a trθ − a trθ | + | θ | dr γ ≤ exp 2a′ λ′ r0 ( ' # r′ %1 $ (10.54) × r′ | ∇ / log a | + | ∇ / ′ log a′ | ′ dr′ λ r0 . r0 ≤ c(A + A′ ) 1 − ′ . (10.55) r That is, we have r0 . 1 (10.56) sin ϕ ≤ c(A + A′ ) 1 − ′ . r r′ Observe that if Iˆ ̸= I, then Iˆ has a boundary point u1 ∈ I. And at some point p ∈ Su1 , it is ϕ(p) = π2 . But if A and A′ are sufficiently small, this contradicts the estimate. So, it is Iˆ = I. Next, for any vector X tangent to Su at a point it is (10.57)

X = X ′ + (X, Y )N ′

with X ′ being the projection of X onto Su′ ′ through that point. Then, we have / log a′ = ∇ / ′X ′ log a′ + (X, Y )∇ /N ′ log a′ ∇ /X log a′ = X∇ = X ′∇ / ′ log a′ + (X, Y )∇ /N ′ log a′ ,

256

10. THE LAST SLICE

and therefore, as | (X, Y ) | ≤ | X || Y |, it is

|∇ / log a′ | ≤ | ∇ / ′ log a′ | + | Y || ∇N ′ log a′ | .

Together with the estimate for sin ϕ, the inequality (10.40) and the assumptions, we conclude that 3

|∇ / log a′ | ≤ c(A′ + B ′ )r− 2 .

(10.58)

Thus, we obtain (considering the assumption on the diameter of Su ): ( ' supSu a′ 1 ≤ c(A′ + B ′ )r− 2 . (10.59) log ′ inf Su a

It is

(10.60)

log

'

supSu a inf Su a

(

1

≤ cAr− 2 .

Finally, the oscillation of u′ on Su in inequality (10.28) is estimated as follows: ' ( # u 1 a ′ (10.61) oscSu u ≤ oscSu ′ du ≤ c(A + A′ + B ′ ) | u | r− 2 . a 0 To estimate r′ on Su , note that

J = [inf u′ , sup u′ ] Su

Su

as well as | J | = oscSu u′ .

We observe that ′

oscSu r =

#

Eventually, by (10.47), this yields (10.62)

J

dr′ ′ du ≤ oscSu u′ . du′

sup | r − r′ | ≤ oscSu r′ ≤ oscSu u′ . Su

This proves the lemma. Also, let us show here the following lemma, which we shall use in this chapter. Lemma 10. Let H, S0 , u and U be as in lemma 9. Suppose that for all u ∈ I, it is: diam(Su ) ≤ 2πr, a ≤ 2, r r inf trθ ≥ 2, sup trθ ≤ 2, Su 2 Su 2

and 3

sup | trθ − trθ | ≤ cr− 2 Su

3

sup sup r 2 | ∇ / log a | ≤ c. u∈I Su

10.3. COMPARING NULL FRAMES

257

Let V and F be k-covariant tensorfields on H tangent to the surfaces Su , verifying in U the equation ∇ /N V + λ0 trθ V = F

with λ0 being a nonnegative real. Set, for any given p ≥ 2: 2 2 λ1 = 2λ0 − = λ2 − . p p Then, it is for all u ∈ I: H# u H( ' H H λ2 λ2 λ2 ′ ′H H ∥ − ∥p,S (u) ≤ c r − ∥V− ∥ p,S (0) + H r − ∥ F− ∥ p,S (u )du H . r −V 0

Proof. First, compute

d λ2 du (r

−V ∥ − ∥p,Su

)p :

%p d d −1 d $ λ2 ∥ − ∥p,S = −V ∥ − ∥pp,S rλ2 p + rλ2 p | V |pp,S r −V A du du du d (10.63) + rλ2 p A−1 | V |pp,S . du The first term on the right-hand side of (10.63), we calculate as follows, r atrθ: recalling that ∇N r = 2a d λ2 p 1 = −V ∥ − ∥ pp,S λ2 prλ2 p−1 a∇N r = −V ∥ − ∥ pp,S λ2 prλ2 p atrθ r du 2 For the second term on the right-hand side of (10.63), we use the fact that # d A(u) = a trθ dµγ = 4πr2 atrθ = A atrθ, du Su −V ∥ − ∥ pp,S

and obtain

rλ2 p | V |pp,S

d −1 dA A = −A−2 rλ2 p | V |pp,S = −A−1 rλ2 p | V |pp,S atrθ du du = −rλ2 p −V ∥ − ∥pp,S atrθ.

Now, we focus on the third term on the right-hand side of (10.63): # d (ap | V |p−1 ∇N | V | + a tr θ | V |p ) rλ2 p A−1 | V |pp,S = rλ2 p A−1 du #Su = rλ2 p A−1 a | V |p−1 (p∇N | V | + tr θ | V |). Su

As, writing the equation for λ1 in the form 1 = − λ21 p + λ0 p, this reads as # λ2 p −1 a | V |p−1 (p∇N | V | + trθ | V |) r A Su # = rλ2 p A−1 pa | V |p−1 (∇N | V | + λ0 trθ | V |) Su # λ1 p λ2 p −1 − | V |p atrθ. r A 2 Su

258

10. THE LAST SLICE

Therefore, as λ22 p − 1 = λ0 p − 1 = λ21 p , equation (10.63) becomes ( ' %p λ2 p d $ λ2 ∥ − ∥ p,S = ∥ − ∥ pp,S atrθ r −V − 1 rλ2 p −V du 2 # λ2 p −1 +r A pa | V |p−1 (∇N | V | + λ0 trθ | V |) Su # λ1 p λ2 p −1 | V |p atrθ r A − 2 Su # λ2 p −1 pa | V |p−1 (∇N | V | + λ0 trθ | V |) =r A Su # λ1 p λ2 p −1 r A − (10.64) | V |p (atrθ − atrθ). 2 Su

Let us consider the first term on the right-hand side of equation (10.64). # λ2 p −1 pr A a | V |p−1 | ∇N V + λ0 trθV | Su # p−1 1 λ2 p −1 = pr A a p | ∇N V + λ0 trθV | a p | V |p−1 Su

≤ pr (10.65)

λ2 p

≤ cpr

A

−1

'#

Su

λ2 p

A

−1

a | ∇N V + λ0 trθV |

| F |p,S | V

p

|p−1 p,S =

cpr

λ2 p

−1

(1 ' # p

Su

p

a|V |

−F ∥ − ∥ p,S −V ∥ − ∥ p−1 p,S ,

( p−1 p

− p−1

the last equality holding in view of A−1 = A p A p . Now, in view of our assumptions and the last estimate, we obtain from (10.64): %p d $ λ2 ∥ − ∥ p,S ≤ cprλ2 p − ∥ F1 − ∥ p,S −V ∥ − ∥ p−1 r −V p,S du # λ1 p λ2 p −1 (10.66) +c | V |p | atrθ − atrθ | . r A 2 Su As the left-hand side is

% d $ λ2 ∥ − ∥ p,S , r −V du the following inequality holds in view of the assumption supSu | trθ − trθ |≤ 3 cr− 2 : % 3 d $ λ2 ∥V− ∥ p,S ≤ crλ2 − ∥ F− ∥ p,S + cr− 2 rλ2 − ∥V− ∥ p,S . r − (10.67) du ∥V− ∥ p−1 prλ2 p−1 − p,S

1

∥ − ∥ p,S with respect Integrating (10.67) over u, and neglecting cr− 2 rλ2 −V to the left-hand side after the integration, yields for all u ∈ I: H# u H( ' H H λ2 λ2 λ2 ′ ′H H ∥ − ∥ p,S (u) ≤ c r −V ∥ − ∥ p,S (0) + H r −F ∥ − ∥ p,S (u )du H , (10.68) r −V 0

which proves the lemma.

10.4. MAIN THEOREM

259

In view of the proof of the main theorem, we remind ourselves, here, of the Gronwall lemma: Lemma 11. Let Φ ≥ 0 and Ψ ≥ 0 be continuous functions on t0 ≤ t ≤ t1 such that # t Ψ(s)Φ(s) ds Φ(t) ≤ K0 + K1 t0

on t0 ≤ t ≤ t1 , where K0 and K1 are positive constants. Then on t0 ≤ t ≤ t1 it is K1

Φ(t) ≤ K0 e

!t

t0

Ψ(s)ds

.

10.4. Main Theorem The purpose of this section is to estimate on Ht∗ the curvature and geometric components of the foliation by the function u by corresponding quantities with respect to the background foliation given by u′ . That is, we assume estimates for the curvature components relative to a null frame defined by the level surfaces of the function u′ on Ht∗ . This function has been constructed in the previous chapters, and we have shown that the curvature components with respect to this foliation are small. The extension of this function u′ onto the last slice Ht∗ gives the surfaces St∗ ,u′ on Ht∗ . We shall use lemma 9 to compare the curvature and geometrical quantities with respect to the two foliations. In the ‘Introduction’ we explained that the trace lemma yields L4 bounds on the curvature components in St∗ ,u , and that in order to apply this lemma, we have to first overcome the following difficulty: By assumption, the curvature components of the background foliation lie in H 1 (U ′ ). (Notation given in the ‘Introduction’ and in this chapter.) How can they be bounded in H 1 (U )? The solution is as follows: As the transformation formulas between the two foliations are shown to be bounded, we obtain the curvature components to be in H 1 (U ), which allows us to apply the trace lemma to derive for these components to lie in L4 (Su ). We re-emphasize that this is a fundamental difference to the situation in [19], where the curvature components lie in L∞ and therefore can be bounded by straightforward estimates also in the surfaces of the foliation by the function u. In our case, as the curvature components of the background foliation are not in L∞ , but only in H 1 , we have to proceed as explained above. Let us recall that as a consequence from controlling one less derivative of the curvature components, the derivatives of these are not bounded in the surfaces St,u . At a certain point of the proof, we will have to transform from one null frame into the other, where the transformations will involve first derivatives. The calculations for these transformation formulas are carried out in [19]. In [19], the authors showed their estimates to hold for higher weights. In

260

10. THE LAST SLICE

our different setting, we shall obtain the main estimates for lower weights in L4 -norms. They will later on be integrated along the integral curves of N . Actually, the theorem holds for Lp -norms with 2 < p ≤ 4. The upper bound 4 is given by the trace lemma, whereas the lower bound 2 comes from the fact that at certain levels of the proof, in the surfaces St∗ ,u′ , we need to bound the L∞ -norms of the quantities we estimate, and we only have them in Lp up to their first derivatives, we have to require p > 2. This p 4→ L∞ for is necessary in view of the fact that for the said surfaces: Wm mp > 2. Let us already mention, here, that the estimates of the higher derivatives in the next subsection will be in L2 (Ht∗ ). In the following theorem we shall use dimensionless L4 -norms. We recall, here, the dimensionless Lp -norms for a tensorfield t on the surface Ss , for p ≥ 2: (10.69) − ∥ t(s)− ∥ Lp (S) =

'

1 A(Ss )

#

Ss

|t

|pγ

dµγ

(1

p

− p1

= (4πr2 )

∥ t ∥Lp (Ss ,γ) .

Theorem 14. We make the following sets of assumptions: S0 : 1 < r02 −K∥ ∥ −4,S0 < 2, 2 1 r0 r0 inf trχ > , sup trχ < 2. 2 S0 2 2 S0 S1 : 5

∥ / tr χ − ∥ 4,S0 = | r02 ∇ / trχ |4,S0 ≤ ϵ0 . y(0) = r02 −∇ Let u′ be a smooth function without critical points, defined in a tubular neighbourhood U ′ of S0 , including the whole exterior of S0 , extending in the interior up to a level surface Su′ ′ of u′ with area A′0 = 4π( r40 )2 . Also, let u′ vanish on S0 and tend to infinity at infinity. It is 4 U′ = Su′ ′ u′ ∈(u′0 ,∞)

with

Su′ ′

being the level sets of

u′ .

As

r′

is defined by

suppose in addition U0′ and U1′ to hold: U′0 : For all u′ ∈ (u′0 , ∞), diam(Su′ ′ ) ≤ 2πr′ ,

1 ≤ a′ ≤ 2. 2

r′

=

'

Area(Su′ ′ ) 4π

(1 2

,

10.4. MAIN THEOREM

U′1 : For all u′ ∈ (u′0 , ∞), inf ′ Su ′

r′ ′ 1 trθ ≥ , 2 2

r′ ′ trθ ≤ 2, 2

sup Su′ ′

3

3

sup | trχ′ − trχ′ | ≤ cr′− 2 , Su′ ′

3

sup | χ ˆ′ | ≤ cr′− 2 1

sup r′ 2 | ∇ / ′ log a′ | ≤ c,

261

Su′ ′

sup r′ 2 τ−′ | ∇N ′ log a′ | ≤ c

Su′ ′

Su′ ′

1

with τ−′ = (1 + u′2 ) 2 . Denote by (δ ′ , ϵ′ , η ′ ) the decomposition of the second fundamental form of H and by (α′ , β ′ , ρ′ , σ ′ , β ′ , α′ ) the null decomposition of the spacetime curvature on H relative to the surfaces Su′ ′ . And by ′

K0∞

we denote the maximum of the three quantities: 3

sup r′ 2 | δ ′ |, U′

3

sup r′ 2 | ϵ′ |, U′

′1

sup r′ τ−2 | ηˆ′ | . U′

2: Also, denote by R[1] ′

2 =R2+R2 R[1] 0 1 ′

′

′

with R02 and R12 being the quantities: # # # # ′2 ′2 ′ 2 ′2 ′ 2 ′2 ′ 2 R0 = τ− | α | + r |β | + r |ρ | + r′2 | σ ′ |2 ′ ′ ′ ′ U U U U # # ′2 ′ 2 ′2 ′ 2 + r |β | + r |α | ′

′

U′

and

′2

R1 =

#

U′

#

U′

τ−′2 r′2

′ ′ 2

#

|∇ / α | + # r′4 | ∇ / ′ σ ′ |2 +

′4

′ ′ 2

#

r |∇ / β | + # r′4 | ∇ / ′ β ′ |2 +

U′

U′

r′4 | ∇ / ′ ρ′ |2

r′4 | ∇ / ′ α′ |2 # # # ′4 ′ 2 ′2 ′2 ′ 2 + τ− | ∇ /N ′ α | + τ− r | ∇ /N ′ β | + r′4 | ∇ /N ′ ρ′ |2 U′ U′ U′ # # # ′4 ′ 2 ′4 ′ 2 + r |∇ /N ′ σ | + r |∇ /N ′ β | + r′4 | ∇ /N ′ α′ |2 . +

U′

U′

U′

U′

U′

U′

262

10. THE LAST SLICE

In addition, suppose that we have: H0 : ′ ′2 K0∞ ≤ ϵ0 , R[1] ≤ ϵ0 .

Then, if ϵ0 is sufficiently small, the inverse lapse problem (10.2), (10.3) has a solution that is global in the exterior of S0 and extends in the interior at least up to a level surface Su0 of u of area A0 = 4π( r40 )2 . .1 u) 2 Let also r(u) = Area(S . 4π Then for all u ∈ (u0 , ∞) the level surfaces Su of the function u satisfy (10.70)

(10.71) (10.72)

1

2 2) −Kr ∥ − 1∥ −4,Su ≤ cr− 2 (y(0) + K0∞ + R[1] 1 r ′ ′2 − ∥ trχ − 1∥ ) −4,Su ≤ cr− 2 (y(0) + K0∞ + R[1] 2

3

5

′

′

5

2 ), r2 − ∥ χ∥ ˆ−4,S + r 2 −∇ ∥ /χ ˆ− ∥ 4,S + r 2 −∇ ∥ / trχ∥ −4,S ≤ c(y(0) + K0∞ + R[1] ′

′

that is

2) ˆ |4,S + | r2 ∇ (10.73) | rχ /χ ˆ |4,S + | r2 ∇ / trχ |4,S ≤ c(y(0) + K0∞ + R[1] ′

′

with c a numerical constant. Furthermore, we have (10.74) that is (10.75)

3

5

2 ), r 2 −ζ∥ ∥ −4,S + r 2 −∇ ∥ /ζ∥ −4,S ≤ c(y(0) + K0∞ + R[1] ′

′

2 ). /ζ |4,S ≤ c(y(0) + K0∞ + R[1] | rζ |4,S + | r2 ∇ ′

′

Proof. First, we are going to give a set of properties of Su . Then we will show that they are valid on Su for u ≥ 0. Finally, the statements of the theorem will follow. The first set of properties of Su is P0 (u): 1 ∥ −4,Su < 2, < r2 −K∥ 2 r r 1 inf trθ > , sup trθ < 2, Su 2 2 Su 2 3

sup | trθ − trθ |< r− 2 , Su

1 inf a > , Su 2

−1

sup | θˆ |< r−1 τ− 2 , Su

sup a < 2, Su 3

sup | ∇ / log a |≤ r− 2 . Su

The following property is: P1 (u): 5

/ tr χ |4,S (u) = r 2 −∇ ∥ / tr χ∥ −4,S (u) < 1. y(u) =| r2 ∇

10.4. MAIN THEOREM

263

Also, denote by P the property P0 and P1 . In step 1, we will show P to hold for u = 0, whereas in step 2, it will be proven by the method of continuity, that P in fact holds for u ≥ 0. ′2 ≤ ϵ0 , and by the trace lemma (see Observe that in view of H0 , i.e. R[1] ′ ′ lemma 3), for all u ∈ (u0 , ∞) in the surfaces Su′ ′ the following holds: ′3

′1

∥ ′− ∥ 4,S ′ ′ ≤ cϵ0 , r′ τ−2−α

r′2 τ−2−β ∥ ′− ∥ 4,S ′ ′ ≤ cϵ0

u

′ 52

′

′ 52

′

u

r −ρ ∥ − ∥ 4,S ′ ′ ≤ cϵ0 ,

′ 52

′

′ 52

′

r −σ ∥ − ∥ 4,S ′ ′ ≤ cϵ0

u

r −β ∥ − ∥ 4,S ′ ′ ≤ cϵ0 ,

u

r −α ∥ − ∥ 4,S ′ ′ ≤ cϵ0 .

u

u

Step 1: u = 0: Let us first focus on S0 and show that P is true for u = 0. As S0′ = S0 , we already have: − 32

sup(| δ |, | ϵ |) ≤ ϵ0 r0

,

S0

− 52 − 52

( since τ− = 1 on S0′ ) ,

S0

−ρ ∥ − ∥ 4,S0 ≤ c ϵ0 r0

−β ∥ − ∥ 4,S0 ≤ c ϵ0 r0

sup | ηˆ | ≤ ϵ0 r0−1

,

− 52

−σ ∥ − ∥ 4,S0 ≤ c ϵ0 r0

,

.

ˆ a and The parts of P (0) that have to be shown are those involving θ, ∇ / log a. To estimate them, first, recall the Codazzi equation (10.14) 1 1 div /χ ˆ− ∇ / tr χ + ϵ · χ ˆ − ϵtrχ = −β. 2 2

Now, we consider the assumption on the Gauss curvature of S0 as well as on y(0) and we apply the 2-dimensional elliptic estimates to (10.14) on S0 to obtain | rχ ˆ |4,S (0)+ | r2 ∇ /χ ˆ |4,S (0) ≤ cϵ0 ,

(10.76) i.e.

3

(10.77)

5

r 2− ∥χ ˆ− ∥ 4,S (0) + r 2 − ∥∇ /χ ˆ− ∥ 4,S (0) ≤ cϵ0 .

This can be seen as follows: we have, in view of the 2-dimensional elliptic estimates: # # 4 −4 4 |∇ /χ ˆ | +r | χ ˆ| ≤c | div /χ ˆ |4 . S0

S0

Therefore, we calculate # # 5 3 4−2 4 4−2 4 r2 |∇ /χ ˆ | +r 2 |χ ˆ| ≤c S0

S0

5

r 2 4−2 | div /χ ˆ |4 .

264

10. THE LAST SLICE

So, we obtain 3

5

| rχ ˆ |4,S (0) + | r2 ∇ /χ ˆ |4,S (0) = r 2 − ∥ χ∥ ˆ−4,S (0) + r 2 −∇ ∥ / χ∥ ˆ−4,S (0) 5

≤ cr 2−div ∥ / χ∥ ˆ−4,S (0) 5

1

= cr 2 A(S0 )− 4 | div /χ ˆ |4 (0) 5$ ˆ |∞,S0 ≤ Cr 2 −∇ ∥ / tr χ∥ −4,S (0) + | ϵ |∞,S0 · | χ % + | ϵ |∞,S0 · | trχ |∞,S0 +−β ∥ − ∥4,S (0) .

Additionally considering the estimates from above for the components of the second fundamental form and the curvature on S0 , this yields inequality (10.76). In order to obtain an estimate for the supremum of | χ ˆ |, we will need the Sobolev inequalities discussed earlier. Especially, we shall use, here and furtheron in this proof, the following Sobolev inequality: '# (1 4 − 12 4 4 4 | ξ | +r | ∇ /ξ| . (10.78) sup | ξ | ≤ cr Su

Su

Going back to (10.76) and (10.77), the Sobolev inequality on S0 finally yields −3

ˆ |≤ cϵ0 r0 2 . sup | χ

(10.79)

S0

Then, as it is

θAB = χAB + ηAB ,

we conclude (10.80)

sup | θˆ | ≤ c(sup | χ ˆ | + sup | ηˆ |) S0

S0

S0

− 32

≤ cϵ0 (r0 (10.81) (10.82) And from

−1

+ r0−1 τ− 2 )

−1

≤ cϵ0 r0−1 τ− 2 ≤ cϵ0 r0−1

(since τ− = 1 on S0′ ). − 32

∥ trχ − trχ ∥L∞ (S) ≤ cϵ0 r0 as well as from the result for δ, we obtain (10.83)

−3

sup | trθ − trθ | ≤ cϵ0 r0 2 . S0

Now, we discuss the system (10.6, 10.7) on S0 : div / ζ = f − f¯, curl / ζ = σ − ηˆ ∧ χ. ˆ

From (10.4) and (10.9), write (10.84)

f = −ρ −

1 1 |χ ˆ |2 − ηˆ · χ ˆ − δtrχ. 2 2

10.4. MAIN THEOREM

265

Thus, in view of the above, we obtain −5

(10.85) Therefore,

∥ − ∥4,S0 ≤ cϵ0 r0 2 . | f |4,S0 ≤ cϵ0 r0−2 , i.e. −f 5

| r2 div / ζ |4,S (0) = r 2−div ∥ / ζ∥ −4,S (0) ≤ cϵ0 , 5

∥ / ζ∥ −4,S (0) ≤ cϵ0 . | r2 curl / ζ |4,S (0) = r 2−curl

We apply the 2-dimensional L4 -estimates to obtain 3

(10.86)

5

| rζ |4,S (0) + | r2 ∇ ∥ / ζ∥ −4,S (0) / ζ |4,S (0) = r 2−ζ∥ ∥ −4,S (0) + r 2−∇ ≤ cϵ0 .

And thus, it is by the Sobolev inequality, −3

(10.87)

sup | ζ | ≤ cϵ0 r0 2 . S0

Then, as we have (recalling from previous chapters) /A a + ϵA , ζA = a−1 ∇

that is,

ζ=∇ / log a + ϵ,

we conclude (10.88)

−3

sup | ∇ / log a | ≤ cϵ0 r0 2 . S0

Let us state, here, also the equation for △ / log a, which follows directly. We do not need it in this proof, but nevertheless, it shall be used at the end of this chapter. div /ζ=△ / log a + div / ϵ. Also, it follows from (10.88), that 1 inf a > , sup a < 2. S0 2 S0 So, the property P has thus been shown to hold at u = 0. Step 2: u ≥ 0: We shall use the method of continuity to prove this step. The local existence theorem was discussed in a separate chapter. The property P remains true in an open interval containing 0, by the local existence theorem and continuity. There are two possibilities, either the inverse lapse problem (10.2, 10.3) has a global solution in the exterior of S0 and P holds for all u ≥ 0, or there exists a u1 > 0 such that P is true for all u ∈ [0, u1 ) but that P is false at u = u1 . We will show that the second case is impossible. Denote 4 Su . I = [0, u1 ), U = u∈I

266

10. THE LAST SLICE

Now, P is true in I. Therefore, we can apply lemma 9 in I to obtain the following: (10.89) (10.90)

c−1 u ≤ u′ ≤ cu,

c−1 r′ ≤ r ≤ cr′ , 1

oscSu u′ ≤ c | u | r− 2 , 3

sin ϕ ≤ c | u | r− 2 .

(10.91)

In the sequel, we will use the transformation formulas for first derivatives between the two null frames with respect to the foliations by u′ and u. They depend on θ′ and θ (in L∞ ). In view of θ′ and θ, which appear in these formulas, note that: For θ′ we have L∞ -bounds in Su′ ′ by the hypotheses U ′ , whereas for θ the L∞ -bounds in Su are assured by the continuity argument for P0 for all u in I. In addition to lemma 9, these transformation formulas for first derivatives also involve the first derivatives of the corresponding terms in lemma 9. The derivatives of cos ϕ and Y are easily shown to be small. The exact calculation can be found in [19] at the beginning of the proof of proposition 14.0.2. Therefore, all the terms in the transformation formulas for first derivatives are sufficiently bounded. Next, we are going to express the components δ, ϵ, η of the second fundamental form k of H relative to N in terms of the components relative to N ′ . For this purpose, recall, here, the formulas (10.48): N ′ = cos ϕN + Y,

N = cos ϕN ′ + Y ′ .

Let us compute it explicitly for δ. The remaining components are calculated similarly. δ = kNN = kij N i N j = kij (N ′i cos ϕ + Y ′i )(N ′j cos ϕ + Y ′j ) = cos2 ϕδ ′ + cos ϕϵ′ (Y ′ ) + cos ϕϵ′ (Y ′ ) + η ′ (Y ′ , Y ′ ) = cos2 ϕδ ′ + 2 cos ϕϵ′ (Y ′ ) + η ′ (Y ′ , Y ′ ). Therefore, considering the assumptions as well as the results (10.89) and (10.91), this yields in U : (10.92)

3

′

sup r 2 (| δ |, | ϵ |) ≤ cK0∞ , U

(10.93)

1

′

sup rτ−2 | ηˆ | ≤ cK0∞ , U

We are also going to express the components of the spacetime curvature tensor on H relative to l = T + N , l = T − N in terms of the components relative to l′ = T + N ′ , l′ = T − N ′ . Observe that 1 ′ (l + l′ + cos ϕl′ − cos ϕl′ ) = T + cos ϕN ′ , 2 1 ′ (l + l′ − cos ϕl′ + cos ϕl′ ) = T − cos ϕN ′ . 2

10.4. MAIN THEOREM

267

At this point, we are going to show that the curvature components, i.e. (10.94)

α, β, α, β, ρ, σ,

lie in L4 (Su ) uniformly in u,

for all u ∈ I.

In view of the fact that we do not have any L∞ -bounds on the corresponding quantities of the background foliation (as we have for the second fundamental form above), we have to proceed in a different way. First, we are going to show that the curvature components and their first derivatives with respect to the u-foliation are in L2 (U ). Then by the trace lemma, (10.94) will follow. (Note that this trace lemma requires also the first derivatives of the curvature components to be in L2 (U ).) The argument is as follows. By assumption, the terms α′ , β ′ , α′ , β ′ , ρ′ , σ ′ as well as their derivatives ′ /N ′ are in L2 (U ′ ). We now need the transformation formulas (also ∇ / and ∇ for first derivatives) between the null frames with respect to u′ and u. The components α, β, α, β, ρ, σ relative to the u-foliation can be expressed by α′ , β ′ , α′ , β ′ , ρ′ , σ ′ relative to the u′ -foliation. Now, we calculate ρ as an example. For the other curvature components, the procedure is similar. 1 1 1 ρ = W3434 = Wαβγδ eα3 eβ4 eγ3 eδ4 = Wαβγδ lα lβ lγ lδ 4 4 4 1 = Wαβγδ (T α + N α )(T β − N β )(T γ + N γ )(T δ − N δ ) 4 1 = Wαβγδ ([T α + N ′α cos ϕ] + Y ′α )([T β − N ′β cos ϕ] − Y ′β ) 4 ×([T γ + N ′γ cos ϕ] + Y ′γ )([T δ − N ′δ cos ϕ] − Y ′δ ) ' ( 1 ′α 1 ′α ′α ′α ′α [l + l + cos ϕl − cos ϕl ] + Y = Wαβγδ 4 2 ' ( 1 ′β ′β ′β ′β ′β × [l + l − cos ϕl + cos ϕl ] − Y 2 ' ( 1 ′γ ′γ ′γ ′γ ′γ [l + l + cos ϕl − cos ϕl ] + Y × 2 ( ' 1 ′δ ′δ ′δ ′δ ′δ × [l + l − cos ϕl + cos ϕl ] − Y 2 1 1 = cos2 ϕρ′ − cos ϕ(β ′ (Y ′ ) + β ′ (Y ′ )) + α′ (Y ′ , Y ′ ) + α′ (Y ′ , Y ′ ). 4 4 By lemma 9, the transformation terms on the right-hand side are suitably bounded. Therefore, considering the assumptions as well as the results (10.89) and (10.91), we conclude: (10.95)

α, β, α, β, ρ, σ

lie in L2 (U ′ )

and therefore in L2 (U ).

To estimate the derivatives ∇ / and ∇ /N of these components by the corresponding terms of the background foliation, we use the transformation

268

10. THE LAST SLICE

formulas for first derivatives discussed above. Then we obtain by assumptions: / β, ∇ / ρ, ∇ /σ ∇ / α, ∇ / β, ∇ / α, ∇

/N β, ∇ /N α, ∇ /N β, ∇ /N ρ, ∇ /N σ ∇ /N α, ∇

lie in L2 (U ′ ) and therefore in L2 (U ).

(10.96)

/ as well as ∇ /N So, we have shown α, β, α, β, ρ, σ and their derivatives ∇ to be bounded in L2 (U ). That is, we have 2. R2[1] ≤ cR[1] ′

(10.97)

Now, we can apply the trace lemma to obtain (10.94). That is, we apply lemma 3. (The general trace lemma 2 is given in the chapter about 3d-results.) We will give now the computation for α. The other terms are estimated similarly. In lemma 3 we take F = rα and conclude: '# (1 '# (1 '# (1 4 2 2 8 4 2 2 4 2 r |α| ≤ c1 r |α| + c2 r |∇ /α| Su

H

+ c3

'#

H

r4 | ∇ /N α |2

(1 2

H

.

Observe that the right-hand side is bounded by assumption. Summarizing, we obtain the following estimates for all u in I: 5

2 r 2− ∥ α− ∥ 4,Su ≤ cR[1]

(10.98)

′

5

′

5

′

5

′

2 ∥ β− ∥ 4,Su ≤ cR[1] r 2−

(10.99)

2 ∥ ρ− ∥ 4,Su ≤ cR[1] r 2−

(10.100) (10.101)

2 ∥ σ− ∥ 4,Su ≤ cR[1] r 2−

(10.102)

2 rτ−2− ∥ α− ∥ 4,Su ≤ cR[1]

(10.103)

2. ∥ β− ∥ 4,Su ≤ cR[1] r2 τ−2−

3

′

1

′

Next, we focus on trχ. Considering the assumption P0 , 3

sup | trθ − trθ | < r− 2 , Su

and as (10.92) guarantees 3

′

sup r 2 | δ | ≤ cK0∞ , U

we obtain by subtracting trχ from (10.104)

trχ = trθ + δ

the inequality

$ % | trχ − trχ |∞,Su ≤ c | trθ − trθ |∞,Su + | δ − δ¯ |∞,Su ,

10.4. MAIN THEOREM

269

In view of (10.92) and P0 , this is bounded and it becomes: (10.105) (10.106)

3

3

′

r 2 | trχ − trχ |∞,Su ≤ c(r 2 | trθ − trθ |∞,Su + K0∞ ) ′

≤ c′ + cK0∞

for constants c′ and c. Further, by assumption P0 , r 1 < inf trθ, Su 2 2

r sup trθ < 2, Su 2

by (10.104), and by (10.92), we obtain for a constant c: (10.107)

r | trχ |∞,Su ≤ c.

The same, of course, is implied for trχ. In the sequel, we shall use the following

Remark: 1. Note that for a given tensorfield ξ tangential to the St,u , − ∥ ξ− ∥ p,St,u

is a non-decreasing function of p. Furtheron, let /χ ˆ |4,S (u) + | rχ ˆ |4,S (u) x(u) = | r2 ∇ 5

3

= r 2−∇ ∥ / χ∥ ˆ−4,S (u) + r 2 − ∥χ ˆ− ∥ 4,S (u).

(10.108)

We apply the 2-dimensional L4 -estimates to the null Codazzi equation (10.14) on Su for all u ∈ I. Recall, the null Codazzi equation: 1 1 / trχ + ϵ · χ ˆ − ϵ trχ = −β. div /χ ˆ− ∇ 2 2 /χ ˆ |4,S (u) + | rχ ˆ |4,S (u) x(u) = | r2 ∇ 5

(10.109)

3

∥ / χ∥ ˆ−4,S (u) + r 2− ∥ χ∥ ˆ−4,S (u) = r 2−∇ 5 5$ ∥ div /χ ˆ− ∥ 4,S (u) ≤ Cr 2 −∇ ∥ / tr χ∥ −4,S (u) ≤ cr 2−

% ˆ |∞,Su + | ϵ |∞,Su · | trχ |∞,Su +−β∥ ∥ −4,S (u) . + | ϵ |∞,Su · | χ

Observe that the term involving χ ˆ on the right-hand side of (10.109) is multiplied by the small coefficient | ϵ |∞,Su , which by (10.92) and by H0 is less than ϵ0 . Therefore, taking | ϵ |∞,Su | χ ˆ |∞,Su on the left-hand side and using Sobolev inequality to estimate the supremum-norm by the 3 dimensionless L4 -norms, it can be neglected with respect to r 2− ∥ χ∥ ˆ−4,S (u) 5

and r 2−∇ ∥ / χ∥ ˆ−4,S (u). This yields

(10.110) % 5$ ∥ β− ∥ 4,S (u) + | ϵ |∞,Su · | trχ |∞,S (u) . ∥ / tr χ− ∥ 4,S (u) + − x(u) ≤ cr 2 −∇

270

10. THE LAST SLICE

To estimate the quantity | ϵ |∞,Su · | trχ |∞,Su on the right-hand side is straightforward in view of remark 1 above. Thus, the right-hand side of (10.110) can be bounded on Su . And we conclude 2 ). x(u) ≤ c(y(u) + K0∞ + R[1] ′

(10.111)

′

We obtain by the Sobolev inequality, 3

ˆ | ≤ cx(u). sup r 2 | χ

(10.112)

Su

Recall again (10.84): 1 1 |χ ˆ |2 − ηˆ · χ ˆ − δtrχ. 2 2 Considering the assumptions on the components of the second fundamental form and the curvature, we deduce in view of remark 1: $ ∥ −4,Su + | χ ˆ |2∞,Su + | ηˆ |∞,Su | χ ˆ |∞,Su −f ∥ − ∥ 4,Su ≤ c −ρ∥ % + | δ |∞,Su | trχ |∞,Su , f = −ρ −

which yields (10.113)

Thus, as

it follows

$ 5 ′ ′2 % ∥ − ∥ 4,Su ≤ c y(u) + K0∞ + R[1] . r 2−f div / ζ = f − f¯, curl / ζ = σ − ηˆ ∧ χ, ˆ 5

2 ), | r2 div / ζ |4,S (u) = r 2−div ∥ / ζ∥ −4,S (u) ≤ c(y(u) + K0∞ + R[1] ′

5

′

2 ). / ζ |4,S (u) = r 2−curl ∥ / ζ∥ −4,S (u) ≤ c(y(u) + K0∞ + R[1] | r2 curl ′

′

Applying the 2-dimensional L4 -estimates, this yields (10.114) (10.115)

z(u) = | rζ |4,S (u) + | r2 ∇ / ζ |4,S (u) 3

5

= r 2−ζ∥ ∥ −4,S (u) + r 2−∇ ∥ / ζ∥ −4,S (u)

2 ). ≤ c(y(u) + K0∞ + R[1] ′

′

And by the Sobolev inequality on Su , we conclude (10.116)

3

sup r 2 | ζ | ≤ cz(u). Su

Next, we are going to estimate | r2 F1 |4,S (u) for all u in I. In fact, what we are going to estimate is r3−F ∥ 1− ∥ 4,S (u). For, the integrating factor is r3 . This follows from equation (10.10) and lemma 10. The goal is, to find an inequality for the norm of ∇ / tr χ using this lemma. And, in fact, for λ0 = 32 ,

10.4. MAIN THEOREM

271

we obtain the exponent λ2 = 2λ0 = 3 for the integrating factor. To do so, recall equations (10.10) and (10.12) from the beginning of this chapter: 3 / tr χ) + trθ ∇ / tr χ = −F1 ∇ /N (∇ 2 with F1 = (χ ˆ + ηˆ) · ∇ / tr χ + δ∇ / tr χ + ∇ / F0 +

'

( 1 2 ¯ (trχ) + F0 + f (ζ − ϵ). 2

Observe that trθ and trχ are of order r−1 by P0 and the leading term from the point of view of decay in F1 is 12 (trχ)2 (ζ − ϵ) in (10.10), 7 being of order r− 2 by P and by assumptions. The integral over u of 5 5 | r2 F1 |4,S (u) = r 2−F ∥ 1− ∥ 4,S (u), which is associated to r 2 −∇ ∥ / tr χ∥ −4,S , 1

would yield a logarithmic term. Now, we multiply this by r 2 , the integrating factor being actually r3 , integrate it over u, allowing the resulting integral 1 1 to grow like r 2 , and divide by r 2 at the end, which yields a bound by a constant. Thus, we compute, applying what we have shown before, and in view of the assumption that property P holds in I: $ ∥ F1 − ∥ 4,S (u) ≤ cr3 | δ |∞,S (u)−∇ ∥ / tr χ− ∥ 4,S (u) r3− + (| χ ˆ |∞,S (u) + | ηˆ |∞,S (u))∥ −∇ / tr χ − ∥ 4,S (u)

(10.117)

∥ 4,S (u) + | (trχ)2 |∞,S (u)∥ −ζ − ϵ − ∥ 4,S (u) + −∇ ∥ / F0 − % −ζ − ϵ− ∥ 4,S (u) . + (| F0 |∞,S (u) + | f¯ |∞,S (u))∥

To estimate the term −∇ ∥ / F0− ∥ 4,S (u) on the right-hand side, we recall equation (10.11): 1 F0 = | χ ˆ |2 + | ζ |2 . 2 We calculate $ −∇ ∥ / F0 − ∥ 4,S (u) ≤ c | χ ˆ |∞,S (u) −∇ ∥/ χ ˆ− ∥ 4,S (u) % + 2 | ζ |∞,S (u)−∇ ∥ / ζ− ∥ 4,S (u) $ % ≤ cr−4 x(u) + z(u) $ ′ ′2 % ≤ cr−4 y(u) + K0∞ + R[1] (10.118) , where the last inequality holds in view of (10.109), (10.111) and (10.115). With the results from above, we obtain (10.119) (10.120)

$ 1 −1 ′ ′2 ∥ 1− ∥ 4,S (u) ≤ c r−1 τ− 2 y(u) + r− 2 (K0∞ + R[1] ) r3−F % 1 ′ ′2 ) . ≤ cr− 2 (y(u) + K0∞ + R[1]

272

10. THE LAST SLICE

Applying lemma 10 to equation (10.10), and considering (10.120), we obtain # u $ 1 % 1 2 2 r3−F ∥ 1− ∥ 4,S (u′ )du′ (10.121) r (u)y(u) ≤ c r (0)y(0) + #0 u $ 1 % 1 ′ ′2 r− 2 (y(u′ ) + K0∞ + R[1] )du′ . ≤ c r 2 (0)y(0) + (10.122) 0

dr < 4 and recall the Gronwall Observe that in view of P , it is 14 < du lemma 11. Noting that # u 1 1 (10.123) r− 2 du′ ≤ cr 2 (u), 0

¯ we obtain for all u ∈ I: (10.124)

$ 1 1 1 ′ ′2 % ) r 2 (u)y(u) ≤ c r 2 (0)y(0) + r 2 (u)(K0∞ + R[1]

1 ¯ Dividing by r 2 (u) yields for all u ∈ I: $ ′ ′2 % , (10.125) y(u) ≤ c y(0) + K0∞ + R[1]

in view of the fact that

for u ≥ 0

r(u) ≥ r(0)

and we have imposed the restriction

r(u) ≥ c−1 r(0)

By (10.111) and (10.115) this yields

for u ≤ 0.

2 ). x(u), z(u) ≤ c(y(0) + K0∞ + R[1] ′

(10.126)

′

Therefore, we obtain, using Sobolev inequality 3

2 ), sup r 2 | χ ˆ | ≤ c(y(0) + K0∞ + R[1]

(10.127)

′

′

Su

(10.128)

3

2) sup r 2 | trχ − trχ | ≤ c(y(0) + K0∞ + R[1] ′

′

Su

(10.129)

3

2 ). sup r 2 | ζ | ≤ c(y(0) + K0∞ + R[1] ′

Su

That is, in particular at u = u1 , we have (10.130) (10.131)

x(u1 ), y(u1 ), z(u1 ) ≤ cϵ0 , 3

sup r 2 | χ ˆ | ≤ cϵ0 , Su1

(10.132)

3 2

sup r | trχ − trχ | ≤ cϵ0 Su 1

(10.133)

3

sup r 2 | ζ | ≤ cϵ0 . Su1

′

10.4. MAIN THEOREM

273

Finally, the results for the components of the second fundamental form and the curvature ((10.92), (10.93) and (10.99), (10.100), (10.101)) yield at u = u1 : 3

sup | trθ − trθ | ≤ cϵ0 r− 2

(10.134)

Su 1

−1

sup | θˆ | ≤ cϵ0 r−1 τ− 2

(10.135)

Su 1

3

/ log a | ≤ cϵ0 r− 2 . sup | ∇

(10.136)

Su1

From the last inequality, it follows (by requiring that log a = 0): 1

sup | log a |≤ cϵ0 r− 2 .

(10.137)

Su 1

And by the Gauss-Bonnet formula in view of (10.4), one obtains 1 1 f¯ = 2 − (trχ)2 . r 4

(10.138) Therefore, as it is 1 4πr2 we write

(10.139)

#

S

(trχ − trχ)2 = (trχ)2 − (trχ)2 ,

# 4 2 ¯+ 1 − (trχ) = 4 f (trχ − trχ)2 r2 4πr2 Su = 4f¯ + (trχ − trχ)2 .

The left-hand side of (10.139) can also be written as ' (' ( 4 2 2 2 − (trχ) = − trχ + trχ , r2 r r

which yields

(10.140)

2 4f¯ + (trχ − trχ)2 − trχ = 2 r r + trχ r ≤ (4 | f¯ | + sup | trχ − trχ |2 ). 2 Su

as trχ ≥ 0. Next, write

2 2 − trχ = − trχ − (trχ − trχ) r r

to conclude (10.141)

H H H H H2 H H H H − trχH ≤ H 2 − trχH + sup | trχ − trχ |, Hr H Hr H Su

274

10. THE LAST SLICE

which, after considering (10.113), (10.128) and (10.140), yields the following ¯ estimate for all u ∈ I: . 3 2 ′ ′2 −4,Su ≤ c y(0) + K0∞ + R[1] ∥ − trχ∥ . (10.142) r 2− r Also, as 1 K = (trχ)2 + f, 4 one computes 1 1 1 − K = 2 − (trχ)2 − f r2 r 4 1 1 = f¯ − f − (trχ)2 + (trχ)2 , 4 4 ¯ hence, for all u ∈ I, it is: 1 −4,Su ≤ −f ∥ − f¯ − ∥ 4,Su + sup | (trχ)2 − (trχ)2 | (10.143) − ∥ 2 − K∥ r Su and therefore, we have 5 1 ′ ′2 ∥ 2 − K∥ −4,Su ≤ c(y(0) + K0∞ + R[1] ). (10.144) r 2− r In particular, for u = u1 we obtain 3 2 ∥ 4,Su ≤ cϵ0 , (10.145) ∥ − trθ − r2 − 1 r 5 1 (10.146) ∥ 2 −K− ∥ 4,Su ≤ cϵ0 . r2 − 1 r It follows that 1

2 2 ), −Kr ∥ − 1− ∥ 4,Su ≤ cr− 2 (y(0) + K0∞ + R[1] 1 r ′ ′2 ∥ 4,Su ≤ cr− 2 (y(0) + K0∞ + R[1] − ∥ trχ − 1 − (10.148) ). 2 We now focus on (10.130), (10.136), (10.137), (10.145), (10.146), observing that if ϵ0 is sufficiently small, then property P holds at u = u1 as well. That means, P is true for all u ≥ 0. So, the estimates (10.145), (10.146), (10.125), (10.126) also hold for all u ∈ [0, ∞). Therefore, it follows that the inverse lapse problem (10.2), (10.3) has a global solution in the exterior of S0 . This proves the theorem.

(10.147)

′

′

10.4.1. Integrating over u in H t ∗ . In the next proposition, we inteˆ and ζ estimated in theorem 14. grate in Ht∗ over u the norms involving χ Proposition 18. Let theorem 14 hold. Then it is: 3

(10.149) (10.150)

7

7

| r4χ ˆ |4,Ht∗ + | r 4 ∇ /χ ˆ |4,Ht∗ + | r 4 ∇ / tr χ |4,Ht∗ 2) ≤ c(y(0) + K0∞ + R[1] ′

3

′

7

2 ). / ζ |4,Ht∗ ≤ c(y(0) + K0∞ + R[1] | r 4 ζ |4,Ht∗ + | r 4 ∇ ′

′

10.4. MAIN THEOREM

275

Proof. The Lp -norm of a tensor F on Ht∗ can be written as '#

p

Ht∗

|F |

(1

p

=

'#

∞

u0

'# a

p

Su

|F |

(

du

′

(1

p

.

Therefore, we obtain for χ ˆ by a straightforward computation: (1 ' # ∞ ' # ( (1 '# 4 4 3 3 4 3 4 | r4χ ˆ |4,Ht∗ = r |χ ˆ| ≤ a r |χ ˆ | du′ 0

Ht∗

'# ≤c

∞

r5− ∥ χ∥ ˆ−44,S ′ dr′ u

r0

(1 4

Su

2 ). ≤ c(y(0) + K0∞ + R[1] ′

′

The remaining estimates are shown in the same way. Remarks: 1. In the proof of theorem 14, it is shown that for all u in [0, ∞) it is: 3

′

3

′

2) sup r 2 | χ ˆ | ≤ c(y(0) + K0∞ + R[1]

(10.151)

′

Su

2) sup r 2 | ζ | ≤ c(y(0) + K0∞ + R[1]

(10.152)

′

Su

2 ). sup r | trχ | ≤ c(y(0) + K0∞ + R[1] ′

(10.153)

′

Su

From proposition 18 it follows by Sobolev inequality that the terms χ, ˆ ζ and trχ belong also to L∞ (Ht∗ ). 2. Note that, if trχ and ∇ / tr χ lie in L4 (Su ), then they are also in 2 L (Su ), and the corresponding terms lie in L2 (Ht∗ ). 3. Similarly, if χ ˆ and ζ belong to L4 (Su ) and u ∈ [u0 , ∞), this yields (10.154) (10.155)

3

5

3

5

2) ∥ χ∥ ˆ−2,Su + r 2−∇ ∥ / χ∥ ˆ−2,Su ≤ c(y(0) + K0∞ + R[1] r 2− ′

′

2 ). ∥ −2,Su + r 2−∇ ∥ / ζ∥ −2,Su ≤ c(y(0) + K0∞ + R[1] r 2−ζ∥ ′

′

4. Then χ ˆ and ζ also are in L2 (Ht∗ ). Let us state the corresponding L2 -estimates in the last slice as follows: Proposition 19. Let the inequalities (10.154) for χ, ˆ ∇ /χ ˆ and (10.155) for ζ, ∇ / ζ hold for all u ∈ [u0 , ∞). Then it is:

(10.156) (10.157)

|χ ˆ |2,Ht∗ + | r∇ /χ ˆ |2,Ht∗ + | r∇ / tr χ |2,Ht∗ 2) ≤ c(y(0) + K0∞ + R[1] ′

′

2 ). | ζ |2,Ht∗ + | r∇ / ζ |2,Ht∗ ≤ c(y(0) + K0∞ + R[1] ′

′

276

10. THE LAST SLICE

Proof. For χ ˆ we calculate (1 ' # '# 2 2 |χ ˆ |2,Ht∗ = |χ ˆ| =

0

Ht∗

'# =c

∞

r0

∞

r2− ∥ χ∥ ˆ−22,Su dr′

(1

'# a

2

Su

|χ ˆ|

(

du

′

(1 2

2

2 ). ≤ c(y(0) + K0∞ + R[1] ′

′

Analogously, the other estimates of the proposition are shown. 10.5. Higher Derivatives In this subsection, we are going to estimate the second tangential derivatives of χ and ζ. We will estimate the second derivative of χ ˆ by the corresponding derivative of trχ. First, we shall stress that we do not have / 2 trχ to be of additional assumptions on ∇ / 2 trχ, that is, we only assume r2 ∇ the same size as r∇ / tr χ on St∗ ,0 which from the previous chapter is known − 32 to be of order r . So, in view of the result (10.72) in theorem 14 for the first derivative of trχ, we have for the second derivative: 7

(10.158) y1 (0) := r 2−∇ ∥ / 2 trχ∥ −2,S0 ≤ c 7

4 +R2) (10.159) y1 (u) := r 2−∇ ∥ / 2 trχ∥ −2,Su ≤ c(y1 (0) + K0∞ + K[1] [1] ′

′

′

for a constant c, which shall be shown in detail afterwards. / 2χ ˆ and ∇ / 2 ζ in Ht∗ , where one will We shall estimate the L2 -norms of ∇ also see that these are not bounded in the surfaces Su themselves, but only in Ht∗ . This, in fact, is what we need. Therefore, to control the corresponding quantities on Ht∗ , we shall integrate the squares of their norms for St∗ ,u in the last slice over u ∈ [0, ∞). The procedure for ∇ / 2 ζ is similar but more direct. In the following, denote by Su the intersection St∗ ,u = Ht∗ ∩ Cu . Theorem 15. Let the hypotheses of theorem 14 be satisfied. Further, the assumptions on the components of the curvature and the second fundamental form with respect to the foliation by u′ hold. Assume ′4 ≤ c, K[1] and

Then, it is (10.160) (10.161) (10.162)

R′2 [1] ≤ c. ′4 + K′∞ + R′2 ) / 2 trχ |2,Ht∗ ≤ c(y1 (0) + K[1] | r2 ∇ 0 [1]

′4 + K ∞ + R′2 ) / 2χ ˆ |2,Ht∗ ≤ c(y1 (0) + K[1] | r2 ∇ 0 [1] ′

′4 + K ∞ + R′2 ). | r2 ∇ / 2 ζ |2,Ht∗ ≤ c(y1 (0) + K[1] 0 [1] ′

10.5. HIGHER DERIVATIVES

277

Proof: The proof will follow the ideas outlined above. First, observe that the quantities in theorem 14 which are in L4 (Su ) for u ∈ [0, ∞) also belong to L2 (Su ). The corresponding, of course, holds with respect to Ht∗ . (See also remarks and the introductory paragraph before the present theorem.) To compare the null frames, here, we need the derivatives (i.e. one derivative) of the corresponding terms in lemma 9. We have already used this in theorem 14. Nevertheless, let us remind ourselves of the fact that the derivatives of cos ϕ and Y are easily shown to be small. The precise calculation is carried out in [19] at the beginning of the proof of proposition 14.0.2. Therefore, the terms calculated relative to the foliation u can be estimated by the ones relative to u′ . This yields 4 ≤ cK′4 , K[1] [1]

(10.163) and

R2[1] ≤ cR′2 [1] .

(10.164)

In order to estimate ∇ / 2 χ, ˆ we focus on ∇ / 2 trχ. From (10.158) it will follow by lemma 10 and a similar argument as for ∇ / tr χ in the main theorem that (10.159) holds. Actually, ∇ / 2 trχ lies also in L4 (Su ) for all u ∈ [0, ∞). Afterwards, we shall show the corresponding in Ht∗ , that is inequality (10.160). All the other terms that will appear in the estimates for (10.161) and (10.162), are under control in view of the previous subsection and the assumptions. Now, in Su for any u in [0, ∞), we can apply the estimates for Hodge systems in 2 dimensions from the corresponding chapter to the null Codazzi equation: 1 1 / tr χ − ϵ · χ ˆ + ϵtrχ − β. div /χ ˆ= ∇ 2 2

(10.165)

We differentiate once tangentially to Su and obtain: 1 1 1 2 / trχ − ∇ / ϵ · trχ + ϵ · ∇ / tr χ − ∇ / β. / ϵ·χ ˆ−ϵ·∇ /χ ˆ+ ∇ (10.166) ∇ / div /χ ˆ= ∇ 2 2 2 From the chapter about Hodge systems on S we have: (10.167)

#

Su

'# |∇ / χ ˆ| ≤c 2

2

Su

2

|∇ / div /χ ˆ | +r

−2

2

| div /χ ˆ|

(

.

278

10. THE LAST SLICE

That is in view of (10.166) and integrating with respect to u, we obtain: # # ∞ # 2 2 2 2 4 2 (10.168) / χ ˆ |2,Ht∗ = r |∇ / χ ˆ| = a r4 | ∇ / 2χ ˆ |2 du |r ∇ Ht∗ Su 0 # # ≤c r4 | ∇ / β |2 + c r4 | ∇ / 2 trχ |2 Ht∗ Ht∗ # $ +c r4 | ∇ ˆ |2 + | ϵ |2 | ∇ /χ ˆ |2 / ϵ |2 | χ Ht∗

% +|∇ / ϵ |2 | trχ |2 + | ϵ |2 | ∇ / trχ |2 # $ +c r2 | ∇ ˆ |2 / tr χ |2 + | ϵ |2 | χ Ht∗ 2

(10.169) (10.170)

+ | ϵ | | trχ |2 + | β |2 # ≤c r4 | ∇ / β |2 Ht∗ # +c r4 | ∇ / 2 trχ |2

(10.172)

#

$ r4 | ∇ ˆ |2 + | ϵ |2 | ∇ /χ ˆ |2 / ϵ |2 | χ Su 0 % +|∇ / ϵ |2 | trχ |2 + | ϵ |2 | ∇ / trχ |2 du # ∞ # $ +c a r2 | ∇ ˆ |2 / tr χ |2 + | ϵ |2 | χ Su 0 % 2 + | ϵ | | trχ |2 + | β |2 du. +c

(10.171)

Ht∗ ∞

#

%

a

The term (10.169) is bounded according to our assumptions on the curvature / β is bounded in Ht∗ but not components. More precisely, the L2 -norm of ∇ in the surfaces Su . Further, the integral (10.170) we are going to show to be controlled right below. Finally, the quantities (10.171) and (10.172) have been proven to be bounded in the previous subsection. Now, let us estimate the second tangential derivative of trχ in the surfaces Su for all u ∈ [0, ∞), that is, to show (10.159). To do so, we continue from the case for ∇ / tr χ in the main theorem. Therefore, we consider again the propagation equation (10.10) for trχ. Applying the commutator / ] to (10.10) yields: [∇N , △

(10.173) with

/ trχ + 2trθ △ / trχ = −F2 ∇N △

ˆ F2 = 2(χ ˆ + ηˆ)∇ / 2 trχ + δ△ / trχ + 2 | ∇ / tr χ |2

+ (2P − ∇ / δ)∇ / tr χ + 2trχ(ζ − ϵ)∇ / tr χ + 2(ζ − ϵ)(χ ˆ + ηˆ)∇ / tr χ ' ( 1 / F0 + △ / F0 , (trχ)2 + F0 + f¯ + 2(ζ − ϵ)∇ + (f − f¯+ | ζ − ϵ |2 ) 2 (10.174)

10.5. HIGHER DERIVATIVES

279

where

1 1 Pm = Πi m N j Rij = − (β m + βm ) + ηˆmn ϵn + δϵm . 2 2 Observe that we have

(10.175) (10.176)

ˆ·△ /χ ˆ+ | ∇ /χ ˆ |2 + 2ζ · △ / ζ +2|∇ / ζ |2 . △ / F0 = χ

Next, we are going to eliminate the terms χ ˆ·△ /χ ˆ and 2ζ · △ / ζ. Therefore, in view of the first term, we go back to the Codazzi equation. That is, we apply D / ∗2 to the null Codazzi equation (10.14). In view of the identity 1 D / ∗2 D /2=− △ / +K 2 and D / 2χ ˆ = div /χ ˆ we obtain: ' ( 1 ˆ2 ∗ (10.177) △ /χ ˆ − 2K χ ˆ=∇ / trχ + 2D ˆ − ϵtrχ . / 2 β + ϵχ 2 Whereas in view of △ / ζ we apply D / ∗1 to the ζ-system (10.6, 10.7). Taking into account the identity

and we deduce: (10.178)

D / ∗1 D / 1 = −△ / + K, / ζ, curl / ζ) D / 1 ζ = (div △ / ζ − Kζ = −∇ / f −∗ ∇ / (σ − ηˆ ∧ χ). ˆ

We substitute for △ /χ ˆ and △ / ζ in △ / F0 the corresponding quantities from (10.177) and (10.178), respectively. As we want to estimate ∇ / 2 trχ, we derive from (10.173) the following equation: (10.179) where

/ 2 trχ + 2trθ∇ / 2 trχ = −F3 , ∇N ∇

1 (F3 )ij := γij F2 . 2 Note that the derivatives of the curvature components β and σ in (10.177) and (10.178) disappear. It still remains the derivative of the curvature component ρ in ∇ / f , which is only controlled in Ht∗ , but not in the surfaces Su . Therefore, we substitute from (10.4) for ∇ / f . Let us remind ourselves of the equations (10.84) and (10.4) for f : 1 1 ˆ |2 − ηˆ · χ f = −ρ − | χ ˆ − δtrχ 2 2 and 1 f = K − (trχ)2 . 4 (10.180)

280

10. THE LAST SLICE

Thus, the term involving ∇ / ρ in F3 is eliminated. And the corresponding derivative of the Gauss curvature K is zero in view of the uniformization and the Gauss-Bonnet theorems. Therefore, all the terms involving derivatives of curvature components disappeared. One can easily check now that all the other terms in F3 are controlled in the required way. In the study of r2 ∇ / 2 trχ below we will estimate explicitly the terms for which this is less obvious. First, we apply lemma 10 to equation (10.179). With λ0 = 2 and λ2 = 4 we obtain for all u′ ∈ [0, u): 1

∥ / 2 trχ∥ −2,S (u) r 2 y1 (u) = r4−∇ H# ' H 2 4 (10.181) ∥ / trχ∥ −2,S (0) + HH ≤ c r −∇

0

u

H( H r −F ∥ 3− ∥ 2,S (u )du H . 4

′

1

′H

The integral on the right-hand side is growing like r+ 2 . Dividing both 1 sides by r+ 2 yields a constant bound on the right-hand side. We obtain: $ % 7 ′ ′4 + R′2 . ∥ / 2 trχ∥ −2,S (u) ≤ c y1 (0) + K0∞ + K[1] (10.182) y1 (u) = r 2−∇ [1]

This proves (10.159). / 2 trχ is bounded in Ht∗ . For this, Next, we are going to show, how r2 ∇ we integrate the square of the norm for ∇ / 2 trχ on Sr′ over r′ ∈ [r0 , ∞) as follows: # # ∞ | r2 ∇ / 2 trχ |22,Ht∗ = r4 | ∇ / 2 trχ |2 = r′4 | ∇ / 2 trχ |22,Sr′ dr′ =c

Ht∗ # ∞ r0

Now, we observe that r−1 y12

6

= r −∇ ∥/

2

trχ∥ −22,S

r0

′6

r −∇ ∥/

≤ cr

−2

≤ cr6

'#

'#

trχ∥ −22,S ′ dr′ r r

4

=c

r −F ∥ 3− ∥ 2,S ′ dr r

r0 r

r0

6 −2

≤ cr r

2

#

r0

′

r

r0

| F3 |2,Sr′ dr

r′−1 y12 dr′ .

(2

r−1 | F3 |2,Sr′ dr′

'#

∞

′

(2

(2

≤ cr6 (| F3 |2,S )2 .

In view of the fact that for a function Φ, where Φ denotes the mean value of Φ: # c r Φ= Φdr, r r0 the following inequality holds, # ∞ # ∞ 2 ′ Φ dr ≤ c Φ2 dr′ , r0

r0

10.5. HIGHER DERIVATIVES

we conclude 2

2

|r ∇ / trχ

|22,Ht∗

(10.183)

=c ≤c

#

∞

r0 ∞

#

r0

r′−1 y12 dr′

≤c

#

281

∞

r0

r′6 (| F3 |2,Sr′ )2 dr′

r′6 | F3 |22,Sr′ dr′ .

Now, let us come back to what we mentioned above and estimate explicitly the special terms on the right-hand side of (10.183). First, we note / 2 trχ times a term which that the quantities in F3 which are of the form ∇ is small, can be taken to the left-hand side of (10.183) and neglected, as the factor can be made arbitrarily small. Next, we focus on the product of first derivatives, that is, we consider ∇ / δ·∇ / tr χ. In the following, denote by I the interval [r0 , ∞). So, we have # # ′6 2 ′ / δ |24,Sr′ · | ∇ / tr χ |24,Sr′ dr′ r |∇ / δ·∇ / tr χ |2,Sr′ dr ≤ r′6 | ∇ I I # = | r′ ∇ / δ |24,Sr′ · | r′2 ∇ / tr χ |24,Sr′ dr′ I

≤ sup | r2 ∇ / tr χ |24,Sr · | r∇ / δ |L2 (I,L4 (Sr )) ,

(10.184)

r∈I

which is bounded according to what we have already shown. The computa/χ ˆ |2 and | ∇ / ζ |2 work similarly. tions for | ∇ / trχ |2 , | ∇ We consider the term β · ∇ / tr χ. # # r′6 | β · ∇ / tr χ |22,Sr′ dr′ ≤ | r′ β |24,Sr′ · | r′2 ∇ / tr χ |24,Sr′ dr′ I

I

≤ sup | r2 ∇ / tr χ |24,Sr · | rβ |L2 (I,L4 (Sr )) .

(10.185)

r∈I

For δ ·

(trχ)3

#

I

(10.186)

′6

we have:

r |δ·

(trχ)3 |22,Sr′

′

dr ≤

#

I

r

′6

#

Sr ′

| δ |2 · | trχ |6 dr′

≤ sup | rtrχ |6 · | δ |22,Ht∗ . H

All the remaining terms in F3 are estimated in a similar way. It follows that the right-hand side of (10.183) and therefore | r2 ∇ / 2 trχ |22,Ht∗ is bounded as (10.187)

′4 + K′∞ + R′2 ) | r2 ∇ / 2 trχ |2,Ht∗ ≤ c(y1 (0) + K[1] 0 [1]

From the inequality for | r2 ∇ / 2χ ˆ |22,Ht∗ , that is ((10.169)–(10.172)), and (10.187) we deduce (10.188)

′4 + K ∞ + R′2 ). / 2χ ˆ |2,Ht∗ ≤ c(y1 (0) + K[1] | r2 ∇ 0 [1]

This proves (10.161).

′

282

10. THE LAST SLICE

To estimate | r2 ∇ / 2 ζ |2,Ht∗ , consider the system (10.6, 10.7) for ζ: div / ζ = f − f¯ curl / ζ = σ − ηˆ ∧ χ. ˆ

Using the results of the chapter about Hodge systems in 2 dimensions, we can write: # # $ 2 2 |∇ / ζ| ≤c / ζ |2 |∇ / div / ζ |2 + r−2 | div Su Su % +|∇ / curl / ζ |2 + r−2 | curl / ζ |2 . (10.189)

In view of the first term on the right-hand side of (10.189), we calculate 1 1 / δ · trχ − δ · ∇ / tr χ. ∇ / f = −∇ / ρ− | χ ˆ|∇ /χ ˆ−∇ / ηˆ · χ ˆ − ηˆ · ∇ /χ ˆ− ∇ 2 2 (10.190) We deduce, with I = [r0 , ∞): # # 2 2 4 2 / f |2,Ht∗ = r |∇ /f| = | r2 ∇ / f |22,Sr′ dr′ |r ∇ H

I

$ 3 1 ≤ c | r2 ∇ / ρ |22,Ht∗ + | r 2 χ ˆ |2∞,Ht∗ · | r 2 ∇ /χ ˆ |22,Ht∗ 3

1

+ | r2χ ˆ |2∞,Ht∗ · | r 2 ∇ / ηˆ |22,Ht∗ + sup | r2 ∇ /χ ˆ |24,Sr · | ηˆ |L2 (I,L4 (Sr )) r∈I % + | rtrχ |2∞,Ht∗ · | r∇ / δ |22,Ht∗ + sup | r2 ∇ / tr χ |24,Ht∗ · | δ |22,Ht∗ . r∈I

So, the right-hand side is bounded in view of our assumptions on the curvature components and our previous results. /div / ζ |2,Ht∗ and | r2 ∇ /curl / ζ |2,Ht∗ are controlled We observe that also | r2 ∇ by curvature assumptions and previous results. Whereas by the main theorem the terms | rdiv / ζ |2,Ht∗ and | rcurl / ζ|2,Ht∗ are estimated in the appropriate way. Therefore, we conclude for | r2 ∇ / 2 ζ |2,Ht∗ : (10.191)

′4 + K ∞ + R′2 ), / 2 ζ |2,Ht∗ ≤ c(y1 (0) + K[1] | r2 ∇ 0 [1] ′

which proves (10.162). This closes the proof of the theorem.

APPENDIX A

Curvature Tensor – Components In order to enhance reading, we collected, here, some notation for the curvature components, which was given in the chapter ‘Comparison’. This notation is used throughout the whole work. The null curvature components of W are: αAB = WA4B4 2βA = WA434

αAB = WA3B3 2β A = WA334

4ρ = W3434

2σϵAB = WAB34

∗

−ϵAB βC = WABC4

−ϵAB ϵCD ρ = WABCD

ϵAB ∗ β C = WABC3

−ρδAB + σϵAB = WA3B4

For the spacetime dual of W it is: −∗ αAB = ∗ WA4B4

∗

−2 ∗ βA = ∗ WA434

2 ∗ β A = ∗ WA334

−2ρϵAB = ∗ WAB34

−ϵAB βC = ∗ WABC4

ϵAB ϵCD σ = ∗ WABCD

αAB = ∗ WA3B3 4σ = ∗ W3434

−ϵAB β C = ∗ WABC3

−σδAB − ρϵAB = ∗ WA3B4

283

APPENDIX B

Uniformization Theorem: Standard Situation, Cases 1 and 2 In this section, we write down the proof of the standard uniformization theorem 9 from section 8.1 for the cases 1 and 2. We refer to standard literature cited at the beginning of section 8.1. To prove cases 1 and 2, we recall equation (8.7): E 2u = K. △g u + Ke

Case 1: We then have the linear equation (B.1)

△g u = K.

In general, the linear equation (Poisson equation) (B.2)

△g u = f,

where f is a given function on a compact manifold M (without boundary), is solvable if and only if # f dµg = 0. (B.3) M

The solution is then unique up to an additive constant. In our case the integrability condition reads: # Kdµg = 0, (B.4) M

which is precisely the Gauss-Bonnet formula in case 1. Case 2: E = −1 and the equation reads Here, we have K (B.5)

△g u − e2u = K.

This is a non-linear elliptic partial differential equation. The Gauss-Bonnet formula is written in its usual form: # Kdµg = 2πχ. M

285

286

APPENDIX B: UNIFORMIZATION THEOREM

For any function f on M let us denote by f¯ the mean value of f on M . # 1 ¯ (B.6) f= f dµg . Area ((M, g)) M

In particular, by the Gauss-Bonnet formula we have 2πχ ¯ = (B.7) K . Area ((M, g)) ¯ = −1. Let us define: By an initial constant rescaling we can set K f = K + 1.

Then it is

f¯ = 0

and the equation (B.5) becomes △g u − e2u + 1 = f.

(B.8)

So, let us state the following

Theorem 16. In fact, we shall show, more generally, that this equation (B.8) has a unique smooth solution u for any given smooth function f such that f¯ = 0. Proof. Uniqueness: Let u′ be another solution correspondig to the same f . Subtracting the equation for u from that for u′ yields: △g (u′ − u) − (e2u − e2u ) = 0. ′

Multiply by (u′ − u) and integrate over M . Integrating by parts the first term, we obtain: # A B ′ | ∇(u′ − u) |2 + (u′ − u)(e2u − e2u ) dµg = 0. M

Now, it is

(e2u according to

Hence, one obtains that

′

⎧ ⎨>0 2u −e ) =0 ⎩ 0 ′ (u − u) = 0 ⎩ 0.

Recall that 2-d Hodge systems are defined in the Main Introduction to this monograph. We refer the reader to Chapter 2 of [5] for the Lp theory regarding Hodge systems that is used to derive estimates for the second fundamental, whose components with respect to the frame {N, eA ; A = 1, 2} are given as follows: kN N = δ kAN = ϵA kAB = η AB .

(1.7)

Note that δ is a function, ϵA is a 1-form tangent to the St,u and η AB is a 2-form tangent to the St,u . This process allows us to rewrite the elliptic system for kij above in terms of operators on the tangent spaces of the St,u . 1.5.2. The Lapse Function. Again analogous to [5], we use elliptic theory relating to the Poisson equation to estimate the lapse function. Components of the second fundamental form and the lapse function satisfy a Poisson equation of the form △φ = f

where f is a scalar function on the level surface Σ. In [5] the authors develop an L2 theory for estimating the solutions of the Poisson equation assuming the Dirichlet boundary condition at infinity: sup |φ| → 0 Su

as u → ∞.

318

1. INTRODUCTION

There are three main propositions that are stated and proved in [5], which we utilize and include below. For each proposition, Christodoulou and Klainerman assume that f ∈ H0,1 and there exists a solution φ, such that φ ∈ H2,−1 . Proposition 1. &Let φ be a solution of the Poisson equation. (i) Assume that Σ r2 |f |2 is finite. There exists a constant E c1 such that #

2

Σ

(ii) Assume that

&

|∇φ| ≤ E c1

2 2 Σ τ − |f |

#

Σ

r2 |f |2 .

is finite. There exists a constant E c2 such that 2

Σ

#

|∇φ| ≤ E c2 r0

#

Σ

τ 2− |f |2 .

Proposition % the fundamental constants are finite. Also as$ 2. Assume / is finite and φ is a solution of the Poisson equasume that supΣ ra−1 |▽a| tion. & c3 such that (i) Assume that Σ r2 |f |2 is finite. There exists a constant E #

Σ

(ii) Assume that stant E c4 such that

&

H

H2 r ∇ φH ≤ E c3 2H

2 2 Σ {τ − |f |

2

#

Σ

r2 |f |2 .

+ τ 2− r2 |∇f |2 } is finite. There exists a con-

# A H 2 H2 B / 2 φ|2 + r2 |∇ /∇ / N φ|2 + w12 H∇ r2 |∇ / N φH Σ # ≤E c4 r0 {τ 2− |f |2 + τ 2− r2 |∇f |2 } Σ

where w1 is defined according to

B A 1/2 wp = min τ p− r0 , rp .

Proposition 3. Assume the fundamental constants are finite. Also $& %1/4 assume that supΣ (ra−1 |∇a|) / and Σ r5 |P |4 is finite where Pi = Πji Rjk N k and φ is a solution of the Poisson equation. B & A 2 2 / |2 is finite. There exists a constant (i) Assume that Σ r |f | + r4 |∇f E c5 depending on the fundamental constants and quantities supΣ

1.5. THE GEOMETRY OF THE SPACE-TIME

319

-& .1/4 % $& %1/4 $& 5 H −1 H%1/4 / , Σ r5 |∇trθ| / , Σ r Ha ∇a / H and Σ r5 |P |4 such ra−1 |▽a| that CH D # H H H2 H H2 H 3 H2 H 2 H / φH + H∇ r 4 H∇ / ∇N φH + H∇∇ / 2N φH Σ # A B ≤E c5 r2 |f |2 + r4 |∇f / |2 . Σ C H H D & H 2 H2 2 2 2 2 2 2 4 / fH is finite. / | + τ − r H∇ (ii) Assume that Σ τ − |f | + τ − r |∇f

$

on the quantities in part (i) and the There exists a-constant E c6 depending . −1/2 quantity supΣ r3/2 τ − a−1 |∇a| / such that # A B # H H2 r4 |∇ / 2N ∇φ| / 2 + r2 w12 H∇∇ / 3 φ|2 + |∇ / 2N φH Σ Σ # A B 2 2 2 2 / |2 + τ 2− r4 |∇ / 2 f |2 . τ − |f | + τ − r |∇f ≤E c6 r0 Σ

We will also use the theories described above to derive estimates for the components of the electromagnetic field and the components of the Weyl tensor.

CHAPTER 2

Norms and Notation This chapter contains the description of norms and notation used in the statement of the Existence Theorem. We follow the conventions and notation adopted in [5] and expand these as necessary to describe terms involving the electromagnetic field. Let T = φ−1 ∂t be the unit normal to Σt and N = a−1 ∂u , where φ is the lapse function of the t-foliation and a is the lapse function of the u-foliation, respectively. The vectors (e4 = T + N, e3 = T − N, eA : A = 1, 2) where (eA : A = 1, 2) is an orthonormal frame for St,u make up the standard null frame for (M, g). With this frame, the null components of the Weyl tensor are written as WA3B3 = α(W )AB WA334 = 2β(W )A W3434 = 4ρ(W )

(2.1)

WA4B4 = α(W )AB WA434 = 2β(W )A WAB34 = 2σ(W )ϵAB .

The null components of the electromagnetic field are written as FA3 = α(F )A F34 = 2ρ(F )

(2.2)

FA4 = α(F )A F12 = σ(F ).

The corresponding null decomposition {×α(F ),×α(F ),×ρ(F ),×σ(F )} of ∗ F is given by ×

(2.3)

×

α(F )A = −α(F )B ϵBA × ρ(F ) = σ(F )

α(F )A = α(F )B ϵBA × σ(F ) = −ρ(F )

where the Hodge dual of a tensor u tangent to St,u , is defined by ∗

uA = ϵB A uB .

The components of the second fundamental form k are written with respect to the orthonormal frame (N, eA : A = 1, 2):

(2.4)

kN N = δ kAN = ϵA kAB = η AB .

Denoting χAB as the second fundamental form of St,u with respect to the null normal leads to the following identity (2.5)

θAB = χAB + η AB .

In this case θAB are components of the second fundamental form of St,u with respect to N . 321

322

2. NORMS AND NOTATION

Furthermore as in [5], the covariant derivative of the space-time is denoted D; its projection onto the tangent space of Σt is denoted ∇; and its / For an orthonormal projection onto the tangent space of St,u is denoted ∇. frame (T, e1, e2 , e3 ) the following formulas hold Di ej = ∇i ej − kij T Di T = −kij

$ % DT ei = Pr oj [DT ei ] + φ−1 ∇ /iφ T $ % / i φ ei DT T = φ−1 ∇

∇N eA = ∇ /N eA + a−1 (∇ /A a) N ∇A N = θAB eB ∇A eB = ∇ /A eB − θAB N ∇N N = −a−1 ∇ /A aeA .

Again following [5], the Ricci Coefficients of the null standard frame are given by the following χAB = θAB − η AB χAB = −θAB − η AB

ξ A = φ−1 ∇ /A φ − a−1 ∇ /A a

/A φ − ϵA ζ A = φ−1 ∇

ζ A = a−1 ∇ /A a + ϵA

ν = −φ−1 ∇ /N φ + δ ν = φ−1 ∇ /N φ + δ

(2.6) and

ω = δ − a−1 ∇ /N a. The terms χAB , ζ A , and ω are components of the Hessian of the optical function. These terms are discussed further in the chapter on the optical function, in which they are defined with respect to a normalized frame. We use the same basic norms for the curvature tensor R, the second fundamental from k, the lapse function φ, and the optical quantities χ, ζ, ω that are introduced in [5]. For an S-tangent tensor field V , define

|V |p,S = (2.7)

0#

St,u

p

|V |P dµγ

= sup |V | St,u

11

if p = ∞.

if 1 ≤ p < ∞

2.1. NORMS INVOLVING THE ELECTROMAGNETIC FIELD F

323

Also define in the interior and exterior regions Σit and Σet of each slice: 0# 11 p

∥V ∥p,i =

Σit

|V |P

= sup |V | Σit

∥V ∥p,e (t) =

0#

if 1 ≤ p < ∞

if p = ∞ 11

p

Σet

|V |P

= sup |V | Σet

if p = ∞

|||V ||||p,e (t) = sup |V |p,S u≥u0 (t)

= sup |V | Σet

if 1 ≤ p < ∞

if 1 ≤ p < ∞

if p = ∞,

where Σit = I is defined as all of the points for which r ≤ includes those points for which r ≥ r02(t) . Furthermore define the local L2 norm 0# 11 2 2 (loc) ∥V ∥2,e (t) = sup |V | λ≥u0 (t)

r0 (t) 2

and Σet = E

D(λ,t)

with D (λ, t) defined as the annulus λ ≤ u ≤ λ + 1 on Σt . Lastly, define the weights ! τ − = 1 + u2 G (2.8) τ + = 1 + (2r − u)2 A B 1 (2.9) wp = min τ p− r02 , rp . 2.1. Norms involving the Electromagnetic Field F Analogous to the definition of norms for the Weyl tensor in [5], in this section we define the interior and exterior norms i F[q] and e F[q] where $ % F[q] = max i F[q] ,eF[q] for q = 0, 1, 2, 3. To define the interior norms, we set . i Fq = r01+q ∥Dq E(F )∥2,i + ∥Dq H(F )∥2,i

for q = 0, 1, 2 and

i

. F3 = r03 ∥Dq E(F )∥2,i + ∥Dq H(F )∥2,i .

324

2. NORMS AND NOTATION

We then set i i i i

F[0] = i F0

F[1] = i F[0] + i F1 F[2] = i F[1] + i F2

F[3] = i F[2] + i F3 .

We define the exterior quantities e Fq as follows: # # 2 e 2 2 F0 = τ − |α(F )| + τ 2+ |α(F )|2 Σet

+ e

F12

=

#

+ + + + e

F22

=

Σet

Σet

+

#

#

+ + + + +

τ 2+ |ρ(F )|2

#

+

τ 2+ |σ(F )|2

Σet

/ )|2 τ 2+ τ 2− |∇α(F

+

#

#

Σet Σet

/ 3 α(F )|2 + τ 4− |D

Σet

τ 4+ |D / 3 α(F )|2

Σet

τ 2− τ 2+ |D / 3 ρ(F )|2

+

Σet

τ 2− τ 2+ |D / 3 σ(F )|2

+

# # # # #

H

H2

H 2 H / α(F )H τ 4+ τ 2− H∇

Σet

#

τ 4+ |∇σ(F / )|2

Σet

#

Σet

τ 2+ τ 2− |D / 4 α(F )|2

Σet

τ 4+ |D / 4 α(F )|2

#

+

τ 4+ |∇α(F / )|2

Σet

τ 4+ |∇ρ(F / )|2 +

Σet

+

Σet

#

+

#

#

Σet

τ 4+ |D / 4 ρ(F )|2

Σet

τ 4+ |D / 4 σ(F )|2

#

Σet

H H2 # H 6 H 2 ∇ / τ + H ρ(F )H +

Σet

Σet Σet

τ 6+ |∇ /D / 3 α(F )|2

Σet

τ 2− τ 4+ |∇ /D / 3 ρ(F )|2

+

Σet

τ 2− τ 4+ |∇ /D / 3 σ(F )|2

+

# # #

Σet

H

H2

H H 2 τ 6− HD / 3 α(F )H

#

+

+

#

Σet

Σet

#

Σet

H

H2

H 2 H / σ(F )H τ 6+ H∇

τ 2+ τ 4− |∇ /D / 3 α(F )|2 +

#

H2 H H 2 H / α(F )H τ 6+ H∇

#

τ 4+ τ 2− |∇ /D / 4 α(F )|2

τ 6+ |∇ /D / 4 α(F )|2

Σet

τ 6+ |∇ /D / 4 ρ(F )|2

Σet

τ 6+ |∇ /D / 4 σ(F )|2

#

τ 2+ τ 4− |D / 3D / 4 α(F )|2

2.1. NORMS INVOLVING THE ELECTROMAGNETIC FIELD F

+

#

Σet

#

τ 2+ τ 4− |D / 4D / 3 α(F )|2 +

#

H H2 H 2 H / 4 α(F )H τ 2− τ 4+ HD

Σet

H2 # H H 2 4 H 2 / 3 α(F )H + + τ − τ + HD τ 6+ |D / 3D / 4 α(F )|2 Σet Σet # # H2 H H H 2 2 6 / 4 α(F )H + τ + |D / 4D / 3 α(F )| + τ 6+ HD Σet

+ + + (2.10)

+

#

Σet

#

Σet

#

Σet

#

Σet

Σet

τ 2− τ 4+ |D / 4D / 3 ρ(F )|2 + H2 # H H 4 2 H 2 τ − τ + HD / 3 σ(F )H + τ 2− τ 4+ |D / 4D / 3 σ(F )|2

+ r0−δ

#

Σet

#

+

Σet

# H H2 H −δ 6 H 3 / ρ(F )H + r0 τ + H∇

τ 2− τ 4+ |D / 3D / 4 ρ(F )|2

#

Σet

Σet

We define e F3 as follows # # H2 H H H 3 e τ 6+ H∇ / α(F )H + r0−δ F3 = r0−δ Σet

Σet

H2 # H H 4 2 H 2 / 3 ρ(F )H + τ − τ + HD

H H2 H H 2 τ 6+ HD / 4 ρ(F )H

τ 2− τ 4+ |D / 3D / 4 σ(F )|2

#

Σet

H2 H H H 2 τ 6+ HD / 4 σ(F )H .

H2 H H H 3 τ 6+ H∇ / α(F )H

Σet

H H2 H 3 H / σ(F )H τ 6+ H∇

# H2 H2 H H H H H 2 −δ −δ 4 2 H 2 + r0 τ + τ − H∇ / 3 α(F )H + r0 τ 6+ H∇ / 3 α(F )H / D / D Σe Σet # t # H2 H2 H H H 2 H 2 H H / D / D + r0−δ τ 6+ H∇ / 3 ρ(F )H + r0−δ τ 6+ H∇ / 3 σ(F )H Σet

Σet

Σet

Σet

#

# H2 H2 H H H 2 H H −δ −δ 6 H 2 / D / D + r0 τ + H∇ / 4 α(F )H + r0 τ 6+ H∇ / 4 α(F )H e e Σ Σ # t # t H H H2 H2 H H −δ −δ 6 H 2 6 H 2 ∇ / ∇ / + r0 τ+ H D / 4 ρ(F )H + r0 τ+ H D / 4 σ(F )H #

# H H H2 H2 H H H H /D / 23 α(F )H + r0−δ /D / 23 α(F )H + r0−δ τ 2+ τ 4− H∇ τ 6+ H∇ e e Σ Σ # t # t H H H2 H2 H H H H /D / 23 ρ(F )H + r0−δ /D / 23 σ(F )H + r0−δ τ 4+ τ 2− H∇ τ 4+ τ 2− H∇ Σet

+

r0−δ

+

r0−δ

+

r0−δ

#

Σet

Σet

τ 4+ τ 2− |∇ /D / 3D / 4 α(F )|2

Σet

τ 4+ τ 2− |∇ /D / 3D / 4 ρ(F )|2

Σet

τ 4+ τ 2− |∇ /D / 4D / 3 α(F )|2

# #

+

r0−δ

+

r0−δ

+

r0−δ

#

Σet

τ 6+ |∇ /D / 3D / 4 α(F )|2

Σet

τ 4+ τ 2− |∇ /D / 3D / 4 ρ(F )|2

Σet

τ 6+ |∇ /D / 4D / 3 α(F )|2

#

#

325

326

2. NORMS AND NOTATION

+

r0−δ

+

r0−δ

#

Σet

#

Σet

#

τ 4+ τ 2− |∇ /D / 4D / 3 ρ(F )|2 H2

H

H H /D / 24 α(F )H τ 6+ H∇

+

+

r0−δ

#

r0−δ

#

τ 4+ τ 2− |∇ /D / 4D / 3 σ(F )|2

Σet

H2 H H H /D / 24 α(F )H τ 6+ H∇

Σet

# H2 H2 H H H H H 2 −δ −δ 6 H /D / 4 ρ(F )H + r0 /D / 24 σ(F )H + r0 τ + H∇ τ 6+ H∇ e e Σ Σ # t # t H H2 H2 H H H −δ −δ 6 H 3 4 2 H 3 / 3 α(F )H + r0 + r0 τ − HD τ + τ − HD / 3 α(F )H Σet

+ r0−δ

#

Σet

#

Σet

# H2 H H H 3 τ 2+ τ 4− HD / 3 ρ(F )H + r0−δ

Σet

H2 H H H 3 τ 2+ τ 4− HD / 3 σ(F )H

# H2 H2 H H H H H 2 −δ −δ 2 4 H 2 + r0 τ + τ − HD / 4 α(F )H + r0 τ 4+ τ 2− HD / 4 α(F )H / 3D / 3D e e Σ Σ # t # t H2 H2 H H H H H 2 H 2 + r0−δ τ 4+ τ 2− HD / 4 ρ(F )H + r0−δ τ 4+ τ 2− HD / 4 σ(F )H / 3D / 3D Σet

+

r0−δ

+

r0−δ

+

r0−δ

#

Σet

τ 2+ τ 4− |D / 3D / 4D / 3 α(F )|2

+

r0−δ

+

r0−δ

Σet

#

τ 4+ τ 2− |D / 3D / 4D / 3 ρ(F )|2

Σet

#

Σet

#

H

H2

H H τ 2+ τ 4− HD / 23 α(F )H / 4D

+

r0−δ

#

#

#

τ 4+ τ 2− |D / 3D / 4D / 3 α(F )|2

Σet

τ 4+ τ 2− |D / 3D / 4D / 3 σ(F )|2

Σet

Σet

H

H2

H H τ 4+ τ 2− HD / 23 α(F )H / 4D

# H2 H2 H H H H H H + r0−δ τ 4+ τ 2− HD / 23 ρ(F )H + r0−δ τ 4+ τ 2− HD / 23 σ(F )H / 4D / 4D e e Σ Σ # t # t H2 H2 H H H H H 2 H 2 / 4D / 4D + r0−δ τ 4+ τ 2− HD / 3 α(F )H + r0−δ τ 6+ HD / 3 α(F )H Σet

Σet

#

# H2 H2 H H H H H 2 −δ −δ 6 H 2 / 4D / 4D + r0 τ + HD / 3 ρ(F )H + r0 τ 6+ HD / 3 σ(F )H Σet Σet # # H2 H2 H H H H H 3 −δ −δ 6 H 3 + r0 τ + HD τ 6+ HD / 4 α(F )H + r0 / 4 α(F )H Σet

(2.11)

+ r0−δ

#

Σet

# H2 H H −δ 6 H 3 τ + HD / 4 ρ(F )H + r0

Here δ is less than one-quarter.

Σet

Σet

H2 H H H 3 τ 6+ HD / 4 σ(F )H .

2.2. Norms Involving the Weyl Tensor W This chapter defines the interior and exterior norms i W[q] and e W[q] and then sets the norm 2 3 W[q] = max i W[q] ,e W[q] for q = 0, 1, 2. The definitions of these quantities are the same as in [5], / 23 α(W ) and D / 24 α(W ) are allowed to except that the weighted L2 -norm of D

2.2. NORMS INVOLVING THE WEYL TENSOR W

grow slightly in t. # # e W02 = τ 4− |α(W )|2 + Σet

+ e

W12

=

#

+ + + + + e

W22

=

#

Σet

#

+ + +

r2 τ 4− |∇α(W / )|2

r |∇σ(W / )| +

Σet

τ 6− |α(W )3 |2

# #

Σet

#

#

+

6

#

Σet

#

Σet

#

Σet

#

+

H H2 r8 H∇β(W )4 H + /

Σet

r8 |∇β(W / )4 |2 +

+

#

Σet

#

Σet

#

Σet

#

#

Σet

#

+

+

+

2

τ − r |ρ(W )34 | +

#

Σet

r6 |β(W )3 |2

#

r6 |α(W )4 |2

Σet

H2

#

+

r6 τ 2− |∇α(W / )4 |2

τ 2− r6 |∇ρ(W / )3 |2 +

Σet

r |∇σ(W / )4 | +

r8 |∇α(W / )3 |2 +

#

Σet

Σet Σet

Σet

#

r4 τ 4− |α(W )34 |2 H

H

2 τ 6− r2 Hβ(W )33 H

H H2 r8 Hβ(W )44 H + 7

2

+ #

Σet

#

τ − r |ρ(W )43 | +

#

Σet

H H2 τ 4− r4 H∇β(W )3 H /

Σet

r8 |∇ρ(W / )4 |2

Σet

r8 |∇β(W / )3 |2

r8 |∇α(W / )4 |2

+

+

H H2 H 2 H / ρ(W )H r 8 H∇

H2 H H H 2 r 8 H∇ / α(W )H

#

2

8

#

#

r6 |ρ(W )4 |2

Σet

Σet

Σet

H H2 τ 2− r6 Hβ(W )43 H + 7

#

Σet

H2 # H H 8H 2 r H∇ / β(W )H +

Σet

Σet

τ 8− |α(W )33 |2

r6 τ 2− |α(W )44 |2

#

+

#

H

#

H H2 τ 4− r2 Hβ(W )3 H

Σet

H 2 H / β(W )H τ 2− r6 H∇

Σet

Σet

Σet

+

r6 |α(W )3 |2 +

H2 # H H 8H 2 r H∇ / σ(W )H + r2 τ 6− |∇α(W / )3 |2

r6 |σ(W )4 |2 +

Σet

Σet

τ 2− r6 |∇σ(W / )3 |2

r0−δ

#

r6 |∇α(W / )|2

#

#

r4 |α(W )|2

r6 |∇ρ(W / )|2

Σet

Σet

+

Σet

#

+

#

τ 2− r4 |ρ(W )3 |2 +

#

H2

2

Σet

Σet

H

H

Σet

r6 |β(W )4 |2 +

H 2 H / α(W )H r4 τ 4− H∇

H

r4 τ 2− |α(W )4 |2

#

#

r4 |β(W )|2 +

r |∇β(W / )| +

Σet

#

Σet

r4 |ρ(W ) − ρ(W )|2

Σet

Σet

#

H H2 r6 Hβ(W )4 H +

#

#

2 τ 2− r4 H∇β(W )H /

Σet

τ 2− r4 |σ(W )3 |2 +

#

+

#

+

2

6

+

+

r4 |σ(W ) − σ(W )|2 +

Σet

Σet

+

Σet

#

Σet

H H2 τ 2− r2 Hβ(W )H +

327

#

#

Σet

Σet

Σet

#

r4 τ 4− |α(W )43 |2

H H2 τ 2− r6 Hβ(W )34 H

τ 4− r4 |ρ(W )33 |2

Σet

r8 |ρ(W )44 |2

328

2. NORMS AND NOTATION

+ + + + + (2.12)

#

Σet

#

τ 4− r4 |σ(W )33 |2

+

2

7

Σet

τ − r |σ(W )43 | +

Σet

τ 2− r6 |β(W )33 |2

# #

2

8

+ #

#

τ − r7 |σ(W )34 |2

Σet

r8 |σ(W )44 |2

#

#

2

8

Σet

r |β(W )34 | + 2

8

Σet

r |β(W )44 | +

Σet

r8 |α(W )43 |2 + r0−δ

#

Σet

Σet

r |α(W )33 | +

#

Σet

#

#

Σet

r8 |β(W )43 |2

r8 |α(W )34 |2

Σet

r8 |α(W )44 |2 .

In this case, δ is less than one-half. The definitions of α(W )4 , etc. are given in the chapter on the Comparison Theorem for the Weyl tensor. 2.3. Norms Involving the Second Fundamental Form of the Foliation Norms for the second fundamental form are defined as in [5], with one exception. Notably, in our work N η 4 includes a term involving a component of the electromagnetic field, namely |α(F )|2 . However, this difference does not affect the decay of N η 4 . Following [5], the expressions i Kqp and e Kqp represent the interior and exterior weighted Lp norms of the q-covariant derivatives of the second fundamental form. Let Kq denote 2 3 Kqp = max i Kqp ,e Kqp . Define i Kqp for 1 ≤ p ≤ ∞ as i

2+q− p3

Kqp = r0

∥Dq k∥p,i .

To describe the exterior expression, use the radial decomposition of k together with a decomposition of η into its traceless part N η and trace tr(η) = −δ to define / 3δ δ3 = D / 3ϵ ϵ3 = D

δ4 = D / 4δ ϵ4 = D / 4ϵ

1 1 η − |α(F )|2 . N η4 = D / 4N η + tr(χ)N 2 4

1 N η3 = D / 3N η + tr(χ)N η 2 Then define e

Kqp (δ)

e

=

−1 r0 2 − 12

Kqp (ϵ) = r0

Q Q " # Q 5 − 3 +q q Q Qr 2 p ∇ / δQ Q Q Q Q " # Q 5 − 3 +q q Q Qr 2 p ∇ / ϵQ Q Q

p,e

p,e

2.3. NORMS INVOLVING THE SECOND FUNDAMENTAL FORM OF FOLIATION 329 e

Kqp (N η)

=

−1 r0 2

Q Q " # Q Q 1− 2 +q q " # p Qr w 3−1 ∇ / N ηQ Q Q 2

Q " Q # 1 Q 5 3 Q − +q q e p 2 Q Kq+1 (δ 4 ) = r0 Qr 2 p ∇ / δ4Q Q

p

p,e

p,e

Q " Q # Q 1− 2 +q Q q Q e p " # p Q Kq+1 (δ 3 ) = r0 Qr w 3−1 ∇ / δ3Q 1 2

2

e

e

e

e

− 12

p Kq+1 (ϵ4 ) = r0

− 12

p Kq+1 (ϵ3 ) = r0

− 12

p Kq+1 (N η 4 ) = r0 p Kq+1 (N η3) =

−1 r0 2

p

p,e

Q " Q # Q 7 − 3 +q q Q Qr 2 p ∇ / ϵ4 Q Q Q

p,e

Q " Q # Q 2− 2 +q Q q Q " # p Qr w 3−1 ∇ / ϵ3 Q Q 2

p

p,e

Q " Q # Q 7 − 3 +q q Q Qr 2 p ∇ / N η4Q Q Q

p,e

Q " Q # Q 1− 2 +q Q q Q " # p Qr w 5−1 ∇ / N η3Q Q 2

p

p,e

Q " Q # 1 Q 5 Q − p3 +q q e p 2 Q 2 Kq+2 (DS δ 4 ) = r0 Qr ∇ / DS δ 4 Q Q

p,e

Q " Q # Q 1− 2 +q Q q e p p Kq+2 (DS δ 3 ) = r0 Q w" 3 − 1 # ∇ / DS δ 3 Q Qr Q 1 2

2

e

e

e

(2.13)

e

− 12

p Kq+2 (D / S ϵ4 ) = r0

− 12

p Kq+2 (D / S ϵ3 ) = r0

− 12

p Kq+2 (D / SN η 4 ) = r0 p Kq+2 (D / SN η3) =

−1 r0 2

p

p,e

Q " Q # Q 7 − 3 +q q Q Qr 2 p Q ∇ / D / ϵ 4 S Q Q

p,e

Q " Q # Q 2− 2 +q Q q " # p Qr Q w ∇ / D / ϵ 3 3 1 S Q Q − 2

p

Q " Q # Q 7 − 3 +q q Q Qr 2 p ∇ / D / SN η4Q Q Q

p,e

p,e

Q " Q # Q 1− 2 +q Q q " # p Qr w 5−1 ∇ / D / SN η3Q Q Q 2

p

.

p,e

The primary case of interest is p = 2. Let Kq = Kq2 and set K0 [δ] = e K0 (δ) e K1 [δ] = e K1 (δ) + e K1 (δ 3 ) + e K1 (δ 4 ) e K2 [δ] = e K2 (δ) + e K2 (δ 3 ) + e K2 (δ 4 ) e K3 [δ] = e K3 (δ) + e K3 (δ 3 ) + e K3 (δ 4 ) e

Define similar expressions for ϵ and N η. 0, 1, 2, 3 by the expression e

Then e Kq are defined for all q =

Kq = e Kq [δ] + e Kq [ϵ] + e Kq [N η] .

330

2. NORMS AND NOTATION

Furthermore set e

K2∗ = e K2 (DS δ 4 ) + e K2 (DS δ 3 )

/ S ϵ4 ) + e K2 (D / S ϵ3 ) + e K2 (D

+ e K2 (D / SN η 4 ) + e K2 (D / SN η 3 ).

In addition, in the exterior region define: Kqp,S

(δ) =

−1 r0 2 |||r −1

Kqp,S (ϵ) = r0 2 |||r −1

Kqp,S (N η ) = r0 2 |||r p,S Kq+1 (δ 4 )

= |||r

p,S Kq+1 (δ 3 ) = |||r p,S Kq+1 (ϵ4 )

"

5 − p2 +q 2

− 12 −1

−1 r0 2 |||r −1

p,S Kq+1 (N η 3 ) = r0 2 |||r

# #

" # 1− p2 +q

3 − p2 +q 2

p,S Kq+1 (ϵ3 ) = r0 2 |||r

(2.14)

"

5 − p2 +q 2

# " 3− p2 +q

= r0 |||r

p,S Kq+1 (N η4) =

"

"

q

∇ / q ϵ|||p,e

w3 ∇ / qN η |||p,e 2

∇ / δ 4 |||p,e

#

w3 ∇ / q δ 3 |||p,e 2

7 − p2 +q 2

#

" # 2− p2 +q "

∇ / q δ|||p,e

7 − p2 +q 2

and let

w3 ∇ / q ϵ3 |||p,e

#

" # 1− p2 +q

∇ / q ϵ4 |||p,e 2

∇ / qN η 4 |||p,e

w5 ∇ / qN η 3 |||p,e 2

K0p,S [δ] = K0p,S (δ)

K1p,S [δ] = K1p,S (δ) + K1p,S (δ 2 ) + K1p,S (δ 4 )

Define similar expressions for ϵ and N η . Then for q = 0, 1 define Kqp,S = Kqp,S [δ] + Kqp,S [ϵ] + Kqp,S [N η ].

Finally, the basic norms for the second fundamental form become i i i i e e

K[0] = i K0

K[1] = i K[0] + i K1 K[2] = i K[1] + i K2 K[3] = i K[2] + i K3 K[0] = e K0

K[1] = e K[0] + e K1 + K03,S

2.4. NORMS INVOLVING THE LAPSE FUNCTION φ e e e

331

K[2] = e K[1] + e K2 + K13,S K[3] = e K[2] + e K3 + K23,S ∗ K[3] = e K[3] + e K2∗

K[q] = i K[q] + e K[q] .

2.4. Norms Involving the Lapse Function φ Following [5], let the expressions i Lpq and e Lpq represent the interior and exterior weighted Lp norms of the q + 1-covariant derivatives of the ϕ, where ϕ is the logarithm of the lapse function. Then let Lqp denote 2 3 Lqp = max i Lqp , eLqp . Define i Lqp as

i

2+q− p3

Lqp = r0

Q q+1 Q QD ϕQ . p,i

To define the exterior expression decompose ∇ϕ as follows: φ/A = ∇ /A ϕ ϕN = ∇N ϕ and define e

Lpq (φ) /

=

−1 r0 2

Q Q " # Q 5 − 3 +q q Q Qr 2 p ∇ / φ/Q Q Q

p,e

Q " Q # Q 3 − 3 +q q Q e p 2 p Q Lq (ϕN ) = r0 Qr ∇ / ϕN Q Q 1 2

p,e

e

e

Lpq+1 (D4 φ) / = Lpq+1 (D3 φ) / =

−1 r0 2 −1 r0 2

Q Q " # Q Q 7 − 3 +q q Qr 2 p ∇ / D / 4 φ/Q Q Q

p,e

Q Q " # Q Q 7 − 3 +q q Qr 2 p ∇ / D / 3 φ/Q Q Q

p,e

Q " Q # Q 3 − 3 +q q Q e p 2 p Lq+1 (D4 ϕN ) = r0 Q ∇ / D4 ϕN Q Qr Q 3 2

p,e

Q " Q # Q 3 − 3 +q Q q e p " # 2 p Lq+1 (D3 ϕN ) = r0 Q w 3−1 ∇ / D3 ϕN Q Qr Q 3 2

2

e

− 12

Lpq+1 (DS φ) / = r0

p

p,e

Q Q " # Q 5 − 3 +q q Q Qr 2 p ∇ / φ/Q Q Q

p,e

Q " Q # Q 3 − 3 +q q Q e p 2 p Q Lq+1 (DS ϕN ) = r0 Qr ∇ / ϕN Q Q 1 2

p,e

332

2. NORMS AND NOTATION e

e

Lpq+2 (DS D4 φ) /

=

−1 r0 2 − 12

L2q+1 (DS D3 φ) / = r0

Q Q " # Q Q 7 − 3 +q q 2 p Qr ∇ / D / 4 φ/Q Q Q

p,e

Q Q " # Q Q 7 − 3 +q q Qr 2 p ∇ / D / 3 φ/Q Q Q

p,e

Q " Q # Q 3 − 3 +q q Q e p 2 p Q Lq+2 (DS D4 ϕN ) = r0 Q ∇ / D ϕ r 4 NQ Q 3 2

p,e

(2.15)

Q " Q # Q 3 − 3 +q Q q e p " # 2 p Q Lq+2 (DS D3 ϕN ) = r0 Qr w 3−1 ∇ / D3 ϕN Q Q 3 2

2

Then define

p

Lp0 [φ] / = e Lp0 (φ) / e p e p L1 [φ] / = L1 (φ) / + e Lp1 (D / 3 φ) / + e Lp1 (D / 4 φ) / e p e p e p e p L2 [φ] / = L2 (φ) / + L2 (D / 3 φ) / + L2 (D / 4 φ) / e p e p e p e p L1 [φ] / = L3 (φ) / + L3 (D / 3 φ) / + L3 (D / 4 φ). /

.

p,e

e

Define similar expressions ϕN . Then e Lpq are defined for all q = 0, 1, 2, 3 by the expression e p Lq = e Lpq [φ] / + e Lpq [ϕN ]. Furthermore set e Lq = e L2q and define e

Also define

L∗2 = e L2 (DS D / 4 φ) / + e L2 (DS D / 3 φ) /

+ e L2 (DS D4 ϕN ) + e L2 (DS D3 ϕN ).

Lp,S q

(φ) / =

Lp,S q (ϕN ) = Lp,S / 4 φ) / = q+1 (D Lp,S / 3 φ) / = q+1 (D Lp,S q+1 (D4 ϕN ) = (2.16)

Lp,S q+1 (D3 ϕN ) =

" # 5 − 12 − p2 +q 2 r0 |||r ∇ / q φ||| / p,e " # 2− p2 +q |||r ∇ / q ϕN |||p,e " # 7 − 12 − p2 +q 2 r0 |||r ∇ / qD / 4 φ||| / p,e " # 7 −1 − 2 +q r0 2 |||r 2 p ∇ / qD / 3 φ||| / p,e " # 2− p2 +q r0 |||r ∇ / q D4 ϕN |||p,e " # 1 − 2 +q r0 |||r 2 p w3 ∇ / q D3 ϕN |||p,e , 2

and set / = Lp,S / Lp,S 0 [φ] 0 (φ)

Lp,S / = Lp,S / + Lp,S / 3 φ) / + Lp,S / 4 φ), / 1 [φ] 1 (φ) 1 (D 1 (D

Define similar expressions ϕN . Then for q = 0, 1 set Lp,S / = Lp,S / + Lp,S q [φ] q [φ] q [ϕN ].

2.5. NORMS INVOLVING THE HESSIAN OF THE OPTICAL FUNCTION

333

Finally, the basic norms for the lapse function become i i i i e e e e e

L[0] = i L0

L[1] = i L[0] + i L1 L[2] = i L[1] + i L2 L[3] = i L[2] + i L3 L[0] = e L0

L[1] = e L[0] + e L1 + L3,S 0 L[2] = e L[1] + e L2 + L3,S 1 L[3] = e L[2] + e L3 L∗[3] = e L[3] + e L∗2

with L[q] = i L[q] + eL[q] . Lastly, define L∞ norms as follows 2i ∞ e ∞ 3 L∞ 0 = max L0 , L0 2i ∞ e ∞ 3 L∞ 1 = max L1 , L1 . 2.5. Norms Involving the Hessian of the Optical Function Following [5], let the expressions i Oq and e Oq represent the interior and exterior norms for components of the Hessian of the optical function. Define Oq as 2 3 max i Oq ,e Oq . In the interior region decompose the Hessian of u with respect to the standard null frame and define 3 " Q % %Q $ +q− p3 Q Q i j k$ i q / D3 D4 trχ − trχ Q Op trχ − trχ = r02 Q∇ i+j+k=q

i

3

Opq (N χ) = r02

i

i

3

Opq (ζ) = r02 3

Opq (ω) = r02

+q− p3

"

i+j+k=q +q− p3

"

i+j+k=q +q− p3

"

i+j+k=q

p,i

Q Q Q i j k Q / D3 D4 χ NQ Q∇

p,i

Q Q Q i j k Q / D3 D4 ζ Q Q∇

p,i

Q Q Q i j k Q / D3 D4 ω Q . Q∇ p,i

334

2. NORMS AND NOTATION

In the exterior region decompose the Hessian of u with respect to the l-pair null frame and set trχ3 = D3 trχ + 12 trχtrχ / 3χ N + trχN χ χ N3 = D ζ3 = D / 3 ζ + trχζ ω 3 = D3 ω

trχ4 = D4 trχ + 12 trχtrχ χ N4 = D / 4χ N + trχN χ ζ4 = D / 4 ζ + trχζ ω 4 = D4 ω.

Then define % % $ (2+q− p2 ) q $ ∇ / trχ − trχ |||p,e Oqp,S trχ − trχ = |||r Oqp,S (N χ) = |||r

(2+q− p2 )

Oqp,S (ζ) = |||r

(2+q− p2 )

∇ / q (N χ) |||p,e 2 % 3 $ Oqp,S (χ) = max Oqp,S trχ − trχ , Oqp,S (N χ)

Oqp,S (ω) = |||r Oqp,S (trχ3 )

= |||r

Oqp,S (trχ4 )

= |||r

Oqp,S

(N χ3 ) = |||r

Oqp,S (N χ4 ) = |||r Oqp,S (ζ 3 ) = |||r Oqp,S (ζ 4 )

= |||r

Oqp,S (ω 3 ) = |||r (2.17)

Oqp,S (ω 4 )

= |||r

∇ / q (ζ) |||p,e

( 12 +q− p2 )

w3 ∇ / q (ω) |||p,e 2

( 32 +q− p2 )

w3 ∇ / q (trχ3 )|||p,e 2

(3+q− p2 )

w3 ∇ / q (trχ4 )|||p,e 2

(3+q− p2 ) (3+q− p2 )

∇ / qχ N 3 |||p,e ∇ / qχ N 4 |||p,e

( 32 +q− p2 )

w3 ∇ / q ζ 3 |||p,e 2

(3+q− p2 )

q

∇ / ζ 4 |||p,e

( 12 +q− p2 )

w5 ∇ / q ω 3 |||p,e 2

(3+q− p2 )

q

∇ / ω 4 |||p,e .

Furthermore define O0p,S [trχ] = O0p,S (trχ)

O1p,S [trχ] = O1p,S (trχ) + O1p,S (trχ3 ) + O1p,S (trχ4 )

O2p,S [trχ] = O2p,S (trχ) + O2p,S (trχ3 ) + O2p,S (trχ4 ) In addition, define similar expressions for χ N , ζ, ω. Then for q = 0, 1, 2 set Oqp,S = Oqp,S [trχ] + Oqp,S [N χ] + Oqp,S [ζ] + Oqp,S [ω].

Furthermore, for q = 1, 2, 3 set Q Q Oq (χ) = (loc) Qr1+q ∇ / q χQ2,e Q Q e Oq (ζ) = (loc) Qr1+q ∇ / q ζ Q2,e . e

2.5. NORMS INVOLVING THE HESSIAN OF THE OPTICAL FUNCTION

Then for q = 1, 2 define e

Q Q 1 Q Q Oq (ω) = (loc) Qr− 2 +q w 3 ∇ / q ωQ

2,e

2

and for q = 3 set

e

− 12

Oq (ω) = r0

For q = 0, 1, 2 define

Q Q Q 3 Q / 3ωQ Qr w1 ∇

2,e

.

Q Q 1 Q Q Oq (trχ3 ) = (loc) Qr 2 +q w 3 ∇ / q (trχ3 )Q 2 2,e Q Q (loc) Q 2+q q e Oq (trχ4 ) = / (trχ4 )Q2,e r ∇ Q Q e Oq (N χ3 ) = (loc) Qr2+q ∇ / q (N χ3 )Q2,e Q Q e Oq (N χ4 ) = (loc) Qr2+q ∇ / q (N χ4 )Q2,e .

e

(2.18) For q = 0, 1 define

Q 1 Q Q Q Oq (ζ 3 ) = (loc) Qr 2 +q w 3 ∇ / q ζ 3Q 2 2,e Q Q (loc) Q 2+q q Q e r ∇ Oq (ζ 4 ) = / ζ 4 2,e Q 1 Q Q Q e Oq (ω 3 ) = (loc) Qr− 2 +q w 5 ∇ / q ω3Q 2 2,e Q Q e (loc) Q 2+q q Oq (ω 4 ) = / ω 4 Q2,e , r ∇ e

(2.19) and for q = 2 define

(2.20)

Q Q Q 3 Q / 2ζ 3Q Qr w1 ∇ Q2,e 1 Q Q 2 e O (ω ) = r − 2 Qr 2 w ∇ / ω 3 3 2 3Q 0 Q

eO

3 (ζ 3 )

− 12

= r0

2,e

Allow

Q Q Q Q = (loc) Qr4 ∇ / 2ζ 4Q Q Q2,e Q 2 e O (ω ) = (loc) Qr 4 ∇ / ω Q 3 4 4Q .

eO

3 (ζ 4 )

O0 [trχ] = e O0 (trχ) e O1 [trχ] = e O1 (trχ) + e O1 (trχ3 ) + e O1 (trχ4 ) e O2 [trχ] = e O2 (trχ) + e O2 (trχ3 ) + e O2 (trχ4 ) e O3 [trχ] = e O3 (trχ) + e O3 (trχ3 ) + e O3 (trχ4 ) e

Define similar expressions for χ N , ζ, ω. Then for q = 0, 1, 2 define e

Oq = e Oq [trχ] + e Oq [N χ] + e Oq [ζ] + e Oq [ω].

2,e

335

336

2. NORMS AND NOTATION

Finally the basic norms for the Hessian of H 3 H i ∞ i ∞ 2 O[0] = O0 + r0 sup HHtrχ − i i

i

∞ O[1]

O[2] = O[2] = =

O∞ 0

=

e

O[2] =

e

O∞ [0]

∞ O[0]

e

∞ O[1]

e

∞ O[0]

e

e

e

=

I ∞ + O1 ∞ i O2 + i O[1] ∞ i O3 + i O[2] + i O[1] i

O∞ [1]

i

H H + sup r Htrχ − r0 r≥

+

e

2

O∞ 1

2H

e + e O4,S 1 + O2

u are defined as follows: H 1 2 HH 2 sup |a − 1| + r 0 rH I

H 2 HH + sup r |a − 1| r H r≥ r0

e 4,S O[3] = e O[2] + e O3 + e O∞ [1] + O 2 ∞

∞ O[q] = i O[q] + e O∞ [q]

and

O[q] = i O[q] + e O[q] e

∞,S O∞ . q = Oq

2

CHAPTER 3

Existence Theorem The Existence Theorem for the solutions of the Einstein-Maxwell equations is stated below and is a generalization of the Main Theorem in [5]. The Existence Theorem mirrors the structure of Christodoulou and Klainerman’s Main Theorem; however, we change the nature of the field equations and incorporate terms that include components of the electromagnetic field. Theorem 1. Any strongly asymptotically flat, maximal initial data set that satisfies the global smallness assumption leads to a unique, globally hyperbolic, smooth, and geodesically complete solution of the EinsteinMaxwell equations, which is foliated by a normal maximal time function t, defined for all t ≥ −1. Moreover, there exists a global, smooth, exterior optical function u, namely a solution of the Eikonal equation defined everywhere in the exterior region r ≥ r20 , with r0 (t) representing the radius of the 2-surfaces of intersection between the hypersurfaces Σt and a fixed null cone C0 with vertex at a point on Σ−1 , with respect to which e ∞ W0∞ , e F0∞ , e K0∞ , e L∞ 0 , O0 ≤ ε 0 e W[2] , e F[3] , e K[3] , e L[3] , e O[3] ≤ ε0 .

e

Moreover, in the complement of the exterior region, i i

W0∞ , i F0∞ , i K0∞ , i L∞ 0 , ≤ ε0

W[2] , i F[3] , i K[3] , i L[3] , i O[3] ≤ ε0 .

The structure of the proof of the Existence Theorem is the same as the proof of the Main Theorem in [5]. The differences arise due to the extra terms introduced by the presence of the electromagnetic field F . As we discussed in the “Time Foliation” section above, the results of Theorem 5.0.1 in [5] still hold and F and k have same regularity and asymptotic behavior on the initial slice. Therefore the initial data set Σ, g, k, F satisfies the smallness assumptions: 0# 3 " Σ l=0

σ 2l+2 |∇l k|2 0 337

1 12

≤ε

338

3. EXISTENCE THEOREM

0# 3 " Σ l=0

σ 2l+2 |∇l F |2 0

1 12

≤ε

D0 ≤ ε

where d0 represents the distance from a point O in Σ and defines the radial $ %1 foliation. Furthermore, σ 0 = 1 + d20 2 and Q2 Q Q $ Q2 %Q2 Q D02 = Qσ 20 Q − Q Q2 + Qσ 30 QN Q2 + Qσ 30 ∇Q / Q2 Q2 Q Q2 Q Q Q2 + Qσ 40 ∇N QN Q2 + Qσ 40 ∇Q / N Q2 + Qσ 40 ∇ / 2 QQ2 Q Q2 Q Q2 Q Q2 +Qσ 20 P Q2 + Qσ 30 ∇P / Q2 + Qσ 30 ∇ / N P Q2 Q Q2 Q Q2 Q Q2 +Qσ 40 ∇ / 2 P Q2 + Qσ 40 ∇ /∇ / N P Q2 + Qσ 40 ∇ / 2N P Q2 Q Q2 Q Q2 Q Q2 +Qσ 20 SNQ2 + Qσ 30 ∇ / SNQ2 + Qσ 30 ∇ / N SNQ2 ' (2 Q2 Q Q2 Q Q2 Q +Qσ 40 ∇ / 2 SNQ2 + Qσ 40 ∇ /∇ / N SNQ2 + Qσ 40 ∇ / 2N SNQ2 + sup σ 30 Q Σ

where

@N N , Q = Ric

@ AN , PA = Ric

1 SAB = SNAB − Qγ AB . 2 @ with respect The latter set of equations represent the decomposition of Ric to the foliation. Define r to be the function that describes the area, namely 4πr2 , of foliation’s level-surfaces. Below we state the local existence theorem, which mirrors the local existence theorem cited in [5]; however, in this case it refers to the EinsteinMaxwell equations instead of the Vacuum-Einstein equations. @ AB , SAB = Ric

Theorem 2. Let (Σ, g0 , k0 , F0 ) be an initial data set verifying the following conditions: (1) (Σ, g0 ) is a complete Riemannian manifold diffeomorphic to R3 . (2) The isoperimetric constant I (Σ, g0 ) is finite, where I is defined to be V(S) sup 3/2 A(S) S where S is an arbitrary surface in Σ, A(S) its area, and V (S) the enclosed volume. (3) The Ricci curvature Ric (g0 ) verifies, relative to the distance function d0 from a given point O, Ric (g0 ) ∈ H2,1 (Σ, g0 ).

(4) k is a 2-covariant symmetric trace-free tensor field on Σ verifying k ∈ H3,1 (Σ, g0 )

3. EXISTENCE THEOREM

339

where for a given tensor field h, ∥h∥Hs,δ (Σ,g0 ) denotes the norm ∥h∥Hs,δ (Σ,g0 ) =

0# s " Σ i=0

H H2 σ 2i+2 H∇i0 hH dµg 0

0

11 2

$ %1 and σ 0 = 1 + d20 2 . (5) F |Σ is a 2-covariant anti-symmetric tensor field on Σ verifying F |Σ ∈ H3,1 (Σ, g0 ).

(6) (g0 , k0 , F0 |Σ ) verify the constraint equations on Σ. Then there exists a unique, local-in-time smooth development, foliated by a normal, maximal time function t with range in some interval [0, t∗ ] and with t = 0 corresponding to the initial slice Σ. Moreover, g (t) − g0 ∈ C 1 ([0, t∗ ] ; H3,1 (Σ, g0 )) k(t) ∈ C 0 ([0, t∗ ] ; H3,1 (Σ, g0 ))

F (t) |Σ ∈ C 0 ([0, t∗ ] ; H3,1 (Σ, g0 )) .

Furthermore,

Ric (t) ∈ C 0 ([0, t∗ ] ; H2,1 (Σ, g0 )) .

The proof follows the reasoning used in [5], which in turn is similar to the proof of the existence result in [2], with a few modifications. In our case we consider the following hyperbolic-elliptic system of equations . △φ = |k|2 + RT T φ $

− φ−2 ∂t

%2

∂t gij = −2φkij

. kij + △kij = Nij + ∂t Rij + ∇(i φ2 RTj) !F = 0.

where Nij depends on k, the first derivatives of k, Ric, φ, and ∂t φ together with the space derivatives of order ≤ 2 for the last two terms ([2]). As in [5], the worst term on the right-hand side of the wave equation for k is −∇∇∂t φ. Therefore estimates for k follow from [5]. That this system is equivalent to the Einstein-Maxwell equations and gives a unique solution is proved in [2]. We provide an outline of the proof. Following [5], employ an iteration argument with respect to (gn , kn , ∂t kn , Fn , ∂t Fn ) defined on the product manifold [0, t∗ ] × Σ. Associate the lapse function φn to these terms such that . △n φn = |kn |2n + |En |2n + |Hn |2n φn . and φn tends to 1 at ∞ on Σ at each t. Let gn+1 satisfy the following ∂t gn+1 = −2φn kn ,

340

3. EXISTENCE THEOREM

where gn+1 |t=0 is prescribed by the initial data for g. Then let kn+1 be a solution of .. $ %2 2 0 ∂ k + △ k = N + ∂ R + ∇ φ R − φ−2 t n+1 n n+1 n t ij (i n j) n

where kn+1 |t=0 , ∂t kn+1 |t=0 are prescribed by the initial data for k and ∂t k. Finally define Fn+1 to be a solution of the wave equation !n Fn+1 = 0

such that Fn+1 |t=0 , ∂t Fn+1 |t=0 agree with the given initial data for F and ∂t F . The proof then follows from [5] with the exception that we must provide estimates for F . To estimate Fn+1 , we employ classical energy estimates as in [6] and find that Fn+1 ∈ H2,1 (Σ, gn ). We prove that Fn+1 ∈ H3,1 (Σ, gn ) in Chapter 7 (see section 7.2). We now return to the proof of the Existence Theorem. Following [5], let S be the set of all t ≥ 0 for which there exists a space-time slab ∪t′ ∈[0,t] Σt′ endowed with an exterior optical function such that the bootstrap assumptions BA0 , BA1 , and BA2 hold: BA0 : For all t′ ∈ [0, t], % $ % 3$ % 1$ 1 + t′ ≤ r0 t′ ≤ 1 + t′ . 2 2 ′ BA1 : For all t ∈ [0, t], e ∞ W0∞ , F0∞ , K0∞ , L∞ 0 , O0 ≤ ε 0 .

BA2 : For all t′ ∈ [0, t],

∗ e ∗ e W[2] , F[3] , K[3] , L[3] , O[3] ≤ ε0 .

Also define the following Auxiliary Assumption: Let −1

5

/ log a| ≤ ε0 sup r0 2 r 2 |∇ Σt

on the last slice Σt of the space-time slab. Next we outline the five-step structure of the proof, which mirrors the structure of the proof of the Main Theorem in [5]. The details of the proof are discussed in the remaining chapters of this work. Step 1: The set S defined above is not empty; it contains t = 0. This follows from the local existence theorem, which allows for the construction of ∪t′ ∈[−1,0] Σt′ and thus also the construction of an initial cone C0 , which has a vertex on Σ−1 . By solving the inverse lapse problem, construct an exterior optical function u on t = 0. Let the distance function from S0,0 be the radial function. For small ε, which bounds the initial data, check that the norms e ∞ ∗ e ∗ e W0∞ , F0∞ , K0∞ , L∞ 0 , O0 and W[2] , F[3] , K[3] , L[3] , O[3] are arbitrarily small. These steps verify assumptions BA1 and BA2 . The auxiliary assumption follows from an analogue of Proposition 14.0.1 in [5].

3. EXISTENCE THEOREM

341

Step 2: Define t∗ = sup S. If t∗ = ∞, then the proof is complete. Therefore let t∗ < ∞. This implies that t∗ ∈ S. Extend the exterior optical function into the interior region. Step 3: Using BA0 , BA1 and BA2 , show that magnitudes of W0∞ , F0∞ , ∞ e ∞ ∗ e ∗ e K0 , L∞ 0 , O0 and W[2] , F[3] , K[3] , L[3] , O[3] cannot be greater than a constant multiple of ε. Choose ε and ε0 sufficiently small so that 1 e ∞ W0∞ , F0∞ , K0∞ , L∞ 0 , O0 ≤ ε 0 2 1 ∗ e ∗ e W[2] , F[3] , K[3] , L[3] , O[3] ≤ ε0 . 2 Step 3a : Use the Bootstrap Assumptions to verify the assumptions of the Comparison Theorems and show that quantities T and Q (defined in subsequent chapters) are bounded by the initial data. Then by the Comparison Theorems, F[3] and W[2] can be bounded by the initial data. Also show on the null hypersurface C0 that # O / ) |2 + r8 | ∇ / 2 α(W ) |2 r4 | α(W ) |2 + r6 | ∇α(W A= C0 P + r2 | α(F ) |2 + r4 | ∇α(F / ) |2 + r6 | ∇ / 2 α(F ) |2 + r6−δ | ∇ / 3 α(F ) |2 % $ ≤ c W[2] (0) + F[3] (0) R# .S 12 2 4 2 6 2 8 2 B= / ) | +r | ∇ / β(W ) | r | β(W ) | +r | ∇β(W C0 $ % ≤ c W[2] (0) + F[3] (0) . Therefore,

W[2] + F[3] + A + B ≤ cε.

Using BA1 , apply the Sobolev inequalities and conclude that W0∞ , F0∞ ≤ cε. For ε sufficiently small, conclude that 1 W[2] + F[3] + A + B ≤ ε0 2 1 W0∞ , F0∞ ≤ ε0 . 2 Step 3b : Using BA0 , BA1 and BA2 , and the results of step 2, show that $ % ∗ K[3] , L∗[3] ≤ c W[2] + F[3] .

Applying the Sobolev inequalities, conclude that $ % K0∞ , L∞ 0 ≤ c W[2] + F[3] .

342

3. EXISTENCE THEOREM

Thus for ε sufficiently small, 1 K0∞ , L∞ 0 ≤ ε0 2 1 ∗ K[3] , L∗[3] ≤ ε0 . 2 Step 3c : Using BA0 , BA1 and BA2 , show that O0∞ ≤ cε e O[3] ≤ cε. e

Thus for ε sufficiently small,

1 O0∞ ≤ ε0 2 1 e O[3] ≤ ε0 . 2 Step 4: Combining the local existence theorem and the last step above, extend the space-time slab from t∗ to t∗ + δ. In order to expand the exterior optical function u, which exists on the slab ∪t∈[0,t∗ ] Σt , extend the hypersurface Cu along its null geodesic generators into the future up to t∗ +δ. Choose e ∞ ∗ e ∗ δ sufficiently small so that W0∞ , F0∞ , K0∞ , L∞ 0 , O0 and W[2] , F[3] , K[3] , L[3] , e

−1/2

remain strictly less than ε0 . Also, the quantity supΣt∗ +δ r0 r5/2 |∇ / log a|, appearing in the auxiliary assumption, is strictly smaller than ε0 . On Σt∗ +δ construct a new optical function and extend it to the spacee ∞ time slab. Again show that the norms W0∞ , F0∞ , K0∞ , L∞ 0 , O0 and W[2] , ∗ e ∗ e F[3] , K[3] , L[3] , O[3] can be made arbitrarily small so that the bootstrap and auxiliary assumption hold. This implies that t∗ + δ ∈ S. However, this last step contradicts the assumption that t∗ < ∞. Step 5: Prove that the exterior optical function u, which is defined on the space-time slab, converges to a global exterior optical function as t → ∞. This follows from [5] and the properties of F . eO

[3]

CHAPTER 4

The Electromagnetic Field Exterior and interior estimates for F are performed separately. This division is necessary because the optical function is defined separately for the exterior and the interior regions and the resulting weights are different on the two regions. These differences are later smoothed over when we glue the two optical functions together. To obtain exterior estimates for the electromagnetic field F , consider the field equations Dα Fαβ = 0 Dα∗ Fαβ = 0

(4.1)

in conjunction with the energy-momentum tensor 1 (Fα ρ Fβρ + ∗Fα ρ ∗Fβρ ) . 8π The tensor ∗F is the Hodge dual of F and is defined as follows Tαβ =

1 Fαβ = F µν ϵµναβ 2 where ϵµναβ are coefficients of the volume form of the space-time slab (M, g), and ∗ (∗ F ) = −F . ∗

4.1. The Energy Momentum Tensor To prove the Existence Theorem, we need to obtain energy estimates for the electromagnetic field. These estimates are derived by using the stress energy tensor in conjunction with the Field equations for the electromagnetic field F . The stress energy tensor Tαβ associated with F satisfies the following well-known proposition. Proposition 4. The energy momentum tensor is symmetric, traceless and satisfies the following positivity condition: for any non-space-like futuredirected vector fields X and Y, T (X, Y ) ≥ 0.

For F , a solution of the Maxwell equations, T(F ) is divergence free. In the linear case for F , a solution of Maxwell’s equations, and X, a conformal Killing vector field, the Lie derivative LX F is also a solution of Maxwell’s equations. Therefore in the flat case we could derive energy estimates for F and all iterates of the Lie derivative of F by using Stokes’ Theorem. 343

344

4. THE ELECTROMAGNETIC FIELD

In the general case there are no killing or conformal killing vector fields. To combat this problem in [5], Christodoulou and Klainerman define almost conformal killing vector fields which are analogous to conformal killing vector fields in Minkowski Space. We use the same construction. Note that the quantity LX F for X an almost conformal killing field is no longer a solution of the homogeneous field equations 4.1 but satisfies an inhomogeneous equation. Also LX ∗F ̸= ∗LX F . For this reason we define a modified Lie derivative for F as follows (X) T] N X F = LX F − 1 [F L 2

where

and

(X) π N

(X) T

Nµα F µ β + (X) π Nµβ Fα β [F ]αβ = (X) π

is the traceless part of

(X) π.

Lemma 1. Let F be an electromagnetic field. Then, N X F is traceless and antisymmetric. (1) L (2) The modified Lie derivative commutes with the Hodge dual of F , that is, N X F. N X ∗F = ∗L L

N X F behaves like an electromagnetic field, i.e. Notice that the quantity L an anti-symmetric, traceless, 2-covariant tensor. It satisfies the following equations (4.2)

Dα Fαβ = Jβ Dα∗Fαβ = Jβ′

where the inhomogeneous terms arise from commuting the covariant derivative with the Lie derivative. We use the modified Lie derivatives of F to obtain estimates for the covariant derivatives of F . From now on we conN X F to be electromagnetic fields in the sense that they sider the quantities L are traceless, anti-symmetric covariant 2-tensors. Proposition 5. Let F be an electromagnetic field satisfying 4.2. Then Dα Tαβ = Proof. Since Tαβ =

% 1 $ µ Fβ Jµ + ∗Fβ µ Jµ′ . 8π

1 (Fα µ Fβµ + ∗Fα µ ∗Fβµ ), 8π

4.2. NULL DECOMPOSITION OF THE ELECTROMAGNETIC FIELD

345

it follows that 8πDα Tαβ = Jµ Fβ µ + Jµ′ ∗Fβ µ + F αµ Dα Fβµ + ∗F αµ Dα ∗Fβµ 1 = Jµ Fβ µ + Jµ′ ∗Fβ µ + F αµ (Dα Fβµ − Dµ Fβα ) 2 1 ∗ αµ ∗ + F (Dα Fβµ − Dµ ∗Fβα ) 2 1 1 = Jµ Fβ µ + Jµ′ ∗Fβ µ + F αµ Dβ Fαµ + ∗F αµ Dβ ∗Fαµ 2 2 1 ′∗ µ αµ µ = Jµ Fβ + Jµ Fβ + Dβ (F Fαµ + ∗F αµ ∗Fαµ ) 4 = Jµ Fβ µ + Jµ′ ∗Fβ µ "

since ⟨F, F ⟩ + ⟨∗F, ∗F ⟩ = 0.

We use this proposition to describe approximate conservation laws as Christodoulou and Klainerman do for the Bel-Robinson tensor in [5]. Assume (M, g) is foliated by the time function t. For an arbitrary vector field X, define the momentum Pα = T(F )αβ X β . With this definition . 1 DivP = X β (DivT )β + T αβ (X) π αβ . 2

In this case (X) π represents the deformation tensor of X. Integrating along the slab ∪t′ ∈[t0 ,t] Σt′ yields the following result:

Corollary 1. Let T(F ) be the energy momentum tensor of a solution of the inhomogeneous field equations 4.2 in a space-time foliated by a time function t. Then for a vector field X, # # T(F ) (X, T ) = T(F ) (X, T ) Σt

(4.3)

Σt0

+

#

t

t0

dt

′

J#

Σt′

R 1 (DivT)β X β + Tµν 2

(X)

S

K

π µν φdµg .

4.2. Null Decomposition of the Electromagnetic Field As in the case of the second fundamental form, we project the components of the electromagnetic field onto the surfaces of the 2-foliation. Estimating these null components provides information on the asymptotic decay of the electromagnetic field. Using the standard null pair e4 and e3 in [5], define a projection tensor 1 Πµν = gµν + (eν3 eµ4 + eµ4 eν3 ). 2 This tensor projects from the tangent space of M to the tangent space of St,u . With this projection and the vectors e4 and e3 , the null components

346

4. THE ELECTROMAGNETIC FIELD

of F are defined as follows α(F )µ = Πλµ Fλν eν3 α(F )µ = Πλµ Fλν eν4 1 ρ = Fµν eµ3 eν4 2 1 µν σ = ϵ Fµν 2 µν where ϵ is the area form of the St,u . Clearly, α(F ) and α(F ) are 1-forms tangent to St,u . Given an orthonormal from {eA : A = 1, 2} on St,u the following identities hold α(F )A = FA3 = EA − ϵB A HB , α(F )A = FA4 = EA + ϵB A HB , 1 ρ(F ) = F34 = −EN, 2 σ(F ) = F12 = −HN.

(4.4)

The corresponding null decomposition of {× α(F ),× α(F ),× ρ(F ),× σ(F )} of ∗F is given by ×

(4.5)

× α(F )A = −α(F )B ϵBA α(F )A = α(F )B ϵBA × × ρ(F ) = σ(F ) σ(F ) = −ρ(F ).

Recall that for a tensor u tangent to St,u , its Hodge dual ∗ u is defined by ∗

uA = ϵB A uB.

With this notation for an electromagnetic field F , the following relations hold for the energy-momentum tensor T (e3 , e3 ) = 2 | α(F ) |2 ,

T (e4 , e4 ) = 2 | α(F ) |2 , $ % T (e3 , e4 ) = 2 ρ(F )2 + σ(F )2 .

Given these relations together with the vector fields T and K = 12 (τ 2+ e4 + τ 2− e3 ), the following formula holds C $ % 1 2 τ | α(F ) |2 + τ 2+ | α(F ) |2 T(F ) K, T = 2 − D $ 2 %$ % 2 2 2 (4.6) + τ − + τ + ρ(F ) + σ(F ) . 4.3. Lie Coefficients of the Almost Conformal Killing Fields

To estimate $the derivatives of % $ %$the electromagnetic % $ %$field, %we consider $ the N N NX L N NY quantities T(F ) K, T , T LX F K, T , T LX LY F K, T , and T L %$ % N Z F K, T for X, Y, Z almost conformal killing fields. Commuting the Lie L derivative of an almost conformal killing field with the covariant derivative

https://doi.org/10.1090/amsip/045/02

4.3. LIE COEFFICIENTS OF THE ALMOST CONFORMAL KILLING FIELDS

347

gives rise to deformation tensors, which we will need to estimate. Analogous to Proposition 7.3.1 in [5], we use the commutators for a vector field X and the standard null frame to compute the formulas for Lie derivatives of F and its null decomposition. Let X be an arbitrary vector field. The following formulas represent the commutators of X with respect to the null frame e1 , e2 , e3 , e4 : [X, e3 ] = (X) P A eA + (X) M e3 + (X) N e4 [X, e4 ] = (X) PA eA + (X) N e3 + (X) M e4 1 1 [X, eA ] = Π [X, eA ] + (X) QA e3 + (X) QA e4 2 2 where (X)

P A = g (DX e3 , eA ) − D3 XA 1 1 (X) M = − g (DX e3 , e4 ) + D3 X4 2 2 (4.7) 1 (X) N = D3 X3 2 (X) QA = g (DX e3 , eA ) + DA X3

(X)

PA = g (DX e4 , eA ) − D4 XA 1 1 (X) M = − g (DX e4 , e3 ) + D4 X3 2 2 1 (X) N = D4 X4 2 (X) QA = g (DX e4 , e3 ) + DA X4

represent the Lie coefficients of X with respect to the null frame. Note that N44 4(X) N = (X) π

4(X) N = (X) π N33

(X) QA − (X) PA = (X) π N4A QA − (X) P A = (X) π N3A . 1 2 (X) M + (X) M = (X) π 34 = (X) π N34 − tr(X) π. 2 (X)

Using these formulas, we derive the relationship between the null components of the Lie derivatives of F and the Lie derivatives . of- a null.components . NX F , ρ L NX F , N of F . Let α(F ), α(F ), ρ(F ), σ(F ), and α LX F , α L . N X F , respectively. Let N X F denote the null decompositions of F and L σ L L /X α(F ), L /X α(F ), L /X ρ(F ), L /X σ(F ) denote the projection onto St,u of LX α(F ), LX α(F ), LX ρ(F ), LX σ(F ). Then -

NX F α L

.

A

' ( 1 (X) 1 (X) (X) =L /X α(F )A − M− M − tr π α(F )A 2 4 . . 1 -(X) 1 (X) − QA + (X) PA ρ(F ) − QB + (X) PB σ(F )ϵAB 2 2 1 (X) − π NAB α(F )B 2

348

4. THE ELECTROMAGNETIC FIELD

( 1 (X) NX F α L =L /X α(F )A − M− M − tr π α(F )A 4 A ( ' 1 (X) 3 (X) 1 (X) − π NAB α(F )B + QA − P A ρ(F ) 2 2 2 . 1 -(X) QB + (X) P B σ(F )ϵAB − 2 ' ( . 1 (X) (X) (X) N ρ LX F = L /X ρ(F ) + M+ M + tr π ρ(F ) 2 1 (X) 1 (X) QA α(F )A − QA α(F )A − 2 2 ' ( . 1 (X) (X) (X) N σ LX F = L /X σ(F ) + M+ M + tr π σ(F ) 2 1 1 (4.8) − (X) Q∗A α(F )A − (X) Q∗A α(F )A . 2 2 -

.

'

(X)

(X)

4.3.1. Ricci Rotation Coefficients of the Null Frame. Following [5], write the deformation tensors of the almost conformal killing fields in terms of the ricci coefficients. This provides identities that relate the deformation tensors to the geometry of the St,u foliation. Christodoulou and Klainerman define the Ricci rotation coefficients of a null frame e1 , e2 , e3 , e4 as follows: ⟨DA e3 , eB ⟩ = χAB

⟨DA e4 , eB ⟩ = χAB

⟨D4 e3 , eA ⟩ = 2ζ

⟨D3 e4 , eA ⟩ = 2ζ

⟨D3 e3 , eA ⟩ = 2ξ

⟨D3 e3 , e4 ⟩ = 2ν ⟨DA e4 , e3 ⟩ = ϵA

⟨D3 e4 , eA ⟩ = 0

⟨D4 e4 , e3 ⟩ = 2ν

where χAB = θAB − kAB χAB = −θAB − kAB

ξ A = φ−1 ∇ / A φ − a−1 ∇ / Aa

/ A φ − ϵA ζ A = φ−1 ∇

ζ A = a−1 ∇ / A a + ϵA (4.9)

ν = −φ−1 ∇N φ + δ ν = φ−1 ∇N φ + δ.

In particular, χ is the second fundamental form of the St,u with respect to e4 .

4.3. LIE COEFFICIENTS OF THE ALMOST CONFORMAL KILLING FIELDS

349

Using this notation and following [5], calculate the deformation tensors of T , S, and K. The null components of (T ) π are (T )

π 44 = −2ν π 34 = 2δ (T ) π 33 = −2ν (T )

(T )

π AB = −2η AB

(T )

(T )

(4.10)

π A3 = 2ϵA + φ−1 ∇ / Aφ

π A4 = −2ϵA + φ−1 ∇ / A φ.

The Lie coefficients of T are given by (T )

P A = ζA − ζA 1 (T ) M= ν 2 1 (T ) N = (T ) π 33 4 (T ) QA = ξ A + ζ A + ϵA

(T )

PA = ζ A − ζ A 1 (T ) M= ν 2 1 (T ) N = (T ) π 44 4 (T ) QA = ζ A − ϵA .

To calculate the deformation tensor of S let r Λ = D4 r = φ−1 φtrχ 2 Λ = D3 r = D4 r − 2∇N r % r r $ −1 φ φtrχ − a−1 atrχ = a−1 atrχ + (4.11) 2 2 and D3 u = −2a−1 D4 u = 0 D3 v = 2Λ + 2a−1 D4 v = 2Λ.

Using this notation the deformation tensor of S is defined by the following formulas: (S)

π 44 = 2uν

(S)

π 34 = −2a−1 − 2Λ − uν + vν

(S) (S)

π 33 = −4Λ − 4a−1 − 2vν

π AB = vχAB − uχAB

(S)

(4.12) Using the fact that

π A3 = −uξ A + v (ϵA + ζ A ) . (S) π A4 = u ϵA − ζ A .

tr(S) π = 2a−1 + 2Λ − u(ν − trχ) + v(−ν + trχ),

350

4. THE ELECTROMAGNETIC FIELD

write the traceless components as (S) (S) (S) (S)

(S)

u v (ν + trχ) + (ν + trχ) 2 2 −1 = −4Λ − 4a − 2vν 1 = vN χAB − uN χAB + δ AB(S) π N34 2 = −uξ A + v (ϵA + ζ A ) . = u ϵA − ζ A .

π N34 = −a−1 − Λ − π N33

π NAB

(S)

(4.13)

π N44 = 2uν

π NA3 π NA4

The Lie coefficients of S are (S)

P A = v(ζ A − ζ A ) 1 (S) M = vν − a−1 2 1 (S) N = (S) π 33 4 1 (S) QA = v (ϵA + ζ A ) − uξ A 2

(S)

PA = u(ζ A − ζ A ) 1 (S) M = − uν − Λ 2 1 (S) N = (S) π 44 4 1 (S) QA = (uϵA + vζ A ). 2

Finally, the null components of the traceless part of the deformation tensor of K are given by (K)

π N44 = −2u2 ν . (K) π NA4 = u2 ζ A − ϵA π N34

(K)

π N33 = −8v(Λ + a−1 ) − 2v 2 ν

(K)

(4.14)

' ( u2 2 trχ + = 2u(a − 1) − 2v(Λ − 1) − 2 r ' ( 2 2 1 v trχ − + u2 ν + v 2 ν + 2 r 2

(K)

−1

π NA3 = u2 ξ A + v 2 (ϵA + ζ A ) 1 (K) π NAB = v 2 χ N AB + u2 χ N AB + δ AB(K) π N34 . 2

4.3.2. Null Decomposition of the Maxwell Equations. In this section, we describe the null decomposition of the Maxwell Equations. Using the field equations D[λ Fµν] = 0 and D[λ∗ Fµν] = 0,

4.4. THE COMPARISON THEOREM FOR F

351

we calculate the null decomposition of the Maxwell Equations as follows / A ρ(F ) + ϵAB ∇ / B σ(F ) D / 4 α(F )A = −∇ ( ' 1 1 − ζ A + ζ A + ζ B ϵAB + ζ B ϵAB ρ(F ) 2 2 3 1 + ζ B ϵAB σ(F ) + ωα(F )A − trχα(F )A − χ (4.15) N AB α(F )B 2 2

/ A ρ(F ) + ϵAB ∇ / B σ(F ) D / 3 α(F )A = ∇ ( ' 1 1 + ζ A + ζ A − ζ B ϵAB − ζ B ϵAB ρ(F ) 2 2 ( ' 1 1 1 (4.16) N AB α(F )B ζ B ϵAB − ζ B ϵAB σ(F ) + trχα(F )A + χ − 2 2 2 $ % D4 ρ(F ) = −divα(F / ) − trχρ(F ) − ζ − ζ · α(F ) (4.17) $ % (4.18) / ) − trχσ(F ) + ζ − ζ · ∗ α(F ) D4 σ(F ) = −curlα(F $ % (4.19) D3 ρ(F ) = divα(F / ) + trχρ(F ) + ζ − ζ · α(F ) $ % (4.20) / ) − trχσ(F ) + ζ − ζ · ∗ α(F ). D3 σ(F ) = −curlα(F 4.4. The Comparison Theorem for F In this section we show how to bound the first and second covariant derivatives of the components of F by the Lie derivatives of the components. Recall in Chapter 2, we defined the weighted L2 norms for F . We now introduce the following notation involving the stress-energy tensor T of an electromagnetic field F and the vector fields T , K, S, and O. #

# .$ % ˆ S F K, T dµ + T L

.$ % ˆ O F K, T dµ T L Σt #Σt # .$ .$ % % ˆ ˆ S F K, T dµ ˆ ˆ OL T LO LO F K, T dµ + T L T2 (F ) = Σt Σt # .$ % ˆ S F K, T dµ. ˆS L + (4.22) T L

(4.21) T1 (F ) =

Σt

Analogous to the Comparison Theorem in [5], the following Comparison Theorem for F provides bounds for F0 , F1 , F2 in terms of T1 (F ) and T2 (F ). Theorem 3. Assume that given a space-time (M, g) is endowed with a (t, u)-foliation. Consider an arbitrary electromagnetic field F that satisfies the homogeneous equations 4.1. Assume that the space-time verifies

352

4. THE ELECTROMAGNETIC FIELD

Assumptions 0 and 1 below and that there exist angular momentum vector fields (i) Ω. Then there exits a constant c such that if T1 (F ) is finite, 1

F0 + F1 ≤ cT1 (F ) 2 and

. 1 1 F[2] ≤ c T1 (F ) 2 + T2 (F ) 2 .

There are three assumptions for this theorem and these overlap with Bootstrap Assumptions for the Existence Theorem. Assumption 0: Assume that at each t, r is a function of u and satisfies dr ≤ 32 . Also assume that for all values of t, the constants the following: 23 ≤ du km , kM , am , aM , a1 , hm , hM , and φm of each slice Σt are uniformly bounded. Also define φm = (inf Σt φ)−1 . Assumption 1: Everywhere in the exterior region of the space-time let H$ %H sup r2 H ξ, ζ, ζ, ν, ν H ≤ ε0 Σet

H H% sup r |rtrχ − 2|, Hrtrχ − 2H ≤ ε0 Σet

$

sup r2 |N χ| ≤ ε0 Σt H H N H ≤ ε0 . sup rτ − Hχ

(4.23)

let

Σet

Assumption 2: Everywhere in the exterior region of the space-time $ % 5 / ξ, ζ, ζ, ν, ν |||4,e ≤ ε0 ||| r 2 ∇ $ % 5 ||| r 2 ∇ / trχ, trχ |||4,e ≤ ε0 5

(4.24)

||| r 2 ∇N / χ |||4,e ≤ ε0 3

||| r 2 τ − ∇N / χ |||4,e ≤ ε0 .

Besides these assumptions we use the following lemma, which follows directly from the Bootstrap Assumptions and the Sobolev inequalities. Lemma 2. The null components of satisfy the following: H H supΣet r H(S) P A H ≤ ε0 H H supΣet H(S) M H ≤ c H H H H supΣet r H(S) QA H ≤ ε0

(S) π N

and the Lie coefficients of S

H H supΣet r H(S) PA H ≤ ε0 H H supΣet H(S) M H ≤ c H H supΣet r H(S) QA H ≤ ε0 .

4.4. THE COMPARISON THEOREM FOR F

Also |||r∇ /

-

(S)

(S)

M,

(S)

353

.

(S)

π NAB |||4,e ≤ c . |||r∇ / (S) Q,(S) P |||4,e ≤ c

M , tr

-

π,

|||D / S(S) P |||4,e ≤ c

|||D / S(S) Q|||4,e ≤ c

|||D / S(S) M |||4,e ≤ c |||D / S(S) M |||4,e ≤ c

|||D / S(S) trπ|||4,e ≤ c

N|||4,e ≤ c |||D / S(S) π sup |rD / S Λ| ≤ c. Σet

We need other properties of the Lie coefficients of S and introduce these as we derive the estimates below. We also use Proposition 7.5.3 from [5], which describes the properties of the rotation vector fields (i) Ω. This proposition is reproduced below for convenience. Proposition 6. Assume there exists rotation vector fields (i) Ω that span the tangent space of St,u at every point in the space-time and that verify the following properties: Property 0: Given and S-tangent covariant tensor field on M there exists a constant cO such that # # # 2 2 2 c−1 r | ∇f / | dµ ≤ |L f | dµ ≤ c (|f |2 + r2 |∇f / |2 )dµγ O O γ γ O St,u

St,u

St,u

where γ is the metric induced on St,u ; dµγ is the corresponding area element on St,u ; and |LO f |2 = Σi=1,2,3 |L(i) Ω f |2 . Moreover, if f is either a 1-form or a symmetric 2-covariant traceless tensor tangent to the surfaces St,u , # # 2 −1 cO |f | dµγ ≤ |LO f |2 dµγ . St,u

St,u

Property 1: The Lie coefficients (O) N , (O) P , (O) Q, (O) Q, of (O) Ω are identically zero everywhere in the exterior region. The Lie coefficients (O) M are given by the formula ((i) Ω) M = −a−1(i) Ω (a) in the exterior region. Also, (O)

π 44 = (O) π 4A = 0;

A = 1, 2,

354

4. THE ELECTROMAGNETIC FIELD

and, for a given, small, positive number h, H.H 1 H H sup r 2 +h H (O) π 33 ,(O) π 34 H ≤ ε0 Σet

⎞1 2 H H " 2 1 H(O) H ⎠ +h ⎝ 2 sup r ≤ ε0 H π AB H Σet

⎛

A,B

in the exterior region. Property 2:

. ||| r1+h ∇ / (O) π 33 ,(O) π 34 |||4,e ≤ c " ||| r1+h(O) π AB |||4,e ≤ c A,B

everywhere in the exterior region. Now we begin the proof of the Comparison Theorem for F . Proof. By direct calculation we see that # C # H H $ H $ %$ % %H 1 N O F H2 N N O F )H2 + τ 2 Hα L τ 2− Hα(L T LO F K, T dµg ≥ + 2 Σet Σet D H $ H $ %H2 %H2 2H N 2H N H H dµg . + τ + ρ LO F + τ + σ LO F

From the commutation formulas, it follows that H .H2 H H N Hα LO F H H .H2 H H N Hα LO F H H .H2 H H N Hρ LO F H H .H2 H H N Hσ LO F H

. ≥ c−1 |L /O α(F )|2 − cε0 |α(F )|2 . ≥ c−1 |L /O α(F )|2 − cε0 |α(F )|2 + |ρ(F )|2 + |σ(F )|2 . ≥ c−1 |L /O ρ(F )|2 − cε0 |ρ(F )|2 . ≥ c−1 |L /O σ(F )|2 − cε0 |σ(F )|2 .

By choosing ε0 sufficiently small, and using Proposition 6 as well as the Poincare inequality for ρ and σ, # # 2 2 r |ρ − ρ| ≤ c r4 |∇ρ| / 2 #

Σet

Σet

Σet

r2 |σ − σ|2 ≤ c

#

Σet

r4 |∇σ| / 2

https://doi.org/10.1090/amsip/045/03

4.4. THE COMPARISON THEOREM FOR F

(note ρ = σ = 0) it follows that R# # .$ % −1 ˆ O F K, T dµg ≥ c T L Σet

+ + +

#

Σet

τ 2− |α(F )|2 + #

Σet

τ 2+ |ρ(F )|2

Σet

τ 2− τ 2+ |∇α(F / )|2

Σet

τ 4+ |∇ρ(F / )|2

# #

+

#

Σet

Σet

+

355

τ 2+ |σ(F )|2

+

#

τ 2+ |α(F )|2

#

Σet

Σet

τ 4+ |∇α(F / )|2

τ 4+ |∇σ(F / )|2

S .

To check that the first derivatives in the null directions are bounded in the weighted L2 -norm, we use the null decomposition of the Maxwell Equations. The only derivatives that we are missing are D / 4 α(F ) and D / 3 α(F ). In order N to estimate these we consider LS F . As before # C H # H .$ .H2 .H2 % 1 H H ˆ ˆ S F HH + τ 2 HHα L ˆ τ 2− Hα L T LS F K, T dµg ≥ F H S + e e 2 Σt Σt H H .H2 .H2 D H H 2 H 2 ˆ S F H + τ + Hσ L ˆ S F HH dµg . + τ + Hρ L Using Lemma 2 we have H(S) H supΣet rH P A H ≤ ε0 H(S) H supΣet H M H ≤ c H(S) H supΣet rH QA H ≤ ε0

H(S) H supΣet rH PA H ≤ ε0 H(S) H supΣet H M H ≤ c H(S) H supΣet rH QA H ≤ ε0 .

Furthermore by the commutation formulas H .H2 . H H N /S α(F )|2 − c |α(F )|2 + |ρ(F )|2 + |σ(F )|2 Hα LS F H ≥ c−1 |L H .H2 . H H N /S α(F )|2 − c |α(F )|2 + |ρ(F )|2 + |σ(F )|2 . Hα LS F H ≥ c−1 |L Therefore

#

Σet

# .$ % ˆ S F K, T dµg + T L −1

≥c

Y#

Σet

Σet

τ 2− |L /S α(F )|2

+

#

.$ % ˆ O F K, T dµg T L

Σet

τ 2+ |L /S α(F )|2

Z

.

Note L /S α(F )A = ΠρA LS α(F )ρ =D / s α(F )A −

. 1vχAB − uχAB α(F )B. 2

356

4. THE ELECTROMAGNETIC FIELD

Thus

#

Σet

τ 2+ |D / S α(F )|2

'# ≤c +

Σet

#

Σet

-

ˆS F T L -

ˆ OF T L

.$

.$

% K, T dµg

( K, T dµg . %

Using the null decomposition of the Maxwell Equations to bound the D / 3 α(F ) part of D / S α(F ), we find that '# # .$ % 2 2 2 ˆ S F K, T dµg τ + v |D / 4 α(F )| ≤ c T L Σet

+

Σet

#

Σet

-

.$

-

.$

ˆ OF T L

%

(

K, T dµg ,

which yields the desired result. We use the same method to show that following holds '# # .$ % 2 2 2 ˆ S F K, T dµg τ − u |D / 3 α(F )| ≤ c T L Σet

+

#

Σet

Σet

ˆ OF T L

( K, T dµg . %

The first part of the theorem is proved. To prove the second part of the theorem, we start by calculating # # C H H .$ .H2 .H2 % 1 H H 2 2 ˆ ˆ 2 F HH + τ 2 HHα L ˆ τ 2− Hα L T LO F K, T dµg ≥ F H + O O 2 Σet Σet H H .H2 .H2 D H H 2 H 2 2 H 2 ˆ ˆ + τ + Hρ LO F H + τ + Hσ LO F H dµg .

From the commutation formulas and the steps in the first part of this proof, the following holds # C H H .H2 .H2 H ˆ O F HH ˆ O F HH + τ 2+ HHL τ 2− HL /O α L /O α L Σet

+ 0#

≤c

H

H τ 2+ HL /O ρ Σet

T

-

-

H .H2 .H2 D H H 2 H ˆ ˆ LO F H + τ + HL /O σ LO F H dµg

ˆ2 F L O

.$

%

K, T dµ +

#

Σet

-

ˆ OF T L

.$

%

1

K, T dµg .

Using the commutation formulas again, we calculate ' ' ( . 1 (O) 1 (O) 2 (O) ˆ L /O α LO F =L /O α(F )A − L /O M − M − tr π α(F )A 2 4 A ( 1 + (O) π NAB α(F )B . 2

4.4. THE COMPARISON THEOREM FOR F

Therefore by Proposition 6, # H .H2 # H 2 H ˆ τ + HL /O α LO F H ≥ Σet

Σet

−c

H2

H

H H 2 τ 2+ HL /O α(F )H

#

Σet

− cε0

τ 2+ |α(F )|2 r2

#

Σet

357

τ 2+ |L /O α(F )|2

'H H H H H (O) H2 H (O) H2 / MH / M H + H∇ H∇

H H2 H H2 ( H H (S) H (O) H / / π + H∇tr π H + H∇ NAB H .

We estimate the last term by 'H # H2 H H H2 H H H2 ( H H H (O) H2 H H (S) 2 2 H (O) 2 (O) H / M H + H∇ τ + |α(F )| r H∇ π H + H∇ NAB H / M H + H∇tr / / π Σet

≤c

#

∞

τ 2+ r2

ro 2

0#

4

|α(F )|

Sr ′

11 # 2

H4 D 12 H H4 H H H H (S) (O) H + H∇tr / π H + H∇ / π NAB H

Sr ′

CH H H H H (O) H4 H (O) H4 / MH / M H + H∇ H∇

≤ cr0−1 ||| r2 α(F )|||24,e (||| r∇ / (O) M |||24,e + ||| r∇ / (O) M |||24,e + ||| r∇tr / (O) π|||24,e + ||| r∇ / (S) π NAB |||24,e ).

Given the non-degenerate form of the global Sobolev inequality, # . ||| r2 α(F )|||24,e ≤ c / )|2 + r2 |∇ / N α(F )|2 r2 |α(F )|2 + r2 |∇α(F Σet

≤c

#

Σet

' r2 |α(F )|2 + r2 |∇α(F / )|2 + r2 |D / 4 α(F )|2 2

2

+ r |D / 3 α(F )|

(

.

Therefore, 'H # H H H H2 H H H2 ( H H (S) H H (O) H2 H (O) H2 H τ 2+ |α(F )|2 r2 H∇ NAB H / M H + H∇tr / (O) π H + H∇ / π / M H + H∇ Σet

≤ cr0−1 ε0 T1 (F ).

Proceeding in the same manner it follows that # H2 # H2 # H H H H H 2 H 2 /O α(F )H + /O α(F )H + τ 2+ HL τ 2− HL Σet

+

#

Σet

Σet

H H2 H 2 H /O σ(F )H ≤ c τ 2+ HL

By Proposition 6, #

Σet

H

0#

H2

H H 2 τ 2+ HL /O α(F )H

Σet

Σet

≥c

-

ˆ2 F T L O

#

Σet

H2 H H H 2 /O ρ(F )H τ 2+ HL

.$

%

1

K, T dµ + T1 (F ) .

τ 4+ |∇ /L /O α(F )|2

358

4. THE ELECTROMAGNETIC FIELD

Commuting the Lie derivative and the tangential covariant derivative, it follows that # # # H2 H2 H H H H H 2 2 H 2 4 τ + HL τ + |∇ /L /O α(F )| − c τ 2+ H(O) ΓH |α|2 /O α(F )H ≥ c Σet

Σet

Σet

$ (O) % where (O) ΓABC = 12 ∇ / A(O) π BC − ∇ / C(O) π AB is the error that / B π AC + ∇ arises. Using Property 0 and Property 1 of Proposition 6, we conclude 1 0# # H2 H .$ % H 4 H 2 2 ˆ F K, T dµ + T1 (F ) . T L / α(F )H ≤ c τ + H∇ Σet

Σet

O

Estimates for the other terms follow from analogous steps. Using the null decomposition of the Maxwell Equations, we estimate the mixed tangential-null derivatives of the null components of F in terms of T1 (F ) and T2 (F ) with the exception of # # 2 2 4 τ + τ − |∇ /D / 3 α(F )| + τ 6− |∇ /D / 4 α(F )|2 Σet

≤c

0#

Σet

Σet

-

ˆS F ˆ OL T L

.$

%

K, T dµg +

#

Σet

ˆ2 F T L O

To estimate these other two terms, we write # .$ % ˆ S F K, T dµg ˆ OL T L Σet

≥

#

Σet

H -

H τ 2+ Hα

.H2 # H ˆ ˆ LO LS F H +

-

Σet

.$

%

1

K, T dµ + T1 (F ) .

H .H2 H ˆ S F HH . ˆ OL τ 2− Hα L

Using the commutation formulas again and Proposition 6, it follows that # H .H2 H ˆ OL ˆ S F HH τ 2+ Hα L Σet # # H .H2 .$ % H ˆ S F HH − cε0 ˆ S F K, T . ≥ τ 2+ HL T L /O α L Σet

Σet

Now,

-

ˆS F L /O α L

.

A

' ( 1 (S) 1 (S) (S) =L /O (L /S α(F ) − M − M − tr π α(F )A 2 4 . . 1 -(S) 1 (S) − QA + (S) PA ρ(F ) − QB + (S) PB σ(F )ϵAB 2 2 1(S) − π NAB α(F )B ). 2

4.4. THE COMPARISON THEOREM FOR F

Hence, # # 2 2 τ + |L /O L /S α(F )| ≤ Σet

Σet

#

-

ˆ OL ˆS F T L

.$

%

K, T + cε0

#

Σet

359

.$ % ˆ S F K, T T L

H .H2 H H +c τ 2+ HL NAB H |α(F )|2 /O (S) M,(S) M , tr(S) π,(S) π Σe # t # H H H2 H2 H H H 2 2 H (S) (S) +c τ + HL τ 2+ HL /O ( Q, P )H |ρ(F )| + c /O ((S) Q,(S) P )H |σ(F )| Σet

Σet

Σet

Σet

#

H.H2 H 2 H (S) +c τ+ H M,(S) M , tr(S) π,(S) π NAB H |L /O α(F )|2 e Σ # t # H2 H H H2 H H H H +c τ 2+ H((S) Q,(S) P )H |L /O ρ(F )|2 + c τ 2+ H((S) Q,(S) P )H |L /O σ(F )| .

.$ % & ˆ O F K, T using the The last four integrals can be bounded by c Σe T L t sup-norm estimates on the Lie coefficients of S. Using Assumption 1, the / |||4,e ≤ ε0 ), and the fact Bootstrap Assumption (||| r∇φ / |||4,e ≤ ε0 , ||| r∇a that r Λ = D4 r = φ−1 φtrχ 2 % r $ −1 −1 Λ = D3 r = a atrχ + φ φtrχ − a−1 atrχ 2 it follows that # H .H2 H H τ 2+ HL NAB H |α(F )|2 /O (S) M,(S) M , tr(S) π,(S) π Σet

. ≤ cr0−1 ||| r∇ / (S) M,(S) M , tr(S) π, (S) π NAB |||24,e ||| r2 α(F )|||24,e # H2 H H H τ 2+ HL /O ((S) Q,(S) P )H |ρ(F )|2 Σet

. ≤ cr0−1 ||| r∇ / (S) Q,(S) P |||24,e ||| rρ(F )|||24,e # H2 H H 2 H (S) (S) /O ( Q, P )H |σ(F )|2 τ + HL Σet

≤ cr0−1 ||| r∇ /

-

(S)

. Q,(S) P |||24,e ||| rσ(F )|||24,e

and all of these terms can be bounded above by cT1 (F ). Therefore # # .$ % 2 2 ˆ S F K, T + cT1 (F ). ˆ OL τ + |L /O L /S α(F )| ≤ T L Σet

Σet

. Recall that L /S α(F )A = D / s α(F )A − 12 vχAB − uχAB α(F )B . Using & /D / S α(F )|2 can be bounded by Proposition 6, we conclude that Σe τ 6+ |∇ t the right-hand side of the above inequality. By the Maxwell Equations

360

&

4. THE ELECTROMAGNETIC FIELD

6 /D / 3 α(F )|2 ≤ c Σe τ + |∇ t

#

Σet

&

.$ % ˆ ˆ L T L F K, T + cT1 (F ). Therefore, O O Σe t

τ 6+ |∇ /D / 4 α(F )|2

≤

#

Σet

+c

.$ % ˆ OL ˆ S F K, T T L

#

Σet

.$ % ˆ OL ˆ O F K, T + cT1 (F ). T L

& Note Σe τ 2− τ 4+ |∇ /D / 3 α(F )|2 can be bounded by the same quantity using the t same steps. H2 H2 H H & & H H H 2 H 2 / 3 α(F )H and Σe τ 6+ HD / 4 α(F )H . To Lastly, we need to estimate Σe τ 6− HD t t do so we calculate #

Σet

-

ˆS L ˆS F T L

.$

%

K, T ≥

#

Σet

H -

H τ 2+ Hα

.H2 # H ˆ ˆ LS LS F H +

Σet

Using the commutation formulas it follows that -

ˆS F ˆS L α L

.

A

H .H2 H ˆS L ˆ S F HH . τ 2+ Hα L

( . 1' . 1 (S) (S) (S) ˆ ˆS F =L /S α LS F − M − M − tr π α L 2 4 A . . 1 -(S) ˆS F − QA + (S) PA ρ L 2 . . . 1ˆS F . ˆ S F ϵAB − 1(S) π NAB α L − (S) QB + (S) PB σ L 2 2 B -

Then using the sup-norm estimates for the Lie coefficients of S, it follows that #

Σet

H -

H τ 2+ Hα

.H2 # H ˆ ˆ LS LS F H ≤

Σet

H

H /S α τ 2+ HL

-

# .H2 H ˆ LS F H − c

Σet

.$ % ˆ S F K, T . T L

Again applying the commutation formulas, it is apparent that -

ˆS F L /S α L

.

A

' ( 1 (S) 1 (S) (S) /S α(F ) − =L /S L M − M − tr π α(F )A 2 4 . . 1 -(S) 1 (S) − QA + (S) PA ρ(F ) − QB + (S) PB σ(F )ϵAB 2 2 ( 1(S) − π NAB α(F )B . 2 '

4.4. THE COMPARISON THEOREM FOR F

Therefore # # τ 2+ |L /S L /S α(F )|2 ≤ Σet

Σet

#

# .$ % ˆ S F K, T + c ˆS L T L

Σet

361

.$ % ˆ S F K, T T L

H .H2 H H +c /S (S) M,(S) M , tr(S) π,(S) π τ 2+ HL NAB H |α(F )|2 e Σ # t # H2 H H H2 H H H 2 2 H (S) (S) τ 2+ HL /S ( Q, P )H |ρ(F )| + c /S ((S) Q,(S) P )H |σ(F )| +c τ + HL Σet

Σet

Σet

Σet

#

H.H2 H H +c NAB H |L τ 2+ H (S) M,(S) M , tr(S) π,(S) π /S α(F )|2 Σe # t # H2 H2 H H H H 2 2 H (S) (S) 2 H (S) (S) τ + H( Q, P )H |L /S σ(F )|. +c τ + H( Q, P )H |L /S ρ(F )| + c

The last four integrals can be bounded by cT1 (F ). We need to estimate the L /S of the Lie coefficients of S. Since L /S(S) PA = ΠρA LS(S) Pρ

1 1 =D / S(S) PA + vχAB(S) PB − uχAB(S) PB 2 2

and D / S(S) PA

- .. =D / S u ζA − ζA . . / S ζA − ζA =D / S u · ζ A − ζ A + uD . . / S ζA − ζA , = au ζ A − ζ A + uD

it follows from Assumptions 1 and 2 that

|||D / S(S) P |||4,e ≤ c.

Similar reasoning leads us to deduce that

|||D / S(S) Q|||4,e ≤ c.

/ 3a = D / 4 a − 2∇N a, we have from the Bootstrap Since D / 4 a = −ν and D Assumptions that |||D / S(S) M |||4,e ≤ c. We also have

|||D / S(S) M |||4,e ≤ c.

This estimate follows from the fact that

sup |rD / S Λ| ≤ c Σet

(see 5.80 and 5.54). The estimates |||D / S(S) trπ|||4,e ≤ c

362

4. THE ELECTROMAGNETIC FIELD

and

N|||4,e ≤ c |||D / S(S) π follow from the calculations we have just made and the assumptions for the Comparison Theorem for F . From the previous calculation, L /S α(F )A = D / s α(F )A − 12 (vχAB − uχAB ) α(F )B . Therefore using Assumption 2, we deduce that # H2 # H .$ % H 2 H 2 ˆ S F K, T + cT1 (F ) ˆS L τ + HD T L / S α(F )H ≤ Σet

Σet

and consequently, #

Σet

H

H2

H H 2 τ 2+ v 4 HD / 4 α(F )H

≤ c (T2 (F ) + T1 (F )).

Proceeding in the same manner, we can show that # H2 H H 2 4H 2 τ − u HD / 3 α(F )H ≤ c (T2 (F ) + T1 (F )). Σet

. & ˆT L ˆT F Because u goes to zero in the wave zone, we use the integral Σe T L t H2 H % $ & H H 2 2 / 3 α(F )H . Therefore we can conclude that K, T to estimate Σe τ − HD t # H2 H H 6 H 2 τ − HD / 3 α(F )H ≤ c (T2 (F ) + T1 (F )), Σet

which ends the proof of the theorem in the exterior region. We will show how to estimate the derivatives of the components of the electromagnetic field on the interior in Chapter 6. "

CHAPTER 5

Error Estimates for F This section estimates the error terms that arise when deriving energy estimates for F . These error terms occur for two reasons: (1) the almost conformal killing fields have nonzero deformation tensors, and (2) the Lie derivative does not commute with the covariant derivative. To calculate ν these error terms, recall that Pµ = Tµν K is the momentum tensor associated to K and that . 1 β (5.1) DivP = K (DivT )β + Tαβ (K ) π αβ . 2 Using this formula, we see that for a vector field X, # # .$ .$ % % ˆ ˆ X F K, T T LX F K, T = T L Σt

Σt0

+

#

t

dt

′

t0

J#

Σt′

R . ˆX F Kµ φDivT L

S D .µν 1 -ˆ K) ( + T LX F π µν dµg . 2 We calculate (5.2) where

. ˆ X Fµ ν J ′ ˆX F ˆ X Fµ ν Jν + ∗L DivT L =L ν µ

ˆ X Fµν Jν = Dµ L = (5.3)

1 (X) λµ 1 π ˆ Dµ Fλν + Dµ (X) π ˆ µλ F λ ν 2 2 . 1(X) + Dν(X) π ˆ µλ − Dλ π ˆ νµ F µλ . 2

and Jν′ is just Jν with F replaced by ∗ F . From the definition of T1 (F ) in 4.21, we have (5.4)

T1 (F, t) ≤ T1 (F, 0) + E1 (F, t), 363

µ

364

5. ERROR ESTIMATES FOR F

where the error # E1 (F, t) = (5.5)

term is given by H H # H H . . H H H H ˆ OF K µH + ˆ S F K µH HφDivT L HφDivT L H H H H µ µ Vt H H Vt # H # H . . H H H H 1 ˆ O F (K )π µν H + 1 ˆ S F (K ) πµν H . HφT L HφT L + H H H H 2 2 µ µ Vt

Vt

Similarly, from 4.22 we have, (5.6)

T2 (F, t) ≤ T2 (F, 0) + E2 (F, t),

where the error term is given by H # H H # H . . H H H H ˆ OL ˆ OL ˆ OF K µH + ˆ S F K µH HφDivT L HφDivT L E2 (F, t) = H H H H µ µ Vt Vt H H H H # # . . H H 1 H H µ K ( ) ˆS L ˆ OL ˆS F K H + ˆ OF HφDivT L HφT L π µν HH + H H H 2 Vt µ µ Vt H H # H # H . . H H 1 H H 1 K) K) ( ( ˆ ˆ ˆ ˆ H H H + π µνH + π µνHH. (5.7) φT LO LS F φT LS LS F H H 2 2 µ µ Vt

Vt

When bounding these error terms, we must ensure that each term decays quickly enough so that integration with respect to time does not produce logarithmic divergence. As in [5], to avoid these log terms, we sometimes first integrate along the light cones Cu and then in the spatial direction. This process is useful because the weight τ − , which involves only u, does not contribute decay in the time-like direction in the Wave Zone. Therefore we define the following quantities for the exterior region: & & ˆ O F )(K, e4 ) + ˆ Definition 2. TE1 (F ; u, t) = Cu T(L Cu T(LS F )(K, e4 ) $ % & & ˆ O F ) K, e4 + ˆ OL ˆ ˆ Definition 3. TE2 (F ; u, t) = Cu T(L Cu T(LO LS F ) . % & $ ˆ S F (K, e4 ) ˆS L K, e4 + Cu T L Definition 4. T1 ∗ = sup T1 (F, t) + sup sup TE1 (F ; u, t) [0,t∗ ]

[0,t∗ ]u≥u0

Definition 5. T2 ∗ = sup T2 (F, t) + sup sup TE2 (F ; u, t) [0,t∗ ]

[0,t∗ ]u≥u0

We also note that $ % 1A 2 $ %B T(F ) K, e4 = τ + |α(F )|2 + τ 2− ρ(F )2 + σ(F )2 . 2 The above definitions together with the inequalities 5.4 and 5.6 lead to the following theorem in the exterior region: Theorem 4. 1. T1 ∗ ≤ c(T1 (0) + E1 (t∗ )) 2. T2 ∗ ≤ c(T2 (0) + E2 (t∗ ))

The remainder of this chapter will be devoted to proving the next result.

5. ERROR ESTIMATES FOR F

365

Theorem 5. 1. T1∗ ≤ cT1 (0) + c1 ε0 (T1∗ + T2-∗) . . 2. T2∗ ≤ cT1 (0) + cT2 (0) + c2 ε0 T1∗ + T2 ∗ + r0−δ T3 ∗ .

3. r0−δ T3 ∗ ≤ c (T1 (0) + T2 (0) + T3 (0)) + c2 ε0 (T1∗ + T2 ∗).

(See equation 5.123 for the definition of T3 (t).) To derive estimates for the two error terms, we write them with respect to the null components. This step is necessary because of the different weights for the different components. Using the standard null frame, the divergence of the stress energy tensor becomes . ˆ X Fµ ν Jν + K µ∗ L ˆ X Fµ ν J ′ ˆ X F K µ = K µL DivT L ν µ

1 ˆ 1 2ˆ 4 B = τ 2+ L X F4 J4 + τ + LX F4 JB 2 2 1 ˆ 1 2ˆ 3 B (5.8) + τ 2− L X F3 J3 + τ − LX F3 JB + dual. 2 2 B ˆ X F4 4 J4 and 1 τ 2 L ˆ The first two terms, 12 τ 2+ L 2 + X F4 JB , are the most difficult to estimate because they involve the largest weights in the Wave Zone. Therefore we focus on these. (X) λµ (1) We label the first term of Jν in 5.3 as Jν = 12 -π ˆ Dµ Fλν , the second . (2)

(3)

= 12 Dµ (X) π ˆ µλ F λ ν , and the third as Jν

as Jν

=

1 2

F µλ . Clearly,

(X)

Dν

(X)

π ˆ µλ − Dλ π ˆ νµ

Jν = Jν(1) + Jν(2) + Jν(3) .

(5.9) (1)

With this notation, J4 (X) = (X) λµ

π ˆ

Dµ Fλ4 =

(5.10) (1)

Similarly JB (X) = (X) λµ

π ˆ

Dµ FλB =

(5.11)

1 (X) λµ π ˆ Dµ Fλ4 2

where

1(X) 1 1 π ˆ 44 D3 F34 − (X) π ˆ C3 D4 FC4 − (X) π ˆ C4 D3 FC4 4 2 2 1 − (X) π ˆ C4 DC F34 + 2N π CD DC FD4 . 2 1 (X) λµ π ˆ Dµ FλB 2

where

1(X) 1 1 π ˆ 44 D3 F3B + (X) π ˆ 33 D4 F4B + (X) π ˆ 34 D4 F3B 4 4 4 1 1 1 + (X) π ˆ 34 D3 F4B − (X) π ˆ C3 D4 FCB − (X) π ˆ C4 D3 FCB 4 2 2 1 1 − (X) π ˆ C4 DC F3B − (X) π ˆ C3 DC F4B + 2N π CD DC FDB . 2 2

(2)

ˆ µλ F λ 4 where Also J4 (X) = 12 Dµ (X) π Dµ (X) π ˆ µλ F λ (5.12)

4

1 1 1 = D4(X) π ˆ 43 F34 + D3(X) π ˆ 44 F34 − D4(X) π ˆ 3C FC4 4 4 2 1 − D3(X) π ˆ 4C FC4 + 4DD(X) π ˆ CD FC4 . 2

366

5. ERROR ESTIMATES FOR F (2)

Similarly JB (X) = 12 Dµ (X) π ˆ µλ F λ B where 1 1 1 = D3(X) π ˆ 34 F4B + D3(X) π ˆ 44 F3B + D4(X) π ˆ 33 F4B 4 4 4 1 1 1 + D4(X) π ˆ 34 F3B − D4(X) π ˆ 3C FCB − D3(X) π ˆ 4C FCB 4 2 2 1 1 (5.13) − DC(X) π ˆ 3C F4B − DC(X) π ˆ 4C F3B + 4DD(X) π ˆ CD FCB . 2 2 . (3) (X) (X) ˆ µλ − Dλ π ˆ 4µ F µλ where Finally, J4 (X) = 12 D4 π Dµ (X) π ˆ µλ F λ

-

(X)

D4

B

(X)

π ˆ µλ − Dλ

(5.14) (3)

and JB (X) = -

(5.15)

(X)

1 2

-

(X)

. . 1 - (X) (X) D4 π π ˆ 4µ F µλ = ˆ 34 − D3 π ˆ 44 F34 4 . 1 - (X) (X) − ˆ 34 − D3 π ˆ C4 FC4 DC π 2 . 1 - (X) (X) D4 π + ˆ C4 − DC π ˆ 44 FC3 . -2 (X) (X) + D4 π ˆ CD − DD π ˆ C4 FCD . (X)

ˆ µλ − Dλ DB π (X)

ˆ µλ − Dλ DB π

. π ˆ Bµ F µλ where

. 1 (X) π ˆ Bµ F µλ = (D4 π ˆ B3 )F34 4 . 1 - (X) (X) DC π − ˆ B3 − D3 π ˆ BC FC4 2 . 1 - (X) (X) − ˆ B4 − D4 π ˆ BC FC3 DC π -2 . (X) (X) + DB π ˆ CD − DD π ˆ BC FCD .

These terms are not yet written with respect to the null components of F . To accomplish this, we need to project DF and Dπ onto the tangent / 4 α(F )B , whereas D3 FB4 = spaces of the St,u . For example D4 FB4 = D D / 3 α(F )B + 2ζ B ρ(F ) + ζ A ϵBA σ(F ) − 2ωα(F )B . In following sections the J’s are written in terms of tensors tangent to the 2-foliation. Following [5], define notation for the null components of the deformation tensors. Definition 6. Given the deformation tensor (X) π of an arbitrary vector field X, the null decomposition of its traceless part (X) π N is defined as (5.16)

(X) i (X) π NAB ; AB = (X) m = (X) π N A 4A ; (X) n = (X) π N44 ;

(X) j

= (X) π N34 (X) π = N3A A (X) n = (X) π N33 . (X) m

5.1. PROOF OF THE FIRST PART OF THEOREM 5

367

5.1. Proof of the First Part of Theorem 5 & N O F )µ K µ , which imTo prove Theorem 5, we must bound Vt DivT(L & & 2 ˆ B ˆ O F4 4 J4 and plies that we need to bound Vt τ 2+ L Vt τ + LO F4 JB . We also Nµν and its first derivatives. From the Bootstrap Asneed estimates on (O) π sumptions it follows that (O)

m,(O) n = 0 Q Q Q (O) (O) (O) (O) Q ≤ ε0 Qr( i, j, m, n)Q

(5.17)

∞,e

and

. 3 |||r 2 ∇ / (O) i,(O) j,(O) m,(O) n |||4,e ≤ ε0 . 1 |||r 2 D / 3 (O) i,(O) j, (O)m,(O) n |||4,e ≤ ε0 . 3 |||r 2 D / 4 (O) i,(O) j,(O) m,(O) n |||4,e ≤ ε0 .

(5.18)

From formula 5.9, we have # # # (1) 2 ˆ 4 2 ˆ ˆ O F )J (2) | τ + |LO F4 J4 | ≤ c τ + |ρ(LO F )J4 | + c τ 2+ |ρ(L 4 Vt Vt Vt # ˆ O F )J (3) |. +c (5.19) τ 2+ |ρ(L 4 Vt

Using the co-area formula, we see that the first term in 5.19 can be bounded by # t# ˆ O F )J (1) |φdµg dt, τ 2+ |ρ(L 4 0

where

#

Σet

(5.20)

Σet

Q Q Q ˆ O F )J (1) | ≤ c Q ˆ τ 2+ |ρ(L rρ( L F ) Q Q O 4 1

≤ cT12

Q Q Q (1) Q QrJ4 Q

2,e

2,e

.

Q Q Q (1) Q rJ Q 4 Q

2,e

Using 5.10, we have 'Q Q Q Q Q Q Q 2 Q 2 Q Q Q Q Q (1) Q −2 Q (O) Q Qr D Qr ∇α(F / 4 α(F )Q2,e + Qr(O) iQ / )Q2,e Qr mQ QrJ4 Q ≤ cr0 2,e ∞,e ∞,e Q Q Q (O) Q + Qr iQ ∥rχ∥∞,e ∥rρ(F )∥2,e ∞,e Q Q Q Q + Qr(O) iQ ∥rχ∥∞,e ∥rσ(F )∥2,e ∞,e ( Q Q Q 2 Q Q (O) Q Qr ζ Q (5.21) + Qr iQ ∥rα(F )∥2,e . ∞,e ∞,e

The extra terms in the last three lines of 5.21 come from projecting the covariant derivative of the null components of the electromagnetic field onto the 2-surfaces.

368

5. ERROR ESTIMATES FOR F

Now we estimate #

(5.22)

Vt

(2)

ˆ O F )J |. τ 2+ |ρ(L 4

We have #

Σet

(5.23)

Q Q ˆ O F )J (2) | ≤ c Q ˆ O F )Q τ 2+ |ρ(L rρ( L Q Q 4 1 2

≤ cT1

Q Q Q (2) Q QrJ4 Q

2,e

2,e

.

Q Q Q (2) Q QrJ4 Q

2,e

In order to estimate these terms, we need to utilize the formulas for the projections of the covariant derivatives of the components of the deformation tensors of the almost conformal Killing fields, which are computed in [5] on pages 230–231. These are written as follows:

(5.24)

O P P O (X) (X) D3 π N43 = D / 3 j + Qr[ν; j] + Qr ζ;(X) m + Qr ξ;(X) m O P O P (X) (X) N33 = D / 4 n + Qr ν;(X) n + Qr ζ;(X) m D4 π O P O P (X) (X) N33 = D / 3 n + Qr ν;(X) n + Qr ξ;(X) m D3 π O P (X) (X) N44 = D / 4 n + Qr ν;(X) n D4 π P O P O (X) (X) N44 = D / 3 n + Qr ζ;(X) m + Qr ν;(X) n D3 π O P O P (X) (X) N43 = D / 4 j + Qr ν;(X) j + Qr ζ;(X) m . D4 π

Furthermore,

O .P P O P O (X) (X) N3A = D / 3 m + Qr ν;(X) m + Qr ζ;(X) n + Qr ξ; (X) i,(X) j D3 π O .P P O (X) (X) N3A = D / 4 m + Qr ν;(X) m + Qr ζ; (X) i,(X) j D4 π O P O P (X) (X) N4A = D / 4 m + Qr ν;(X) m + Qr ζ;(X) n D4 π O P P O (X) (X) N4A = D / 3 m + Qr ν;(X) m + Qr ν;(X) m D3 π .P O P O + Qr ζ; (X) i,(X) j + Qr ξ;(X) n O P P O (X) (X) NAB = D / 3 iAB + Qr ζ;(X) m + Qr ξ;(X) m D3 π O P (X) (X) NAB = D / 4 iAB + Qr ζ;(X) m . D4 π

(5.25)

5.1. PROOF OF THE FIRST PART OF THEOREM 5

369

Also, O P O P (X) (X) N33 = ▽ / A n − trχ(X) mA + Qr χ N ;(X) m + Qr ϵ;(X) n DA π . 1 - (X) (X) (X) DA π N34 = ▽ /A j− trχ mA + trχ(X) mA 2 P O P O (X) + Qr χ N; m + Qr χ N ;(X) m O P O P (X) (X) N44 = ▽ / A n − trχ(X) mA + Qr χ N ;(X) m + Qr ϵ;(X) n DA π

(5.26)

. 1 1 (X) (X) DA π N3B = ▽ / A mB − trχ(X) iAB − δ AB trχ(X) j + trχ(X) n 2P O O - 4 .P O P (X) + Qr χ N; n + Qr χ N ; (X) i,(X) j + Qr ϵ;(X) m . 1 1 (X) (X) DA π N4B = ▽ / A mB − trχ(X) iAB − δ AB trχ(X) j + trχ(X) n 2P O O - 4 .P O P (X) + Qr χ N; n + Qr χ N ; (X) i,(X) j + Qr ϵ;(X) m . 1 (X) (X) (X) (X) DA π NBC = ▽ / A iBC − trχ δ AC mB + δ AB mC 4 P . O P O 1 - (X) (X) (X) (X) − trχ δ AC mB + δ AB mC + Qr χ (5.27) m + Qr χ m . N; N; 4 Next we calculate Q Q Q (2) Q QrJ4 Q

2,e

≤

'

3 2

(5.28)

3

(O)

|||r 2 D / 4 m|||4,e |||r2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr ν,(O) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ, (O) i,(O) j Q ∥rα(F )∥2,e ∞,e Q Q Q (O) Q + Qr3 D / 3 mQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(O) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (O) i,(O) j Q ∥rα(F )∥2,e cr0−2

∞,e (O) 2 2 ▽ / C i|||4,e |||r α(F )|||4,e

+ |||r Q Q Q Q + Qr2 trχ(O) mQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rα(F )∥2,e ∞,e ( Q O PQ Q Q 3 (X) + Qr Qr χ N; m Q ∥rα(F )∥2,e . ∞,e

370

5. ERROR ESTIMATES FOR F

Now we must estimate (5.29)

#

Vt

We have

#

Σet

(5.30)

ˆ O F )J (3) |. τ 2+ |ρ(L 4

Q Q ˆ O F )J (3) | ≤ c Q ˆ O F )Q τ 2+ |ρ(L rρ( L Q Q 4 1 2

≤ cT1

Q Q Q (3) Q QrJ4 Q

2,e

2,e

.

Q Q Q (3) Q QrJ4 Q

2,e

In the estimates that follow we will often use the co-area formula: ( # ∞ '# # af dµy du f dµg = Σext

u0

and the sup norm on the spheres: |||f |||p,e = sup

u≥u0

We calculate that Q Q Q (3) Q QrJ4 Q

2,e

'#

p

|f | dµγ

(1/p

.

3

/ (O) j|||4,e |||r2 α(F )|||4,e ≤ cr0−2 (|||r 2 ▽ Q Q Q (O) Q + Qr3 D / 3 mQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν,(O) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ, (O) i,(O) j Q ∥rα(F )∥2,e ) ∞,e

+

3 3 −3 (O) cr0 2 (|||r 2 D / 4 j|||4,e |||r 2 ρ(F )|||4,e

Q O PQ Q Q + Qr3 Qr ν;(O) j Q 3

(O)

3

(O)

∞,e

∥rρ(F )∥2,e

3

+ |||r 2 D / 4 i|||4,e |||r 2 σ(F )|||4,e

(5.31)

3

+ |||r 2 ▽ / A m|||4,e |||r 2 σ(F )|||4,e Q Q Q Q + Qr2 trχ(O) iQ ∥rσ(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(O) j Q ∥rσ(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr χ N ; (X) i,(X) j Q ∥rσ(F )∥2,e ). ∞,e

Putting estimates 5.21, 5.28 and 5.31, we find that # ˆ O F4 4 J4 | ≤ cε0 T1 ∗ . τ 2+ |L (5.32) Vt

5.1. PROOF OF THE FIRST PART OF THEOREM 5

371

& ˆ O F4 B JB |. We have We must show the same estimate holds for Vt τ 2+ |L # # # (1) 2 ˆ B 2 ˆ ˆ O F )J (2) | τ + |LO F4 JB | ≤ c τ + |α(LO F )JB | + c τ 2+ |α(L B Vt Vt Vt # ˆ O F )J (3) |. +c (5.33) τ 2+ |α(L B Vt

As before, we estimate the first term in 5.33 as follows # Q Q Q Q Q (1) Q ˆ O F )Q ˆ O F )J (1) | ≤ c Q rα( L rJ τ 2+ |α(L Q Q Q B B Q Σet

(5.34)

1 2

≤ cT1

Q Q Q (1) Q QrJB Q

2,e

2,e

2,e

.

From formula 5.11, we can estimate 'Q Q Q Q Q Q Q 2 Q 2 Q Q Q Q Q (1) Q −2 Q (O) Q Qr D Qr D / 4 α(F )Q2,e + Qr(O) j Q / 3 α(F )Q2,e QrJB Q ≤ cr0 Qr nQ 2,e ∞,e ∞,e Q Q Q Q Q 2 Q Q Q Q Q (O) Q Q Qr ∇α(F Qr2 D + Qr mQ / )Q2,e + Qr(O) mQ / 4 σ(F )Q2,e ∞,e ∞,e Q Q Q 2 Q Q (O) Q Qr ∇σ(F + Qr iQ / )Q2,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(O) mQ ∥τ − α(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ξ Q + Qr(O) mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(O) mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q Q + Qr(O) mQ ∥rχ∥∞,e ∥rσ(F )∥2,e ∞,e Q Q Q Q + Qr(O) mQ ∥rχ∥∞,e ∥rρ(F )∥2,e ∞,e Q Q Q 2 Q Q Q Qr ξ Q + Qr(O) mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q Q Q Q QrχQ + Qr(O) iQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(O) j Q ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(O) j Q ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(O) j Q ∥rσ(F )∥2,e ∞,e ∞,e Q Q Q Q + r0 Qr(O) j Q ∥rτ − D / 4 α(F )∥2,e ∞,e ( Q Q Q (O) Q + r0 Qr iQ ∥rχ∥∞,e ∥τ − α(F )∥2,e . ∞,e (5.35)

372

5. ERROR ESTIMATES FOR F

The last two terms do not decay fast enough in the Wave Zone. If we estimate them as above, we will obtain logarithmic divergence in t. Instead, to estimate these terms we integrate over the null hypersurfaces Cu . We derive # ˆ O F )(O) iχα(F )| τ 2+ |α(L Vt

≤c ≤c

#

∞ '#

Cu

u0

Q Q Q Q ≤ c Qr(O) iQ ×

(5.36)

∞,e

0#

t

t0 (u)

Q Q Q Q 3 2 Q ∥rχ∥∞,e Qrτ − α(F )Q Q

∞,e

11

≤c ≤c ≤c

#

Cu

∞ '#

Cu

u0

#

× (5.37)

∞ '#

u0

#

∞

u0

'#

#

Su

#

∞

u0

H2 H H H τ 2+ H(O) iχα(F )H −3 τ−2

'#

−3 τ−2

Su

H

H2 ( 12 '#

Cu

0 H2 ( 12 # H H H ˆ O F )H τ 2+ Hα(L

t

H

Cu

Cu

τ 2+

H2 ( 12 R # H 2 H ˆ O F )HH τ + Hα(L

r6 τ 6− |D / 4 α(F )|4

≤ cε0 (T1 ∗ + T2 ∗) .

(1 S1 2

H2 ( 12

H H τ 2+ H(O)j D / 4 α(F )H

t0 (u)

'#

#

t

t0 (u)

Su

r0−2

'#

Su

2

Vt

(5.39)

#

Vt

ˆ O F )J (2) |. τ 2+ |α(L B

H2 ( 12

H2 H H H(O) / 4 α(F )H H jD

Therefore, we have shown that # ˆ O F )J (1) | ≤ cε0 (T1∗ + T2 ∗) . (5.38) τ 2+ |α(L B Now we estimate

2

H

≤ cε0 (T1∗ + T2 ∗) .

H ˆ H τ 2+ Hα(L O F )H

11

H H ˆ τ 2+ Hα(L O F )H

2

r0−2

Furthermore, # ˆ O F )(O) j D τ 2+ |α(L / 4 α(F )| Vt

t

t0 (u)

Cu

H2 ( 12

H

H H τ 2+ H(O) iχα(F )H

Cu

0 H2 ( 12 # H H 2 H ˆ O F )H τ + Hα(L

# ∞ '# u0

H2 ( 12 '#

H

H H ˆ τ 2+ Hα(L O F )H

H r H

2 H(O)

11 2

H4 ( 12 H jH

https://doi.org/10.1090/amsip/045/04

5.1. PROOF OF THE FIRST PART OF THEOREM 5

We have #

Σet

(5.40)

Q Q ˆ O F )J (2) | ≤ c Q ˆ O F )Q τ 2+ |α(L rα( L Q Q B 1 2

≤ cT1

Furthermore, Q Q Q (2) Q QrJB Q

2,e

−3

1

Q Q Q (2) Q QrJB Q

2,e

2,e

.

Q Q Q (2) Q QrJB Q

2,e

(O)

≤ cr0 2 (|||r 2 τ − D / 3 j|||4,e |||r2 α(F )|||4,e Q O PQ Q 3 (O) Q + Qr Qr ν; j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(O) m Q ∥rα(F )∥2,e 3 2

∞,e (O) D / 4 n|||4,e |||r2 α(F )|||4,e

3 2

∞,e 3 (O) D / 4 m|||4,e |||r 2 σ(F )|||4,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(O) n Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(X) m Q ∥rα(F )∥2,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(O) m Q ∥rσ(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (O) i,(O) j Q ∥rσ(F )∥2,e ∞,e Q Q Q (O) Q + Qr3 D / 3 mQ ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(O) m Q ∥rσ(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (O) i,(O) j Q ∥rσ(F )∥2,e ∞,e

3 2

+ |||r ▽ / (O)m|||4,e |||r2 α(F )|||4,e Q Q Q Q + Qr2 trχ(O) iQ ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(O) j Q ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(O) nQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(O) n Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr2 τ − Qr χ N ; (O) i,(O) j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(O) m Q ∥rα(F )∥2,e ∞,e

373

374

5. ERROR ESTIMATES FOR F 3

3

+ |||r 2 ▽ / (O) i|||4,e |||r 2 σ(F )|||4,e Q Q Q Q + Qr2 trχ(O)mQ ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rσ(F )∥2,e 1 2

3 2

∞,e

(O)

3

1

+ r0 |||r D / 4 j|||4,e |||r 2 τ −2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr ν;(O)j Q ∥τ − α(F )∥2,e ).

(5.41)

∞,e

Notice that the second to last term does not decay fast enough in the Wave Zone. The second to last term is estimated as follows # H H H (O) 2 H ˆ τ + Hα(LO F )D4 jα(F )H Vte

≤c

≤c ≤c

#

Cu

u0

# ∞ '#

Cu

u0

#

∞

u0

× (5.42)

∞ '#

0#

−3

τ−2 t

H2 ( 12 '#

H

H H ˆ τ 2+ Hα(L O F )H

t

#

t0 (u)

'#

H H2 ( 12 H 2 H ˆ τ + Hα(LO F )H

Cu

3 2

t0 (u)

Cu

0 H2 ( 12 # H H 2 H ˆ O F )H τ + Hα(L

1 2

(O)

H ( 12

H

H2 H (O) τ 2+ HD4 jα(F )H Su

H2 H H H (O) τ 2+ HD4 jα(F )H

3 2

r0−2 |||r D4 j|||24,e |||r τ − α(F )|||24,e

11 2

≤ cε0 (T1 ∗ +T2 ∗) .

Thus we have shown that # ˆ O F )J (2) | ≤ cε0 (T1∗ + T2 ∗) . (5.43) τ 2+ |α(L B Vt

Now we estimate #

(5.44)

Vt

ˆ O F )J (3) |. τ 2+ |α(L B

We have #

Σet

(5.45)

Q Q ˆ O F )J (3) | ≤ c Q ˆ O F )Q τ 2+ |α(L rα( L Q Q B 1 2

≤ cT1

Q Q Q (3) Q rJ Q B Q

2,e

2,e

.

Q Q Q (3) Q QrJB Q

2,e

11 2

5.1. PROOF OF THE FIRST PART OF THEOREM 5

375

Furthermore, Q Q Q (3) Q QrJB Q

2,e

≤

−3 cr0 2

'

3

|||r 2 τ − ▽ / (O) m|||4,e |||r2 α(F )|||4,e Q Q Q Q Q Q Q Q + Qr2 trχ(O) iQ ∥rα(F )∥2,e + Qr2 trχ(O)j Q ∥rα(F )∥2,e ∞,e ∞,e Q Q Q Q + Qr2 trχ(O) nQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(O) n Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr χ N ; (O) i,(O) j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(O) m Q ∥rα(F )∥2,e 1 2

∞,e (O) τ −D / 3 i|||4,e |||r2 α(F )|||4,e

+ |||r Q O PQ Q Q 3 (O) + Qr Qr ζ; m Q 3

(O)

∞,e

∥rα(F )∥2,e

3

+ |||r 2 D / 4 m|||4,e |||r 2 ρ(F )|||4,e Q O PQ Q Q + Qr3 Qr ν,(O) m Q ∥rρ(F )∥2,e 3 2

∞,e 3 (O) ▽ / C i|||4,e |||r 2 σ(F )|||4,e

+ |||r Q Q Q Q + Qr2 trχ(O)mQ ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rσ(F )∥2,e ∞,e

1 2

3

(O)

3

1

3

(O)

1

3

1

3

1

3

1

3

1

+ r0 |||r 2 D / 4 i|||4,e |||r 2 τ −2 α(F )|||4,e 1

3

+ r02 |||r 2 ▽ / A m|||4,e |||r 2 τ −2 α(F )|||4,e

+ r02 |||r 2 trχ(O) i|||4,e |||r 2 τ −2 α(F )|||4,e (5.46)

+ r02 |||r 2 trχ(O)j|||4,e |||r 2 τ −2 α(F )|||4,e ( Q O .PQ Q 3 Q (X) (X) + Qr Qr χ N; i, j Q ∥τ − α(F )∥2,e . ∞,e

1

The terms with the r02 in front do not decay fast enough in the Wave Zone. But they can be estimated precisely as in 5.42. Thus we have shown that (5.47)

#

Vt

ˆ O F )J (3) | ≤ cε0 (T1 ∗ +T2 ∗). τ 2+ |α(L B

376

5. ERROR ESTIMATES FOR F

. N S F . To prove the first part of the theo5.1.1. Bounds for DivT L & N S F )µ K µ . This process includes boundrem, we must also bound Vt DivT(L & & 2 ˆ B ˆ S F4 4 J4 and ing Vt τ 2+ L Vt τ + LS F4 JB . From formula 5.9, we have #

Vt

ˆ S F4 4 J4 | τ 2+ |L

≤c

#

Vt

+c

(5.48)

#

ˆ S F )J (1) | τ 2+ |ρ(L 4

Vt

+c

#

Vt

ˆ S F )J (3) |. τ 2+ |ρ(L 4

ˆ S F )J (2) | τ 2+ |ρ(L 4

Furthermore, #

Σet

(5.49)

Q Q ˆ S F )J (1) | ≤ c Q ˆ S F )Q τ 2+ |ρ(L rρ( L Q Q 4

2,e

1 2

≤ cT1

Q Q Q (1) Q To estimate QrJ4 Q [5]

2,e

Q Q Q (1) Q · QrJ4 Q

Q Q Q (1) Q QrJ4 Q

2,e

2,e

.

, we will use the following estimates from p. 224 of

Q Q Q (S) (S) (S) (S) Q ≤ ε0 Qr( i, j, m, n)Q ∞,e Q Q Q 2 −1 (S) (S) Q ≤ ε0 , Qr τ − ( m, n)Q ∞,e

which also follow from our Bootstrap Assumptions. We find that 'Q Q Q Q Q Q Q 2 Q 2 Q Q Q (1) Q Q Q −2 Q (S) Q Qr D Qr D / 4 α (F )Q2,e + Qr(S)mQ / 3 α(F )Q2,e QrJ4 Q ≤ cr0 Qr mQ 2,e ∞,e ∞,e Q Q Q Q Q 2 Q Q 2 −1(S) Q Q (S) Q Qr ∇ρ(F + Qr τ − nQ ∥rτ − D / 3 ρ(F )∥2,e + Qr mQ / )Q2,e ∞,e ∞,e Q Q Q Q Q Q Q Q (S) Q Q Q Q (S) Qr2 ξ Q + Qr iQ Qr2 ∇α(F / )Q2,e + Qr2 τ −1 nQ ∥rα(F )∥2,e − ∞,e ∞,e ∞,e Q Q Q 2 Q Q (S) Q Qr ζ Q + Qr2 τ −1 nQ ∥τ − α(F )∥2,e − ∞,e ∞,e Q Q Q Q Q Q 1 1 Q 2 −1(S) Q 2 Q Q 2 + Qr τ − nQ r τ ω ∥rρ(F )∥2,e r − 0 Q Q ∞,e

∞,e

Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)mQ ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)mQ ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q Q + Qr(S)iQ ∥rχ∥∞,e ∥rρ(F )∥2,e ∞,e Q Q Q Q + Qr(S)iQ ∥rχ∥∞,e ∥rσ(F )∥2,e ∞,e

5.1. PROOF OF THE FIRST PART OF THEOREM 5

(5.50)

Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)iQ ∥rα(F )∥2,e ∞,e ∞,e ( Q Q Q 2 −1(S) Q + Qr τ − mQ ∥rχ∥∞,e ∥τ − α(F )∥2,e . ∞,e

Therefore, we have

#

Σet

Now we estimate (5.51)

ˆ S F )J (1) | ≤ cε0 T1 ∗ . τ 2+ |ρ(L 4 #

Vt

We have

#

Σet

(5.52)

ˆ S F )J (2) |. τ 2+ |ρ(L 4

Q Q ˆ S F )J (2) | ≤ c Q ˆ S F )Q τ 2+ |ρ(L Qrρ(L Q 4

2,e

1 2

≤ cT1

Furthermore, Q Q Q (2) Q QrJ4 Q

2,e

3

Q Q Q (2) Q · QrJ4 Q

Q Q Q (2) Q QrJ4 Q

2,e

2,e

.

(S)

≤ cr0−2 (|||r 2 D / 4 m|||4,e |||r2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (S)i,(S)j Q ∥rα(F )∥2,e 3 2

∞,e (S) D / 3 m|||4,e |||r2 α(F )|||4,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (S)i,(S)j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ξ;(S) n Q ∥rα(F )∥2,e 3 2

∞,e

+ |||r ∇ /(S) i|||4,e |||r2 α(F )|||4,e Q Q Q Q + Qr2 trχ(S)mQ ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(S)mQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(S) m Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(S) m Q ∥rα(F )∥2,e ) ∞,e

377

378

5. ERROR ESTIMATES FOR F −3

1

3

(S)

+ cr0 2 (|||r 2 r0 D / 3 n|||4,e |||r 2 ρ(F )|||4,e Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rρ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(S) n Q ∥rρ(F )∥2,e 1 2

+ |||r Q O PQ Q Q + Qr3 Qr ν;(S) i Q ∥rρ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rρ(F )∥2,e ).

(5.53) 3 2

∞,e 3 (S) r0 D / 4 j|||4,e |||r 2 ρ(F )|||4,e

∞,e

(S) D4 m|||4,e

3

(S)

≤ cε0 and |||r 2 D3 m|||4,e ≤ cε0 follow from The estimates |||r the Bootstrap Assumptions and the definition of the deformation tensor of 3 S. The fact that |||r 2 ∇ / (S)i|||4,e ≤ cε0 can be see from the definitions 1 i = vN χAB − uN χAB + δ AB(S) π N34 2

(S)

and

u v (ν + trχ) + (ν + trχ) 2 2 / 2r φtrχ and and the Bootstrap Assumptions using the fact that ∇Λ / = φ−2 ∇φ Hr H 1 (S) sup H φtrχH ≤ c. To show |||r 2 r0 D j|||4,e ≤ cε0 we must consider D4 Λ (S)

j = −a−1 − Λ −

4

2

r Λ D4 Λ = −φ−2 D4 φ φtrχ + φ−1 φtrχ − φ−1 Λφtrχ 2 2 # # . r 2 −1 −1 r +φ φ D4 trχ + (trχ) + φ D4 φtrχ 2r2 Su 2r2 Su r Λ = −φ−2 D4 φ φtrχ − φ−1 φtrχ 2 ( ' #2 # 1 2 −1 r −1 r φ D4 trχ + (trχ) + φ aφ(trχ)2 +φ 2r2 Su 2 4r2 Su # −1 r (5.54) +φ D4 φtrχ. 2r2 Su Now,

−1

φ

r 4r2

(5.55)

#

Su

2

# r φtrχ = φ φ[(trχ)2 − trχtrχ] + l.o.t 2 2 4r Su # −1 r φtrχ[trχ − trχ] + l.o.t, =φ 4r2 Su

−1 Λ

φ(trχ) − φ

−1

where the lower order terms areH bounded Hby cε0 r−2 since r |a − 1| ≤ ε0 so |a| ≤ 1 + ε0 r−1 . Furthermore r2 Htrχ − trχH ≤ ε0 . In particular, H H sup Hr2 D4 ΛH ≤ cε0 Σet

5.1. PROOF OF THE FIRST PART OF THEOREM 5

or

379

3

(S)

|||r 3 D4 Λ|||4,e ≤ cε0 .

We can now estimate D4 j. We calculate u u (S) D4 j = a−2 D4 a − D4 Λ − D4 ν − D4 trχ + Λν 2 2 v v + D4 ν + D4 trχ + Λtrχ. (5.56) 2 2 Since v = 2r − u, we can combine u2 D4 trχ and part of v2 D4 trχ to obtain −uD4 δ. Recall that trχ = trθ + δ and trχ = −trθ + δ. Furthermore, D4 a = −ν. So we have u (S) D4 j = a−2 D4 ν − D4 Λ − D4 ν − uD4 δ + Λν 2 (5.57) rD4 ν + rD4 trχ + Λtrχ. However, (5.58)

D4 rtrχ = rD4 trχ + Λtrχ.

Recall that

d 1 χ|2 − R44 , trχ + (trχ)2 = − |N ds 2 where the right-hand side of the equation is bounded in absolute value by cε0 r−4 . We can write ( ' d d 1 2 rtrχ = r trχ + (trχ) − rtrχtrχ − Λtrχ ds ds 2 ( ' $ % 1 d 2 (5.59) trχ + (trχ) − rtrχ trχ − trχ + l.o.t. =r ds 2 Using the Bootstrap Assumptions, we have 1

(5.60)

(S)

|||r 2 r0 D4 j|||4,e ≤ cε0 .

Therefore, we have shown that # ˆ S F )J (1) | ≤ cε0 (T1 ∗ + T2 ∗) (5.61) τ 2+ |ρ(L 4 Vte

Now we estimate

(5.62)

#

Vt

We have

#

Σet

(5.63)

ˆ S F )J (3) |. τ 2+ |ρ(L 4

Q Q ˆ S F )J (3) | ≤ c Q ˆ S F )Q τ 2+ |ρ(L rρ( L Q Q 4

2,e

1 2

≤ cT1

Q Q Q (3) Q · QrJ4 Q

Q Q Q (3) Q QrJ4 Q

2,e

2,e

.

380

5. ERROR ESTIMATES FOR F

Furthermore, Q Q Q (3) Q QrJ4 Q

2,e

3

(S)

≤ cr0−2 (|||r 2 ▽ / A j|||4,e |||r2 α(F )|||4,e Q Q Q Q + Qr2 trχ(X) mQ ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(X) mQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(X) m Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr2 Qr χ N ;(X) m Q ∥rα(F )∥2,e 3 2

∞,e (S) D / 3 m|||4,e |||r2 α(F )|||4,e

+ |||r Q PQ O Q Q + Qr3 Qr ν;(S) m Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (S)i,(S) j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ξ;(S) n Q ∥rα(F )∥2,e ) ∞,e

+

1 3 −3 (S) cr0 2 (|||r 2 r0 D / 4 j|||4,e |||r 2 ρ(F )|||4,e

Q O PQ Q Q + Qr3 Qr ν;(S) j Q ∥rρ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rρ(F )∥2,e 1 2

∞,e 3 (S) r0 D / 3 n|||4,e |||r 2 ρ(F )|||4,e

1 2

∞,e 3 (S) r0 D / 4 i|||4,e |||r 2 σ(F )|||4,e

+ |||r Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rρ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(S) n Q ∥rρ(F )∥2,e + |||r Q O PQ Q Q + Qr3 Qr ζ;(S) m Q 3

(S)

∞,e

3

∥rσ(F )∥2,e

+ |||r 2 ▽ / A m|||4,e |||r 2 σ(F )|||4,e Q Q Q Q + Qr2 trχ(S)iQ ∥rσ(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(S)j Q ∥rσ(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(S)nQ ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr2 Qr χ N ;(S) n Q ∥rσ(F )∥2,e ∞,e

5.1. PROOF OF THE FIRST PART OF THEOREM 5

381

Q O .PQ Q Q + Qr3 Qr χ N ; (S)i,(S) j Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(S) m Q ∥rσ(F )∥2,e 5 2

∞,e

3

1 (S) D / 4 m|||4,e |||r 2 τ −2 α(F )|||4,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥τ − α(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(S) n Q ∥τ − α(F )∥2,e 5 2

∞,e

(S)

1

3

+ |||r ▽ /A π NB4 |||4,e |||r 2 τ −2 α(F )|||4,e 1

3

1

3

3

1

+ r02 |||r 2 trχ(S)π NAB |||4,e |||r 2 τ −2 α(F )|||4,e 3

1

+ r02 |||r 2 trχ(S)π N34 |||4,e |||r 2 τ −2 α(F )|||4,e −1

5

3

1

+ |||r 2 τ − 2 trχ(S)π N44 |||4,e |||r 2 τ −2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr χ N ;(S) π N44 Q ∥τ − α(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr χ N ; (S)π NAB ,(S) π N34 Q ∥τ − α(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(S) π N4A Q ∥τ − α(F )∥2,e

(5.64)

∞,e

1

The terms with the r02 need to be estimated as in 5.42. Once that is accomplished, we have #

(5.65)

Vt

(3)

ˆ S F )J | ≤ cε0 (T1∗ + T2 ∗). τ 2+ |ρ(L 4

We must show the same estimate holds for #

Vt

ˆ S F4 BJB | ≤ c τ 2+ |L

#

Vt

(1)

Vt

+c

(5.66)

&

#

ˆ S F )J | + c τ 2+ |α(L B

Vt

ˆ S F )J (3) |, τ 2+ |α(L B

ˆ S F4 B JB |. We have τ 2+ |L #

(2)

Vt

ˆ S F )J | τ 2+ |α(L B

where #

Σet

(5.67)

Q Q ˆ S F )J (1) | ≤ c Q ˆ S F )Q τ 2+ |α(L rα( L Q Q B

2,e

1 2

≤ cT1

Q Q Q (1) Q · QrJB Q

2,e

Q Q Q (1) Q QrJB Q

2,e

.

382

5. ERROR ESTIMATES FOR F

Furthermore, Q Q Q (1) Q QrJB Q

2,e

'Q Q Q 2 Q Q (S) Q Qr D ≤ / 4 α(F )Q2,e Qr nQ ∞,e Q Q Q 2 Q Q (S) Q Qr D + Qr j Q / 3 α(F )Q2,e ∞,e Q Q Q 2 Q Q (S) Q Qr ∇α(F + Qr mQ / )Q2,e ∞,e Q Q Q 2 Q Q (S) Q Qr D + Qr mQ / 4 σ(F )Q2,e ∞,e Q Q Q 2 Q Q (S) Q Qr ∇σ(F + Qr iQ / )Q2,e ∞,e Q Q Q 2 −1(S) Q + Qr τ − mQ ∥rτ − D / 3 σ(F )∥2,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)mQ ∥τ − α(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ξ Q + Qr(S)mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q Q + Qr(S)mQ ∥rχ∥∞,e ∥rσ(F )∥2,e ∞,e Q Q Q Q + Qr(S)mQ ∥rχ∥∞,e ∥rρ(F )∥2,e ∞,e Q Q Q 2 Q Q Q Qr ξ Q + Qr(S)mQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q Q Q Q QrχQ + Qr(S)iQ ∥rα(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)j Q ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)j Q ∥rρ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q Q Qr ζ Q + Qr(S)j Q ∥rσ(F )∥2,e ∞,e ∞,e Q Q Q 2 Q Q (S) Q Qτ − D + r0 Qr2 τ −1 nQ / 3 α(F )Q2,e − ∞,e Q Q Q Q + r0 Qr(S)j Q ∥rτ − D / 4 α(F )∥2,e cr0−2

Q Q Q Q + r0 Qr(S)iQ

(5.68)

∞,e

∞,e

( ∥rχ∥∞,e ∥τ − α(F )∥2,e .

The last three terms do not decay fast enough in the Wave Zone. But we can estimate them in the same manner as in 5.42. Thus, we obtain # ˆ S F )J (1) | ≤ cε0 (T1∗ + T2 ∗) . τ 2+ |α(L (5.69) B Vt

5.1. PROOF OF THE FIRST PART OF THEOREM 5

Now we estimate

#

(5.70)

Vt

We have (5.71)

#

Σet

ˆ S F )J (2) |. τ 2+ |α(L B

Q Q ˆ S F )J (2) | ≤ c Q ˆ S F )Q τ 2+ |α(L rα( L Q Q B

2,e

1 2

≤ cT1

where

Q Q Q (2) Q QrJB Q

2,e

≤

−3 cr0 2

'

1

(S)

Q Q Q (2) Q · QrJB Q

2,e

Q Q Q (2) Q QrJB Q

2,e

,

|||r 2 τ − (D / 3 j − rD / 3 ν)|||4,e |||r2 α(F )|||4,e

1

5

+ ||r02 w1 D / 3 ν||2,e ||r 2 α(F )||∞,e Q O PQ Q Q + Qr3 Qr ν;(S) n Q ∥rα(F )∥2,e ∞,e Q PQ O Q Q + Qr3 Qr ξ;(S) m Q ∥rα(F )∥2,e 1 2

∞,e (S) r0 D / 4 n|||4,e |||r2 α(F )|||4,e

3 2

∞,e 3 (S) D / 4 m|||4,e |||r 2 σ(F )|||4,e

3 2

∞,e 3 (S) D / 3 m|||4,e |||r 2 σ(F )|||4,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(S) n Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rα(F )∥2,e

+ |||r Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rσ(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (S)i,(X) j Q ∥rσ(F )∥2,e + |||r Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rσ(F )∥2,e 3 2

+ |||r ∇ / 3

(S)

∞,e

m|||4,e |||r2 α(F )|||4,e

+ |||r 2 trχ(S) i|||4,e |||r2 α(F )|||4,e 3

+ |||r 2 trχ(S) j|||4,e |||r2 α(F )|||4,e 3

+ |||r 2 trχ(S)n|||4,e |||r2 α(F )|||4,e O .P 3 + |||r 2 Qr χ N ; (S)i,(S) j |||4,e |||r2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr χ N ;(S) n Q ∥rα(F )∥2,e ∞,e

383

384

5. ERROR ESTIMATES FOR F

Q O PQ Q Q + Qr3 Qr ϵ;(S) m Q

∞,e

3

∥rα(F )∥2,e

3

+ |||r 2 ∇ / (S)i|||4,e |||r 2 σ(F )|||4,e 3

3

3

3

+ |||r 2 trχ(S)m|||4,e |||r 2 σ(F )|||4,e

+ |||r 2 trχ(S)m|||4,e |||r 2 σ(F )|||4,e Q PQ O Q Q + Qr3 Qr χ N ;(S) m Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(S) m Q ∥rσ(F )∥2,e ∞,e

5 2

3

1

+ |||r τ −1 / (S)m|||4,e |||r 2 τ −2 α(F )|||4,e − ∇ 1

3

1

3

1

3

1

3

+ r02 |||r 2 trχ(S) i|||4,e |||r 2 τ −2 α(F )|||4,e

+ r02 |||r 2 trχ(S) j|||4,e |||r 2 τ −2 α(F )|||4,e 5

−1

3

1

+ |||r 2 τ − 2 trχ(S)n|||4,e |||r 2 τ −2 α(F )|||4,e Q O PQ Q Q + Qr3 Qr χ N ;(S) n Q ∥τ − α(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr χ N ; (S)i,(S) j Q ∥τ − α(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(S) m Q ∥τ − α(F )∥2,e 1 2

1 2

∞,e

+ r0 |||r Q O PQ Q Q + Qr3 Qr ν;(S) j Q

∞,e

Q O PQ Q Q + Qr3 Qr ζ;(S) m Q

(5.72)

3

1 (S) r0 D4 j|||4,e |||r 2 τ −2 α(F )|||4,e

∥τ − α(F )∥2,e

∞,e

(

∥τ − α(F )∥2,e .

A B 1/2 Recall that wp = min τ p− r0 , rp and r ≤ c(rtrχ, a)τ − r0 everywhere on Σ. 1/2

1

/ 3 ν||2,e ≤ cε0 by the Bootstrap AssumpTherefore r ≤ cr0 w1 and ||r02 w1 D tions. The terms involving α(F ), which do not decay fast enough for this type of estimate, can be estimated as in 5.42 to avoid logarithmic divergence. We must show the first term is less than cε0 . To accomplish this, we calculate (S)

D3 j = a−2 D3 a − D3 Λ + a−1 (ν + trχ) + (Λ + a−1 )(ν + trχ) u v u − D3 ν + D3 ν − D3 δ + rD3 trχ 2 2 2 u v u −2 −1 = a D3 a − D3 Λ + a ν + Λν + a−1 δ − D3 ν + D3 ν − D3 δ 2 2 2 (5.73) + Λtrχ + rD3 trχ.

5.1. PROOF OF THE FIRST PART OF THEOREM 5

385

The last two terms can be written as ( ' 1 1 (5.74) Λtrχ + rD3 trχ = r D3 trχ + trχtrχ − trχrtrχ + Λtrχ. 2 2 Now D3 Λ = D3 D4 r = D4 D3 r = D4 Λ. By (5.75)

r r Λ = a−1 atrχ + (φ−1 φtrχ − a−1 atrχ), 2 2

and what we already have shown for Λ, it is enough to bound D4 a−1 2r atrχ. We find that r Λ r D4 a−1 atrχ = −a−2 D4 a atrχ − a−1 atrχ 2 2 2 # # $ % −1 r −1 r +a a D trχ + trχtrχ D4 atrχ + a 4 2r2 Su 2r2 Su r Λ 1 = −a−2 D4 a atrχ − a−1 atrχ + a−1 aφtrχtrχ 2 2 (4r ' # # r 1 −1 −1 r (5.76) +a a D4 trχ + trχtrχ + a D4 atrχ. 2r2 Su 2 2r2 Su Adding Λ 1 −a−1 atrχ + a−1 aφtrχtrχ, 2 4r we derive (5.77)

−1

a

r 4r2

Furthermore,

#

Su

$ % trχ trχ − trχ + l.o.t. D4 a = ν

(5.78) and (5.79) Altogether we have

Q ' (Q Q Q 3 Qr D4 trχ + 1 trχtrχ Q ≤ cε0 . Q Q 2 ∞,e Q Q 2 Qr D3 ΛQ ≤ cε0 . ∞,e

(5.80)

(S)

Therefore the only term in D3 j that cannot be bounded in the ||| |||4,e 1

1

norm is rD3 ν. The second term, is bounded by cε0 T12 ∗ since r02 ||w1 D3 ν||2,e ≤ cε0 . To bound the last term # (S) ˆ S F )D τ 2+ |α(L /4 π N43 α(F )|, (5.81) Vt

386

5. ERROR ESTIMATES FOR F

we consider # (S) ˆ S F )D τ 2+ |α(L / 4 jα(F )| Vt

≤ ≤

#

∞ '#

Cu

u0

#

∞

u0

×

'#

−3 τ−2

'#

H

Cu

Cu

≤ cε0 (T1 ∗ + T2 ∗) .

t

t0 (u)

( 1 1 12

H

H ( 12

H (S) H2 τ 2+ |α(F )|2 HD / 4 jH

0#

H H2 ( 12 H 2 H ˆ τ + Hα(LS F )H

τ 6− r2 |α(F )|4

Su

H2 ( 12 '#

H H ˆ τ 2+ Hα(L S F )H

r0−2

2

'#

Su

H

H ( 12

H (S) H4 r2 r04 HD4 j H

(5.82)

(S)

1

Recall that |||r 2 r0 D4 j|||4,e ≤ cε0 for all t and c; in particular it does not depend on t. Altogether we estimate # ˆ S F )D(S)jα(F )| ≤ cε0 (T1 ∗ + T2 ∗). (5.83) τ 2+ |α(L 4 Vt

Now we estimate

#

(5.84)

Vt

We have

#

Σet

Q Q ˆ S F )J (3) | ≤ c Q ˆ S F )Q τ 2+ |α(L rα( L Q Q B 1 2

(5.85) and

ˆ S F )J (3) |. τ 2+ |α(L B

≤ cT1

Q Q Q (3) Q QrJB Q

2,e

−3

3

Q Q Q (3) Q QrJB Q

2,e

2,e

Q Q Q (3) Q QrJB Q

≤ cr0 2 (|||r 2 ∇ / (S) m|||4,e |||r2 α(F )|||4,e Q Q Q Q + Qr2 trχ(S)iQ ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(S)j Q ∥rα(F )∥2,e ∞,e Q Q Q Q + Qr2 trχ(S)nQ ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(S) n Q ∥rα(F )∥2,e ∞,e Q O .PQ Q Q + Qr2 Qr χ N ; (S)i,(S) j Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(S) m Q ∥rα(F )∥2,e ∞,e

2,e

5.1. PROOF OF THE FIRST PART OF THEOREM 5

387

1 1 (S) + |||r 2 τ − (D / 3 i − δ AB rD / 3 ν)|||4,e |||r2 α(F )|||4,e 2 Q O PQ Q Q + Qr3 Qr ζ;(S) m Q ∥rα(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ξ;(S) m Q ∥rα(F )∥2,e

∞,e

1 2

5

+ ||r0 w1 D / 3 ν||2,e ||r 2 α(F )||∞,e 3

(S)

3 2

(S)

3

+ |||r 2 D / 4 m|||4,e |||r 2 ρ(F )|||4,e Q O PQ Q Q + Qr3 Qr ν;(S) m Q ∥rρ(F )∥2,e ∞,e Q O .PQ Q Q + Qr3 Qr ζ; (S)i,(S) j Q ∥rρ(F )∥2,e ∞,e

3 2

+ |||r ∇ / i|||4,e |||r σ(F )|||4,e 3

3

3

3

+ |||r 2 trχ(S)m|||4,e |||r 2 σ(F )|||4,e

+ |||r 2 trχ(S)m|||4,e |||r 2 σ(F )|||4,e Q O PQ Q Q + Qr3 Qr χ N ;(S) m Q ∥rσ(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr χ N ;(S) m Q ∥rσ(F )∥2,e ∞,e 1

3

3

1

3

3

1

3

3

1

3

3 2

+ |||r ∇ / (S)m|||4,e |||r 2 τ −2 α(F )|||4,e

+ |||r 2 trχ(S)i|||4,e |||r 2 τ −2 α(F )|||4,e

+ |||r 2 trχ(S)j|||4,e |||r 2 τ −2 α(F )|||4,e

(5.86) Since

+ |||r 2 trχ(S)n|||4,e |||r 2 τ −2 α(F )|||4,e O P 3 1 3 + |||r 2 τ − Qr χ N ;(S) n |||4,e |||r 2 τ −2 α(F )|||4,e Q O .PQ Q Q + Qr3 Qr χ N ; (S)i,(S) j Q ∥τ − α(F )∥2,e ∞,e Q O PQ Q Q + Qr3 Qr ϵ;(S) m Q ∥τ − α(F )∥2,e 1 2

1 2

(S)

∞,e

1

3

+ r0 |||r r0 D4 i|||4,e |||r 2 τ −2 α(F )|||4,e ).

1 i = vN χAB − uN χAB + δ AB(S)j, 2 the last term can be estimated as in 5.82. Finally, we have shown that & (S) 2 ˆ Vt τ + |α(LS F )D4 jα(F )| ≤ cε0 (T1 ∗ + T2 ∗). Therefore, it follows that # ˆ S F )J (3) | ≤ cε0 (T1 ∗ + T2 ∗) , (5.87) τ 2+ |α(L B (S)

Vt

which finishes the proof of the first part of Theorem 5.

388

5. ERROR ESTIMATES FOR F

5.2. Proof of the Second Part of Theorem 5 The second part of Theorem 5 involves two Lie derivatives of the electromagnetic field. To find the error terms that we must estimate, we calculate the following divergence. . ˆX F ˆ X Fµ ν J (X, Y ) + ∗L ˆY L ˆ X Fµ ν J ′ (X, Y ) , ˆY L ˆY L =L DivT L ν ν µ

where

ˆY L ˆ X Fµν J (X, Y )ν = Dµ L R (X) 1 ˆY 1 π =L ˆ λµ Dµ Fλν + Dµ (X) π ˆ µλ F λ ν 2 2 S . 1 - (X) (X) µλ Dν π + ˆ µλ − Dλ π ˆ νµ F 2

1 (Y ) λµ ˆ X F λν ˆ X Fλν ) + 1 Dµ (Y ) π π ˆ Dµ (L ˆ µλ L 2 2 . 1 - (X) (X) ˆ X F µλ . + ˆ µλ − Dλ π ˆ νµ L Dν π 2 and Jν′ is just Jν with F replaced by ∗ F . +

& 5.2.1. Bounds for the Second Derivatives. The bounds for Vt & N S F )µ K µ are estimated precisely as NO L NO L N O F )µ K µ and DivT(L DivT(L V t & & µ µ N N Vt DivT(LO F )µ K and Vt DivT(LS F )µ K are estimated in the first part of Theorem 5. This follows from the uniform behavior of the tangential derivatives of all of the null components of the covariant quantities appearing in these integrals. More precisely, each tangential derivative of a null component improves the asymptotic behavior by order O(r−1 ) in the exte& N S F )µ K µ . NS L rior region. Therefore it is enough to bound Vt DivT(L To proceed with this bound we denote . ˆ S J (1) (S)ν = L ˆ S (S) π L ˆ λµ Dµ Fλν . ˆ S Dµ (S) π ˆ S J (2) (S)ν = L ˆ µλ F λ ν L ' ( . (S) ˆ S 1 Dν(S) π ˆ S J (3) (S)ν = L ˆ µλ − Dλ π ˆ νµ F µλ L 2 ˆ S Fλν ) ˆ λµ Dµ (L J (1) (S, S) =(S) π ν

ˆ S F λν (S, S)ν = Dµ (S) π ˆ µλ L . (S) ˆ S F µλ . J (3) (S, S)ν = Dν(S) π ˆ µλ − Dλ π ˆ νµ L J

(5.88)

(2)

Estimates for the first three terms will lead to estimates for the last three terms by using the commutation formulas for the Lie Derivatives. Therefore, in the next three sections, we derive estimates for the first three terms.

5.2. PROOF OF THE SECOND PART OF THEOREM 5

389

ˆ S J(1) (S). Recall from the previous section that 5.2.2. Estimates for L we must estimate . # .- .. ˆ S F4 4 L ˆS L ˆ S (S) π ˆ S J (1) (S)4 = τ2 L ˆ λµ Dµ Fλ4 (5.89) E L Vt

and

(5.90)

+

. # .- .. B (1) 2 ˆ ˆ (S) λµ ˆ ˆ E LS J (S)B = τ + LS LS F4 LS π ˆ Dµ FλB . -

Vt

As before, we calculate . ˆ S (S) π ˆ S J (1) (S)4 = 1 L ˆ 44 D3 F34 L 4 . 1 . 1 ˆ -(S) ˆ S (S) π − L π ˆ C3 D4 FC4 − L ˆ C4 D3 FC4 S 2 2 . . 1 ˆ -(S) ˆ S (S) π π ˆ C4 DC F34 + 2L NCD DC FD4 . (5.91) − LS 2 From our Bootstrap Assumptions, we find that ' HH HH 1 3 3 − 32 HH ˆ (1) HH |||r 2 r0 D J (S) ≤ cr / S(S)n|||4,e |||r 2 τ −2 D3 ρ(F )|||4,e r L HH S 4 HH 0 2,e

1

+ |||r 2 τ − D / S(S) m|||4,e |||r3 D4 α(F )|||4,e 1

1

5

+ |||r 2 τ − D / S(S) m|||4,e |||r 2 τ −2 D3 α(F )|||4,e 3

5

1

+ |||r 2 D / S(S) m|||4,e |||r 2 τ −2 ∇ρ(F / )|||4,e 1

+ |||r 2 D / S((S)i − δ AB(S)j)|||4,e |||r3 ∇α(F / )|||4,e ( 1 5 / )|||4,e + |||r02 D / S(S) j)|||4,e |||r 2 ∇α(F 0 Q Q Q 2 Q Q Q Qr D / D / α(F )Q + cr−2 Qr(S)mQ 0

(5.92) Therefore, (5.93) Now we estimate

∞,e

4

S

2,e

Q Q Q 2 Q Q Q Qr D + Qr(S)mQ / SD / 3 α(F )Q2,e ∞,e Q Q Q 2 −1(S) Q + Qr τ − nQ ∥rτ − D / SD / 3 ρ(F )∥2,e ∞,e Q Q Q 2 Q Q Q Qr D + Qr(S)mQ / S ∇ρ(F / )Q2,e ∞,e 1 Q Q Q 2 Q Q (S) Q Qr D + l.o.t. + Qr iQ / ∇α(F / )Q ∞,e

S

2,e

. ˆ S J (1) (S)4 ≤ cε0 (T1∗ + T2 ∗) . E L . ˆ S J (1) (S)B . E L

390

5. ERROR ESTIMATES FOR F

Taking the Lie derivative of 5.11 with respect to S, we find that . 1 . 1 ˆ -(S) ˆ S (S) π ˆ S (S) π L ˆ λµ Dµ FλB = L π ˆ 44 D3 F3B + L ˆ 33 D4 F4B S 4 4 . 1 . 1 ˆ -(S) ˆ S (S) π π ˆ 34 D4 F3B + L ˆ 34 D3 F4B + LS 4 4 . 1 . 1 ˆ -(S) ˆ S (S) π π ˆ C3 D4 FCB − L ˆ C4 D3 FCB − LS 2 2 . 1 . 1 ˆ -(S) ˆ S (S) π π ˆ C4 DC F3B − L ˆ C3 DC F4B − LS 2 . 2 (S) ˆS × 2L π NCD DC FDB .

1/2 r L ˆ (S) ˆ 44 ||| ≤ ε0 , we calculate Keeping in mind that |||τ −1 0 S π − r

HH HH HH ˆ (1) HH HHrLS J (S)B HH

2,e

−3

1

5

≤ cr0 2 (|||r02 D / S(S)n|||4,e |||r 2 D / 4 α(F )|||4,e 1

1

+ |||r02 D / S(S)j|||4,e |||r2 τ −2 D / 3 α(F )|||4,e 1

/ )|||4,e + |||r 2 D / S(S)m|||4,e |||r3 ∇α(F 1

+ |||r 2 D / S(S)m|||4,e |||r3 D / 4 σ(F )|||4,e 1

1

3

+ |||r 2 r0 D / S(S) m|||4,e |||r 2 τ −2 D / 3 σ(F )|||4,e 1

5

1

+ |||r02 D / S (S) i|||4,e |||r 2 τ −2 ∇σ(F / )|||4,e 1

1

1

1

1

5

+ r02 |||r 2 r0 D / S(S)n|||4,e |||r 2 τ −2 D / 3 α(F )|||4,e 3

3

3

3

+ r02 |||r02 D / S(S)j|||4,e |||r 2 τ −2 D / 4 α(F )|||4,e 1

+ |||r 2 r0 D / S(S) n|||4,e |||r 2 τ −2 ∇α(F / )|||24,e ) 0 Q Q Q 2 Q Q Q Qr D / SD / 4 α(F )Q2,e + cr0−2 Qr(S)nQ ∞,e

Q Q Q 2 Q Q (S) Q Qr D + Qr j Q / SD / 3 α(F )Q2,e ∞,e Q Q Q 2 Q Q Q Qr D + Qr(S)mQ / S ∇α(F / )Q2,e ∞,e Q Q Q 2 Q Q (S) Q Qr D + Qr mQ / SD / 4 σ(F )Q2,e ∞,e Q Q Q 2 Q Q (S) Q Qr D + Qr iQ / S ∇σ(F / )Q2,e ∞,e Q Q Q 2 −1(S) Q + Qr τ − mQ ∥rτ − D / SD / 3 σ(F )∥2,e ∞,e

5.2. PROOF OF THE SECOND PART OF THEOREM 5

391

Q Q Q 2 Q Q (S) Q Qτ − D + r0 Qr2 τ −1 n / SD / 3 α(F )Q2,e Q − ∞,e Q Q Q (S) Q / SD / 4 α(F )∥2,e + r0 Qr j Q ∥rτ − D ∞,e Q Q Q Q + r0 Qr(S)iQ ∥rD / S χ∥∞,e ∥τ − α(F )∥2,e ∞,e

Q Q Q (S) Q + r0 Qr iQ

(5.94)

∞,e

1

1

∥rχ∥∞,e ∥τ − D / S α(F )∥2,e + l.o.t. .

/ S(S) n|||24,e ≤ cε0 , we must bound D3 a and D3 Λ. We can To see that |||r02 D 1

1

write D3 a as D4 a − 2∇N a. By the Bootstrap Assumptions, |||r 2 r02 τ − ∇N a|||4,eQ ≤ cε0 . Furthermore, D4 a = −ν. We have shown in 5.80 that Q Qr2 D3 ΛQ ≤ cε0 . Thus to estimate D3 Λ, it is sufficient to estimate ∞,e H H2 $ %2 . $ %2 r 1 1 Hχ a D trχ + trχ + = − by 5.75. However, D trχ trχ NH − 3 3 2 2 2 % $ |α(F )|2 , where the right-hand side of this equation decays like O r−2 τ −2 − . Therefore we deduce that ∥rτ − D3 Λ∥∞,e ≤ cε0 . To bound the terms that do not decay fast enough in the Wave Zone, we use the method of 5.82. The last four terms of 5.94 do not decay fast enough in the Wave Zone to be bounded in this manner without obtaining a logarithmic divergence. The last two of these are estimated using the methods in 5.42 and 5.82. We now estimate the remaining two. We use the fact that we can bound 5

1

3

3

/ 24 α(F )|||24,e , r0−δ |||r 2 τ −2 D / 3D / 4 α(F )|||24,e , r0−δ |||r 2 τ −2 D 1

5

/ 23 α(F )|||24,e , r0−δ |||r 2 τ −2 D

$ where δ is %less than 14 , by r0−δ T3 ∗ + T1 ∗ + T2 ∗ and that r0−δ T3 ∗ ≤ c T3 (0) + T1 ∗ + T2 ∗ . We prove the latter inequality in the next section. Thus # H2 # t H 1 5 3 −1(S) H H ˆS D r2 H(S)nL / 3 α(F )H ≤ r0−2+δ |||r 2 τ −2 D / 23 α(F )|||24,e |||r 2 τ − n|||24,e Cu

t0 (u)

+ l.o.t.

(5.95)

≤ cε0 (T3 (0) + T1 ∗ + T2 ∗) .

The remaining terms are estimated the same way. In conclusion, we have shown that . . ˆ S J (1) (S)4 + E L ˆ S J (1) (S)B ≤ cε0 (T3 (0) + T1 ∗ + T2 ∗). (5.96) E L ˆ S J(2) (S). Now we estimate 5.2.3. Estimates for L . # .- .. (2) ˆ S F4 4 L ˆ ˆS L ˆ S Dµ (S) π (5.97) E LS J (S)4 = τ 2+ L ˆ µλ F λ 4 Vt

392

5. ERROR ESTIMATES FOR F

and . # .- .. (1) ˆ ˆ S F4 B L ˆS L ˆ S Dµ (S) π E LS J (S)B = τ 2+ L ˆ µλ F λ B .

(5.98)

Vt

To estimate the first quantity, we take the Lie derivative of 5.12, . 1 . 1 . ˆ S D4(S) π ˆ S D3(S) π ˆ S Dµ (S) π L ˆ µλ F λ 4 = L ˆ 43 F34 + L ˆ 44 F34 4 4 . 1 . 1 ˆ - (S) ˆ S D3(S) π ˆ 3C FC4 − L ˆ 4C FC4 − LS D4 π 2 . 2 (S) ˆ S DD π ˆ CD FC4 . + 4L

We find that

HH HH HH ˆ (2) HH HHrLS J (S)4 HH

2,e

0Q Q Q Q Q Q Q 3 −1 Q 2 1 (S) Q 2 2 ˆ Q Q Q r r L / ≤ τ ρ(F ) r D / j − S 4 Q Q Q 0 Q ∞,e 2,e Q Q Q Q Q 2 1 Q Q Q 3 −1 2 ˆ / (S)nQ Q Qr 2 τ L +Q 3 Qr τ − ρ(F )Q Q Q 0 − /S D −3 cr0 2

∞,e

Q 1 Q Q 5 Q Q 2ˆ Q Q (S) Q L / r + Qr 2 α(F )Q D / m Q Q S 4 ∞,e 2,e Q 5 Q Q Q Q 2 Q Q Q −1 ˆ + Qr α(F )Q / 3(S)mQ /S D Qrτ − L ∞,e 2,e 1 Q 5 Q Q Q Q Q Q ˆ (S) Q + Qr 2 α(F )Q / iQ /S ▽ Qr0 L ∞,e

2,e

2,e

ˆ/ α(F )|||4,e / 4 m|||4,e |||r2 L + cr0−2 (|||r 2 D S 3

(S)

ˆ/ α(F )|||4,e + |||r 2 D / 3 m|||4,e |||r2 L S 3

(S)

ˆ/ α(F )|||4,e ) + |||r 2 ∇ / (S)i|||4,e |||r2 L S 3

3

1 3 − (S) ˆ/ ρ(F )|||4,e / 3 n|||4,e |||r 2 L + cr0 2 (|||r 2 r0 D S

1 3 (S) ˆ/ ρ(F )|||4,e ) + l.o.t. + |||r 2 r0 D / 4 j|||4,e |||r 2 L S

(5.99) Q 3 Q Q ˆ S D3(S)nQ Here r02 Qτ −1 L Q −

2,e

≤ ε0 since D3 ν = D4 ν − 2∇N ν and ∇N ν behaves

like D4 δ by formula (11.4.3) in [5]. Thus we find that

(5.100) Now we estimate

. ˆ S J (2) (S)4 ≤ cε0 (T1∗ + T2 ∗). E L . ˆ S J (2) (S)B . E L

5.2. PROOF OF THE SECOND PART OF THEOREM 5

393

Taking the Lie derivative of 5.13, we find that ˆ S Dµ (X) π ˆ µλ F λ L

(5.101)

B

. 1 . 1 ˆ - (S) ˆ S D3(S) π = L ˆ 34 F4B + L ˆ 44 F3B S D3 π 4 4 . 1 . 1 ˆ - (S) ˆ S D4(S) π ˆ 33 F4B + L ˆ 34 F3B + LS D4 π 4 4 . 1 . 1 ˆ - (S) ˆ S D3(S) π ˆ 3C FCB − L ˆ 4C FCB − LS D4 π 2 2 . 1 . 1 ˆ - (S) ˆ S DC(S) π ˆ 3C F4B − L ˆ 4C F3B − LS DC π 2 . 2 (X) ˆ S DD π + 4L ˆ CD FCB .

Then we estimate HH HH HH HH ˆ (2) J (S) r L HH S B HH

2,e

Q Q 'Q Q Q Q 3 −3 Q 52 Q (S) 2 ˆ/ D Q Qr r 2 L ≤ Qr α(F )Q 0 S / 3 jQ Q ∞,e 2,e Q Q Q 5 Q 3 Q Q Q Q (S) Q ˆ/ D Qr 2 r− 32 L + Qr 2 α(F )Q 0 S / 4 nQ ∞,e Q 2,e Q 5 Q Q 1 Q Q 2 Q Q 2ˆ Q + Qr α(F )Q / (S)mQ /S ▽ Qr L 2,e ∞,e Q Q Q Q Q 3 Q Q Q 3 −1 (S) Q 2 2 ˆ Q Q Q /S D + Qrτ − α(F )Q / 3 nQ Qr0 τ − L −3 cr0 2

2,e

∞,e

Q Q Q Q 3 2 Q + r0 Q Qrτ − α(F )Q

∞,e

Q Q Q 3 Q 2 Q +Q Qrτ − α(F )Q

∞,e

Q Q Q 2 1 Q 2 Q +Q Qr τ − σ(F )Q

∞,e ∞,e

− 32

2,e

Q Q Q −1 32 ˆ Q /S ▽ / (S)mQ Qτ − r L

Q Q Q 2 1 Q 2 Q + Qr τ − σ(F )Q Q Q Q Q 2 1 Q 2 Q + Qr τ − σ(F )Q Q

Q Q Q Q 3 −1 (S) 2 ˆ Qr r L /S D / 4 jQ Q Q 0

∞,e

2,e

Q 1 Q Q 2ˆ (S) Q L / D / m r Q Q S 4

2,e

Q Q Q −1 ˆ (S) Q /S D / 3 mQ Qrτ − L Q Q Q ˆ Q /S ▽ / (S)iQ Qr0 L

2,e

(

2,e

ˆ/ α(F )|||4,e / 3 j − rD / 3 ν)|||4,e |||r2 L + cr0 (|||r τ − (D S 1 2

(S)

ˆ/ α(F )||∞,e + ||rD / 3 ν||2,e ||r 2 L S 5

ˆ/ α(F )|||4,e + |||r 2 r0 D / 4 n|||4,e |||r2 L S 1

(S)

ˆ/ σ(F )|||4,e + |||r 2 D / 4 m|||4,e |||r 2 L S 3

(S)

3

ˆ/ σ(F )|||4,e + |||r 2 D / 3 m|||4,e |||r 2 L S 3

(S)

3

394

5. ERROR ESTIMATES FOR F

ˆ/ α(F )|||4,e + |||r 2 ∇ / (S)m|||4,e |||r2 L S 3

3 3 ˆ/ σ(F )|||4,e + |||r 2 ∇ / (S)i|||4,e |||r 2 L S 3

ˆ/ α(F )|||4,e + |||r 2 τ −1 / (S)m|||4,e |||r 2 τ −2 L − ∇ S 5

1

1

3

1 1 (S) ˆ/ α(F )|||4,e ) + l.o.t. + r02 |||r 2 r0 D4 j|||4,e |||r 2 τ −2 L S

(5.102)

Q Q Q Q 3 2 Q The quantity Qrτ − α(F )Q Q

'3Q Q ( Q (S) 2 Q −1 ˆ r0 Qr L does not decay fast enough / 4 jQ /S D 2,e

∞,e

(2)

in the Wave Zone. However, the related term in JB (S) cancels against (3) ˆ AB FB3 of JB . Recall that (S) π ˆ AB = the corresponding term − 12 D4(S) π 1 (S) ˆ 43 . Therefore, we do not need to estimate this vN χAB − uN χAB + 2 δ AB π term. Thus, we have shown that . . ˆ S J (2) (S)4 + E L ˆ S J (2) (S)B ≤ cε0 (T1 ∗ + T2 ∗) (5.103) E L

ˆ S J(3) (S). To estimate 5.2.4. Estimates for L (5.104) . # . - -. .. (S) (3) µλ ˆ ˆ S F4 4 L ˆS L ˆ S D (S) π E LS J (S)4 = τ 2+ L ˆ − D π ˆ F , 4µ µλ 4 λ Vt

we use

(5.105)

-. . (S) ˆ S J (3) (S)4 = 1 L ˆ S D (S) π ˆ 44 F34 L 4 ˆ 34 − D3 π 4 . . 1 ˆ -- (S) (S) − L π ˆ − D π ˆ D F 34 S C4 C4 3 C 2 -. . 1ˆ (S) (S) ˆ C4 − DC π ˆ 44 FC3 D4 π + L S 2 -. . (S) ˆ S D (S) π +L ˆ − D π ˆ F . CD C4 CD 4 D

We find that

HH HH HH HH ˆ (3) HHrLS J (S)4 HH

0Q Q Q Q Q Q Q 3 −1 Q 2 1 (S) Q 2 2 ˆ Q Q Q r r L / ≤ cr0 τ ρ(F ) τ D / n − − 0 S 3 Q Q Q Q ∞,e 2,e Q Q Q Q Q 2 1 Q Q Q 3 −1 (S) Q 2 ˆ/ D Q Qr 2 r L +Q S / 4 jQ Qr τ − ρ(F )Q Q 0 −3 2

2,e

Q 5 Q Q Q + Qr 2 α(F )Q

∞,e

Q 5 Q Q Q + Qr 2 α(F )Q

2,e

∞,e

∞,e

Q Q Q Q 3 −1 (S) Q ˆ/ ▽ Qr 2 r L S / jQ Q 0

2,e

Q Q Q −1 ˆ (S) Q L / rτ D / m Q − S 3 Q

2,e

5.2. PROOF OF THE SECOND PART OF THEOREM 5

Q Q Q 3 Q 2 Q +Q Qrτ − α(F )Q

∞,e

Q Q Q 3 Q 2 Q +Q Qrτ − α(F )Q

∞,e

Q Q Q Q 2 1 2 Q r τ σ(F ) +Q Q Q −

Q 3 Q Q 2 −1 ˆ (S) Q /S D / 4 mQ Qr τ − L '

∞,e

Q Q Q 2 1 Q 2 Q +Q Qr τ − σ(F )Q

∞,e

0

3

395

2,e

Q 2 Q Q (S) Q ˆ/ ▽ r02 Qτ −1 / n L Q − S

'

2,e

Q 2 Q Q (S) Q ˆ/ D r02 Qr−1 L / i Q S 4

(

2,e

Q Q Q −1 32 ˆ (S) Q L / r ▽ / m τ Q Q − S

2,e

(

1

ˆ S α(F )|||4,e / (S) j|||4,e |||r2 L × cr0−2 |||r 2 ▽ + |||r +

3 2

Q

Q

Q Q ˆ (S) D / 3 m|||4,e Qr2 L S α(F )Q 2,e

−3 cr0 2

0

1

1 3 (S) ˆ S ρ(F )|||4,e |||r 2 r0 D / 4 j|||4,e |||r 2 L

1

(S)

1

(S)

3

ˆ S ρ(F )|||4,e + |||r 2 r0 D / 3 n|||4,e |||r 2 L 3

ˆ S σ(F )|||4,e + |||r 2 r0 D / 4 i|||4,e |||r 2 L 3

3

ˆ S σ(F )|||4,e + |||r 2 ▽ / (S)m|||4,e |||r 2 L 5

(S)

5 2

(S)

1

1

ˆ S α(F )|||4,e + |||r 2 D / 4 m|||4,e |||r 2 τ −2 L (5.106)

/ + |||r ▽

1 2

1 2

ˆ S α(F )|||4,e m|||4,e |||r τ − L

1

+ l.o.t.

To estimate (5.107) . # . - -. .. (S) (3) µλ ˆ ˆS L ˆ S D (S) π ˆ S F4 B L E LS J (S)B = τ 2+ L ˆ − D π ˆ F , Bµ µλ B λ Vt

we use . 1ˆ (S) (S) ˆ S D (S) π L ˆ − D π ˆ ˆ B3 )F34 F µλ = L Bµ S (D4 π µλ B λ 4 . 1 ˆ (S) ˆ S D (S) π − ˆ B3 − L ˆ LS DC π BC FC4 3 2 . 1 -ˆ (S) ˆ S D (S) π LS DC π − ˆ B4 − L ˆ FC3 BC 4 . -2 (S) ˆ S D (S) π ˆ (5.108) + L ˆ BC FCD . B ˆ CD − LS DD π

396

5. ERROR ESTIMATES FOR F

We find that HH HH HH ˆ (3) HH HHrLS J (S)B HH

−3 2

2,e

≤ cr0

'

Q 2 1 Q Q Q (S) Q ˆ/ D Qr τ 2 ρ(F )Q Qr 12 L − S / 4 m 2,e ∞,e

Q Q 5 Q Q 1 (S) Q ˆ/ ▽ + Qr 2 α(F )Q∞,e Qr 2 L S / m 2,e ' 3 ( Q Q 5 Q Q 3 (S) − 2 ˆ Q Q Q Q /S D + r 2 α(F ) ∞,e / 3 i 2,e r0 r 2 L

Q Q Q Q 3 3 2 ˆ / (S)mQ2,e /S ▽ + Qrτ −2 α(F )Q∞,e Qτ −1 − r L ' 3 ( Q 3 Q Q Q (S) Q 2 2 Q −1 ˆ Q Q /S D + r0 rτ − α(F ) ∞,e r0 r L / 4 i 2,e ( Q 2 1 Q Q Q (S) 2 ˆ + Qr τ σ(F )Q Qr0 L / iQ / ▽ −

−3

2,e

S

∞,e

3

ˆ S α(F )|||4,e / (S) π N3B |||4,e |||r2 L + cr0 2 (|||r 2 ∇ 1 1 (S) + |||r 2 τ − (D3 π NAB − δ AB rD3 ν)|||4,e 2 3Q Q Q Q ˆ S α(F )|||4,e + r 2 QD3 ν Q Qr 52 L ˆ S α(F )Q |||r2 L 0

+ |||r

(5.109)

3 2 3

2,e 3 (S) ˆ D4 π NB3 |||4,e |||r 2 LS ρ(F )|||4,e

∞,e

3

ˆ S σ(F )|||4,e + |||r 2 ∇ / (S) π NAB |||4,e |||r 2 L 3

1

1

ˆ S α(F )|||4,e ) + l.o.t. + |||r 2 ∇ / (S) π N4B |||4,e |||r 2 τ −2 L

Q . Q -Q 3 Q 3 (S)iQ ˆ/ D / does not decay fast enough The term Qrτ −2 α(F )Q∞,e Qr02 r−1 L S 4 2,e ˆ/ D ˆ/ D in the Wave Zone. But the troublesome part of L / (S)i is L / (S)j and S

4

S

4

this cancels against the same term in 5.102. The estimates of the second derivatives of the components of the deformation tensor follow directly from the Bootstrap Assumptions with (S) ˆ (S) ˆ/ D ˆ/ D /S D / 3(S)j, L a few exceptions, namely estimates involving L S / 4 j, L S / 4 n, (S) (S) (S) ˆ/ D ˆ/ D ˆ/ D L S / 4 i, and L S / 3 i. To estimate L S / 4 j we consider (5.110) $ % ˆ S D4(S) π ˆ S D4 a−1 − L ˆ S D4 Λ − L ˆ S D4 u ν + trχ + L ˆ S D4 v (ν + trχ). L ˆ 34 = −L 2 2 Let us consider this expression term-by-term. After differentiation by D4 , the first term becomes (5.111)$ % LS a−2 ν = −2a−3 (vν − u (ν − 2 (ω + δ))) ν + a−2 (vD4 ν − uD3 ν) .

5.2. PROOF OF THE SECOND PART OF THEOREM 5

397

According ( to the Bootstrap Assumptions, the first term decays like ' 1 − 52 2 O r r0 . The second term can be estimated as follows (5.112)

3 Q Q r02 Qr−1 a−2 (vD4 ν − uD3 ν)Q2,e ≤ ε0 .

ˆ S D4(S) π Therefore, the first term in L ˆ 34 can be estimated as (5.113)

3 Q $ %Q r02 Qr−1 LS a−2 ν Q2,e ≤ ε0 .

ˆ S D4 Λ. We calculate Now we consider the second term L

(5.114)

ˆ S D4 Λ = LS (−φ−1 D4 φ r φtrχ − φ−1 Λ φtrχ L 2 # 2 $ % −1 r 2 +φ φtrχ + φD trχ + φtrχ D ) 4 4 2r2 Su # r −1 −1 r = LS (−φ D4 φ φtrχ + φ D4 φtrχ 2 2r2 Su ( ' # 1 −1 r 2 +φ φ D4 trχ + trχ 2r2 Su 2 # $ $ %% r + φ−1 2 φ trχ trχ − trχ + l.o.t. 2r Su

3Q The first two terms behave nicely because supΣet |D4 φ| ≤ r0−2 ε0 and r02 Qr−1 Q Ls D4 φQ2,e ≤ ε0 (see the chapter on lapse function). Now we consider

' ( ' ( # # $ $ %% r r 1 LS φ−1 2 φ D4 trχ + trχ2 + φ−1 2 φ trχ trχ − trχ 2r Su 2 2r Su ( ' # $ −2 % r 1 2 = v −φ φ D4 trχ + trχ D4 φ 2 2r Su 2 ( ' # r 1 −2 2 + uφ D3 φ 2 φ D4 trχ + trχ 2r Su 2 ( ' # 1 −1 (LS r) 2 +φ φ D4 trχ + trχ 2r2 Su 2 # $ $ %% (LS r) φ trχ trχ − trχ + φ−1 2 2r S ( # u ' r 1 −1 2 + vφ D4 φ D4 trχ + trχ 2r2 Su 2

398

5. ERROR ESTIMATES FOR F

( ' # r 1 2 − uφ D3 φ D4 trχ + trχ 2r2 Su 2 # $ $ %% r D4 φ trχ trχ − trχ + vφ−1 2 2r Su # $ $ %% r D3 φ trχ trχ − trχ − uφ−1 2 2r Su ' ( # 1 −1 r 2 φD4 D4 trχ + trχ + vφ 2r2 Su 2 ( ' # 1 −1 r 2 φtrχ D4 trχ + trχ + vφ 2r2 Su 2 ' ( # r 1 φD3 D4 trχ + trχ2 − uφ−1 2 2r Su 2 ' ( # 1 −1 r 2 φtrχ D4 trχ + trχ − uφ 2r2 Su 2 # %% $ $ r φD4 trχ trχ − trχ + vφ−1 2 2r Su # $ $ %% −1 r φtrχ trχ trχ − trχ + vφ 2 2r Su # %% $ $ −1 r − uφ φD trχ − trχ trχ 3 2r2 Su # %% $ $ −1 r − uφ φtrχ trχ trχ − trχ + l.o.t. 2 2r Su −1

(5.115)

Using the propagation formulas for the components of the Hessian of the optical function and the Bootstrap Assumptions, we see that Q all of these 3 Q $ −2 % Q 2 Q −1 ˆ terms decay like O r . Therefore we have r0 Qr LS D4 ΛQ ≤ cε0 . 2,e

ˆ S D4 v trχ to obtain −L ˆ S D4 (uδ) + ˆ S D4 u trχ and L Now we combine −L 2 2 ˆ S D4 (rtrχ). We see that the first term −L ˆ S D4 (uδ) decays well from 9.40 L ˆ S D4 (rtrχ) can be writand the Bootstrap Assumptions. The second term L ten as LS (Λtrχ + rD4 trχ). We calculate

(5.116)

LS (Λtrχ + rD4 trχ) ' ' ( ( % $ 1 1 2 = LS r D4 trχ + trχ − rtrχ trχ − trχ + l.o.t . 2 2

The first part behaves like

5

N DS χ N ||| ≤ cε0 |||r 2 χ

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

399

(see 11.3). To check the second part, we calculate %% $ $ %% $ % $ $ LS rtrχ trχ − trχ = LS r trχ trχ − trχ + rLS trχ trχ − trχ $ % (5.117) + rtrχLS trχ − trχ + l.o.t.

% $ The first two terms clearly decay like$ O r−2 . In % trying to bound the third term, it is sufficient to consider rD4 trχ − trχ . We have # # % $ $ % rΛ 1 trχ D4 trχ + trχ2 − 3 rD4 trχ − trχ = −rD4 trχ + 2 r Su r Su ( ' ( ' 1 1 2 2 = −r D4 trχ + trχ + r D4 trχ + trχ 2 2 . r- 2 − trχ − trχ2 + l.o.t (5.118) 2 where

# # $ % $ % 1 1 2 (5.119) trχ − = 2 trχ trχ − trχ . trχtrχ − trχ = 2 r Su r Su % $ Therefore rD4 trχ − trχ decays like O(r−2 ). Putting all of these estimates Q Q 3 Q (S) 2 Q −1 ˆ /S D / 4 j Q ≤ cε0 . together, we have r0 Qr L 2

trχ2

2,e

(S) ˆ/ D Estimates for L S / 3 j follow from similar types of calculations and es(S) (S) ˆ/ D ˆ/ D timates. The estimates for L S / 4 i, and L S / 3 i follow from the estimates (S) (S) (S) ˆ/ D ˆ/ D ˆ/ D for L S / 4 j and L S / 3 j, respectively. To estimate L S / 4 n, we notice (S) ˆ/ D that D4 Λ can be estimated by D3 Λ. Then the estimate for L S / 4 n follows (S) ˆ/ D from the estimate for L S / 3 j and the Bootstrap Assumptions. We have shown that . . ˆ S J (3) (S)B ≤ cε0 (T1∗ + T2 ∗) ˆ S J (3) (S)4 + E L (5.120) E L

Putting together 5.96, 5.103 and 5.120, finishes the proof of the second part of Theorem 5.

5.3. Third derivatives for F in the exterior We use the null decomposition of the Maxwell Equations to obtain 2 estimates for the third order derivatives in a weighted % & L norm. $For the N O F )µ T, K µ first and second derivatives, we used the quantities Σt T(L $ % & N 2 F )µ T, K µ to estimate the tangential derivatives of each null and Σt T(L O component. However, we cannot proceed in the same way since we cannot & NO L N O F )µ K µ . Instead by NO L bound the error terms that arise in Vt DivT(L

400

5. ERROR ESTIMATES FOR F

$ % $ % & N2 L N2 N N T(L O T F ) T, K and Σet T(LO LS F ) T, K , we estimate H H H2 H2 & & H 2 H 2 H H / D / D / 4 α(F )H r0−δ Σe τ 6+ H∇ / 3 α(F )H r0−δ Σe τ 6+ H∇ t t H H H2 H2 & & H 2 H 2 H H / D / D r0−δ Σe τ 6+ H∇ / 4 α(F )H r0−δ Σe τ 4+ τ 2− H∇ / 3 α(F )H t t H H H H (5.121) & & H 2 H 2 H2 H2 / D / D r0−δ Σe τ 6+ H∇ / 4 ρ(F )H r0−δ Σe τ 6+ H∇ / 3 ρ(F )H t t H2 H2 H H & & H H H 2 H 2 −δ −δ 6 r0 Σ e τ + H ∇ / 4 σ(F )H r0 Σe τ 6+ H∇ / 3 σ(F )H / D / D t t H2 H & H 2 8 H∇ where δ = 14 . Notice that we can also bound r0−δ Σe τ −2 τ D α(F ) / H , H 4 + − H2 H2 H H t & & H −2 8 H 2 8 H / 2 D ρ(F )H , and r −δ / D4 σ(F )H . The r0−δ r0−δ Σe τ −2 H 4 − τ + H∇ 0 Σet τ − τ + H∇ t arise because the error terms decay too slowly in the Wave Zone, leading to logarithmic divergence in time. Therefore we need to allow for some growth in t. $ % $ % & & N T F ) T, K and e T(L N S F ) T, K , N2 L N2 L Once we have bounded Σe T(L O O Σ t t we use the null decomposition of the Maxwell Equations to estimate the other third derivatives with the exception of H H2 H H & & 6 H 3 6 H / 2 α(F )H2 H 4 Σet τ + ∇D Σet τ + D4 α(F ) H H H H2 & & (5.122) 4 2 H / 2 α(F )H2 4 2 H 3 H e τ + τ − ∇D e τ + τ − D α(F ) . considering

&

Σet

3

Σt

3

Σt

% & NS L N S F ) T, K and e T(L NS NO L NS L We bound these by considering Σe T(L Σt t $ % N T F ) T, K . L Because of logarithmic divergence in t, we use a slightly different technique to bound the error term. First we define J# # $ % $ % NO L N T F ) T, K + NO L N S F ) T, K NO L NO L T3 (t) = T(L T(L (5.123)

+

#

&

Σet

Σet

We then show that

$

%

NS L N S F ) T, K + NO L T(L

(5.124) T3 (t) ≤ T3 (0) + cε0

'#

0

t$

2 + 2t

#

% ′ −1

$

Σet

Σet

$

NS L N T F ) T, K NS L T(L

%

Then we choose ε0 sufficiently small, such that % $ (5.126) T3 (t) ≤ 2c (2 + 2t)δ T3 (0) + T[2] ∗ (5.127)

.

( $ ′% ′ T3 t dt + ln (2 + 2t) T[2] ∗ .

Recall that r0 is like 2 + 2t. Applying Gronwall’s inequality we have % $ (5.125) T3 (t) ≤ c (2 + 2t)cε0 T3 (0) + ε0 ln (2 + 2t) T[2] ∗ . for δ = 14 . From the last section we have

K

. T[2] ∗ ≤ c1 (T1 (0) + T2 (0)) + c2 ε0 T1 ∗ +T2 ∗ +r0−δ T3 ∗ .

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

401

Therefore, we obtain (5.128)

r0−δ T3 ∗ ≤ c1 (T1 (0) + T2 (0) + T3 (0)) + c2 ε0 (T1 ∗ + T2 ∗)

for ε0 sufficiently small. Finally, we conclude (5.129)

T1∗ + T2 ∗ + r0−δ T3 ∗ ≤ c1 (T1 (0) + T2 (0) + T3 (0)) .

& N OL N OL N T F )µ K µ . To achieve 5.3.1. Error estimates for Vt DivT(L the bounds described in the last section we must bound the error terms ˆZ L ˆY L ˆ X F ). We calculate arising from the divergence of T(L (5.130). ˆZ L ˆY L ˆX F ˆY L ˆ X Fµ ν J (X, Y ) +∗ L ˆZ L ˆY L ˆ X Fµ ν J ′ (X, Y ) , ˆZ L DivT L =L ν ν µ

where

ˆZ L ˆY L ˆ X Fµν J(X, Y, Z)ν = Dµ L R (X) 1 ˆZ L ˆY 1 π =L ˆ λµ Dµ Fλν + Dµ (X) π ˆ µλ F λ ν 2 2 S . 1 - (X) (X) µλ + ˆ µλ − Dλ π ˆ νµ F Dν π 2 R (Y ) 1 ˆ ˆ X F λν ˆ X Fλν ) + 1 Dµ (Y ) π + LZ π ˆ λµ Dµ (L ˆ µλ L 2 2 S . 1 - (Z) (Z) ˆ X F µλ ˆ νµ L Dν π + ˆ µλ − Dλ π 2

1 (Z) λµ ˆ X Fλν ) + 1 Dµ (Z) π ˆY L ˆ X F λν ˆY L π ˆ Dµ (L ˆ µλ L 2 2 . 1 - (Z) (Z) ˆY L ˆ X F µλ + ˆ µλ − Dλ π ˆ νµ L Dν π 2

+ (5.131)

and Jν′ is just Jν with F replaced by ∗ F . We have

ˆ OL ˆ OL ˆ T Fµν J (T, O, O)ν = Dµ L R (T ) 1 ˆO 1 π ˆ OL =L ˆ λµ Dµ Fλν + Dµ (T ) π ˆ µλ F λ ν 2 2 S . 1 - (T ) (T ) µλ Dν π + ˆ µλ − Dλ π ˆ νµ F 2 R (O) 1 ˆ T F λν ˆ T Fλν ) + 1 Dµ (O) π ˆ π ˆ λµ Dµ (L ˆ µλ L + LO 2 2 S . 1 - (O) (O) µλ ˆ Dν π + ˆ µλ − Dλ π ˆ νµ LT F 2

402

5. ERROR ESTIMATES FOR F

1 (O) λµ ˆ T Fλν ) + 1 Dµ (O) π ˆ OL ˆ T F λν ˆ OL π ˆ Dµ (L ˆ µλ L 2 2 . 1 - (O) (O) ˆ OL ˆ T F µλ . (5.132) + ˆ µλ − Dλ π ˆ νµ L Dν π 2 . & ˆ OL ˆ OL ˆT F decays at least like r0−1 . It is enough to show that Σe DivT L +

µ

t

This will be true for all of the error terms. We denote . λµ ˆ OL ˆ O J (1) (T ) = L ˆ O (T ) π ˆ OL L ˆ D F µ λν ν . λ ˆ OL ˆ OL ˆ O J (2) (T ) = L ˆ O Dµ (T ) π ˆ F L ν µλ ν . . -(T ) µλ ˆ OL ˆ O J (3) (T ) = L ˆ O Dν(T ) π ˆ OL L ˆ − D π ˆ F νµ µλ ν λ . ˆ T Fλν ) ˆ O J (1) (O, T ) = L ˆ O (O) π ˆ λµ Dµ (L L ν . ˆ O Dµ (O) π ˆ O J (2) (O, T ) = L ˆ T F λν L ˆ L µλ ν -. . (O) ˆ O J (3) (O, T ) = L ˆ O Dν(O) π ˆ T F µλ L ˆ − D π ˆ L νµ µλ ν λ ˆ T Fλν ) ˆ OL Dµ (L ˆ OL ˆ T F λν J (2) (O, O, T )ν = Dµ (O) π ˆ µλ L . (O) ˆ OL ˆ T F µλ J (3) (O, O, T )ν = Dν(O) π ˆ µλ − Dλ π ˆ νµ L J (1) (O, O, T )ν =

(5.133)

(O) λµ

π ˆ

ˆ O J (1) (T ). We need to show that ˆ OL 5.3.1.1. Error estimates for L (5.134) . # (1) ˆ ˆ E LO LO J (T )4 =

... ˆ OL ˆ T F4 4 L ˆ O (T ) π ˆ OL ˆ OL τ 2+ L ˆ λµ Dµ Fλ4

(5.135) . # ˆ OL ˆ O J (1) (T ) E L B =

... ˆ OL ˆ OL ˆ OL ˆ T F4 B L ˆ O (T ) π τ 2+ L ˆ λµ Dµ FλB

Σet

and

Σet

decay at least like O(r0−1 ). First, we consider

ˆ O J (1) (T ) . ˆ OL L 4

From 5.10, we calculate . 1 ˆ ˆ (T ) ˆ OL ˆ O (S) π L ˆ λµ Dµ Fλ4 = L ˆ 44 D3 F34 O LO π 4 1 ˆ ˆ (T ) 1 ˆ ˆ (T ) − L ˆ C3 D4 FC4 − L ˆ C4 D3 FC4 O LO π O LO π 2 2 1 ˆ ˆ (T ) ˆ O(T ) π ˆ OL − L ˆ C4 DC F34 + 2L NCD DC FD4 . O LO π 2

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

403

% % $ (b) ΩB ∇ / B f = (a) ΩAH ∇ / B f +(b) ΩB ∇ / A∇ / Bf /A Ω B∇ H H H H (b) H ˆO for a function f . We have Hr−1(a) ΩH ≤ c and H∇ / ΩH ≤ c. Therefore, the L do not contribute weights. We estimate Now L(a) Ω L(b) Ω f = L(a) Ω

#

Σet

$(b)

. ˆ OL ˆ OL ˆ T F44 L ˆ O (T ) π ˆ OL τ 2+ L ˆ λµ Dµ Fλ4 −3 2

≤ cr0

Q 'Q Q 2 1 Q Qr τ 2 D Q / ρ(F ) − 3 Q Q

∞,e

Q 5 Q Q Q + Qr 2 D / 4 α(F )Q

∞,e

Q Q Q 2 1 Q 2 Q + Qr τ − D / 3 α(F )Q Q

Q .Q Q ˆ OL ˆT F Q ˆ OL Q Qτ + ρ L

2,e

Q .Q Q ˆ OL ˆT Q ˆ OL Q Qτ + ρ L

∞,e

2,e

Q .Q Q ˆ OL ˆT Q ˆ OL Q Qτ + ρ L

Q Q Q Q 1 (T ) Q Qr 2 L / L / n Q Q 0 O O

Q Q Q 1 Q N/ L N/ (T ) mQ Qr 2 L Q 0 O O Q

2,e

2,e

2,e

Q Q Q Q 1 N/ L N/ (T ) mQ Qr 2 L Q Q 0 O O

2,e

Q Q 5 Q Q .Q Q Q Q N N (T ) Q Q Q ˆ OL ˆT Q ˆ OL /O iQ + Qr 2 ∇α / Q /O L Q QL Qτ + ρ L ∞,e 2,e 2,e Q .Q 1 1 5 Q N/ L N (T ) ˆ OL ˆT Q ˆ OL 2 2 / + Qτ + ρ L Q |||rτ −2 L 4 + l.o.t. O /O m|||4 |||r τ − ∇ρ||| 2,e

−3 2

≤ cϵ0 r0 (T1∗ + T2 ∗ + T3 ∗) .

(5.136)

The only other term in #

Vt

ˆ OL ˆ OL ˆ T F4 4 L ˆ OL ˆO τ 2+ L

-

(T ) λµ

π ˆ

. Dµ FλB .

that could cause trouble is # . N/ L N / ˆ OL ˆ T F (T ) iL ˆ OL τ 2+ ρ L ). O /O ∇α(F Σet

To bound this term it is enough to bound #

Σet

. N/ L N / ˆ OL ˆ T F (T ) π ˆ OL τ 2+ ρ L NCD L ) O /O ∇α(F

H H H (T ) H ≤ c sup Hr π ˆ CD H r0−1 Σet

×

(5.137)

0#

Σet

0#

Σet

H -

H τ 2+ Hρ

H2 H H Hˆ ˆ τ 2+ HL / )H O LO ∇α(F

≤ cϵ0 r0−1 (T1 ∗ + T2 ∗ + T3 ∗),

11 2

.H2 H ˆ ˆ ˆ LO LO LT F H

+ l.o.t.

11 2

404

5. ERROR ESTIMATES FOR F

where #

Σet

H

H2

Hˆ ˆ H τ 2+ HL / )H O LO ∇α(F

0#

≤c

+ +

(5.138)

+

#

Σet

Σet

#

Σet

#

Σet

H .H2 H H ˆ ˆ L τ 2+ Hρ L L (F ) H O O S H -

H τ 2+ Hρ

.H2 ˆ OL ˆ O LT (F ) HH L

H .H2 H H ˆ ˆ τ 2+ Hσ L L (F ) L H O O S H -

H τ 2+ Hσ

.H2 ˆ O LT (F ) HH + T1 + T2 ˆ OL L

1

by the null . of the Maxwell Equations. Therefore, we can - decomposition ˆ OL ˆ O J (1) (T ) . bound E L 4 . ˆ OL ˆ O J (1) (T ) , we consider To bound E L B ˆ OL ˆ O J (1) (T ) . L B

We calculate -

. 1 ˆ ˆ (T ) 1 ˆ ˆ (T ) ˆ OL ˆ O (S) π L ˆ λµ Dµ FλB = L ˆ 44 D3 F3B + L ˆ 33 D4 F4B O LO π O LO π 4 4 1 ˆ ˆ (T ) 1 ˆ ˆ (T ) + L ˆ 34 D4 F3B + L ˆ 34 D3 F4B O LO π O LO π 4 4 1 ˆ ˆ (T ) 1 ˆ ˆ (T ) − L ˆ C3 D4 FCB − L ˆ C4 D3 FCB O LO π O LO π 2 2 1 ˆ ˆ (T ) 1 ˆ ˆ (T ) − L ˆ C4 DC F3B − L ˆ C3 DC F4B O LO π O LO π 2 2 ˆ O (T ) π ˆ OL × 2L NCD DC FDB .

ˆ/2 (T ) j D ˆ/2 (T ) i∇σ(F ˆ/2 (T ) nD / 3 α, L / 4 α and L / ). We The troublesome terms are L O O O see that # .. - 2 . - ˆ/ (T ) n (D ˆ OL ˆT F ˆ OL τ 2+ ρ L L / 3 α) O Σet

≤ ≤

#

Σet

0#

H H .H H. HH ˆ ˆ H ˆ ˆ ˆ H L L L L τ 2+ Hρ L F ν (D α(F )) HH O O H + l.o.t. 3 O O T

Σet

H .H2 H H ˆ ˆ ˆ L L τ 2+ Hρ L F H O O T

+ l.o.t.

1 1 0# 2

Σet

HH2 . H ˆ ˆ H L τ 2+ H L ν (D α(F )) H 3 O O

11 2

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

≤

0#

Σet

×

'#

≤

(5.139)

∞

u0

≤ r0−1 ×

H .H2 H H ˆ ˆ ˆ τ 2+ Hρ L O LO LT F H

0#

'#

∞

11 2

5

3 1 2 2 ˆ 2 ˆ 2 2 r0−2 τ −5 − |||r LO LO ν|||4 |||r τ − D3 α(F )|||4

Σet

H .H2 H H ˆ ˆ ˆ τ 2+ Hρ L O LO LT F H

11

(1 2

2

3

3 1 2 2 ˆ 2 ˆ 2 2 τ −3 − |||r LO LO ν|||4 |||r τ − D3 α(F )|||4

u0 −1 cϵ0 r0 (T1 ∗

405

+ T2 ∗ + T3 ∗).

(1 2

ˆ/2 (T ) j D / 4 α can be estimated in the same way. Now The term L O # .- 2 . ˆ/ (T ) i (∇σ(F ˆ OL ˆT F ˆ OL τ 2+ ρ L L / )) O Vt

≤ ≤

# t# 0

Σet

# t 0#

(5.140)

0

H H .H H- 2 . H H ˆ (T ) H H ˆ ˆ ˆ τ 2+ Hρ L /O i (∇σ(F / ))H + l.o.t. O LO LT F H H L

Σet

H .H2 H H ˆ ˆ ˆ τ 2+ Hρ L O LO LT F H

1 1 0# 2

Σet

H2 H- 2 . H H ˆ (T ) τ 2+ H L / ))H /O i (∇σ(F

11 2

and #

Σet

(5.141)

H-

H τ 2+ H

H2 . 2 (T ) H ˆ / ))H L /O i (∇σ(F

≤ cr0−4

#

∞

u0

5

3

5

1

2 2 2 / ∇N τ −4 / η |||24,e |||r 2 τ −2 ∇σ||| / 4,e . − |||r τ − ∇

Combining 5.137, 5.139, 5.140 and 5.140, we see that

(5.142)

. . ˆ OL ˆ OL ˆ O J (1) (T ) + E L ˆ O J (1) (T ) E L 4 B $ % −1 ≤ cε0 2 + 2t′ (T3∗ + T[2] ∗).

ˆ O J (2) (T ). Recall from the previous ˆ OL 5.3.1.2. Error estimates for L section that we need to estimate . # . (2) ˆ ˆ OL ˆ OL ˆ T F4 4 L ˆ OL ˆ O Dµ (T ) π ˆ τ 2+ L ˆ µλ F λ4 (5.143) E LO LO J (T )4 = Vte

406

5. ERROR ESTIMATES FOR F

and (5.144) . # (2) ˆ ˆ E LO LO J (T )B =

Σet

. ˆ OL ˆ OL ˆ T F4 B L ˆ OL ˆ O Dµ (T ) π τ 2+ L ˆ µλ F λB .

The estimates for these terms follow directly from the Bootstrap Assumptions without difficulties. ˆOL ˆ O J (3) (T ). Recall that we also need 5.3.2. Error estimates for L to estimate . ˆ OL ˆ O J (3) (T ) E L 4 # -. . (T ) µλ ˆ OL ˆ OL ˆ T F4 4 L ˆ OL ˆ O D (T ) π τ 2+ L ˆ − D π ˆ (5.145) F = 4µ µλ 4 λ Σet

and

(5.146)

. ˆ OL ˆ O J (3) (T ) E L B # -. . (T ) µλ ˆ OL ˆ OL ˆ T F4 B L ˆ OL ˆ O D (T ) π ˆ − D π ˆ F . τ 2+ L = Bµ µλ B λ Σet

The only term the cannot from the Bootstrap . - be estimated directly (T ) (T ) 2 ˆ ˆ µλ − Dλ π ˆ Bµ LO F µλ . The most deliAssumptions comes from DB π .. (T ) N N/ L cate term is D /3 i L O /O α(F ) . We can estimate it using the fact that H H H 3 H supΣet HHrτ −2 N η HH ≤ cε0 in the following manner # ... (T ) N/ L N/ α(F ) ˆ OL ˆT F D ˆ OL τ 2+ α L /3 i L O O Σet

0# H H H −1 H 5 2 ≤ sup HHrτ − D3 N η HH r0 e Σt

×

(5.147)

Σet

0#

Σet

H -

H τ 2+ Hα

H .H2 H ˆ OL ˆ OL ˆ T F HH τ 2+ Hα L

.H2 ˆ OL ˆ O F HH L

11

11 2

2

+ l.o.t.

≤ cε0 r0−1 (T3 (t) + T[2] (t)). &

N OL N OL N S F )µ K µ . DivT(L ˆ O J (1) (S). Recall that we need to ˆ OL 5.3.3.1. Error estimates for L estimate . # . (1) ˆ ˆ ˆ OL ˆ OL ˆ S F4 4 L ˆ OL ˆ O (S) π (5.148) E LO LO J (S)4 = τ 2+ L ˆ λµ Dµ Fλ4 5.3.3. Error estimates for

Vte

Σe t

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

407

and (5.149) . # (1) ˆ ˆ E LO LO J (S)B =

Vte

ˆ OL ˆ OL ˆ S F4 B L ˆ OL ˆO τ 2+ L

-

(S) λµ

π ˆ

. Dµ FλB .

First, we consider ˆ O J (1) (S)4 ˆ OL L Using 5.10, we calculate . 1 ˆ ˆ (S) ˆ O (S) π ˆ OL ˆ λµ Dµ Fλ4 = L ˆ 44 D3 F34 L O LO π 4 1 ˆ ˆ (S) 1 ˆ ˆ (S) − L ˆ C3 D4 FC4 − L ˆ C4 D3 FC4 O LO π O LO π 2 2 1 ˆ ˆ (S) ˆ O(S) π ˆ OL (5.150) − L ˆ C4 DC F34 + 2L NCD DC FD4 . O LO π 2 -

With this calculation, we estimate # . ˆ OL ˆ OL ˆ S F4 4 L ˆ O (S) π ˆ OL τ 2+ L ˆ λµ Dµ Fλ4 Σet

Q .Q Q 1 Q Q Q −1 2 N N (S) Q ˆ ˆ ˆ Q ≤ LO LO LS F Q Qτ − r0 L /O L / O nQ 2,e 2,e Q 5 Q Q Q .Q Q 1 Q Q N N (S) Q Q Q ˆ OL ˆS Q ˆ OL /O L /O mQ + Qr 2 D4 α(F )Q Q Qr 2 L Qτ + ρ L ∞,e 2,e 2,e Q Q Q Q .Q Q 1 Q 2 1 Q Q Q 2 N N (S) Q Q 2 ˆ ˆ ˆ Q L / L / L L +Q r τ D α(F ) ρ L m τ r Q Q Q Q + O O S O O Q − 3 Q 2,e 2,e ∞,e Q Q 5 Q .Q .Q Q Q 1 Q Q Q Q 2 Q (S) N/ L N/ L N N ˆ OL ˆ S Q Qr 2 L ˆ OL Q + Qr ∇α / Q Qτ + ρ L 0 O /O i − L O /O Λ Q Q ∞,e 2,e 2,e Q Q Q .Q Q Q Q ˆ ˆ Q Q Q ˆ OL ˆ OL ˆ S Q Qr2 ∇α + QrLO LO ΛQ / Q2,e Qτ + ρ L ∞,e 2,e Q Q Q Q .Q Q 1 Q 2 1 Q Q Q 2 N N (S) Q 2 ˆ OL ˆS Q ˆ OL Q τ r + Qr τ − ∇ρ / Q ρ L m L / L / Q + l.o.t. Q Q Q + O O Q Q Q

Q Q

1

Q

Q 2 2 Q cr0−2 Q Qτ + ρ Qr τ − D3 ρ(F )Q ∞,e

≤

cϵ0 r0−2 (T1 ∗

(5.151)

∞,e

-

2,e

2,e

+ T2 ∗ + T3 ∗)

ˆ OL ˆ O J (1) (S)4 , in which we take the If we consider the second part of L Lie derivatives of the terms with components of the electromagnetic field instead of the deformation tensor, then we see that all the of terms in # ˆ OL ˆ OL ˆ S F4 4(S) π ˆ OL ˆ O Dµ Fλ4 r0−δ τ 2+ L ˆ λµ L Σet

408

5. ERROR ESTIMATES FOR F

Q Q can be estimated in the same manner because Qr(S) π ˆ λµ Q∞,e ≤ ε0 . FurtherQ Q Q ˆ (S) λµ Q ˆ Q ≤ ε0 holds. Therefore, the terms in more, QrLO π ∞,e # ˆ OL ˆ OL ˆ S F4 4 L ˆ O (S) π ˆ O Dµ Fλ4 r0−δ τ 2+ L ˆ λµ L Σet

ˆ O J (1) (S)4 ) are also bounded. ˆ OL (the last term when we apply Leibniz rule to L Now we consider # . ˆ OL ˆ OL ˆ S F4 B L ˆ OL ˆ O (S) π τ 2+ L ˆ λµ Dµ FλB Σet

to estimate 5.149. Using 5.11, we calculate . 1 ˆ ˆ (S) 1 ˆ ˆ (S) ˆ OL ˆ O (S) π L ˆ λµ Dµ FλB = L ˆ 44 D3 F3B + L ˆ 33 D4 F4B O LO π O LO π 4 4 1 ˆ ˆ (S) 1 ˆ ˆ (S) + L ˆ 34 D4 F3B + L ˆ 34 D3 F4B O LO π O LO π 4 4 1 ˆ ˆ (S) 1 ˆ ˆ (S) − L ˆ C3 D4 FCB − L ˆ C4 D3 FCB O LO π O LO π 2 2 1 ˆ ˆ (S) 1 ˆ ˆ (S) − L ˆ C4 DC F3B − L ˆ C3 DC F4B O LO π O LO π 2 2 ˆ O (S) π ˆ OL + 2L NCD DC FDB .

The first term is the most sensitive term and can be estimated in the following manner # ˆ OL ˆ OL ˆ S F4 4 L ˆ OL ˆ O(S) π τ 2+ L ˆ 44 D3 F3B Σte # H H .H H . H H N/ L N/ ν (D ˆ OL ˆ OL ˆ S F HH HHτ − L ≤ τ 2+ Hα L / α(F )) H + l.o.t O O 3 Σte

0#

≤ r0−1 ×

(5.152)

Σte

'#

∞

u0

−3

H .H H ˆ S F HH ˆ2 L τ 2+ Hα L O

11 2

2 3 1 5 2N τ −3 / 3 α(F )|||24 /O ν|||24 |||r 2 τ −2 D − |||r L

(1 2

+ l.o.t.

≤ cϵ0 r02 (T1∗ + T2 ∗ + T3 ∗) .

The other terms can be estimated in a similar manner. As for the terms in # . ˆ OL ˆ OL ˆ S F4 B L ˆ OL ˆ O (S) π τ 2+ L ˆ λµ Dµ FλB Σet

that arise when we apply Leibniz rule, the only one that is estimated differently is (S) N N / D nL /O L / 3 α(F ). Q 2 O−1(S) Q In this case, we use the fact that Qr τ − nQ∞,e ≤ ε0 .

https://doi.org/10.1090/amsip/045/05

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

409

In conclusion, we find that (5.153) . . $ % ˆ OL ˆ OL ˆ O J (1) (S)4 + E L ˆ O J (1) (S)B ≤ cε0 2 + 2t′ −1 (T3 ∗ +T[2] ∗) E L

ˆ O J (2) (S) and L ˆ O J (3) (S). These ˆOL ˆOL 5.3.4. Error estimates for L & N OL N S F )µ K µ . follow from the corresponding estimates for Vt DivT(L &

N OL NSL N S F )µ K µ . DivT(L ˆ S J (1) (S) and L ˆ S J (2) (S) and ˆ OL ˆ OL 5.3.5.1. Error estimates for L ˆ OL ˆ S J (3) (S). Recall that we need to estimate L . # . (1) ˆ OL ˆ S J (S)4 = ˆ OL ˆS L ˆ S F4 4 L ˆ OL ˆ S (S) π (5.154) E L τ 2+ L ˆ λµ Dµ Fλ4 5.3.5. Error estimates for

Σe t

Vte

and

. # (1) ˆ ˆ (5.155) E LO LS J (S)B =

Vte

. ˆ OL ˆS L ˆ S F4 B L ˆ OL ˆ S (S) π τ 2+ L ˆ λµ Dµ FλB .

. ˆ S J (1) (S)4 follow from the estimates for ˆ OL The estimates for E L # . 4ˆ ˆ 2 ˆ ˆ ˆ (S) λµ τ + LO LO LS F4 LO LO π ˆ Dµ Fλ4 Σet

(see 5.148) and

#

. ˆS L ˆ S F4 4 L ˆ S (S) π τ 2+ L ˆ λµ Dµ Fλ4

#

. ˆS L ˆ S F4 B L ˆ S (S) π τ 2+ L ˆ λµ Dµ FλB

Σet

(see 5.89). . ˆ OL ˆ S J (1) (S)B follow from the estimates for The estimates for E L # . ˆ OL ˆ OL ˆ S F4 B L ˆ OL ˆ O (S) π τ 2+ L ˆ λµ Dµ FλB Σet

(see 5.149) and

Σet

(see 5.90). Next we need to estimate . # (2) ˆ ˆ (5.156) E LO LS J (S)4 =

. ˆ OL ˆS L ˆ S F4 4 L ˆ OL ˆ S Dµ (S) π τ 2+ L ˆ µλ F λ4

. # (2) ˆ ˆ (5.157) E LO LS J (S)B =

. ˆ OL ˆS L ˆ S F4 B L ˆ OL ˆ S Dµ (S) π τ 2+ L ˆ µλ F λB .

Vte

and

Vte

410

5. ERROR ESTIMATES FOR F

The estimates for these quantities follow directly from the Bootstrap Assumptions and the estimates for # . ˆ OL ˆ OL ˆ S F4 B L ˆ OL ˆ O Dµ (S) π τ 2+ L ˆ µλ F λB Σet

(see 5.149) and

#

Σet

. ˆS L ˆ S F4 B L ˆ S (S) π τ 2+ L ˆ λµ Dµ FλB

(see 5.90). . ˆ OL ˆ S J (3) (S) are very similar to those proThe error estimates for E L . ˆ OL ˆ O J (3) (S) . duced for E L

5.3.6. Other terms in J (O, S, T ). We do not- have difficulty esti. λµ ˆ O J (1) (S, S) = L ˆ O (S) π ˆ mating the error terms involving L ˆ D ( L F ) µ S λν , ν '' . ˆ S F λ ν , and L ˆ O Dµ (S) π ˆ O J (3) (S, S) = L ˆ O Dν(S) ˆ O J (2) (S, S) = L ˆ µλ L L ν ν ( ( (S) µλ ˆ because these estimates are very similar to those for π ˆ µλ −Dλ π ˆ νµ LS F & 2 (1) (O, S, S) = (O)π ˆ S Fλν ) ˆ ˆ ˆS L ˆ λµ Dµ (L ν Vt τ + DivT(LO LS F ). Furthermore, J & µ NS L N T F )µ K . However, NS L do not cause difficulties because we bound e T(L Σt (2) we have to be a -little careful with J .(O,S, S)ν (O) (O) ˆ 2 F µλ . J (3) (O,S, S)ν = Dν π ˆ µλ − Dλ π ˆ νµ L S

ˆ 2 F λ ν and = Dµ (O) π ˆ µλ L S

To estimate these terms we use the Sobolev inequalities to obtain 1

1 ˆS L ˆ S ρ(F )|||4,e ≤ T[3] |||r 2 τ −2 L ˆ S α(F )|||4,e ≤ T[3] . ˆS L |||rL

N (see 5.18) we are able Combining these bounds with the estimates for (O) π to bound J (2) (O,S, S) and J (3) (O,S, S). & NSL NSL N T F )µ K µ . 5.3.7. Error estimates for Σe DivT(L t ˆ S J (1) (T ). Recall that we need to estiˆS L 5.3.7.1. Error estimates for L mate . # . . 2 4 ˆ ˆ (T ) λµ ˆ S J (1) (T ) = ˆ ˆ ˆ ˆS L E L L L L τ F π ˆ D F L L µ λ4 S S T 4 S S + 4 Vte

and

. # (1) ˆ ˆ E LS LS J (T )B =

Vte

. ˆS L ˆS L ˆ T F4 B L ˆ OL ˆ S (T ) π τ 2+ L ˆ λµ Dµ FλB .

The estimates for these terms follow from the Bootstrap Assumptions and the methods used in 5.82.

5.3. THIRD DERIVATIVES FOR F IN THE EXTERIOR

411

ˆ S J (2) (T ) and L ˆ S J (3) (T ). Recall ˆS L ˆS L 5.3.7.2. Error estimates for L that we need to estimate . # . 2 ˆ ˆ ˆ 4ˆ ˆ µ (T ) λ ˆ S J (2) (T ) = ˆS L L L L L L τ F π ˆ F D (5.158) E L 4 S S µλ + S S T 4 4 Vte

and

. # ˆ S J (2) (T ) ˆS L (5.159) E L B =

Vte

. ˆS L ˆS L ˆ T F4 B L ˆS L ˆ S Dµ (T ) π τ 2+ L ˆ µλ F λB .

. ˆ S J (2) (T ) from ˆS L In estimating E L 4

. 1 . 1 . ˆS L ˆS L ˆS L ˆ S Dµ (X) π ˆ S D4(T ) π ˆ S D3(T ) π L ˆ 43 F34 + L ˆ 44 F34 ˆ µλ F λ 4 = L 4 4 . 1 ˆ ˆ - (T ) ˆ 3C FC4 − LS LS D4 π 2 . 1 ˆ ˆ - (T ) − L ˆ 4C FC4 S LS D3 π 2 . ˆ S DD(T ) π ˆS L ˆ CD FC4 . (5.160) + 4L

2 ˆ S Dµ (X) π ˆS L ˆ µλ can be bounded it is enough to show that L $ −2by % ε0 in the L e norm on Σt because α(F ) and ρ(F ) decay at least like O r . Remark that when estimating D3 ν, we only need to consider D4 δ and D4 ϕN because ∇N ν can be expressed in terms of D4 δ. These estimates are derived in Chapters 8 and 9. To estimate . ˆS L ˆ S J (2) (T ) , E L B

we calculate

ˆ S Dµ (T ) π ˆS L ˆ µλ F λ L

(5.161)

B

1ˆ ˆ 1ˆ ˆ (T ) (T ) = L ˆ 34 F4B + L ˆ 44 F3B S LS D3 π S LS D3 π 4 4 1ˆ ˆ 1ˆ ˆ (T ) LS LS D4(T ) π ˆ 33 F4B + L ˆ 34 F3B S LS D4 π 4 4 1ˆ ˆ 1ˆ ˆ (T ) (T ) − L ˆ 3C FCB − L ˆ 4C FCB S LS D4 π S LS D3 π 2 2 1ˆ ˆ 1ˆ ˆ (T ) (T ) − L π ˆ 3C F4B − L π ˆ 4C F3B S LS DC S LS DC 2 2 ˆ S DD(T ) π ˆS L + 4L ˆ CD FCB .

The most sensitive terms F3A and in particular α(F ). In terms of r $ involve % this only decays like O r−1 . Therefore, we must ensure that r0 ||D2S D4 ν||2,e ≤ cε0 , i.e. that r0 ||D2S D4 δ||2,e ≤ cε0 and r0 ||D2S D4 ϕN ||2,e ≤ cε0 . The first inequality follows from equation 9.40 and the Bootstrap Assumptions. The second is discussed in the chapter on the lapse function.

412

5. ERROR ESTIMATES FOR F

ˆ 2 J (3) (T ), L S

ˆ2 L S

'

(T )

DC π In estimating the troublesome terms are ˆ B4 ( (T ) −D4 π ˆ BC FC3 . We produce these estimates in the chapter on the sec-

ond fundamental form. The estimates for . ˆ T Fλν ) ˆ S (S) π ˆ S J (1) (S, T ) = L ˆ λµ Dµ (L L ν . λ ˆ ˆ S Dµ (S) π ˆ S J (2) (S, T ) = L L ˆ F L ν µλ T ν -. . (S) µλ ˆ S Dν(S) π ˆ ˆ S J (3) (S, T ) = L ˆ − D π ˆ F L L νµ T µλ ν λ ˆ T Fλν ) ˆS L ˆ λµ Dµ (L J (1) (S, S, T )ν = (S)π ˆS L ˆ T F λν J (2) (S,S, T )ν = Dµ (S) π ˆ µλ L . (S) ˆ T F µλ ˆ OL J (3) (S,S, T )ν = Dν(S) π ˆ µλ − Dλ π ˆ νµ L

follow modifications to the estimates for the corresponding terms & from slight N 2 F )µ K µ . in Vt DivT(L S Finally we have shown that all of the error terms integrated over Σet can be bounded by $ %−1 cε0 2 + 2t′ (T3∗ + T[2] ∗).

CHAPTER 6

Interior Estimates for F In this chapter, we discuss estimates for the electromagnetic field on the interior region. We begin by showing that iF0 is bounded by the initial data. We then show that F3i is bounded by the initial data. The estimates for iF [2] follow from the same methodology and therefore we do not write them out explicitly. 6.1. Estimates for F0i

To show that r02

(6.1) we use

I

C 1 2 τ |α(F )|2 + τ 2+ |α(F )|2 T(F ) K, T = 2 − D $ 2 %$ % 2 2 2 + τ − + τ + ρ(F ) + σ(F ) , %

$

where

# A B |E|2 + |H|2 dµg ≤ cε,

τ 2− = 1 + u2

τ 2+ = 1 + v 2 .

Recall that v = 2r − u, and u is negative in the interior region. Furthermore in the interior # 0 1 dr 1 r0 ≤ r0 − r(u) = ≤ − c(a, trθ)u. 2 du 2 u Therefore τ − ∼ r0 . Collecting these facts together, we infer that # A # B r02 |E|2 + |H|2 dµg ≤ T(F )(K, T ). &

I

I

Now we show that I T(F )(K, T ) can be bounded by the initial data. Recall that # # $ % $ % T(F ) K, T = T(F ) K, T Σt

Σt0

+

#

t

t0

dt

′

J#

Σt′

K R S 1 µ µν (K ) φDivT(F )µ K + T(F ) π Nµν dµg . 2 413

414

6. INTERIOR ESTIMATES FOR F

However, DivT(F ) = 0. Therefore, we only need to show that J# R K S # t # 1 ′ µν (K ) π Nµν dµg ≤ ε0 sup T(F ) dt T(F )(K, T ). Σt′ 2 t0 [0,t∗ ] Σt

Since we already have derived these estimates in the exterior region, it is enough to show that the estimate holds on the interior region. We derive H # H 3 H1 H H T(F )µν (K ) π H dµg ≤ r− 2 ||(K ) π N N||∞,I ||E, H||22,I µν 0 H H I 2 # $ % −3 ≤ ε0 r0 2 T(F ) K, T . I

This follows from the Bootstrap Assumptions

1

N||∞,I ≤ ε0 r02 ||(K ) π

and the calculation # # $ % $ % T(F ) K, T ≤ cr02 ||E, H||22,I ≤ E c T(F ) K, T . I

I

Hence, we conclude that

r02

# A B |E|2 + |H|2 dµg ≤ cε. I

6.2. Estimates for i F[3]

Our goal is to show that the quantity # A 3 B " 2s+2 (6.2) r0 |∇s E|2 + |∇s H|2 dµg s=1

Int

is bounded by the initial data. We assume for the moment that the first and second derivatives are bounded and show that the third derivatives are controlled by the initial data. Estimates for the first and second derivatives follow using the same methodology. For these derivations, we write the Maxwell Equations in the following form 1 DT Hi − curlEi = ϵi jk ∇j ϕEk + kij H j 2 1 jk (6.3) DT Ei + curlHi = ϵi ∇j ϕHk + kij E j 2 These equations are computed with respect to the Fermi-propagated orthonormal frame and D is the restriction of the space-time covariant derivative to the leaves of the Σ-foliation. We treat these equations as firstorder hyperbolic partial differential equations and obtain energy estimates using standard techniques.

6.2. ESTIMATES FOR i F[3]

415

First, we differentiate the equations three times with respect to the induced covariant derivative on Σt . Then, we multiply each equation by the appropriate tensor such that we have equations for DT (∇3 H) and DT (∇3 E). After adding these two equations, we integrate with respect to t. Lastly, we estimate the error terms that arise and apply Gronwall’s Inequality to obtain the desired bounds. To follow this program, we need to commute ∇ with DT . We employ the following commutation lemmas, which come from Lemma 13.1.2 in [5]. Lemma 3. If Ui is a tensor tangent to Σt then

$ % ∇l DT Ui − DT ∇l Ui = RlT is Us + kls ∇s Ui + φ−1 ∇s φ kli Us $ % $ % + φ−1 ∇i φ kls Us + φ−1 ∇l φ DT Ui .

Lemma 4. If Uil is a tensor tangent to Σt then

∇m DT Uil − DT ∇m Uil = R mT is Usl + R mT ls Uis + kms ∇s Uil $ % $ % + φ−1 ∇s φ kmi Usl + φ−1 ∇s φ kml Uis $ % $ % + φ−1 ∇i φ kms Usl + φ−1 ∇l φ kms Uis $ % + φ−1 ∇m φ DT Uil

Lemma 5. If Uilm is a tensor tangent to Σt , then

∇n DT Uilm − DT ∇n Uilm = RnT is Uslm + RnT ls Uism + R nT ms Uils $ % + kns ∇s Uilm + φ−1 ∇s φ kni Uslm $ % $ % + φ−1 ∇s φ knl Uism + φ−1 ∇s φ knm Uils $ % $ % + φ−1 ∇i φ kns Uslm + φ−1 ∇l φ kns Uism $ % $ % + φ−1 ∇m φ kns Uils + φ−1 ∇n φ DT Uilm .

Using these lemmas, we can define Error(DT H) = ∇n ∇m ∇l DT Hi − DT ∇ n ∇ m ∇ l H i . Definition 7. Let

Error(DT H) = ∇n ∇m ∇l DT Hi − DT ∇n ∇m ∇l Hi = RnT is ∇m ∇l Hs + RnT ls ∇m ∇s Hi + RnT ms ∇s ∇l Hi + kns ∇s ∇m ∇l Hi + (φ−1 ∇s φ)kni ∇m ∇l Hs + (φ−1 ∇s φ)knl ∇m ∇s Hi

+ (φ−1 ∇s φ)knm ∇s ∇l Hi + (φ−1 ∇i φ)kns ∇m ∇l Hs $ % $ % + φ−1 ∇l φ kns ∇m ∇s Hi + φ−1 ∇m φ kns ∇s ∇l Hi $ % + φ−1 ∇n φ DT ∇m ∇l Hi + ∇n [RmT is ∇l Hs + RmT ls ∇s Hi + kms ∇s ∇l Hi + (φ−1 ∇s φ)kmi ∇l Hs + (φ−1 ∇s φ)kml ∇s Hi + (φ−1 ∇i φ)kms ∇l Hs + (φ−1 ∇l φ)kms ∇s Hi

+ (φ−1 ∇m φ)DT ∇l Hi ] + ∇n ∇m [RlT is Hs + kls ∇s Hi

+ (φ−1 ∇s φ)kli Hs + (φ−1 ∇i φ)kls Hs + (φ−1 ∇l φ)DT Hi ].

416

6. INTERIOR ESTIMATES FOR F

An analogous definition can be made for Error(DT E) = ∇n ∇m ∇l DT Ei − DT ∇ n ∇ m ∇ l E i . Error terms also arise when commuting ∇ with curl. Definition 8. Define

Error(∇H) = ∇n ∇m ∇l ∇j Hk − ∇j ∇n ∇m ∇l Hk

= ∇n ∇m (Rjlks Hs ) + ∇n (Rjmls ∇s Hk + Rjmks ∇l Hs ) + Rjnms ∇s ∇l Hk + Rjnls ∇m ∇s Hk + Rjnks ∇m ∇l Hs

There is an analogous definition for Error(∇E).

Using these definitions, the Maxwell Equations become ' ( 1 jk 3 j 3 3 DT ∇ Hi − curl∇ Ei = ∇ ϵi ∇j ϕEk + kij H 2 (6.4) + Error(DT H) + Error(∇E) and

( 1 j DT ∇ Ei + curl∇ Hi = ∇ ϵi ∇j ϕHk + kij E 2 + Error(DT E) + Error(∇H). 3

(6.5)

3

3

'

jk

Following [5], consider the interior region V =

∪ Vt

t∈[0,t∗ ]

with the Vt defined as closed balls of radius d1 (t) = 9/20 (1 + t) centered at Pt and P0 ∈ Σ0 is contained in a closed ball of radius ε1 centered at O0 . Let P be the integral curve of T through the point P0 . The Pt are the points of intersection of P with the Σt . Let M0 be the union of the interior of C0 to the past of Σ0 and C0 to the past of Σ0 . Now let uM (t) be the largest value of u so that the past part of the light cone Cu is contained in V ∪ M0 . Define I as (6.6)

∪ Int St,uM (t)

t∈[0,t∗ ]

together with the part of CuM (0) that lies to the past of Σ0 . Furthermore, let tm (u) denote the value of t at the vertex of Cu such that t0 (u) = tm (uM ) i.e. r(uM (t), t0 ) = 0. In our estimates below, we use Lemma 9.2.1 from [5]: 1 2

Lemma 6. If ε0 is sufficiently small, then, for all t ∈ [0, t∗ ], 2 + t0 ≥ (2 + t) .

We also use the fact from [5] that r (uM (t), t) ≥ 14 (1 + t) as well as the following truncation function ⎧ 1 ⎪ ( ' ⎨ 1 f or s ≤ r 4 = h(s) = (6.7) f ⎪ 2+t ⎩ 0 f or s ≥ 9 . 20

6.2. ESTIMATES FOR i F[3]

417

For this choice of f , the following estimates hold c 2+t c |DT f | ≤ , 2+t |∇f | ≤

(6.8)

where the constant depends on the lapse function and the second fundamental form of the St,u -foliation. Now we multiply equation 6.4 by r4 (uM , t)f and commute to obtain DT (r4 (uM , t)f ∇3 Hi ) − curl(r4 (uM , t)f ∇3 Ei ) ' ( 1 4 3 jk j = r (uM , t)f (∇ ϵi ∇j ϕEk + kij H 2

+ r4 (uM , t)f (Error(DT H) + Error(∇E))

− (DT f ) r4 (uM , t)∇3 Hi − 3(DT r)r3 (uM , t)∇3 H + (∇f ) ∧ (r4 (uM , t)∇3 Ei ).

(6.9)

Next, we multiply this equation by r4 (uM , t)f ∇3 Hi to obtain (r4 (uM , t)f ∇3 Hi )DT (r4 (uM , t)f ∇3 Hi )

− (r4 (uM , t)f ∇3 Hi )curl(r4 (uM , t)f ∇3 E)i ' ( R $ 3 1 4 3 4 jk j = (r (uM , t)f ∇ Hi ) · r (uM , t)f ∇ ϵi ∇j ϕEk + kij H 2 + r4 (uM , t)f (Error(DT H) + Error(∇E))

(6.10)

− (DT f ) r4 (uM , t)∇3 Hi − 3(DT r)r3 (uM , t)∇3 H S 4 3 + (∇f ∧ r (uM , t)∇ Ei ) .

The analogous procedure is applied to equation 6.5. Adding 6.10 and the corresponding equation for E, we obtain (r4 (uM , t)f ∇3 Hi )DT (r4 (uM , t)f ∇3 Hi )

+ (r4 (uM , t)f ∇3 Ei )DT (r4 (uM , t)f ∇3 Ei )

− (r4 (uM , t)f ∇3 Hi )curl(r4 (uM , t)f ∇3 E)i

+ (r4 (uM , t)f ∇3 Ei )curl(r4 (uM , t)f ∇3 H)i ' ( R 1 4 3 4 3 jk j = (r (uM , t)f ∇ Hi ) · r (uM , t)f ∇ ϵi ∇j ϕEk + kij H 2

+ r4 (uM , t)f (Error(DT H) + Error(∇E)) − (DT f )r4 (uM , t)∇3 Hi S 3 3 4 3 − 3(DT r)r (uM , t)∇ Hi + (∇f ∧ r (uM , t)∇ Ei )

418

6. INTERIOR ESTIMATES FOR F

' R 4 3 + (r (uM , t)f ∇ Ei ) · r (uM , t)f ∇ ϵi 4

(6.11)

3

jk

1 ∇j ϕHk + kij E j 2

(

+ r4 (uM , t)f (Error(DT E) + Error(∇H)) − (DT f ) r4 (uM , t)∇3 Ei S − 3(DT r)r3 (uM , t)∇3 Ei + (∇f ∧ r4 (uM , t)∇3 Hi ) .

Multiplying by φ, the lapse for the time slices, and applying Leibniz rule, the left-hand side of the equation 6.11 becomes H2 . H2 . ∂ -HH 4 ∂ -HH 4 r (uM , t)f ∇3 H H + r (uM , t)f ∇3 E H ∂t ∂t − 2φr4 (uM , t)f ∇3 H curl(r4 (uM , t)f ∇3 E) + 2φr4 (uM , t)f ∇3 E curl(r4 (uM , t)f ∇3 H).

(6.12)

Integrating by parts on Σt , the curl terms vanish, and no boundary terms appear because of the presence of the truncation function. After integration 6.12 becomes # H2 H H2 . ∂ -HH 4 r (uM , t)f ∇3 H H + Hr4 (uM , t)f ∇3 E H dµg Σt ∂t # (6.13) + 2∇j φ(r4 (uM , t)f ∇3 Hi ϵi jk r4 (uM , t)f ∇3 Ek )dµg . Σt

Integrating 6.13 with respect to t over the interval [t0 , t] and using the fact that r(uM , t0 ) = 0 and trkij = 0 so that the derivative of the metric g by t does not contribute terms, we obtain # H H H . H 4 Hr (uM , t)f ∇3 H H2 + Hr4 (uM , t)f ∇3 E H2 dµg Σt K Y # J# -H t H2 H 4 H2 . 4 ′ 3 ′ 3 ≤c |∇φ| Hr (uM , t )f ∇ H H + Hr (uM , t )f ∇ E H dµg dt′ Σt′

t0

+

# t J# t0

+

t0

+

Σt′

# t J#

Σt′

# tJ# t0

Σt′

-H H2 H H2 . |∇f | Hr4 (uM , t′ )f ∇3 H H + Hr4 (uM , t′ )f ∇3 E H dµg

K

-H H2 H H2 . |DT f | Hr4 (uM , t′ )f ∇3 H H + Hr4 (uM , t′ )f ∇3 E H dµg H H HDT r(uM , t′ )H

H 7 H2 H H + Hr 2 (uM , t′ )f ∇3 E H

1

0

H2 H 7 H H 2 Hr (uM , t′ )f ∇3 H H

K

′

dµg dt +

# t RC # t0

Σt′

dt′

K

|φ(r4 (uM , t′ )f ∇3 Hi )

dt′

'

6.2. ESTIMATES FOR i F[3]

1 j × [r4 (uM , t′ )f (∇3 ϵjk i ∇j ϕEk + kij H 2 K J# +

+ Error(∇E))]|dµg Y

4

′

× r (uM , t )f (∇

3

'

Σt′

ϵjk i ∇j ϕHk

(

419

+ r4 (uM , t′ )f (Error(DT H)

|φ(r4 (uM , t′ )f ∇3 Ei ) 1 + kij E j 2

(

+ r3 (uM , t′ )f (Error(DT E) + Error(∇H))]|dµg

KZ

dt′

Using the estimates for |∇f | and |DT f | and the Bootstrap Assumption |∇φ| ≤ r0−2 ε0 , we obtain # H H H . H 4 Hr (uM , t)f ∇3 H H2 + Hr4 (uM , t)f ∇3 E H2 dµg Σt K Y # J# . t H H H 1 -HH 4 2 2 ≤c r (uM , t′ )f ∇3 H H + Hr4 (uM , t′ )f ∇3 E H dµg dt′ ′ 2 + t Σt′ t0 # t YJ # + |φ(r4 (uM , t′ )f ∇3 Hi ) · [r4 (uM , t′ ) t0

f (∇

3

'

Σt′

ϵjk i ∇j ϕEk

( 1 j + kij H + r4 (uM , t′ )f (Error(DT H) 2 J#

+ Error(∇E))]|dµg } + Y

0

'

Σt′

|φ(r4 (uM , t′ )f ∇3 Ei )·

1 j × r4 (uM , t′ )f ∇3 ϵjk i ∇j ϕHk + kij E 2

(

Z

+ r4 (uM , t′ )f (Error(DT E) + Error(∇H)) |dµg

KZ

dt′ .

(6.14) Our goal is to apply Gronwall’s Inequality to this expression and, consequently, prove that the weighted norm is bounded by the initial data. To accomplish this goal, we first estimate the terms on the right-hand side of 6.14. In particular, we show that all of the terms are of the form J # 0 # t H 4 H 1 Hr (uM , t′ )f ∇3 H H2 c ′ Σt′ t0 2 + t K 1 H 4 H2 ′ 3 ′ + Hr (uM , t )f ∇ E H dµg + cF[2] (t ) dt′ .

420

6. INTERIOR ESTIMATES FOR F

The terms involving the second fundamental form k are treated in the same way as the terms involving ∇φ since they behave the same in the interior region, i.e. i

and i

2+q− 32

Kq = r0

2+q− 32

Lq = r0

∥ Dq k∥2,i

Q q+1 Q Q D ϕQ

2,i

where ϕ is log φ. Using the Gauss-Codazzi equation ∇i kjm − ∇j kim = RmT ij ,

and that fact that Ric behaves like ∇∇φ, it is enough to estimate the terms # t# H 4 HH H Hr (uM , t′ )f ∇3 H H Hr4 (uM , t′ )f ∇3 (kH)H. (6.15) Σt′

t0

We estimate # t# H HH H r8 (uM , t′ ) Hf ∇3 H H Hf ∇3 (kH)H dµg dt′ t0

≤

Σt′

# t#

+ + + (6.16)

Σt′

t0

# t#

-

t0

Σt′

t0

Σt′

t0

Σt′

H H2 . r8 (uM , t′ ) Hf ∇3 H H |k| dµg dt′

# t# # t#

-

H H H2 H2 . r7 (uM , t′ ) Hf ∇3 H H + r9 (uM , t′ ) |∇k|2 Hf ∇2 H H dµg dt′

-

. H H2 H H2 r7 (uM , t′ ) Hf ∇3 H H + r9 (uM , t′ ) H∇3 k H |f H|2 dµg dt′ .

-

. H H2 H H2 r7 (uM , t′ ) Hf ∇3 H H + r9 (uM , t′ ) H∇2 k H |f ∇H|2 dµg dt′

Using the Bootstrap Assumption supI |k| ≤ ε0 r0−2 , the first term on the right-hand side of 6.16 is estimated as follows # t# H H2 . r8 (uM , t′ ) Hf ∇3 H H |k| dµg dt′ Σt′ # t

t0

(6.17)

Notice that

≤

# t# t0

(6.18)

t0

≤

Σt′ # t t0

1 c 2 + t′

-

#

Σt′

H2 H H H 7/2 Hr (uM , t′ )f ∇3 H H dµg dt′ .

H H2 . r7 (uM , t′ ) Hf ∇3 H H dµg dt′

1 c 2 + t′

#

Σt′

-

H H2 . r8 (uM , t′ ) Hf ∇3 H H dµg dt′

6.2. ESTIMATES FOR i F[3]

421

Thus terms such as this in 6.16 are in the correct form. Again using the Bootstrap Assumption supΣt |∇k| ≤ ε0 r0−3 , we have # t#

Σt′

t0

≤ (6.19)

#

-

1 ε0 c 2 + t′

t

t0

#

≤

H H2 . r9 (uM , t′ ) |∇k|2 Hf ∇2 H H dµg dt′

t

ε0 c

t0

0#

Σt′

H 3 H Hr (uM , t′ )f ∇2 H H2 dµg

1

dt′

1 F 2 (t′ )dt′ . 2 + t′ [2]

The next term in 6.16 can be estimated using the Sobolev inequalities from Proposition 3.2.2 in [5] as follows # t# t0

≤ ≤

Σt′

#

t0

#

t

t0

× (6.20) ≤

t

#

-

. H H2 r9 (uM , t′ )f H∇2 k H f |∇H|2 dµg dt′

Σt′

t

ε0 c

H H4 r12 Hf ∇2 k H dµg

Σt′

0#

1 c 2 + t′

0#

t0

0#

1 c 2 + t′

H H2 r4 Hf ∇2 k H dµg

Σt′

r4 |f ∇H|2 dµg

1 0# 1 4

Σt′

11 0# 2

Σt′

1 0# 1 4

Σt′

r8 |f ∇H|4 dµg

H H6 r20 Hf ∇2 k H dµg

r12 |f ∇H|6 dµg

11 4

dt′

1 F2 . 2 + t′ [2]

H 5 H Now using the estimate supI Hr02 H H ≤ F[2] , we have # t# t0

≤ (6.21)

≤

Σt′

#

t

t0

#

-

. H H2 r9 (uM , t′ ) H∇3 k H |f H|2 dµg dt′

1 c F 2 (t′ ) 2 + t′ [2]

t

t0

ε0 c

#

Σt′

1 F 2 (t′ ). 2 + t′ [2]

11

H H2 r7 (uM , t′ ) H∇3 k H dµg

2

11 4

dt′

422

6. INTERIOR ESTIMATES FOR F

Collecting the estimates in 6.17, 6.18, 6.19, 6.20, and 6.21 together, we find that # H 8 H H H . Hr (uM , t)f ∇3 H H2 + Hr8 (uM , t)f ∇3 E H2 dµg Σt J# # t H 4 H C Hr (uM , t′ )f ∇3 H H2 ≤ ε0 2 + t′ Σt′ t0 K H 4 H 2 + Hr (uM , t′ )f ∇3 E H dµg + F 2 (t′ ) dt′ [2]

≤

#

t

t0

+

C ε0 2 + t′

2 cF[2] (0).

J#

Σt′

H 4 H H H Hr (uM , t′ )f ∇3 H H2 + Hr4 (uM , t′ )f ∇3 E H2 dµg

K

dt′

(6.22) 1 We know that 2+t0 ≥ 12 (2 + t) for all t ∈ [0, t∗ ] and thus the sup 2+t ′ ≤ Therefore, # t C t − t0 ε0 dt′ ≤ const. ≤ const. ′ 2+t 2+t t0

c 2+t .

Finally, by Gronwall’s Inequality applied to 6.22, # H 4 H H H . Hr (uM , t)f ∇3 H H2 + Hr4 (uM , t)f ∇3 E H2 dµg ≤ εC. Σt

6.3. Top Derivatives for Local Existence

For the proof of local existence F must be in H3,1 . In this section, we use the methods of the previous section to show that the weighted L2 -norm of the third derivatives tangent to Σ are bounded. However, the previous section only focused on a compact set. As in [5], to extend estimates to all of Σt consider the truncation function fR , defined by ( ' σ 0 (x) , (6.23) fR (x) = h R

where x is an arbitrary point on Σ and h is a real function J 1:s≤E h (s) = 0 : s ≥ 2E

for E a constant. Eventually we will let R → ∞ for R a real number. Consider the quantity # -H H2 H H2 . (6.24) σ 80 HfR ∇3 E H + HfR ∇3 H H . Σt

6.3. TOP DERIVATIVES FOR LOCAL EXISTENCE

423

Using the estimates k ∈ H3,1 , φ ∈ H4,−1 and the Sobolev inequalities for any given tensor field f , we have ∥σ 0 f ∥L6 ≤ c ∥f ∥H1,0 H H H 3/2 H 1/2 1/2 sup Hσ 0 f H ≤ c ∥f ∥H1,0 ∥f ∥H2,0 . Σ

Note that the error terms in the previous section that arise when trying to bound the counterpart of 6.24 can bear the weight σ 80 in the L2 -norm on Σ (see 6.14). However, we cannot yet assume that the slices are maximal. When we take the derivative of # -H H2 H H2 . σ 80 HfR ∇3 E H + HfR ∇3 H H dµg Σt

with respect to t, we encounter a term involving the trace of the second fundamental form and the third derivatives of E and H. These terms will be dealt with by Gronwall’s Inequality. Therefore after applying Gronwall’s Inequality, we have # -H H2 H H2 . σ 80 HfR ∇3 E H + HfR ∇3 H H ≤ cect, Σt

where the constant c is independent of R. Thus we have uniform bounds for all R. This leads us to conclude that F is in H3,1.

CHAPTER 7

Comparison Theorem for the Weyl Tensor In [5] the Weyl tensor, which is the traceless part of the space-time Riemann curvature tensor, completely determines the space-time curvature. In the case of the Vacuum Equations, the Weyl tensor satisfies the homogenous field equations Dα Wαβγδ = 0 (7.1)

Dα∗ Wαβγδ = 0,

where (7.2)

∗

1 Wαβγδ = ϵαβµν W µν γδ 2

and 1 ∗ Wαβγδ = Wαβ µν ϵµνγδ . 2 The field equations are derived from the Bianchi identity for the Riemann curvature tensor. In our case, the Ricci curvature is not zero. Thus, the Weyl tensor satisfies the inhomogeneous equations (7.3)

1 Dα Wαβγδ = (Dγ Rβδ − Dδ Rβγ ) 2 1 (7.5) Dα∗ Wαβγδ = (Dµ Rβν − Dν Rβµ )ϵµν γδ . 2 Nonetheless, the Weyl tensor satisfies the same symmetry properties as in [5]. That is (7.4)

Wαβγδ + Wαγδβ (7.6)

Wαβγδ = −Wβαγδ = −Wαβδγ + Wαδβγ = 0 Wαβγδ = Wγδαβ

and (7.7)

W α βαδ = 0.

Following [5] consider arbitrary Weyl tensors, i.e. tensors with the properties just described, which satisfy the inhomogeneous equations (7.8)

Dα Wαβγδ = Jβγδ

(7.9)

∗ , Dα∗ Wαβγδ = Jβγδ 425

426

7. COMPARISON THEOREM FOR THE WEYL TENSOR

where J ∗ = 12 Jβµν ϵµν γδ . In our case, the J’s are error terms arising from the commutation of the space-time covariant derivative and the Lie derivative plus the inhomogeneous parts in equations 7.4 and 7.5. 7.1. The Bel-Robinson Tensor The estimation of components of the Weyl tensor mirrors the method used in [5]. However, in our case, we also need to bound the extra terms that arise from the inhomogeneous part of the field equations for W . In [5], Christodoulou and Klainerman use the Bel-Robinson tensor, which is a quadratic in the Weyl tensor, to create global energy norms. We also use the Bel-Robinson tensor and obtain similar global energy norms. The Bel-Robinson tensor is defined as follows (7.10)

Q(W )αβγδ = Wαργσ Wβ ρ δ σ + ∗ Wαργσ ∗ Wβ ρ δ σ .

All of the results cited in this section are stated and proven in [5] and are described below for convenience. Lemma 1. Given an arbitrary Weyl tensor W, (1) Q(W ) is symmetric and traceless in all pairs of indices. (2) Q(W ) (X1 , X2 , X3 , X4 ) is positive for any non-spacelike future directed vector fields X1 , X2 , X3 , X4 , whenever two of them at most are distinct. Proposition 7. Let W be a Weyl field verifying equations 7.4 and 7.4. Then ∗ ∗ + ∗ Wβ µ γ ν Jµδν . Dα Qαβγδ = Wβ µ δ ν Jµγν + Wβ µ γ ν Jµδν + ∗ Wβ µ δ ν Jµγν

Now let X, Y , Z represent arbitrary vector fields and define Pα = Q(W )αβγδ X β Y γ Z δ . With this definition DivP = X β Y γ Z δ (DivQ)βγδ . 1 + Qαβγδ (X)π αβ Y γ Z δ + (Y )π αβ Z γ X δ + (Z)π αβ X γ Y δ . 2

In this case (X) π represents the deformation tensor of X. Integrating along the slab ∪t′ ∈[t0 ,t] Σt′ , yields the following corollary. Corollary 2. Let Q(W ) be the Bel-Robinson tensor of a solution of the inhomogeneous Bianchi equations 7.4 and 7.5 in a space-time foliated by a time-function t. Then for any vector fields X, Y , Z, # # Q(W ) (X, Y, Z, T ) dµg = Q(W ) (X, Y, Z, T ) dµg Σt

Σt0

+

#

t

t0

dt

′

Y#

Σ′t

(DivQ)βγδ X β Y γ Z δ φ dµg

7.2. THE ELECTRIC-MAGNETIC DECOMPOSITION OF W

+

1 2

#

Σ′t

Qαβγδ

$(X)

π αβ Y γ Z δ

+ (Y )π αβ Z γ X δ + (Z) π αβ X γ Y

(7.11)

427

% δ

Z

φ dµg .

To bound derivatives of the Weyl tensor, we take Lie derivatives of W and apply the same type of energy estimate as in [5]. Because LX W does not have all of the properties of the original Weyl tensor, Christodoulou and N X W for an arbitrary vector Klainerman define a modified Lie derivative L field X, which has the same properties as W . The modified Lie derivative is defined as follows: N X W = LX W − 1(X) [W ] + 3 tr(X) πW (7.12) L 2 8 and

(X)

[W ]αβγδ = π µ α Wµβγδ + π µ β Wαµγδ

(7.13)

+ π µ γ Wαβµδ + π µ δ Wαβγµ .

With this definition, the Weyl tensor has the following properties due to [5]. Lemma 8. Let W be a Weyl tensor field, and let X be an arbitrary vector field. N X W has all the symmetries of Weyl field and is trace free. (1) L (2) The modified Lie derivative commutes with the Hodge dual of W, that is, N X W. NX ∗ W = ∗ L L 7.2. The Electric-Magnetic Decomposition of W

To obtain estimates for the Weyl tensor on the interior region of each foliation surface, Christodoulou and Klainerman consider the electric-magnetic decomposition of the Weyl tensor with respect to the vector field T . They write E(W )αβ = iiT (W )αβ = Wµανβ T µ T ν (7.14)

H(W )αβ = iiT (∗ W )αβ = ∗ Wµανβ T µ T ν .

Note that E and H are traceless, symmetric 2-tensors. From this we calculate 1 div E(W )a = −ϵa ij H(W )li klj + (Da RT T − DT RaT ) 2 1 div H(W )a = −ϵa ij E(W )li klj + (Dµ RT ν − Dν RT µ ) ϵµν aT (7.15) 2

428

7. COMPARISON THEOREM FOR THE WEYL TENSOR

and $ % N T Hab + curl Eab = −φ−1 ϵa ij ∇i φEjb + ϵb ij ∇i φEja −L ' ( 1 1 ij lm ϵa ϵb kil Hjm + (k · H) gab − 2 3 1 2 + (k · H) gab − (Dµ Raν − Dν Raµ ) ϵµν T b 3 2 $ ij % −1 N ϵa ∇i φHjb + ϵb ij ∇i φHja LT Eab + curl Hab = −φ ' ( 1 1 ij lm ϵa ϵb kil Ejm + (k · E) gab + 3 2 1 2 (7.16) − (k · E) gab − (DT Rab − Db RaT ). 3 2 We can rewrite these equations as 1 (Da RT T − DT RaT ) 2 1 (7.18) div H(W ) = −k ∧ E(W ) + (Dµ RT ν − Dν RT µ ) ϵ µν aT 2 −1 N T H(W ) + curl E(W ) = −φ ∧ E(W ) − 1 k × H(W ) −L 2 1 − (Dµ Raν − Dν Raµ ) ϵ µν T b (7.19) 2 N T E(W ) + curl H(W ) = −φ−1 ∧ H(W ) + 1 k × E(W ) L 2 1 − (DT Rab − Db RaT ). (7.20) 2 (7.17)

div E(W ) = k ∧ H(W ) +

7.3. Null Decomposition of a Weyl Field Following [5], define the null components of W as

(7.21)

α(W )µν = Πρµ Πσν Wργσδ eγ3 eδ3 1 β(W )µ = Πρµ Wρσγδ es3 eγ3 eδ4 2 1 ρ(W ) = Wαβγδ eα3 eβ4 eγ3 eδ4 4 1∗ σ(W ) = Wαβγδ eα3 eβ4 eγ3 eδ4 4 1 β(W )µ = Πρµ Wρσγδ es4 eγ3 eδ4 2 α(W )µν = Πρµ Πσν Wργσδ eγ4 eδ4 .

7.4. DERIVATIVES OF W EXPRESSED IN TERMS OF ITS NULL COMPONENTS429

With respect to the standard null frame the following holds WA3B3 = α(W )AB

WA4B4 = α(W )AB

WA334 = 2β(W )A

WA434 = 2β(W )A

W3434 = 4ρ(W )

WAB34 = 2σ(W ) ∈AB

∗

WABC3 = ∈AB β(W )C

WA3B4 = −ρ(W )δ AB + σ(W ) ∈AB

(7.22)

WABC4 = −∈AB ∗ β(W )C

WABCD = −∈AB ∈CD ρ(W ).

For the dual of the Weyl tensor the following holds ∗

WA3B3 = ∗ α(W )AB

∗

WA334 = 2∗ β(W )A

∗ ∗

WA3B4 = −σ(W )δ AB − ρ(W ) ∈AB

(7.23)

WA4B4 = ∗ α(W )AB

∗ ∗

W3434 = 4σ(W )

WABC3 = ∈AB β(W )C

∗

∗

∗ ∗

WA434 = 2∗β(W )A

WAB34 = −2ρ(W ) ∈AB

WABC4 = −∈AB β(W )C

WABCD = ∈AB ∈CD σ(W ).

7.4. Derivatives of W Expressed in Terms of Its Null Components In this section, we compute the space-time covariant derivatives of the null decomposition of the Weyl tensor. Writing 7.4 and 7.5 in terms of their null decomposition, we obtain the following formulas 3 1 N · α(W ) D4 ρ(W ) = div/β(W ) − trχρ(W ) − χ 2 2 1 (7.24) + ε · β(W ) + 2ζ · β(W ) − (D3 R44 − D4 R34 ) 4 3 1 D3 ρ(W ) = −div/β(W ) − trχρ(W ) − χ N · α(W ) 2 2 $ % 1 (7.25) + ε · β(W ) + 2 ξ · β(W ) + ζ · β(W ) + (D3 R34 − D4 R34 ) 4 3 1 D4 σ(W ) = div/ ∗ β(W ) − trχσ(W ) + χ N · ∗ α(W ) 2 2 1 (7.26) − ε · ∗β(W ) − 2ζ · ∗β(W ) − (Dµ R4ν − Dν R4µ ) ϵµν 34 4 3 1 ∗ N · α(W ) D3 σ(W ) = −div/ ∗β(W ) − trχσ(W ) + χ 2$ 2 % − ε · ∗β(W ) − 2 ξ · ∗ β(W ) + ζ · ∗β(W ) 1 (7.27) + (Dµ R3ν − Dν R3µ ) ϵµν 34 4 / B α(W )AB − 2trχβ(W )A − νβ(W )A D / 4 β(W )A = ∇ $ % 1 (7.28) + 2ϵB + ζ α(W )AB − (DA R44 − D4 R4A ) 2

430

7. COMPARISON THEOREM FOR THE WEYL TENSOR

D / 3 β(W )A = ∇ / A ρ(W )+ ∈AB ∇ / B σ(W ) − trχ β(W )A

+ 2N χAB β(W )B + νβ(W )A + ξ B α(W )AB 1 (7.29) + 3 (ζ A ρ(W ) + ∗ ζ A σ(W )) + (DA R34 − D4 R3A ) 2 / A ρ(W )+ ∈AB ∇ / B σ(W ) − trχ β(W )A D / 4 β(W )A = −∇ . + 2N χAB β(W )B + νβ(W )A − 3 ζ A ρ(W ) − ∗ ζ A σ(W ) 1 (DA R43 − D3 R4A ) 2 D / 3 β(W )A = −∇ / B α(W )AB − 2trχβ(W )A − νβ(W )A . − (−2ϵB + ζ B ) α(W )AB + 3 −ξ A ρ(W ) + ∗ ξ A σ(W )

(7.30)

−

(7.31)

+

1 (DA R33 − D3 R3A ) 2

We also have D / 4 α(W )AB = −(∇ / A β(W )B + ∇ / B β(W )A − divβ(W / )δ AB ) . 1 − trχ α(W )AB + 2να(W )AB − 3 χ N AB ρ(W ) − ∗ χ N AB σ(W ) -2 . . + εA − 4ζ A β(W )B + εB − 4ζ B β(W )A

(7.32) and

$ % 1 − ε − 4ζ · β(W )δ AB − (D3 R34 − D4 R33 ) δ AB 2

D / 3 α(W )AB = (∇ / A β(W )B + ∇ / B β(W )A − divβ(W / )δ AB ) 1 − tr χα(W )AB + 2να(W )AB − 3 (N χAB ρ(W )∗ χ N AB σ(W )) 2 + (εA + 4ζ A ) β(W )B + (εB + 4ζ B ) β(W )A 1 (7.33) − (ε + 4ζ) · β(W )δ AB + (D3 R44 − D4 R43 )δ AB . 2 Notice that the null components that cannot be expressed in this fashion are D / 3 α(W ) and D / 4 α(W ). We make use of the following definition introduced in [5]. Definition 9. Given an arbitrary Weyl tensor W , and u any of its null components, let 3−s trχ u 2 3+s u4 = D trχ u / 4u + 2

u3 = D / 3u +

7.5. THE STATEMENT OF THE COMPARISON THEOREM FOR W

431

where s = s(u) is the signature of that particular component. In addition, define 3 − (s − 1) + 1 trχu3 2 3 + (s − 1) + 1 trχu3 =D / 4 u3 + 2 3 − (s + 1) + 1 =D / 3 u4 + trχu4 2 3 + (s + 1) + 1 =D / 4 u4 + trχu4 . 2

/ 3 u3 + u33 = D u34 u43 u44

Using this definition and equations 7.24 through 7.33, we then define α(W )3 , etc. 7.5. The Statement of the Comparison Theorem for W Using the same notation as in [5], define # # $ % $ % N O W ) K, K, T, T + N T W ) K, K, K, T Q1 (W ) = Q(L Q(L

Q2 (W ) = (7.34)

#Σ

$

%

#Σ

$ % N T W ) K, K, K, T NO L Q(L Σ Σ # # $ % $ % N N N 2T W ) K, K, K, T . + Q(LS LT W ) K, K, K, T + Q(L N2 W ) Q(L O

K, K, T, T +

Σ

Σ

The following statement of the Comparison Theorem for the Weyl tensor mirrors the statement of the Comparison Theorem in [5]; however, we introduce the necessary modifications to reflect the presence of the electromagnetic field. Theorem 6. We assume we are given a space-time (M, g) endowed with a (t, u)-foliation as described in the previous sections and consider an arbitrary Weyl tensor W that satisfies the inhomogeneous equations 7.4 and 7.5. (i) Assume that the space-time verifies the Bootstrap Assumptions and that there exist angular momentum vector fields (i) Ω, i = 1, 2, 3 that satisfy assumptions 7.5.12a and 7.5.13b of Proposition 7.5.3 in [5]. Then there exists a sufficiently small ε0 and a constant c such that if Q1 (W ) is finite, then 1

W0 + W1 ≤ cQ1 (W ) 2 + cF[2]

and

1

sup r3 |ρ, σ| ≤ cε0 Q1 (W ) 2 + cF[2]

r≥r0

1

sup τ − r3 |ρ, σ| ≤ cε0 Q1 (W ) 2 + cF[2] .

r≤r0

432

7. COMPARISON THEOREM FOR THE WEYL TENSOR

In particular, if we let W[1] = W0 + W1 + sup r3 |ρ, σ| + sup τ − r3 |ρ, σ| , r≥r0

then

r≤r0

1

W[1] ≤ cQ1 (W ) 2 + cF[2] .

(ii) Moreover, if the vector fields (i) Ω also satisfy 7.5.13c of Proposition 7.5.3 in [3], there exists a constant c such that, if Q2 (W ) is finite, then . 1 1 W2 ≤ c Q1 (W ) 2 + Q2 (W ) 2 + F[2] .

Proof. The proof of this theorem differs from the proof of the Comparison Theorem in [5] only in that we need to bound extra terms involving F when estimating ρ, σ and the derivatives in the null directions of the null decomposition of W . It follows from Proposition 4.4.4 in [5] that # ' 1 3 − κ(ρ − ρ) 4πr ρ = 2 Br / A a) E(W )AN + r (k ∧ H(W ))N + rN θAB E(W )AB − ra−1 (∇ ( r + (D4 R43 − D3 R44 + D4 R33 − D3 R34 ) 2 # ' 1 4πr3 σ = − κ(σ − σ) 2 Br + rN θAB H(W )AB − ra−1 (∇ / A a) H(W )AN + r (k ∧ E(W ))N ( r + ϵAB (DA R4B − DB R4A + DA R3B − DB R3A ) . 2 From equations 7.7.3c and 7.73d in [5] and our estimates of the electromagnetic field we estimate 1

sup r3 |ρ, σ| ≤ cε0 Q1 (W ) 2 + cF[2]

r≥r0

1

sup τ − r3 |ρ, σ| ≤ cε0 Q1 (W ) 2 + cF[2] .

r≤r0

To estimate the first derivatives of the null components of W in the null directions, we show that the extra terms in the null decomposition of 7.8 and 7.9 can be bounded. In particular, to prove that # 2 r6 |ρ(W )4 |2 ≤ cQ1 (W ) + cF[2] Σet

we must show that

#

Σet

r6 |D3 R44 − D4 R34 |2 ≤ cF[2] ,

7.5. THE STATEMENT OF THE COMPARISON THEOREM FOR W

433

which is equivalent to the following # . r6 |D / 3 α(F )|2 |α(F )|2 + |D4 ρ(F )|2 |ρ(W )|2 + |D4 σ(F )|2 |σ(W )|2 Σet

≤ cF[2] .

This identity holds according to the Comparison Theorem for F . The other estimates we need for the first derivatives of the null components of the Weyl tensor follow from similar arguments. To estimate the first derivatives on the interior region, it is enough to show that # 2 , r06 |Dα Rβγ |2 ≤ cF[2] I$

which bounds the extra terms that appear in equations 7.17 through 7.20. Recall that Rαβ is a quadratic in F . Therefore, # 4 r0 |DF |2 ≤ F12 , I$

and

5

sup r02 |F | ≤ F[2] . I$

Thus we obtain our estimate. To prove the second part of the theorem, we need to show that all of the second derivatives of the null components in the null directions are bounded. To accomplish this goal, we show that the extra terms in the null decomposition of 7.8 and 7.9 can be bounded with the proper weights after differentiation. First, to prove that # . 2 8 2 r |ρ(W )44 | ≤ c Q1 (W ) + Q2 (W ) + F[2] Σet

it is enough to show that # 2 r8 |D3 R44 − D4 R34 |2 ≤ cF[2] , Σet

which is equivalent to # H H2 H H2 H 2 H H 2 H r8 (|D / 4D / 3 α(F )|2 |α(F )|2 + HD / 4 ρ(F )H |ρ(F )|2 + HD / 4 σ(F )H |σ(F )|2 Σet

+ |D / 3 α(F )|2 |D / 4 α(F )|2 + |D / 4 ρ(F )|2 |D / 4 ρ(F )|2 + |D / 4 σ(F )|2 |D / 4 σ(F )|2 ) 2 ≤ cF[2] .

We have

#

Σet

H H2 H H2 H 2 H H 2 H 2 r6 (|D / 4D / 3 α(F )|2 + HD / 4 ρ(F )H + HD / 4 σ(F )H ≤ cF[2]

434

and

7. COMPARISON THEOREM FOR THE WEYL TENSOR

H H H H 2 1 2 sup HHr τ − (ρ(F ), σ(F ))HH ≤ ε0 e Σt

Furthermore, by the co-area formula, # # ∞# 4 8 r |D / 4 ρ(F )| ≤ Σet

≤ ≤

u0 ∞

#

and

Su,t

H 5 H H H sup Hr 2 α(F )H ≤ ε0 . Σet

r8 |D / 4 ρ(F )|4 φ 1

5

−2 2 2 τ −2 / 4 ρ(F )|||44,e − r |||τ − r D

u0 2 cF[2] .

The remaining terms |D / 3 α(F )|2 , |D / 4 α(F )|2 , |D / 4 σ(F )|2 have weights that are at least as great in the ||| |||4,e norm as |D / 4 ρ(F )|2 . Second, to prove that # . 2 τ 4− r4 |ρ(W )33 |2 ≤ c Q1 (W ) + Q2 (W ) + F[2] , Σet

it is enough to show that # 2 τ 4− r4 |D3 D3 R34 − D3 D4 R34 |2 ≤ cF[2] Σet

or that

#

Σet

-H H2 τ 4− r4 HD23 ρ(F )H |ρ(F )|2

. H H2 + HD23 σ(F )H |σ(F )|2 + |D3 ρ(F )|4 + |D3 σ(F )|4

2 . ≤ cF[2]

We have

#

Σet

-H H2 H H2 . τ 4− r2 HD23 ρ(F )H + HD23 σ(F )H ≤ F[2] .

We can use the sup norms of ρ(F ) and σ(F ) to produce the rest of the weight in r. Furthermore # ∞# # τ 4− r4 |D3 ρ(F )|4 ≤ τ 4− r4 |D3 ρ(F )|4 φ Σet

Su,t ∞ 3 3 2 2 × r−2 τ −2 / 4 ρ(F )|||44,e − |||τ − r D u0 2 ≤ cF[2] . u0

#

& The same estimate holds for Σe τ 4− r4 |D3 σ(F )|4 . t Next to prove that # . 2 τ 4− r4 |σ(W )33 |2 ≤ c Q1 (W ) + Q2 (W ) + F[2] , Σet

7.5. THE STATEMENT OF THE COMPARISON THEOREM FOR W

435

it is enough to show #

H H2 τ 4− r4 HD3 DA R3B ϵAB H ≤ cF[2]

Σet

or that

#

τ 4− r4 (|∇ /D / 3 α(F )|2 |σ(F )|2 + |α(F )|2 |∇ /D / 3 σ(F )|2

Σet

+ |∇α(F / )|2 |D / 3 σ(F )|2 + |D / 3 α(F )|2 |∇σ(F / )|2 )

2 ≤ cF[2] .

We have

#

Σet

2 τ 4− r2 |∇ /D / 3 α(F )|2 ≤ cF[2]

Σet

2 τ 2− r4 |∇ /D / 3 σ(F )|2 ≤ cF[2]

#

and the rest of the weights needed are produced from the sup |α(F )| and sup |σ(F )|. Furthermore, # τ 4− r4 |∇α(F / )|2 |D / 3 σ(F )|2 Σet

≤c ≤c

#

∞#

Su,t

u0

#

∞

u0

τ 4− r4 |∇α(F / )|2 |D / 3 σ(F )|2

r−2 τ −2 −

2 ≤ cF[2]

and #

Σet

0#

Su,t

τ 6− r6 |∇α(F / )|4

11 0# 2

Su,t

τ 6− r6 |D / 3 σ(F )|4

11 2

τ 4− r4 |D / 3 α(F )|2 |∇σ(F / )|2

≤c ≤c ≤

#

∞#

Su,t

u0

#

∞

r−2 τ −2 −

u0

2 cF[2] .

To show

τ 4− r4 |D / 3 α(F )|2 |∇σ(F / )|2

#

Σet

0#

Su,t

2 τ 10 / 3 α(F )|4 − r |D

11 0# 2

Su,t

τ 2− r10 |∇σ(F / )|4

. H H2 2 τ 2− r6 Hβ(W )34 H ≤ c Q1 (W ) + Q2 (W ) + F[2] ,

11 2

436

7. COMPARISON THEOREM FOR THE WEYL TENSOR

it is enough to show #

Σet

2 τ 2− r6 |D4 DA R33 − D4 D3 R3A |2 ≤ cF[2]

or #

Σet

τ 2− r6 (|∇ /D / 4 α(F )|2 |α(F )|2 + |D / 3D / 4 α(F )|2 |σ(F )|2 + |D / 3D / 4 σ(F )|2 |α(F )|2 + |D / 4 α(F )|2 |∇α(F / )|2

+ |D / 4 α(F )|2 |D / 3 σ(F )|2 + |D / 3 α(F )|2 |D / 4 σ(F )|2 )

2 . ≤ cF[2]

We have #

#

2 τ 2− r4 (|∇ /D / 4 α(F )|2 ≤ cF[2]

Σet

Σet

2 τ 4− r2 |D / 3D / 4 α(F )|2 ≤ cF[2]

Σet

2 τ 2− r4 |D / 3D / 4 σ(F )|2 ≤ cF[2] ,

#

and we can use the sup norms to produce the rest of the weights. The estimates for the other terms follow from arguments used above. To show # . 2 τ 2− r6 |β(W )33 |2 ≤ c Q1 (W ) + Q2 (W ) + F[2] , Σet

it is enough to show #

Σet

or

#

Σet

2 τ 2− r6 |D3 DA R34 − D3 D4 R3A |2 ≤ cF[2]

2 τ 2− r6 (|∇ /D / 3 ρ(F )|2 |ρ(F )|2 + |D / 3 ρ(F )|2 |∇ρ(F / )|2 ) ≤ cF[2]

since the other terms were bounded above. We have # 2 τ 2− r4 |∇ /D / 3 ρ(F )|2 ≤ cF[2] . Σet

7.5. THE STATEMENT OF THE COMPARISON THEOREM FOR W

437

Using sup ρ(F ), we can bound the first term. The second term can be bounded in the following manner #

Σet

τ 2− r6 |D / 3 ρ(F )|2 |∇ρ(F / )|2

≤c ≤c

#

∞#

u0 # ∞

u0 2 cF[2] .

≤

Su,t

τ 2− r6 |D / 3 ρ(F )|2 |∇ρ(F / )|2 3

1

3

5

−2 2 2 τ −2 / 3 ρ(F )|||24,e |||τ −2 r 2 ∇ρ(F / )|||24,e − r |||τ − r D

Finally, to show #

Σet

. 2 r8 |α(W )33 |2 ≤ c Q1 (W ) + Q2 (W ) + F[2] ,

it is enough to show #

Σet

or #

Σet

H H2 2 r8 HD23 R44 − D3 D4 R43 H ≤ cF[2] ,

H2 -H H 2 H / 3 α(F )H |α(F )|2 + |D r 8 HD / 3D / 4 ρ(F )|2 |ρ(F )|2

+ |D / 3 α(F )|4 + |D / 4 ρ(F )|2 |D / 3 ρ(F )|2

We have

#

Σet

and

#

Σet

.

2 ≤ cF[2] .

H2 H H H 2 2 / 3 α(F )H ≤ cF[2] r4 τ 2− HD

r4 τ 2− |D / 3D / 4 ρ(F )|2 ≤ cF[2] .

We can use the sup norms of α(F ) and ρ(F ) to produce the rest of the weights. Furthermore, #

Σet

8

4

r |D / 3 α(F )| ≤ c ≤

#

∞

u0 2 cF[2]

−2 τ −2 − r

#

Su,t

τ 2− r10 |D / 3 α(F )|4

438

7. COMPARISON THEOREM FOR THE WEYL TENSOR

and

#

Σet

r8 |D / 4 ρ(F )|2 |D / 3 ρ(F )|2

≤c ≤c ≤

#

∞#

u0

#

∞

u0

Su,t

τ −4 −

2 cF[2] .

r8 |D / 4 ρ(F )|2 |D / 3 ρ(F )|2

0#

Su,t

1 1 0# 2

τ 2− r10 |D / 4 ρ(F )|4

Su,t

τ 6− r6 |D / 3 ρ(F )|4

11 2

Lastly, we must bound the extra terms that arise while deriving the interior estimates for the inhomogeneous parts of equations 7.8 and 7.9. In other words we must show that # 2 8 . r0 |DT Dα Rβγ |2 ≤ cF[2] I$

But recall that Rβγ is a quadratic in F , and # H H2 2 r06 HD2 F H ≤ cF[2] . I$

Plus, we have

5

sup r02 |F | ≤ F[2] . I$

Now we only need to show that # 8 2 r0 |Dα F |2 |Dβ F |2 ≤ cF[2] . I$

Using the co-area formula and H¨ older’s Inequality we have, # r08 |Dα F |2 |Dβ F |2 ≤ ≤

I$

'#

I$

'#

I$

r08 |Dα F |4 r04 |Dα F |2

( 1 '# 2

I$

( 1 '# 4

I$

r08 |Dβ F |4

(1

r012 |Dα F |6

2

( 1 '# 4

I$

r04 |Dβ F |2

( 1 '# 4

I$

r012 |Dβ F |6

By the Sobolev inequality '# ( 1 '# (1 1 6 2 6 2 2 12 2 4 r0 |Dβ F | ≤ r0 |Dβ F | + r0 |∇Dβ F | ≤ F22 . I$

I$

(1 4 .

Thus we can estimate the components of the Weyl tensor in the interior. "

CHAPTER 8

Error Estimates for W In the last chapter we proved that the null components of the Weyl tensor and their derivatives can be bounded on Σt by the quantities Q1 (W ) and Q2 (W ). Now we need to show that these objects can be bounded by the initial data. It follows from Corollary 7.1.1.1 in [5] that # # Q(W ) (X, Y, Z, T )dµg = Q(W ) (X, Y, Z, T )dµg Σt

Σt0

+

#

t

dt

t0

1 + 2 (8.1)

+

#

Σ′t

(Y )

′

R#

Σ′t

(DivQ)βγδ X β Y γ Z δ φ dµg

Qαβγδ ((X)π αβ Y γ Z δ

π αβ Z X +

where

γ

δ

(Z)

π αβ X Y )φ dµg γ

δ

S

(DivQ)αβγ = Dα Qαβγδ ∗ ∗ = Wβ µ δ ν Jµγν + Wβ µ γ ν Jµδν + ∗ Wβ µ δ ν Jµγν +∗ Wβ µ γ ν Jµδν .

As in [5], to show that Q1 (W ) and Q2 (W ) are bounded by the initial conditions, we bound the error terms that arise from this computation. However, in [5] there is no electromagnetic field and the J’s only include the error terms that arise from commuting the Lie derivative with the spacetime covariant derivative. In our case the J’s are the sum of such error terms and the Lie derivatives of the inhomogeneous equation 1 (Dγ Rβδ − Dδ Rβγ ) 2 1 Dα∗ Wαβγδ = (Dµ Rβν − Dν Rβµ ) ϵµν γδ . (8.2) 2 We define the extra error term that arises as 1 Iβγδ = (Dγ Rβδ − Dδ Rβγ ). 2 In this chapter, we focus on I because the other parts of J are bounded precisely as in Chapter 8 of [5] with the exception of a few terms that we Dα Wαβγδ =

439

440

8. ERROR ESTIMATES FOR W

discuss below. Like the original error terms in [5], the I’s are Weyl currents. That is they satisfy the following definition from [5]. Definition 10. A tensor field Iβγδ is called a Weyl current if is satisfies the following: Iβγδ + Iγδβ + Iδβγ = 0 Iβγδ + Iβδγ = 0 trIδ = gβγ Iβγδ = 0. . N Using this definition, we compute the null components of DivQ LX W and state the following proposition, which is a direct generalization of Proposition 8.1.3 in [5]. -

Proposition 8. Let W be an arbitrary Weyl . tensor, let X be a given vecN X W . The following formulas hold tor field, and set D (X, W ) = DivQ L everywhere in the exterior region % 1 $ D (X, W ) K, K, T = τ 4+ (D (X, W )444 + D (X, W )344 ) 8 1 + τ 2+ τ 2− (D (X, W )344 + D (X, W )334 ) 4 1 4 + τ − (D (X, W )334 + D (X, W )333 ) 8 % 1 6 $ 3 D (X, W ) K, K, K = τ + D (X, W )444 + τ 4+ τ 2− D (X, W )344 8 8 3 1 (8.3) + τ 2+ τ 4− D (X, W )334 + τ 6− D (X, W )333 8 8 where . $ $ % % N X W × JA3B + JB3A − δ CD JC3D δ Ab D (X, W )333 = 4α L . N X W · J33A + 4β L . . N X W J343 − 2σ L N X W ∈AB J3AB D (X, W )334 = 2ρ L . N X W · J43A − 4β L . . N X W J434 + 2σ L N X W ∈AB J4AB D (X, W )443 = 2ρ L . N X W · J34A + 4β L . $ $ % % N X W · JA4B + JB4A − δ CD JC4D δ Ab D (X, W )444 = 4α L . N X W · J44A . − 4β L (8.4) The proof follows from the proof of Proposition 8.1.3 in [5] because J is a Weyl current.

8. ERROR ESTIMATES FOR W

441

For the proof of the Boundedness Theorem for the Bel Robinson tensor, define the following quantities in the exterior region as in [5] # .$ % N O W K, K, T, e4 E Q L Q1 (W ; u, t) = Cu (t0 ,t) # .$ % N T W K, K, K, e4 + Q L C (t ,t) # u 0 .$ % E2 (W ; u, t) = N 2 W K, K, T, e4 Q Q L O Cu (t0 ,t) # .$ % NO L N T W K, K, K, e4 Q L + C (t ,t) # u 0 .$ % N T W K, K, K, e4 NS L + Q L C (t ,t) # u 0 .$ % N 2 W K, K, K, e4 . + Q L (8.5) T Cu (t0 ,t)

In this case 0 ≤ t0 = t0 (u) ≤ t is the value of t for which the cone Cu intersects the boundary of the interior region. Moreover, Cu (t0 , t) is defined as the region of Cu that is contained between Σt0 and Σt . Following [5], for a fixed t∗ let Q1 ∗ = sup Q1 (W, t) + sup t∈[0,t∗ ]

(8.6)

Q2 ∗ = sup Q2 (W, t) + sup t∈[0,t∗ ]

E1 (W ; u, t) sup Q

t∈[0,t∗ ] u≥u0 (t)

E2 (W ; u, t) sup Q

t∈[0,t∗ ] u≥u0 (t)

where the value of u that corresponds to the boundary of the interior region r ≤ r20 is defined as u0 (t). Now we state a theorem that is a generalization of the Boundedness Theorem 8.2.1 in [5]. Theorem 7. We assume that a given space-time (M, g) which is endowed with a (t, u)-foliation as described previously and consider an arbitrary Weyl tensor satisfies the equations 7.8 and 7.9. Assume that the assumptions of the comparison theorem are verified in the space-time slab Vt8 = ∪t∈[0,t∗ ] Σt . i. We require the vector fields T , O verify Assumptions 0 and 1. Then there exists a constant c1 such that, with the notation Q1 ∗ and Q2 ∗ 2 . Q1 ∗ ≤ Q1 (0) + c1 ε0 (Q1∗ + Q2 ∗) + F[2]

ii. In addition we also require Assumption 2 is verified. Then 2 + F32 . Q2 ∗ ≤ Q2 (0) + c2 ε0 (Q1∗ + Q2 ∗) + F[2]

Thus, if ε0 is chosen sufficiently small, we infer that

Q1 ∗ +Q2 ∗ ≤ 2 (Q1 (0) + Q2 (0) + F1 (0) + F2 (0) + F3 (0)).

This theorem is proved in the subsequent sections.

442

8. ERROR ESTIMATES FOR W

8.1. Proof of the Generalization of the Boundedness Theorem for W Recall that the Bootstrap Assumptions in this work differ from those η AB in the e4 direction. In our in [5] by the covariant derivative of kAB = N case, this term includes an extra piece that incorporates a component of the null decomposition of F , namely N η4 = D / 4N η + 12 trχN η − 14 |α(F )|2 δ AB . Therefore to prove our generalization of the Boundedness Theorem for W , we only need to control the extra terms involving- |α(F )|.2 that appear $ % & N T W K, K, K, T , in the error terms arising from estimates for Σe Q L .$ . $t % & % & & 2W N N N Q L W K, K, K, T , Q L K, K, K, T and Σe Q L O T T Σet Σet t . .$ % & N N N LS LT W K, K, K, T . It suffices to consider estimates for Σe Q LT W t .$ % % $ & N N K, K, K, T and Σe Q LS LT W K, K, K, T as estimates for the other t quantities follow. The proof of.the theorem then follows from [5]. $ % & N In estimating Σe Q LT W K, K, K, e4 , we need to consider the term t

.P O Qr (ρ(W ), σ(W )) ; Θ (T ) q

on page 243 in equation (8.3.8.c) from [5] since Θ particular, we must prove that

$(T ) % q involves D / 4N η . In

#

H H . H N 2H 2 τ 6+ Hα L W σ(W ) |α(F )| . H ≤ c1 ε0 (Q1∗ + Q2 ∗) + F[2] T

#

H H . H N 2H τ 6+ Hα L W σ(W ) |α(F )| H T

Vt

We have

Vt

H H ≤ sup sup Hr3 σ(W )H t∈[t0, t] E

×

'#

t

t0

r0−2

#

Su

#

∞

u0

τ −3 −

r2 τ 6− |α(F )|4

2 ≤ c1 ε0 (Q1 ∗ +Q2 ∗) + F[2] .

'#

( 12

Cu

H -

H τ 6+ Hα

.H2 ( 12 N T W HH L

.$ % & N T W K, K, K, T , the most difficult terms come NS L In estimating Σe Q L t N/ Θ3 (T, W ). In particular, we must estimate the N/ Θ1 (T, W ) and L from L SS . . NT W L N S |α(F )|2 σ(W ). We find N T W rtrχ |α(F )|2 σ(W ) and α L terms α L

8.1. PROOF OF THE GENERALIZATION OF THE BOUNDEDNESS

that #

Vt

H H . H N 2H τ 6+ Hα L W rtrχσ(W ) |α(F )| H T

H H ≤ sup sup Hr3 σ(W )H sup sup |rtrχ| t∈[t0, t] E

×

'#

t

t0

t∈[t0, t] E

r0−2

#

Su

r2 τ 6− |α(F )|4

( 12

#

∞

u0

τ −3 −

'#

Cu

H -

H τ 6+ Hα

443

.H2 ( 12 N T W HH L

2 ≤ c1 ε0 (Q1 ∗ +Q2 ∗) + F[2] . . NT W L N S |α(F )|2 σ(W ) follows along the same lines. The estimate for α L . & 2 N There is also a term that arises when estimating Σe Q LT W (K, K, R' - t . .( N N K, e4 ) that may cause concern. That term Qr ρ LT W , σ LT W ; ' S . . $ % NT W , N T W and includes |α(F )|2 ρ L comes from Θ3 T, L Θ (T ) q ( . NT W σ L . Therefore, we estimate

#

Vt

H H . . H N 2H N τ 6+ Hα L W σ L W |α(F )| H T T

H2 # H H H −3 2 H ≤ sup sup Hrτ − α(F )HH t∈[t0, t] E

×

'#

Cu

∞

u0

τ −5 −

'#

Cu

H -

H τ 6+ Hα

.H2 ( 12 H N LT W H

H .H2 ( 12 2 4 H 2 N T W HH τ + τ − Hσ L ≤ c1 ε0 (Q1∗ +Q2 ∗) + F[2] .

Now that we have estimated the error terms that arise from commutation, we can focus exclusively on the error terms that arise from the spacetime Ricci curvature. Following [5], define H # H H # H . . H H H H β γ β γ δ δ NO W NT W HφDivQ L HφDivQ L K K T HH + K K K HH E1 = H H βγδ βγδ Vt H Vt # H . H (K ) π αβ K γ T δ HH NO W HφQ L + H H αβγδ V H # tH . H γ δH (T ) αβ NO W HφQ L + π K K HH H αβγδ V H # tH . H H K ) αβ γ δ H ( N H + W K K π φQ L T H H αβγδ Vt

(8.7)

444

8. ERROR ESTIMATES FOR W

and

H # H . H H β γ 2 δ N W HφDivQ L E2 = K K T HH O H βγδ Vt H # H . H H β γ δ N N HφDivQ LO LT W K K T HH + H βγδ V H # tH . H β γ δ HH N N H + HφDivQ LS LT W βγδ K K K H Vt H # H . H β γ δ HH 2 N H + HφDivQ LT W βγδ K K K H Vt H # H . H H γ K αβ δ 2 ( ) N W HφQ L π K T HH + O H αβγδ V H # tH . H γ δ HH 2 (T ) αβ N H π K K H + HφQ LO W αβγδ Vt H # H . H γ δ HH K ) αβ ( N N H π K K H + HφQ LS LT W αβγδ Vt H # H . H γ δ HH K ) αβ ( N N H π K K H + HφQ LS LT W αβγδ Vt H H # . H H γ δ K 2 αβ ( ) N W HφQ L π K K HH . + T H αβγδ

(8.8)

Vt

8.1.1. Estimates for E1 . As in [5], first estimate H # H . H β γ δ HH N H K K T H. IntegralA1 = HφDivQ LO W βγδ

Vt

In view of the Proposition 8, it follows that # 1 τ 4 (D (O, W )444 + D (O, W )344 ) IntegralA1 = 8 Vt + # 1 τ 2 τ 2 (D (O, W )344 + D (O, W )334 ) + 4 Vt + − # 1 + (8.9) τ 4 (D (O, W )334 + D (O, W )333 ). 8 Vt − We only consider the corresponding terms that arise from the Ricci curvature terms, namely # H . $ CD $ % %HH H N N τ 4+ Hα L IC4D δ Ab H O W · LO IA4B + IB4A − δ Vt J # H H # H H . . H H N 4 H 4 N O I44A H + N O I434 HH NO W · L + τ Hβ L τ Hρ LO W L Vt

+

Vt

+

8.1. PROOF OF THE GENERALIZATION OF THE BOUNDEDNESS

H H H H . . H N H H N AB H N N + Hσ L H + Hβ L O W LO I4AB ϵ O W · LO I34A H

+

#

Vt

τ 2− τ 2+

J

H H H H . . H N NO W L N O I434 HH + HHσ L N O I4AB ϵAB HH Hρ LO W L

H H . H H N N + Hβ LO W · LO I34A H

K

+

#

Vt

τ 2− τ 2+

J

H H . H N H N Hρ LO W LO I343 H

H H H H . . H N N AB I3AB HH + HHβ L N O I43A HH NO W · L + Hσ L O W LO ϵ

+

#

Vt

τ 4−

J

K

H H H H . . H N NO W L N O I343 HH + HHσ L N O ϵAB I3AB HH Hρ LO W L

JH H . H N + Hα LO W H Vt 1H K H H H . $ CD % H H N H N O I33A H . + IB3A − δ IC3D δ Ab H + Hβ LO W · L H

H H . H H N N W · L I + Hβ L O O 43A H 0

(8.10)

K

N O IA3B ×L

K

#

τ 4−

The most sensitive terms are # H . $ CD $ % %HH H N N τ 4+ Hα L W · L + I − δ I I δ A4B B4A C4D O O Ab H Vt # H H . H N H N + τ 4+ Hβ L (8.11) O W · LO I34A H . Vt

To bound the first integral in 8.11, it suffices to consider #

Vt

(8.12)

H .H H N H τ 4+ Hα L OW H

J

H H H HN / 4 {α(F ) · α(F )}H HLO D

K H H H H HN H HN 2H / 4 σ(F ) H + HLO ∇ / {α(F )σ(F )}H . + HLO D

The most sensitive term of 8.12 is the first one. We estimate # H H .H H H HN H 4 H N τ + Hα LO W H HLO D / 4 {α(F ) · α(F )}H Vt H0# ∞ H '# H .H2 ( 12 H H 3 − 32 H −1 4 2 N O W HH r τ− τ + Hα L ≤ sup sup HHrτ − α(F )HH ∗ t∈[0,t ] u≥u0 (t)

u0

Cu

445

446

8. ERROR ESTIMATES FOR W

× × +

'#

Cu

0#

t

t0

#

t

r

H

×

− 32

+

∞

u0

#

t

r

×

∞

u0

Σet

0#

Σet

H -

H τ 4+ Hα

1

H H 5 H H sup Hr 2 α(F )H

+ sup

t∈[0,t∗ ] u≥u0 (t)

.H2 N O W HH L

H .H2 H N H τ 4+ Hα L OW H

11

1 1 0# 2 ·

Σet

2

0#

Σet

H .H2 H H N τ 4+ Hα L OW H

11

#

Vt

11 1 H2 2

(1 2

2

3 2N 2 2 τ −2 / 4 α(F )|||24 − ||| r LO α(F )|||4 ||| r τ − D

The estimate for

H

H HN τ 2− τ 2+ HL / 4 α (F )H OD

1 3 N O α(F )|||24 τ −2 / 4 α(F )|||24 ||| r 2 τ − L − ||| r D

− 32

t0

'#

0#

3 r− 2 τ −1 −

t0

'#

H2 ( 12

H HN τ 4+ HL / 4 α(F )H OD

(1 2

2 ≤ cε0 Q1∗ +F[2] .

H H . H N H N τ 4+ Hβ L W · L I O O 43A H ,

the second term in 8.11, is similar once we rewrite the first term in I34A = D4 R3A − DA R34

using the Bianchi identity Dα Rαβ = 0 as

Next estimate

1 D4 R3A = −D3 R4A + γ BC DC RBA . 2

H # H . H β γ δ HH N H IntegralB = HφDivQ LT W βγδ K K K H . Vt

As in [5],

1 8

#

τ 6+ D (T, W )444 +

3 8

#

τ 4+ τ 2− D (T, W )344 Vt Vt # # 3 1 2 4 (8.13) + τ τ D (T, W )334 + τ 6 D (T, W )333 . 8 Vt + − 8 Vt − & The most difficult term to control is Vt τ 6+ D (T, W )444 . We calculate # τ 6+ D (T, W )444 Vt H ' # . H 6H N T IA4B + IB4A NT W · L = τ + Hα L Vt (H # H H . H $ CD % H N H N (8.14) − δ IC4D δ Ab HH + τ 6+ Hβ L T W · LT I44A H . IntegralB =

Vt

8.1. PROOF OF THE GENERALIZATION OF THE BOUNDEDNESS

447

Recall that L /T α(F )A = ΠρA LT α(F )ρ

. 1χAB − χAB α(F )B 2 1 and D / T = 2 (D /4 +D / 3 ). Therefore to estimate 8.14, it suffices to estimate H H # H .H H H 3 H 6 H 2 N H τ + Hα LT W H |D / 3D / 4 {α(F ) · α(F )}| ≤ sup sup Hrτ − α(F )HH =D / T α(F )A −

Vt

×

#

∞

u0

−3 r−1 τ − 2

0 '#

t∈[0,t∗ ] u≥u0 (t)

Cu

H .H2 ( 12 '# H 6 H N τ + Hα LT W H ·

H 5 H# H H sup Hr 2 α(F )H

+ sup

t∈[0,t∗ ] u≥u0 (t)

·

'#

×

Cu

#

∞

u0

u0

τ−

0 '#

Cu

r−1 τ −2 −

(1 1

Cu

H -

H τ 6+ Hα

.H2 ( 12 N T W HH L

H H H H 1−δ 5 2 H + sup sup Hr τ − D / 3 α(F )HH t∈[0,t∗ ] u≥u0 (t) ( (1 1 H .H2 12 '# 2 H 2 6 H 6 NT W H τ + Hα L · r |D / 4 α(F )|4

τ 4− τ 2+ |D / 3D / 4 α(F )|2 5 2

∞

Cu

0 '#

τ 6+ |D / 3D / 4 α(F )|2

2

Cu

(1 1

0 '# H# ∞ H H .H2 ( 12 H H 2−δ 1 H −2 6 H 2 N H H + sup sup Hr τ − D / 4 α(F )H r τ + Hα LT W H Cu u0 t∈[0,t∗ ] u≥u0 (t) (1 1 '# ×

Cu

r6 |D / 3 α(F )|24

2

≤ cε0 Q1∗ +F[2] + F3 .

(8.15)

8.1.2. Estimates for E 2 . First we show # H . H N2 W HφDivQ L integralA2 = O H Vt

(8.16)

H H K K T H βγδ β

≤ c2 ε0 (Q1 ∗ +Q2 ∗) + F[2] + F3 .

As with the estimates for # H . H NO W HφDivQ L H Vt

H H K K T H, βγδ β

γ

δH

γ

δH

the most difficult terms are # H . $ CD $ % %HH 4 H 2 2 N N τ + Hα LO W · LO IA4B + IB4A − δ IC4D δ Ab H Vt # H H . H N2 N 2 I43A HH . (8.17) + τ 4+ Hβ L W · L O O Vt

2

448

8. ERROR ESTIMATES FOR W

As before, it is enough to estimate # H H .H H H H N2 H H N2 (8.18) τ 4+ Hα L / 4 {α(F ) · α(F )}H O W H HLO D Vt

and

#

(8.19)

Vt

H H .H H H H N2 H N2 H τ 4+ Hβ L W D R L H H O O 4 3A H .

The first term 8.18 is estimated as follows # H H .H H H H N2 H H N2 L τ 4+ Hα L W D / {α(F ) · α(F )} H H H O O 4 Vt 0 '# H# ∞ H H .H2 ( 12 3 δ H H 3 − + H 4 H 2 2 2 2 N H H ≤ sup sup Hrτ − α(F )H τ− τ + Hα LO W H Cu u0 t∈[0,t∗ ] u≥u0 (t) 1 1 ( '# H H 5 H2 2 H H 2 H H 4−δ H N 2 + sup × τ −2 τ D / α(F ) sup α(F ) L r H H H H O 4 − + t∈[0,t∗ ] u≥u0 (t)

Cu

×

#

t

r

− 32 +δ

t0

0 0#

Σet

H -

H τ 4+ Hα

N2 W L O

.H2 H H

1 1 0# 2 · 0 '#

Σet

τ 2−δ +

H2 H H H N2 / 4 α(F )H HLO D

11 1

H# ∞ H H .H2 ( 12 H H 3 − 32 H 4 H 2 2 N N H H + sup sup Hrτ − LO α(F )H τ− τ + Hα LO W H Cu u0 t∈[0,t∗ ] u≥u0 (t) H H (1 1 '# 2 H H 2 1 4 N 2 2 H + sup sup Hr τ − D × r |LO D / 4 α(F )| / 4 α(F )HH Cu t∈[0,t∗ ] u≥u0 (t) 0 '# (1 1 # ∞ 3 H .H2 ( 12 '# 2 − 2 +δ H 4 H 2 2 2 2 N α(F )| N W H τ τ + Hα L · r |L × −

u0

+

#

∞

u0

× +

'# #

−3 τ−2

t

t0

∞

u0

'#

'#

Cu

H -

H τ 4+ Hα

3

.H2 ( 12 H 2 N LO W H

'#

Cu

H -

H τ 4+ τ 4+ Hα

3

.H2 ( 12 H 2 N LO W H

3 N 2 α(F )|||2 r0−3+δ |||τ −2 r 2 D / 4 α(F )|||24 ||| r2 L 4 O

× t . - 0 ≤ cε0 Q1 ∗ +Q2 ∗ +r0−δ T[3] ∗ .

(8.20)

O

Cu

3 NO D N O α(F )|||24 r0−3+δ |||τ −2 r 2 L / 4 α(F )|||24 ||| r2 L

−3 τ−2

t

O

Cu

( 12

( 12

+ l.o.t.

2

8.1. PROOF OF THE GENERALIZATION OF THE BOUNDEDNESS

449

To estimate 8.19, we use the Bianchi identity to rewrite 1 D4 R3A = −D3 R4A + γ BC DC RBA . 2 The most difficult remaining term is # H .H H 2 2 3H H H N2 H N2 2 H L τ 4+ Hβ L W D / (F ), ρ(F ) σ H H H. O O A Vt

Using the null decomposition of the Maxwell equations 4.17, 4.18, 4.17 and 4.20, we estimate # H2 # H2 H H H H H N2 2−δ H N 2 L τ + HLO D / 4 α(F )H ≤ τ 2−δ D / α(F ) H . H O 3 + Cu

Cu

However, we showed in Chapter 4 that 0# # H2 H H 2−δ H N 2 −δ τ + HLO D / 3 α(F )H ≤ r0 Cu

Cu

+

#

Cu

$ % NO L N T F ) T, K NO L T(L $

NO L NO L N S F ) T, K T(L

≤ r0−δ T[3] ∗.

%

1

Again using the null decomposition of the Maxwell equations 4.15 and 4.16, we have # H2 H H H N2 L τ 2−δ ▽ / {σ(F ), ρ(F )} H ≤ r0−δ T[3] ∗. H O + Cu

Therefore, we have # H .H H 3HH H H N2 2 2 H N2 L τ 4+ Hβ L W ▽ / σ (F ) H H H O O Vt

H H# H 2 1 H 2 H sup Hr τ − σ(F )HH

≤ sup

t∈[0,t∗ ] u≥u0 (t)

× ×

'#

Cu

'#

t

t0

∞

u0

−3+δ τ−2 2

1 # H H2 ( 12 H 2−δ H N 2 2 + τ + HLO ▽σ / (F )H

0 '#

Cu

∞

u0

3

−3 τ−2

H -

H τ 4+ Hβ

'#

Cu

H -

H τ 4+ Hβ

3 3 N O ▽σ(F N O σ(F )|||2 r0−2+δ |||τ −2 r 2 L / )|||24 ||| r 2 L 4

. ≤ cε0 Q1 ∗ +Q2 ∗ +r0−δ T[3] ∗

.H2 ( 12 H 2 N LO W H

( 12

N2 W L O

+ l.o.t.

(8.21)

Lastly, we estimate (8.22)

intergralC2

H # H . H β γ δ HH N N H = HφDivQ LS LT W βγδ K K K H . Vt

.H2 ( 12 H H

450

8. ERROR ESTIMATES FOR W

It suffices to estimate # H H .H H H HN H 6 H N N τ + Hα LS LT W H HLS D / 3D / 4 {α(F ) · α(F )}H Vt

H # H H H 3 2 sup HHrτ − α(F )HH ·

≤ sup

t∈[0,t∗ ] u≥u0 (t)

× ×

'#

Cu

#

∞

u0

∞

u0

−3 τ−2

'#

Cu

H -

H τ 6+ Hα

×

Cu

#

∞

u0

×

Cu

H -

H τ 6+ Hα

× +

'#

t

t0

#

∞

u0

× +

'#

t0

#

∞

u0

×

'#

2

.H2 ( 12 H N N LS LT W H

H

H2 ( 12

H -

H H H H 2 1 2 H sup Hr τ − D / 4 α(F )HH Cu

∞

(1 2

+

#

∞

u0

−3 τ−2

'#

−1− δ τ− 2

1

0 '#

Cu

H HN τ 2− r2 HL / 3 α(F )H SD H -

H τ 6+ Hα

0 '#

Cu

1

Cu 3

H -

H τ 6+ Hα

.H2 ( 12 H N N LO LT W H

1 NS D r0−1 ||| r 2 τ −2 L / 3 α(F )|||24 ||| r3 D / 4 α(F )|||24

0 '#

Cu

3

3

H -

H τ 6+ Hα

.H2 ( 12 H N N LS LT W H

H -

H τ 6+ Hα

t0

.H2 ( 12 H N N LS LT W H

( 12 1

( 12 1

.H2 ( 12 '# t # NO L N T W HH L ·

N S α(F )|||2 r0−1 ||| r 2 τ −2 D / 3D / 4 α(F )|||24 ||| r2 L 4

≤ cε0 Q1 ∗ +T[2] + T3 .

(8.23)

H2 ( 12

H τ 6+ Hα

3 5 NS D r0−1+δ ||| r 2 τ −2 L / 4 α(F )|||24 ||| r 2 τ −2 D / 3 α(F )|||24

−3 τ−2

t

t0

r |D / 3D / 4 α(F )|

−3 τ−2

t

Cu

.H2 ( 12 '# H N N LS LT W H u0

6

Cu

−5 τ−2

'#

H

HN H τ 2− τ 2+ HL / 3D / 4 α(F )H SD

t∈[0,t∗ ] u≥u0 (t)

'#

t∈[0,t∗ ] u≥u0 (t)

'#

∞

u0

H # H H H 1 2 N H sup Hrτ − LS α(F )HH ·

+ sup

Cu

H2 ( 12 H H 4 HN r HLS D / 4 α(F )H + sup

−3 τ−2

.H2 ( 12 N T W HH NS L L

H 5 H H 2 H sup Hr α(F )H

.H2 ( 12 '# H N N LS LT W H

t∈[0,t∗ ] u≥u0 (t)

×

H -

H τ 6+ Hα

t∈[0,t∗ ] u≥u0 (t)

H # H H H 5 2 H sup Hrτ − D / 3 α(F )HH ·

'#

Cu

H H2 ( 12 H 4 HN τ + HLS D / 3D / 4 α(F )H + sup

−3 τ−2

+ sup

'#

Su

H4 ( 14 H H r HLS α(F )H 8 HN

( 14 1

8.1. PROOF OF THE GENERALIZATION OF THE BOUNDEDNESS

and # Vt

H H .H H H HN H H N N τ 6+ Hα L / 4D / 4 {α(F ) · α(F )}H S LT W H HLS D H # H H H 3 2 H sup Hrτ − α(F )HH ·

≤ sup

t∈[0,t∗ ] u≥u0 (t)

× ×

'#

Cu

#

∞

u0

u0

Cu

H -

H τ 6+ Hα

× +

#

∞

u0

#

∞

u0

× +

'#

t0

#

∞

u0

×

−5

τ−4

'#

t0

τ−

'#

H2 ( 12

Cu

H -

H τ 6+ Hα

.H2 ( 12 H N N LS LT W H

H H H 1 H 2 N H sup Hrτ − LS α(F )HH

t∈[0,t∗ ] u≥u0 (t)

'#

Cu

0 '#

H -

H τ 6+ Hα

Cu

.H2 ( 12 '# NS L N T W HH L

H .H2 ( 12 H 6 H N N τ + Hα LS LT W H

0 '#

Cu

H -

H τ 6+ Hα

Cu

6

.H2 ( 12 H N N LO LT W H

2

r |D / 4D / 4 α(F )|

(1 2

( 12 1

5 1 −3 N S α(F )|||2 r0 2 ||| r 2 τ −2 D / 4D / 4 α(F )|||24 ||| r2 L 4

% ≤ cε0 Q2 ∗ +T[2] + T3 .

(8.24)

$

u0

− 32

H

H HN τ 2− τ 2+ HL / 4D / 4 α(F )H SD

3 1 −3 NS D r0 2 ||| r 2 τ −2 L / 4 α(F )|||24 |||r3 D / 4 α(F )|||24

−5 τ−4

t

.H2 ( 12 H N N LS LT W H

H H 5 H H sup Hr 2 α(F )H Cu

∞

H2 ( 12 H H 4 HN r HLS D / 4 α(F )H + sup

−3 τ−2

t

Cu

H -

H τ 6+ Hα

.H2 ( 12 '# H N N LS LT W H

t∈[0,t∗ ] u≥u0 (t)

Cu

'#

t∈[0,t∗ ] u≥u0 (t)

'#

H # H H H 2 1 2 H sup Hr τ − D / 4 α(F )HH ·

+ sup ×

−3 τ−2

H2 ( 12 H H 4 HN / 4 α(F )H τ + HLS D / 4D + sup

−3 τ−2

'#

∞

451

( 12 1

Recall that T3 is not bounded by the initial data. H We canH only bound & & H2 H 2 / 4 α(F )H and Σe τ 8− r0−δ T3 by the initial data. Thus, we must allow Σe r8 HD t t H H2 H 2 H δ / 3 α(F )H to grow like r0 . This will not affect the sup norm estimate for HD the components of the curvature. Also none of the other estimates we need to make will be adversely affected. In conclusion, the error terms E1 and E2 can be bounded by the initial data. Therefore, the energy norms for the Weyl tensor are bounded by ε.

CHAPTER 9

Second Fundamental Form In this chapter, we estimate the second fundamental form k of the time foliation in terms of norms W and F. Our derivations generalize the estimates in Chapter 11 of [5] by taking into account the presence of the electromagnetic field. The difference between [5] and our case arises from the extra curvature terms that occur in the elliptic system for k. On the leaves of the space-time foliation Σt , k satisfies the elliptic system trk = 0 (9.1)

(curl k)ij = H(W )ij +

(9.2)

(div k)i = R0i .

1 ∈ij l R0l 2

Furthermore, (9.3)

1 1 Rij = kia kja + E(W )ij + gij R00 + Rij . 2 2

Following [5], decompose k with respect to the frame N , eA , A = 1, 2. This decomposition provides information about the weights of k as well as its asymptotic behavior. With respect to this frame, equations 9.2 and 9.1 become

(9.4) (9.5)

(9.6)

(9.7)

3 η ·N θ div / ϵ = −∇N δ − trθδ + N 2 1 1 / · ϵ + |α(F )|2 − |α(F )|2 − 2(a−1 ∇a) 4 4 N curl / ϵ = σ(W ) + θ ∧ N η 1 ∇ / N ϵ + trθϵ = (β(W ) − β(W )) 2 1 + (ρ(F ) (α(F ) − α(F )) + σ(F ) (∗α(F ) − ∗α(F ))) 4 % $ 3 / − ∇δ / −N θ·ϵ+ c·δ−N η · a−1 ∇a 2 % 1$ divN / η= −β(W ) + β(W ) 2 + (ρ(F ) (α(F ) − α(F )) + σ(F ) (∗α(F ) − ∗α(F ))) 1 1 / +N θ · ϵ − trθϵ − ∇δ 2 2 453

454

9. SECOND FUNDAMENTAL FORM

1 1 ∇ / NN η + trθN η = (−α(W ) + α(W )) 2 4 % $ 1 3 N N + δN (9.8) + ∇ / ⊗ϵ θ + a−1 ∇a / ⊗ϵ. 2 2

9.1. Exterior Estimates for kNN

As in [5], the most difficult aspect of the derivations is obtaining good estimates in the Wave Zone. This difficulty arises from the non-linear nature of the space-time. The most sensitive component of the second fundamental form is in the N -N direction, i.e. δ. In the flat case, ∇N δ behaves like ∇N δ = O(r−3 ).

η ·N θ The term ∇N δ cannot be estimated $ −2 % directly from 9.4 because the terms N 2 1 and 4 |α(F )| behave like O r in the Wave Zone. We apply the methods introduced in [5] to obtain better estimates in the Wave Zone. The differences, which we focus on below, again arise from the need to estimate the extra terms due to the presence of the electromagnetic field. Following [5], consider the interior product of k with the position vector field Z = rN . Recall that the traceless part of the deformation tensor of Z is given by (9.9)

(Z)

where

1 π Nij = 2rN θij + κ(gij − 3Ni Nj ) − ra−1 (Ni ∇ / j a + Nj ∇ / i a), 3 κ = ra−1 (2atrθ + atrθ).

(9.10) Furthermore, ∇i Zj = (9.11)

1 (Z) π Nij − ra−1 (Ni ∇ / j a − Nj ∇ / i a) 2 1 + (rtrθ + λa−1 )gij . 3

Decompose iZ k by (9.12) with (9.13)

iZ k = iI Z k + ∇ψ, △ψ = r |N η |2 −

r |α(F )|2 . 4

Then, (9.14) (9.15)

curl iI Z k = curliZ k = F + G div iI Z k = I,

9.1. EXTERIOR ESTIMATES FOR kNN

455

where (9.16) (9.17) and

1 Fi = H(W )ij Z j + ∈ij l R0l Z j , 2 . 1 mn -(Z) Gi = ∈i π Nij − ra−1 (Ni ∇ / j a − Nj ∇ / i a) knj 2

1 (Z) r r π Nij kij + |α(F )|2 − |α(F )|2 − △ψ 2 4 4 r −1 = rN χ·N η − κδ − 2ra ∇a (9.18) / · ϵ + |α(F )|2 . 4 I As in [5], derive a Poisson equation for iZ k by applying the formula below to 9.14 and 9.15. The formula I=

curl(curl v)i = Rij vj − △vi + ∇i (div v),

holds for any 1-form v. Deduce - . - . △ iI = −curl (F + G)i + Rij iI + ∇i I. Zk Zk i

j

To introduce δ, again contract iI Z k with Z and obtain

(9.19)

Thus, we have (9.20)

p = Z · iI Z k = r (rδ − ∇N ψ).

- . - . j I I k + 2∇ Z∇ △p = Z · △ iI j iZ k + △Z · iZ k. Z

Using the formula

1 △Z i + Rij Z j = ∇j π Nij − ∇i (trπ), 6 obtained from differentiating 9.11 and taking the trace, calculate (9.21)

△p = −Z · curl(F + G) + ∇Z I -(Z) . + π Nij − ra−1 (N i ∇ / ja − Nj∇ / i a) ∇i (iZ k)

(9.22)

1 Nij ∇i ∇j ψ + trπI −(Z) π 3 ' ( 1 + ∇j π Nij − ∇i (trπ) ((iZ k)i − ∇i ψ). 6

To understand better the right-hand side of 9.22, calculate % r2 r2 $ div / β(W ) − div / ) (α(F ) − α(F ))) Z · curlF = / β(W ) − div(ρ(F 2 2 r2 − div(σ(F / ) (∗ α(F ) + ∗ α(F ))). 2 The term r2 div / β(W ) does not decay fast enough in the Wave Zone to apply profitably the non-degenerate elliptic estimates in Propositions 1, 2 and 3.

456

9. SECOND FUNDAMENTAL FORM

Following [5], this term is eliminated by utilizing the Bianchi identities. Recall 1 div Ea = −ϵaij Hil klj + (Da R00 − D0 Ra0 ). 2 Contracting this with Na leads to % 1 1$ / β(W ) = ∇N ρ(W ) − χ div / β(W ) + div N · α(W ) 2 4 ' ( 1 3 1 −1 N · α(W ) + trθρ(W ) − a ∇a + χ / + ϵ 4 2 2 ' ( 1 1 / − ϵ · β(W ) − D × β(W ) − a−1 ∇a / |α(F )|2 2 4 3 $ % 1 1 + D / 4 |α(F )|2 + ∇N ρ(F )2 + σ(F )2 . (9.23) 4 2 This produces another expression for div / β(W ). However, slow decaying 2 terms like χ N · α(W ) and D / 4 |α(F )| still exist in 9.23. Furthermore, there does not exist a solution q of △q = −rρ(W ) with the appropriate asymptotic behavior. Following [5], solve these difficulties by defining the mass aspect function (9.24)

µ = −ρ(W ) − χ N·N η.

Then calculate

C % $ 3 3 1 ∇N r µ + κµ = −r div / β(W ) + div / β(W ) 2 2 $ % 1 + a−1 ∇a / · β(W ) + β(W ) 2 D 1 − r−1 a−1 λ |N η |2 + D / 4 |α(F )|2 + J , 4 3

(9.25) (9.26) with

. 1 1 J + a−1 N η·∇ / 2 a = − r−1 κ |N η |2 + χ χ|2 − |N η |2 N · α(W ) + δ 3 |N 2' 2 ( ' ( 1 1 1 1 −1 −1 a ∇a a ∇a + / + ϵ · β(W ) + / − ϵ · β(W ) 2 2 2 2 (9.27)

/ · (N χ−N η) · ϵ −ϵ·N η · ϵ + (N χ−N η ) · ∇ϵ / + 2a−1 ∇a $ % 1 1 / 3 |α(F )|2 − ∇N ρ(F )2 + σ(F )2 . + D 8 4

For the average of µ on St,u , also calculate the following (9.28)

dr3 (µ) r3 η |2 − aD / 4 |α(F )|2 − r2 aκ (µ − µ) − r3 aJ. = λr2 |N du 4

9.1. EXTERIOR ESTIMATES FOR kNN

457

Recall that N = a−1 ∂u . Combining equations 9.25 and 9.28 yields . % M L 1$ / β(W ) = r−1 ∇N r3 (µ − µ) − ra−1 λ |N η |2 − |N η |2 − div / β(W ) + div 2 . 1aD / 4 |α(F )|2 − aD + r2 a−1 / 4 |α(F )|2 4$ % + r2 a−1 aJ − aJ . (9.29)

Now substitute 9.29 into the Poisson equation for p to obtain . η |2 − |N η |2 △p = ∇Z [r (µ − µ) + I] − ra−1 λ |N . 1/ 4 |α(F )|2 + l.o.t. aD / 4 |α(F )|2 − aD + r2 a−1 4 Nonetheless, we still cannot apply the non-degenerate - elliptic.estimates to 1 this Poisson equation because of the decay of ra−1 λ |N η |2 − |N η |2 and r2 a−1 4 . 2 2 / 4 |α(F )| in the Wave Zone. Following [5], introduce q aD / 4 |α(F )| − aD

and ψ ′ that are solutions of the equations (9.30) and

△q = r (µ − µ) + I

. . 1△ψ ′ = −ra−1 λ |N η |2 − |N η |2 + r2 a−1 / 4 |α(F )|2 , aD / 4 |α(F )|2 − aD 4 (9.31) respectively, and vanish at infinity. Now let (9.32)

q ′ = p − ∇Z q − ψ ′ .

Then q ′ satisfies ' ( $ % 1 1 2 ′ 2 2 2 △q = r div / β(W ) + − D / |α(F )| + ∇N ρ(F ) + σ(F ) 4 3 2 ' ( 1 i ij (Z) − π N · ∇∇ (ψ + q) − ∇j π N − ∇ (trπ) (∇i q − iZ ki − ∇i ψ) 6 .. % -(Z) ij $ 2 −1 − r curl / G+r / a π N −ra−1 N i ∇ / j a−N j ∇ / ia aJ − aJ + R S 4 1 × ∇i (iZ k)j + ra−1 3aκµ − aκ (µ − µ) + aκ (µ − µ) 2 3 $ % 1 / · β(W ) + β(W ) . + a−1 ∇a 2 (9.33) The derivations above allow us to generalize Lemma 11.2.1 in [5] as follows: Lemma 9. The component δ = k(N, N ) of the second fundamental form can be decomposed according to the formula % $ δ = r−2 q ′ + ∇Z q + ψ ′ + ∇Z ψ

458

9. SECOND FUNDAMENTAL FORM

where the scalars ψ, q, q ′ , ψ ′ are the solutions, vanishing at infinity, of the following Poisson equations r |α(F )|2 4 △q = r(µ − µ) + I . . 1△ψ ′ = −ra−1 λ |N η |2 − |N η |2 + r2 a−1 / 4 |α(F )|2 aD / 4 |α(F )|2 − aD 4 ( ' $ % 1 1 2 2 2 2 ′ + l.o.t / |α(F )| + ∇N ρ(F ) + σ(F ) △q = r divβ(W / )− D 4 3 2 △ψ = r |N η |2 −

where the error terms in △q and △q ′ are given by equations 9.18 and 9.33. As in [5], estimate ψ. By the Bootstrap Assumptions we have ∥τ − △ ψ∥2 ≤ cε0 K[2] + F[2]

∥τ − r∇ / △ ψ∥2 ≤ cε0 K[2] + F[2] Q Q Q Q / 2 △ ψ Q ≤ cε0 K[2] + F[2]. Qτ − r2 ∇ 2

In the exterior region,

Q 2 Q Qτ − ∇N △ ψ Q ≤ cε0 K[2] + F[2] 2,e 1

3

|||r 3 τ −3 △ ψ|||3,e ≤ cε0 K[2] + F[2]. Similar estimates hold for ψ ′ with the exception that Q Q Q Q / 2 △ ψ Q ≤ cε0 K[2] + F[2] + r0−δ F3 . Qτ − r2 ∇ 2

Now we can apply the degenerate elliptic estimates of Proposition 1, 2 and 3 to △ψ and △ψ ′ to derive the following, which is a generalization of Proposition 11.2.1 in [5]:

Proposition 9. There exists a constant c such that the scalar ψ verifies the estimates − 12

−1

− 12

r0

∥∇ψ∥2,e ≤ cε0 K[2] + F[2] D Q Q 2 Q Q + ∥r∇∇ / N ψ∥2,e + w1 ∇N ψ 2,e ≤ cε0 K[2] + F[2] 2,e D Q Q Q Q Q 2 2 Q 2 Q Q + Qr ∇ / ∇N ψ Q + w1 ∇∇ / N ψ 2,e ≤ cε0 K[2] + F[2]

CQ Q Q 2 Q / ψQ Qr∇

r0 2 CQ Q Q 2 3 Q / ψQ Qr ∇

2,e

r0

2,e

− 12

r0

Q Q Qw2 ∇3N ψ Q ≤ cε0 K[2] + F[2] . 2,e

9.1. EXTERIOR ESTIMATES FOR kNN

459

Also, −1

− 12

r0

5

r0 2 |||r 6 ∇ψ|||3,e ≤ cε0 K[2] + F[2] A 11 B1 11 3 2 3 3 6 6 / ψ|||3,e + |||r ∇∇ / N ψ|||3,e ≤ cε0 K[2] + F[2] |||r ∇ −1

1

r0 2 |||r 3 w 3 ∇3N ψ|||3,e ≤ cε0 K[2] + F[2] . 2

As a consequence, we also have . - 3 −1 r0 2 sup r 2 |∇ψ| ≤ cε0 K[2] + F[2] . E

The scalar ψ verifies the same estimates with r0−δ F3 on the right hand side where appropriate. ′

Next estimate q and q ′ . We find that $ % ∥r △ q∥2 ≤ c ε0 K[2] + W[1] + F[2] Q 2 Q $ % Qr ∇ / △ q Q ≤ c ε0 K[2] + W[1] + F[2] . 2

In the exterior

$ % ∥rτ − ∇ / N △ q∥2,e ≤ c ε0 K[2] + W[1] + F[2] $ % 4 1 |||r 3 τ −2 △ q|||3,e ≤ c ε0 K[2] + W[1] + F[2] .

Using the nondegenerate elliptic estimates of Propositions 1, 2 and 3, we derive the following generalization of Proposition 11.2.2 in [5]. Proposition 10. There exists a constant c such that ⎫1 ⎧ Q Q ⎬2 ⎨ " Q 2 Q2 $ % Q 2 i j Q2 2 Q Q / ∇N q Q ≤ c ε0 K[2] + W[1] + F[2] ∥∇q∥2 + r∇ q 2 + Qr ∇ ⎩ 2⎭ i+j=3,i≥1 Q Q $ % Qrτ − ∇3N q Q ≤ c ε0 K[2] + W[1] + F[2] . 2,e

Also, B1 A 5 $ % 11 11 3 2 3 3 3 6 6 6 / q|||3,e + |||r ∇∇ / N q|||3,e ≤ c ε0 K[2] + W[1] + F[2] |||r ∇q|||3,e + |||r ∇

$ % 4 1 |||r 3 τ −2 ∇2N q|||3,e ≤ c ε0 K[2] + W[1] + F[2] .

As a consequence, we have - 3 . $ % sup r 2 ∇q ≤ c ε0 K[2] + W[1] + F[2] . E

We produce similar estimates up to second derivatives for q ′ . Thus we obtain a generalization of Corollary 11.2.0.1 in [5]: Corollary 3. Given the Bootstrap Assumptions $ % e K (δ) ≤ c ε K + W + F 0 [2] [2] [1] [2] .

460

9. SECOND FUNDAMENTAL FORM

As in [5], we are now in a position to estimate e K[2] (ϵ). Recall 3 η ·N θ div / ϵ = −∇N δ − trθδ + N 2 $ % 1 1 / · ϵ + |α(F )|2 − |α(F )|2 − 2 a−1 ∇a 4 4 N curl / ϵ = σ(W ) + θ ∧ N η ' % 1 1$ ρ(F ) (α(F ) − α(F )) β(W ) − β(W ) + ∇ / N ϵ + trθϵ = 2 2 ( $ % 3 ∗ ∗ / −N θ·ϵ+ c·δ−N / . η · a−1 ∇a + σ(F ) ( α(F ) − α(F )) − ∇δ 2

Following [5], write δ = r−2 p + r−1 ∇N ψ. Use the radial decomposition of △, that is ∇2N = △ − trθ∇N − △ / − a−1 ∇a / ·∇ / to derive ∇N δ − N θ·N η+ (9.34)

1 |α(F )|2 = −2r−3 (∇N r)p + r−2 ∇N p 4 − r−2 (∇N r)∇N ψ + r−1 ∇2N ψ % $ = −N χ·N η − r−1 △ψ / − r−2 rtrθ + a−1 λ ∇N ψ

/ · ∇ψ / + r−2 ∇N p − 2r−3 a−1 λp. − r−1 a−1 ∇a

Using 3, deduce the following estimates Q ' (Q 1 $ Q 2 Q % 1 2 Qr ∇N δ − N θ·N η + |α(F )| Q ≤ cr02 ε0 K[2] + W[1] + F[2] Q Q 4 2,e Q (Q ' 1 $ Q 3 Q % N η + 1 |α(F )|2 Q ≤ cr 2 ε0 K[2] + W[1] + F[2] Qr ∇ 0 Q / ∇N δ − θ · N Q 4 2,e ' ( 1 $ % 17 1 θ·N η + |α(F )| |||3,e ≤ cr02 ε0 K[2] + W[1] + F[2] . |||r 6 ∇N δ − N 4 These estimates imply that

Q 2 Q Qr divϵ / Q2,e Q −1 Q / Q2,e r0 2 Qr2 curlϵ Q −1 Q / divϵ / Q2,e r0 2 Qr3 ∇ Q −1 Q / urlϵ / Q2,e r0 2 Qr3 ∇c − 12

r0

and that

% $ ≤ c ε0 K[2] + W[1] + F[2] $ % ≤ c ε0 K[2] + W[1] + F[2] $ % ≤ c ε0 K[2] + W[1] + F[2] $ % ≤ c ε0 K[2] + W[1] + F[2]

$ % 17 −1 r0 2 |||r 6 divϵ||| / 3,e ≤ c ε0 K[2] + W[1] + F[2] $ % 17 −1 / r0 2 |||r 6 curlϵ||| 3,e ≤ c ε0 K[2] + W[1] + F[2] .

9.3. ESTIMATES FOR THE TIME DERIVATIVES OF k

461

Then applying Proposition 2.3.1 from [5] to the Hodge system described by H1 , it follows that $ % e K (ϵ) ≤ c ε K + W + F (9.35) 0 [2] [2] [1] [2] .

$ % η ) ≤ c ε0 K[2] + W[1] + F[2] follow directly from The estimates for e K[2] (N [5] and our estimates for F . Lastly $ % e K (N (9.36) [2] η ) ≤ c ε0 K[2] + W[1] + F[2] . 9.2. Interior Estimates The interior estimates follow directly from the interior estimates for the second fundamental form in [5] together with our estimates for F . Therefore, we obtain the following generalization of Theorem 11.3.1 in [5]: Theorem 8. Under the Bootstrap Assumptions there exists a numerical constant c such that if ε0 is chosen small enough, then $ % K[2] ≤ c W[1] + F[2] .

The estimates for the third derivatives of the second fundamental form follow from the same steps as before. Thus we can generalize Theorem 11.3.2 in [5]: Theorem 9. Under the Bootstrap Assumptions $ % K[3] ≤ c W[2] + F[3] . 9.3. Estimates for the Time Derivatives of k Recall that on Σ0 , we have the following evolution equation $ % ∂t kij = −∇i ∇j φ + φ −Rij + Rij − 2kia kja .

Following [5], use the radial decomposition of k to derive (9.37)

DT δ = −φ−1 ∇2N φ + ρ(W ) + δ 2 − ζ · ζ + ζ · ϵ − ζ · ϵ % 1 1$ β(W ) + β(W ) + (α(F ) + α(F )) ρ(F ) 2 4 $ −1 % −1 % 1 ∗ 3$ ∗ + ( α(F )+ α(F )) σ(F ) a ∇a / φ ∇N φ− ζ −φ−1 ∇φ / δ 4 2 % $ 1 / +ϵ ·N η + δϵ + ζ − φ−1 ∇φ 2

D / T ϵ = −φ−1 ∇∇ / Nφ +

(9.38)

462

9. SECOND FUNDAMENTAL FORM

. 1 1(α(W ) + α(W )) + |α(F )|2 + |α(F )|2 δ AB 4 4 $ % % 1$ 2 2 N σ(F ) − ρ(F ) δ AB + α(F )⊗α(F + ) 2 $ % N − ζ − φ−1 ∇φ N − δN η + ϵ⊗ϵ / ⊗ϵ.

D / TN η = −φ−1 ∇ / 2φ + (9.39)

Use these three equations to calculate expressions for D4 δ, D / 4 ϵ, D / 4N η. Calculate δ 4 = DT δ + DN δ % $ = ∇N −φ−1 ∇N φ + δ − φ−2 |∇N φ|2 + ρ(W ) + δ 2 − ζ · ζ + ζ · ϵ − ζ · ϵ.

(9.40) Therefore,

δ 4 = ∇N ν − φ−2 |∇N φ|2 + ρ(W ) + δ 2 − ζ · ζ + ζ · ϵ − ζ · ϵ. Using 9.34, calculate % $ / + ρ(W ) + φ−1 trθ∇N φ + a−1 ∇a / · ∇φ / D / 4 δ = δ 4 = φ−1 △φ $ % + ∇N r−2 q ′ + r−1 ∇N q + r−2 ψ ′ − r−2 a−1 λ∇N ψ $ % / · ∇ψ / −ζ ·ζ +ζ ·ϵ−ζ ·ϵ − r−1 trθ∇N ψ + a−1 ∇a 1 − 2 |ϵ|2 − |δ|2 . 2 Now adding equations 9.6, 9.38, 9.8 and 9.39, conclude that 1 (α(F ) + α(F )) ρ(F ) 2 $ % 1 + (∗α(F ) + ∗α(F )) σ(F ) a−1 ∇a / φ−1 ∇N φ 2 % $ % 3 $ −1 / − ζ + φ−1 ∇φ / δ + ζ − φ−1 ∇φ / + a−1 ∇a / ·N η − a ∇a 2 ( ' . 1 δ − trθ ϵ + N η − 2N θ ·ϵ+ 2 1 1 η + trχN η − |α(F )|2 δ AB D / 4N 2 4 1 1 1 −1 2 N + α(W ) + |α(F )|2 δ AB / ⊗ϵ =N η 4 = −φ ∇ / φ+ ∇ 2 2 4 $ $ −1 % % 3 N N + δ θ + ϵ⊗ϵ N N − ζ − φ−1 ∇φ − δN η + a ∇a / ⊗ϵ / ⊗ϵ. 2

/ N φ + ∇δ / + β(W ) + D / 4 ϵ = ϵ4 = −φ−1 ∇∇

(9.41)

(9.42)

Notice that our definition of N η 4 differs from the definition in [5] by the 2 1 term 4 |α(F )| δ AB . We had to handle this term when we estimated the error terms that appear in the energy estimates for the Bel-Robinson tensor.

9.3. ESTIMATES FOR THE TIME DERIVATIVES OF k

463

With these formulas and the Bootstrap Assumptions, we derive the following generalization of Proposition 11.4.2 in [5]: Proposition 11. Under the Bootstrap Assumptions, we have for 0 ≤ q≤1 $ % e Kq+2 (DS δ 4 ) ≤ c e Lq+2 (D / S φ) / + e W[q] + e F[q] $ % e Kq+2 (D / S ϵ4 ) ≤ c e Lq+2 (D / S ϕN ) + e W[q] + e F[q] $ % e Kq+2 (D / SN η 4 ) ≤ c e Lq+2 (D / S φ) / + e W[q] + e F[q] . Also,

$ % Kq+2 (DS δ 3 ) ≤ c e Lq+2 (D / S φ) / + e W[q] + e F[q] $ % e Kq+2 (D / S ϵ3 ) ≤ c e Lq+2 (D / S ϕN ) + e W[q] + e F[q] $ % e Kq+2 (D / SN η 3 ) ≤ c e Lq+2 (D / S φ) / + e W[q] + e F[q] . e

CHAPTER 10

The Lapse Function We use the same method to achieve estimates for the lapse function as in [5]. In particular, we consider the equation . (10.1) △φ = |k|2 + R00 φ. Recall that

R00 =

. 1|α (F )|2 + |α (F )|2 + ρ (F )2 + σ (F ) . 2

We have shown that |α (F )|2 behaves like |N η |2 – the worst decaying term of |k|2 in the Wave Zone. Therefore we derive estimates for the lapse function in the Wave Zone because in the interior and the far exterior, the same estimates as in [5] hold. Following [5], apply the scaling operator S to φ to obtain estimates in the Wave Zone. Recall that S = tT + Z,

(10.2)

with t = (r − u). Also Z represents the position vector field acting on Σt ,

(10.3)

Z = rN.

In [5], Christodoulou and Klainerman find a Poisson equation for △Sφ by first commuting t∂t and △, then commuting ∇Z and △ and lastly adding the two equations together. The result is equation (12.0.5f) in [5]. They then proceed to show that 1

r02 ∥∇Sφ∥2 ≤ cε20

by applying the degenerate elliptic estimates in Proposition 1 for the Poisson equations △φ = f , # # 2 |∇φ| ≤ cr0 τ 2− |f |2 . Σ

Σ

To apply this estimate in our case, we need to control the extra terms due to the presence of the electromagnetic field, which appear in the Poisson equation for △Sφ. These terms include ' ( . 1 2 2 2 |α (F )| + |α (F )| + ρ (F ) + σ (F ) φ. DS 2 465

466

10. THE LAPSE FUNCTION

The worst decaying term in the Wave Zone is DS |α (F )|2 . We can estimate this as follows H2 H # # H H 3 2 2 2 2 τ − |DS α (F )| |α (F )| ≤ sup HHrτ − α (F )HH r0−2 τ 2− |DS α (F )|2 . Σ

Σ

Σ

Therefore, the estimate

1

r02 ∥∇Sφ∥2 ≤ cε20

(10.4)

holds. As in [5], we also need to show 1

r02 ∥∇SSφ∥2 ≤ cε20 .

(10.5)

The extra terms that we obtain from commuting the scaling operator twice include ' ( . 1 2 2 2 DS DS |α (F )| + |α (F )| + ρ (F ) + σ (F ) φ 2 ' ( . 1 2 2 2 |α (F )| + |α (F )| + ρ (F ) + σ (F ) DS φ, + DS 2

where the term ' ( . 1 2 2 2 |α (F )| + |α (F )| + ρ (F ) + σ (F ) D2S φ 2

is controlled in the same manner as |k|2 SSφ. Again the worst decaying terms are D2S |α (F )|2 and DS |α (F )|2 DS φ. The first term is estimated as follows # # H H2 2 2 H 2 H τ − DS α (F ) |α (F )| + τ 2− |DS α (F )|4 Σ Σ H2 H # H H −2 H2 H 3 2 H H ≤ sup Hrτ − α (F )H r0 τ 2− HD2S α (F )H Σ Σ # ∞ # + r0−2 c τ −4 r2 τ 6− |DS α (F )|4 . − u0

Su

Thus, the new terms that appear in our case do not interfere with the original estimates in [5]. Therefore, 1

r02 ∥∇SSφ∥2 ≤ cε20 .

For the error estimates of the third derivatives of the null components of the electromagnetic field, we estimate ∥∇SSSφ∥2 . Calculate SSS = (tT + Z)3

= (tT )3 + (tT )2 Z + (tT ) Z (tT ) + Z (tT ) Z + Z 2 (tT ) + Z (tT )2 + tT Z 2 + Z 3 .

10. THE LAPSE FUNCTION

467

First, let us consider the Poisson equation for (tT )3 φ. [5], the commutator between ∂t and △ is given by

Following

[∂t , △g ] φ = 2k ij (φ∇i ∇j φ + ∇i φ∇j φ).

Therefore, 0 1 0' ( 1 . ' ∂ (3 ∂ 3 2 t φ − |k| + R00 φ △ t ∂t ∂t 0 0' 0' (2 1 (2 11 ∂ ∂ t t φ φ + ∇i φ∇j = 2tk ij φ∇i ∇j ∂t ∂t (( R ' ' ∂φ ∂φ ij φ∇i ∇j t + ∇i φ∇j t + t∂t t 2k ∂t ∂t ( R ' . . % $ ij ∂ -- 2 |k| + R00 φ + t∂t t 2k (φ∇i ∇j φ + ∇i φ∇j φ) + t ∂t ( ' .S ∂φ ∂φ −1 2 −1 −2 +t t △ t − t |∇t| + t ∇t · ∇ t ∂t ∂t 0' 1 0 ( ( 1' .S ∂ 2 ∂ 2 1 2 −1 −2 t φ + t φ t △ t − t |∇t| + ∇t · ∇ t ∂t ∂t 0' ( 1 0' ( 1. ∂ 3 ∂ 3 1 t φ + t φ t−1 △ t − t−2 |∇t|2 . + ∇t · ∇ t ∂t ∂t (10.6) 2

Now, we calculate the Poisson equation for t2 ∇Z ∂∂t2φ . We find that 0 ( 1 ( ' ' . ∂ 2 ∂ 2 2 φ − |k| + R00 ∇Z t φ △ ∇Z t ∂t ∂t 0' 0' (2 1 (2 1 R ∂ 1 ∂ t t φ + trπ |k|2 φ = π Nij ∇i ∇j ∂t 3 ∂t 0' 0' ( 1 ( 1S ∂ 2 1 i ∂ 2 ij t t N ∇i φ − ∇ trπ∇i φ + ∇j π ∂t 6 ∂t ( R ' ∂φ ∂φ ij + ∇i φ∇j t + ∇Z t2k φ∇i ∇j t ∂t ∂t ( R ' . . % $ ij ∂ -- 2 |k| + R00 φ + t∂t t 2k (φ∇i ∇j φ + ∇i φ∇j φ) + t ∂t ( ' .S ∂φ ∂φ −1 2 −1 −2 +t t △ t − t |∇t| + t ∇t · ∇ t ∂t ∂t Z 0' ( 1 0' ( 1. ∂ 2 ∂ 2 1 2 −1 −2 t t (10.7) φ + φ t △ t − t |∇t| . + ∇t · ∇ t ∂t ∂t

468

10. THE LAPSE FUNCTION

$ ∂ %2 $ ∂% $ ∂% ∂ ∂ We also need expressions for ∇2Z t ∂t φ, t ∂t ∇Z φ, t ∂t ∇2Z φ, ∇Z t ∂t φ, t ∂t ∂ ∇Z φ, and ∇3Z φ. We calculate ∇Z t ∂t ' ( . ∂ 2 ∂ △ ∇Z t φ − |k|2 + R00 ∇2Z t φ ∂t ∂t ( ( ' ' R 1 ∂φ ∂φ 2 ij + trπ |k| ∇Z t = π N ∇i ∇j ∇Z t ∂t 3 ∂t ( (S ' ' 1 ∂φ ∂φ − ∇i trπ∇i ∇Z t + ∇j π Nij ∇i ∇Z t ∂t 6 ∂t S R ∂φ 1 ∂φ 1 i ∂φ 2 ∂φ ij ij N ∇i ∇ j t + ∇Z π N ∇i + trπ |k| t + ∇j π − ∇ trπ∇i ∂t 3 ∂t ∂t 6 ∂t R . ∂ + ∇2Z 2tk ij (φ∇i ∇j φ + ∇i φ∇j φ) + φt |k|2 + R00 ∂t ( ' .S 1 ∂ ∂ - −1 2 −2 (10.8) + ∇t · ∇ t φ + t φ t △ t − t |∇t| . t ∂t ∂t Next, we calculate 1 0' ( . ' ∂ (2 ∂ 2 2 ∇Z φ − |k| + R00 ∇Z φ t △ t ∂t ∂t ' ' ' ( (( ∂ ∂ = 2tk ij φ∇i ∇j t ∇Z φ + ∇i φ∇j t ∇Z φ ∂t ∂t R ∂ 2tk ij (φ∇i ∇j (∇Z φ) + ∇i φ∇j (∇Z φ)) +t ∂t R ∂ ij 1 π N ∇i ∇j φ + trπ |k|2 φ + ∇j π +t Nij ∇i φ ∂t 3 ' ( .S 1 ∂ 1 i 2 + ∇t · ∇ t ∇Z φ − ∇ trπ∇i φ + φ∇Z |k| + R00 6 t ∂t 0' 1 (2 S . ∂ ∂ 1 2 t ∇Z φ + ∇t · ∇ + t ∇Z φ t−1 △ t − t−2 |∇t| ∂t t ∂t ( ' . ∂ 2 ∇Z φ t−1 △ t − t−2 |∇t|2 . (10.9) + t ∂t We also have ( ( ( ( ' ' '' .' ∂ ( ∂ ∂ ∂ 2 t ∇Z t φ − |k| + R00 ∇Z t φ △ t ∂t ∂t ∂t ∂t ( ( ( (( ' ' ' ' ' ∂ ∂ ij φ + ∇i φ∇j ∇Z t φ = 2tk φ∇i ∇j ∇Z t ∂t ∂t R ∂ ij ∂ ∂ ∂ 1 π N ∇i ∇j t φ + trπ |k| t φ + ∇j π Nij ∇i t φ +t ∂t ∂t 3 ∂t ∂t

10. THE LAPSE FUNCTION

(10.10)

469

R S 1 i ∂ ∂ − ∇ trπ∇i t φ + t ∇Z 2tk ij (φ∇i ∇j φ + ∇i φ∇j φ) 6 ∂t ∂t ' ( .S ∂ ∂ - −1 1 2 −2 + ∇t · ∇ t φ + t φ t △ t − t |∇t| t ∂t ∂t R ' ( .S ∂ ∂ ∂ ∂ 1 + ∇t · ∇ t ∇Z t φ + t ∇Z t φ t−1 △ t − t−2 |∇t|2 t ∂t ∂t ∂t ∂t

Furthermore, ( ' . ∂ ∂ 2 △ t ∇Z φ − |k|2 + R00 t ∇2Z φ ∂t ∂t % %% $ $ $ = 2tk ij φ∇i ∇j ∇2Z φ + ∇i φ∇j ∇2Z φ R ∂ ij 1 π N ∇i ∇j ∇Z φ + trπ |k| ∇Z φ + ∇j π Nij ∇i ∇Z φ +t ∂t 3 R 1 1 i Nij ∇i ∇j φ + trπ |k|2 φ − ∇ trπ∇i ∇Z φ + ∇Z π 6 3 . SS 1 i 2 ij N ∇i φ − ∇ trπ∇i φ + φ∇Z |k| + R00 + ∇j π 6 ' ( . ∂ ∂ 1 (10.11) + ∇t · ∇ t ∇2Z φ + t ∇2Z φ t−1 △ t − t−2 |∇t|2 . t ∂t ∂t

We also have ( ' . ∂ ∂ 2 △ ∇Z t ∇Z φ − |k| + R00 ∇Z t ∇Z φ ∂t ∂t ∂ 1 ∂ ∂ =π Nij ∇i ∇j t ∇Z φ + trπ |k| t ∇Z φ + ∇j π Nij ∇i t ∇Z φ ∂t 3 ∂t R ∂t S ∂ 1 i ij − ∇ trπ∇i t ∇Z φ + ∇Z 2tk (φ∇i ∇j (∇Z φ) + ∇i φ∇j (∇Z φ)) ∂t 6 R S ∂ ij 1 1 N ∇i ∇j φ + trπ |k| φ + ∇j π + ∇Z t π Nij ∇i φ − ∇i trπ∇i φ ∂t 3 6 R ' ( .S 1 ∂ ∂ 2 −1 −2 (10.12) + ∇Z ∇t · ∇ t ∇Z φ + t ∇Z φ t △ t − t |∇t| . t ∂t ∂t Finally, we have . $ % △ ∇3Z φ − |k|2 + R00 ∇3Z φ

1 Nij ∇i ∇2Z φ =π Nij ∇i ∇j ∇2Z φ + trπ |k| ∇2Z φ + ∇j π 3 R 1 i 1 2 − ∇ trπ∇i ∇Z φ + ∇Z π Nij ∇i ∇j ∇Z φ + trπ |k|2 ∇Z φ + ∇j π Nij ∇i ∇Z φ 6 3

470

10. THE LAPSE FUNCTION

R 1 i 1 Nij ∇i ∇j φ + trπ |k|2 φ + ∇j π − ∇ trπ∇i ∇Z φ + ∇Z π Nij ∇i φ 6 3 . SS 1 i 2 . − ∇ trπ∇i φ + φ∇Z |k| + R00 6 (10.13) To obtain a Poisson equation for SSSφ, we add these equations with the appropriate multiplicity. The resulting equation is . $ % △ S 3 φ − |k|2 + R00 S 3 φ

1 1 Nij ∇i S 2 φ − ∇i trπ∇i S 2 φ =π Nij ∇i ∇j S 2 φ + trπ |k|2 S 2 φ + ∇j π 3 6 R S 1 1 i 2 ij ij + DS π N ∇i ∇j ∇Z φ + trπ |k| ∇Z φ + ∇j π N ∇i ∇Z φ − ∇ trπ∇i ∇Z φ 3 6 R S 1 1 i 2 ij ij 2 N ∇i ∇j φ + trπ |k| φ + ∇j π N ∇i φ − ∇ trπ∇i φ + DS π 3 6 $ $ 2 % $ 2 %% ij + 2tk φ∇i ∇j S φ + ∇i φ∇j S φ R ' ' ' ( (( S ∂ ∂ ij φ∇i ∇j t φ + ∇i φ∇j t φ + DS 2tk ∂t ∂t R S $ % 3 + D2S 2tk ij (φ∇i ∇j (φ) + ∇i φ∇j (φ)) + ∇t · ∇ S 3 φ − S 2 Zφ t ' ( ' ( . $ % 1 ∂ 2 −1 −2 3 2 + 3 t △ t − t |∇t| · S φ − S Zφ + DS ∇t · ∇ t S φ t ∂t ' ( ' ( . ' ∂ ( 1 ∂ 2 −1 −2 2 ∇t · ∇ t φ + DS t △ t − t |∇t| · t S φ + DS ∂t t ∂t . ' ∂ ( + D2S t−1 △ t − t−2 |∇t|2 · t φ + φD3S |k|2 + φD3S R00 + l.o.t. ∂t Using 10.4, 10.5 with Propositions 1 through 3 and the Bootstrap assumptions, we find that (10.14)

∥∇SSSφ∥2 ≤ cε0 .

https://doi.org/10.1090/amsip/045/06

CHAPTER 11

Optical Function In Chapter 1, we described the construction of the optical function and the null decomposition of the second fundamental form of the St,u . In this chapter, we calculate the propagation equations for these terms. 11.1. Propagation Equations for the Components of Hessian of the Optical Function Recall from Chapter 1, that the second fundamental form of St,u relative to the null geodesic normal l is written as χαβ = Πµα Πνβ ∇µ lν

(11.1)

Relative to the normalized null frame this becomes χAB = eµA eνB Dµ lν .

(11.2)

to this last equation and calculate " dχAB 1 =− χAC χCB − α(W )AB − δ AB R44 , ds 2 which can be disaggregated into As in [5], apply

(11.3)

d ds

dtrχ 1 χ|2 − |α (F )|2 = − (trχ)2 − |N ds 2

and dN χAB = −trχN χAB − α(W )AB . ds These equations differ from the corresponding equations in [5] by the extra term |α (F )|2 . Next as in [5], define (11.4)

1 ζ A = eµA lν Dµ lν 2

(11.5) and calculate the identity (11.6)

ζ A = a−1 eA (a) + ϵA .

Use equation 11.5 to calculate (11.7)

dζ a = −χAB ζ B + χAB ζ B − β (W )A − ρ(F )α (F ) . ds 471

472

11. OPTICAL FUNCTION

With these expressions for the components of the Hessian of u, express the Gaussian curvature of the surface St,u as . 1 χ|2 − 2N η·χ N − δtrχ − 2ρ (W ) − ρ (F )2 + σ (F )2 . (11.8) 2K = (trχ)2 − |N 2 Following [5], define the exterior region of the space-time slab. / Definition 11. The exterior region E of the space-time slab t∈[0,t∗ ] Σt is the region defined by the inequality r0 r≥ 2 where r0 (t) = r (t, 0) . Following [5], make the following assumptions: A0 : t 1 + ≤ r0 ≤ 2 + 2t 2 A1 : r |a∗ − 1| ≤ A A2 : H H H 2 H H H ≤ A, − r r2 |N χ∗ | ≤ A H trχ H ∗

/ ≤A r2 a−1 ∗ |∇a| The notation * identifies quantities that are defined on the last slice. Christodoulou and Klainerman start with the last slice because that is where they begin the construction of the optical function. They also assume the following B1 : 1 inf φ ≥ , 2 and then by application of the maximum principle sup φ ≤ 1.

B2 :

sup r2 |δ| ≤ B,

sup r2 |ϵ| ≤ B,

E

E

sup r2 φ−1 |∇φ| / ≤ B, E

B3 : 7

sup r 2 |α (W )| ≤ B, E

5 2

sup r |α (F )| ≤ B, E

sup r |N η | ≤ B, E

sup r2 φ−1 |∇N φ| ≤ B. E

7

sup r 2 |β (W )| ≤ B,

sup r3 |ρ (W )| ≤ B,

sup r2 |σ (F )| ≤ B,

sup r2 |ρ (F )| ≤ B.

E

E

As in [5], define s∗ = r

E

E

11.1. PROPAGATION EQUATIONS FOR THE COMPONENTS OF HESSIAN

473

where s∗ = s|Σt∗ . Also define t1 (u) for Cu as the minimal value of t′ ≥ t0 (u) so that s 1