Black Holes: The Membrane Paradigm 0300037708, 9780300037708

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,/;TBhleackMeHomlbers:ane Paradigm Edited by Kip S. Thorne Richard H. Price Douglas A. Macdonald

Yale University Press New Haven and London

Copyright © 1986 by Yale University. All rights reserved. This book may not be reproduced. in whole or in part, in any form (beyond that copying permitted by Sections l0? and 108 of the U. S. Copyright Law and except by reviewers for the public press). without written permission from the publishers. The paper in this book meets the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources.

Printed in the U.S.A.

Library of Congress Camloguing-in-Publication Dam Black holes; the membrane paradigm. Includes index 1. Black holes (Astronomy) 2. Astrophysrcs. I. Theme, Kip S. ll. Price, Richard H.. I943lll. Macdonald, Douglas A. W. Title: Membrane paradigm. QBS43.BSS.BS9 I986 ISBN 0-300-03769—4 ISBN 0-300—03770-8 (pbk.)

l098765432l

52]

86-50486

Contents

Authors Editors’ Preface

xi

I. Introduction: The Membrane Paradigm by Richard H. Price and Kip S. Thorne ll. Nonrotating and Slowly Rotating Holes by Douglas A. Macdonald, Richard H. Price, Wai-Mo Suen. and Kip S. Thorne

l3

A. The 3+1 Split of Schwarzschild Spacetirne

I3

B. The 3+1 Split of the Laws of Physics outside a Nonrotating Hole

16

1. The electric and magnetic fields measured by FIDOs,

99:55”

Maxwell’s equations, and charge conservation 2. Electric and magnetic field lines, voltages and EMFs, redshifted energy, and Poynting flux FIDOs versus FFOs Boundary conditions on the horizon Effects of spatial curvature The Rindler approximation C. The Stretched Horizon of a Nonrotating or Slowly Rotating Hole 1. The frozen boundary layer 2. The stretched horizon and its membrane-like electrical properties

16 19 22 23 24 26

27 27 30

vi

Contents

3. Entropy, temperature, and ohmic dissipation in the stretched horizon

35

‘7 it; Slowly rotating black holes and electromagnetic torques on the stretched horizon

5. Ingoing time

38 40

. Model Problems for Nonrotating and Slowly Rotating Holes

993‘?!”

1. Vibrating magnetic field: The efficacy of stretching the horizon Point charge at rest outside a nonrotating hole Black hole as a resistor in an electric circuit Black hole as the rotor in an electric motor Point charges moving above the stretched horizon The motion of electric and magnetic field lines near the stretched horizon Ill. Rapidly Rotating Holes by Kip S. Thoma, Richard H. Price, Douglas A. Macdonald. Wai-Mo Suen, and Xiao-He Zhang A. The 3+1 Split of Kerr Spacetime l. The choice of fiducial observers and the 3+1 split of the metric 2. The electric and magnetic fields, gravitational accelerations, and inertial directions measured by HDOs 3. 3+1 physics far outside the horizon 4. The freezing of physics near the horizon . The 3+1 Split of the Laws of Physics outside a Rotating Hole 1. Electric charge and current and their conservation 2. Maxwell’s equations 3. The local laws of energy conservation and force balance 4. The global conservation laws for energy and angular momentum . The Stretched Horizon of a Rotating Hole " 1':- Stretching the horizon 2. The shape, surface gravity, and surface GM fields of the stretched horizon 3. Thermodynamics of the stretched horizon 4. Electrodynamics of the stretched horizon

41 49 52 55 57 64 67

67 67

72 74 80

81 81 83 85 9!

94 94 95 97 100

Contents

vii

D. Model Problems for Rotating Holes 1. Hole immersed in a static, vacuum external magnetic field 2. Hole endowed with a net electric charge 3. Magnetized, rotating hole as a battery for an external circuit 4. Eddy currents in the stretched horizon due to an oblique external magnetic field

102 102 107 110 [15

IV. Astrophysical Applications of Black-Hole

Electrodynamics by Douglas A. Macdonaid. Kip S. Thome. Richard H. Price. and Xiao-He Zhang

l2l

A. Observed Features of Quasars and Active Galactic Nuclei

122

B. Disk and Jet Alignment by Gravitomagnetic Forces

124

C. Qualitative Features of the Black-Hole Magnetosphere

132

D. Structure and Energetics of a Stationary, Axisymmetric Magnetosphere: The Blandford-Znajek Process

138

V. Gravitational Interaction of a Black Hole with Distant Bodies

146

by James B. Harrie, Kip S. Thorne. and Richard H. Price A. 3+1 Split of the Laws of Gravitational Interaction 147 1. 3+1 split of spacetime around a moving, nonrotating hole 147 2. Tidal gravity as viewed in a local inertial frame 152 3. Tidal gravity as viewed by fiducial observers in absolute space

4. Laws of motion and precession for a rotating black hole 5. Equations of motion and precession for a black hole or

157

160

neutron star in a complex external system

164

6. Equations of motion and precession for a system of bodies all small compared with their separations

168

B. Ngudel Problems for the Precession of Black Holes 7 L Black hole at the center of an ellipsoidal star cluster 2. Black hole surrounded by a massive accretion disk @Bladt hole in a binary system

[71 171 l72 175

viii

Contents

VI. Gravitational Interaction of a Black Hole with Nearby Matter

181

by Richard H. Price, Kip S. Theme, and Ian H. Redmount A. Conceptual Foundations

182

B. Gravitational Perturbations outside the Stretched Horizon l. The tidal fields of an unperturbed Kerr black hole

186 186 190

2. The perturbing tidal fields

C. Structure and Evolution of the Stretched Horizon

999%

1. Horizon-fixed spatial coordinates a, 9, 3 2. Behavior of the tidal fields 83,, and 31,, near the stretched horizon 3. Behavior of nongravitational stress, energy, and momentum near the stretched horizon Fiducions and their kinematics: The metric, expansion, and shear of the stretched horizon The tidal force equation and focusing equation The teleological boundary conditions Caustics in the stretched horizon The stretched horizon's surface stress 5"” and surface densities of mass and momentum 2” and fl" The conservation of energy and momentum in the stretched horizon 10. Gauge and slicing changes in the stretched horizon 11. Evolution of the hole’s entropy, angular momentum, and mass

196 197 200 204 205 211 213 216 218 223 226 229

12. Summary of the laws of evolution of the stretched horizon

VII. Model Problems for Gravitationally Perturbed Black Holes by Richard H. Price. [an H. Redmounz, Wai-Mo Suen. Kip S.

234

235

Thome, Douglas A. Macdonald, and Ronald J. Crowley A. Quasistationary Interactions of a Black Hole 1. Compact masses lowered quasistatically toward a nonrotating hole 2. Quasistationary deformations of a rotating hole

235

B. Rotating Perturbations l. Nonaxisymmetric tidal fields moving rigidly around a rotating hole 2. Massive particle moving along the equator of a rotating hole

245

235 243

245 246

Contents

C. Spindown of a Rotating Hole in an External Tidal Field D. Black Hole Processes: Energy Extraction, Superradiance, and Quasinormal Modes l. Gravitational extraction of the rotational energy of a black hole by orbiting bodies 2. Superradiant scattering of gravitational waves 3. Quasinonnal modes of a black hole E. Model Problems with Radially Moving Particles 1. Radial fall of a star into a nonrotating hole 2. Particle radially accelerated above the stretched horizon of a nonrotating hole VII]. The Thermal Atmosphere of a Black Hole by Kip S. Theme. Wojciech H. Zurek, and Richard H. Price

ix

250

261 261 264 265 268 268 275 280

A. Historical Introduction

280

B. The Physical Laws Governing Black Hole Atmospheres I. How a black hole retains its atmosphere 2. Perfectly thermal atmosphere 3. IN and UP modes 4. Superradiant and nonsuperradiant modes 5. Interaction of the atmosphere with the external universe 6, Accelerated observers in flat, empty spacetime — an aside 7. Renormalized angular momentum and redshifted energy in a black-hole atmosphere 8. The entropy of a black—hole atmosphere 9. The second law of thermodynamics

286 286 288 291 295 298

C. Model Problems for Black-Hole Atmospheres l. Evaporation of a black hole into perfect vacuum 2. Interaction of a black hole with an external thermal bath

314 314 320

300 304 308 313

3. The modes of a massless scalar field: Detailed analysis

324

4. Photon and graviton modes: Asymptotic analysis

330

5. Tidal spindown of a slowly rotating hole

334

6. Dumping and mining in the atmosphere of a nonrotating hole 336 References

341

Author Index

349

Subject Index

352

Authors

Kip S. Thome is the William R. Kenan, Jr., Professor and Professor of

Theoretical Physics at the California Institute of Technology, Pasadena. Richard H. Price is Professor of Physics at the University of Utah, Salt Lake City. Douglas A. Macdonald is a member of the technical staff at Rockwell International, Anaheim, California. Wai-Mo Suen is a postdoctoral fellow at the University of Florida, Gainesville. Xiao—He Zhang is a graduate research assistant in physics at the California Institute of Technology, Pasadena. James B. Hartle is Professor of Physics at the University of California at Santa Barbara. Ian H. Redmount is a postdoctoral fellow at Harvard University,

Cambridge, Massachusetts. Ronald J. Crowley is Professor of Physics at California State University at Fullerton.

Wojciech H. Zurek is a J. Robert Oppenheimer Fellow at the Los Alamos National Laboratory, Los Alamos, New Mexico.

Editors’ Preface

This book is a pedagogical introduction to the physics of black holes, with emphasis on a viewpoint that carries the name “membrane paradigm”. The membrane paradigm is a translation of the mathematical and physical formalism for black holes into a form that is specially adapted to astrophysical research.

Whereas the original black-hole for-

malism was accessible only to experts in general relativity, the membrane paradigm should be accessible to any person with a good grounding in nonrelatjvistic physics. Furthermore, it achieves this accessibility without making any approximations or losing any of the content of the original formalism. The key concept in the original black-hole formalism was a hole’s event horizon, viewed as a globally defined null surface in fourdimensional spacetime. By contrast, the membrane paradigm regards the event horizon as a two-dimensional membrane that resides in threedimensional space. Much of the power of the paradigm resides in the simple and familiar physical properties that it attributes to this membrane-like horizon. The horizon is regarded as made from a twodimensional viscous fluid that is electrically charged, electrically conducting and has finite entropy and temperature, but that cannot conduct heat; and the interaction of the horizon with the external universe is described in terms of familiar laws for the horizon’s fluid, e.g. the Navier—Stokes equation, Ohm's law, a tidal-force equation, and the first and second laws of thermodynamics. Because these horizon laws have familiar forms, they are powerful for understanding intuitively and computing quantitatively the behaviors of black holes in complex astrOphysi— cal environments. The membrane paradigm and this book are the result of a collaboration, mainly at Caltech, of the authors of the various chapters. xi

xil

Editors' Preface

Although authorships change from chapter to chapter, this book is by no means simply a collection of closely related papers. Rather, the chapters form a coherent, closely knit whole in which pedagogiml goals are paramount. The physical intuition so central to the membrane paradigm can be built up only by watching the paradigm “in action" in a variety of contexts. This book provides such in-action insight by posing, solving, and discussing a large number of model problems in which black holes interact with the external universe. From these model problems one can infer qualitatively and to order of magnitude how black holes should behave in complicated, realistic astrophysical situations. Roughly 60 percent of the material contained in this book is drawn — with extensive pedagogical changes — from research articles published in journals such as the Physical Review and Monthly Notices of the Royal Astronomical Society. The remaining 40 percent is new material based on the authors’ previously unpublished research.

The editors and the authors gratefully acknowledge helpful discus— sions with Roger Blandford, Thibaut Damour, Igor Novikov, Sterl Phin-

ney, and Roman Znajek. The research reported in this book, and the writing. editing, and phototypesetting of the book, were supported in part by the National Science Foundation [grants AST79-22012, AST82-

14126, and AST85-149ll to the California Institute of Technology] and by funds associated with Kip S. Thome’s William R. Kenan, Jr., Professorship at Caltech. In addition, the contributions of Richard H. Price

were supported in part by the National Science Foundation [grants PHY81-06909 and PHY35-03653 to the University of Utah]; those of Wai—Mo Suen in part by the National Science Foundation [grant

PHY85—00498 to the University of Florida]; those of James B. Hartle in part by the National Science Foundation [grant PHY85-06686 to the University of California at Santa Barbara]; those of [an H. Redmount in part by the National Science Foundation [grant PHY83-06693 to Harvard University] and by 3 Japan Society for the Promotion of Science Long-Tenn Research Fellowship, which supported Redmount's work for one year at Kyoto University; and those of Wojciech H. Zurek in part by a J. Robert Oppenheimer Fellowship in Theoretical Astrophysics at the Los Alamos National Laboratory.

Black Holes: The Membrane Paradigm

I

Introduction: The Membrane Paradigm

by Richard H. Price and Kip S. Thorne

ln theoretical physics a special role is played by the diagrams, pictures, mental images, and descriptive phrases that accompany our equa-

tions — for example, pictures of magnetic field lines threading through a conducting plasma and the corresponding phrase that the field lines are “frozen into the plasma". Such pictures and phrases are no substitute for

the mathematics that validates them, but they play a crucial role in facilitating intuitive leaps of insight and permitting communication both among physicists and between physicists and others. Indeed, the introduc-

tion of a new set of pictures and descriptive phrases can have a profound impact on the subsequent development of a field of research. ‘ Black-hole research is a good example of the importance of pictures and phrases. Before the mid-1960s the object we now call a “black hole" was referred to in the English literature as a “collapsed star" and in the Russian literature as a “frozen star". The corresponding mental picture, based on stellar collapse as viewed in Schwarzschild coordinates (Fig. la), was one of a collapsing star that contracts more and more rapidly as the

grip of gravity gets stronger and stronger, the contraction then slowing because of a growing gravitational redshift and ultimately freezing to a halt at an “infinite-redshift surface" (Schwarzschild radius), there to hover for all eternity. Of course, from the work of Oppenheimer and Snyder (1939) we were aware of an alternate viewpoint, that of an observer on the surface of the collapsing star who sees no freezing but instead experiences collapse to a singularity in a painfully short time. But because nothing inside the infinite-redshift surface can ever influence the external universe, 1

2

Introduction

star 3

Surface

,.---_—.'

1d! FROZEN-STAR VEWPCIN'I’

".

ibl BLACK HCLE VlEWPOlNV

(c) RACK—HOLE VIEWPDWT

Fig. l Diagrams showing qualitatively the gravitational collapse of a star to form a black hole {Figs (a) and (b)! and the collision and coalescence of three black holes [Fig (c)]. Figure (a) is a spacetime diagram of spherical stellar collapse using, at and outside the star’s surface, the Schwarzschild time and radial coordinates t and r. Two “hypersurfaces” of constant time, t - to and t - II, are shown. This diagram illustrates the frozen-star viewpoint. Figure (b) shows the same collapse us-

ing the time coordinate T of Eddington (1924) and Finkelstein (I958); ct". Sec. llCS, below. This diagram, which illustrates the black-hole viewpoint, reveals the collapse-induced formation of the hole's horizon and of a singularity at its center. On this Eddington-Finkelstein diagram is shown, dashed, the surface of constant Schwarzschild time I - 1.. From the past-plunging shape of that surface one can understand why Schwarzschild coordinates are unable to reveal the formation of the horizon and singularity. Figure (c) is an Eddington-Finkelstein type of spacetime diagram showing the highly nonspherical evolution of the horizon as two black holes collide and coalesce and as a third black hole forms and falls into the coalesced hole. Such processes could be studied only after the frozen-star viewpoint was replaced by the black-hole viewpoint.

that “comoving viewpoint” seemed irrelevant for astrophysics. Thus astrophysical theorizing in the early 1960s (e.g., Zel’dovich and Novikov 1964, 1965) was dominated by the "frozen-star viewpoint”. As long as this viewpoint prevailed, physicists failed to realize that black holes can be dynamical, evolving, energy-storing and energy-releasing objects. In the middle and late 19605 the frozen-star viewpoint began to give way to a new set of diagrams, pictures, mental images, and descriptive phrases. Penrose (1965) taught us to use spacetime diagrams based on

l. The Membrane Paradigm

3

Eddington-Finkelstein coordinates (Fig. lb) in which the star’s collapse never slows but instead continues unabated into a singularity, leaving

behind a “horizon” at the Schwarzschild radius; Wheeler (l968a) coined the phrase “black hole” to describe the curved, empty spacetime left behind with the horizon; and Hawking (1971a, 1972, 1973) and others

proved elegant theorems, beautifully illustrated by Penrose’s diagrams (Fig. 1c), about the evolution of the horizon in arbitrary, dynamical situations. The diagrams, pictures, mental images, and descriptive phrases of this new “black-hole viewpoint” meshed beautifully with new mathematical techniques (differential topology, “global analyses of spacetime”) introduced into black-hole research by Penrose (1965, 1966, I968). The mental images and diagrams of the black-hole viewpoint facilitated intuitive leaps of insight, which then were validated by the mathematics. The new

approach produced our present view of black holes as dynamical objects —- objects capable of colliding, coalescing, vibrating wildly, and emitting huge bursts of gravitational waves; objects capable when quiescent of storing 29 percent of their mass as rotational energy and of releasing it to power quasars and galactic nuclei. The frozen-star and black-hole viewpoints might be thought of, in the language of Kuhn ([970), as “paradigms”, and the transition between them might be considered a Kuhnian “scientific revolution".

But the

“frozen-star" and “black-hole” viewpoints are not different paradigms in the same sense, say, that quantum and classical physics are different ways

of viewing the physical world. Both the frozen-star paradigm and the black-hole paradigm agree, after all, that Einstein’s general relativity is the complete mathematical theory of the gravitational field and that it contains within itself the correct answer to any black-hole problem that one may wish to study. The two paradigms may suggest different mathematical approaches to solving that problem (e.g., different spacetime coordinate systems underlying the calculation), but if a calculation is done correctly and exactly within the context of either viewpoint the answer

must be the same. This is not to say that the difference between these two (or any two such) paradigms is trivial. For one thing, the mental images inherent in a paradigm are of enormous importance in a priori intuition about the results of computations and thereby in suggesting what computations should be carried out. For another, the mathematical techniques associ-

ated with the paradigms difl‘er enormously in their suitability for different problems. The frozen-star paradigm makes use of Schwarzschild coordinates, which are well suited to the study of physics outside the horizon of a static black hole. However, Schwarzschild coordinates can be dangerously misleading in highly dynamical situations where the horizon rs crucial, such as the time evolution of the magnetic field of a spherical star that is collapsing to form a horizon and black hole. By contrast, the

4

Introduction

mathematical techniques of the black-hole viewpoint deal well with highly dynamical, horizon-evolving situations but are cumbersome for studies of physics outside the horizon of a static hole. Perhaps the most important reason for making a distinction between paradigms for black holes is that many if not most of the calculations in general relativistic astrophysics (and elsewhere!) are not exact applications of the mathematical theory. The luxury of an exact calculation is usually limited to proofs of general theorems. 1n the more-or-less real world of relativistic astrophysics the mathematical complications of the full general relativistic theory of gravity almost always preclude exact calculation. Even for highly idealized model problems, approximations are either extremely useful or, more usually, indispensable. The power of a paradigm is that it Suggests what approximations are appropriate, or what features of the exact problem can be ignored in an analysis without losing the essence of the problem being studied. For an analogy, think of something as simple as the classical mechanics of a point mass under the joint action of a conservative force (e.g., an idea] spring) and a small but complicated dissipative force. The force viewpoint (Newton's second law) is an exact description but will

probably not lead to an easy estimate of the importance of dissipation. An analysis of the problem in terms of mechanical energy is mathematically equivalent but will likely make much clearer how to deal with the effects of dissipation. In the last two decades, most of the work in black-hole astrophysics

has used the frozen-star viewpoint. Much of this research has been done, in fact, without the inclusion of any real relativistic gravitational efi‘ects. In many of the papers on both disk and spherical accretion into blackhole candidates such as Cygnus X-l, the source of gravitation is treated as a Newtonian (l) monopole and an ad hoc spherical surface is invoked

somewhere near where general relativity would place the event horizon (the “surface" of the black hole); calculations (e.g., of luminous emission)

are simply stopped at this somewhat arbitrarily chosen inner boundary, which is the only non-Newtonian feature of black holes present in the calculation. For examples see Shvartzman (1971), Pringle and Rees (1972), Shakura and Sunyaev (1973), Shapiro (1973a,b), Paczynski and Wiita

([980). For many problems, especially in view of uncertainties in the astrophysics, this approximation is adequate at least for order-ofmagnitude estimates. One knows that in these problems the important effect of the black hole is the “no return” property of its event horizon. The real nature of spacetime near the event horizon does not enter into the problem in a crucial way. Incorporating the black-hole viewpoint would unnecessarily confuse and complicate analyses already mired in questions of radiative transfer, magnetohydrodynamic turbulence, etc.

l. The Membrane Paradigm

5

For some problems in black-hole astrophysics, however, the frozenstar picture is inadequate. The evolving magnetic dipole moment of a collapsing star, for example, would seem in this viewpoint to reach an

asymptotic finite value as the star “freezes" at time 1 — 00. A somewhat more involved frozen-star analysis (Ginzburg and Ozernoy I964) suggested that the dipole moment fell ofl‘ as l/t. The correct answer, an out-

burst of radiation followed by a [/15 fallofl", required a computation that addressed fully the dynamical nature of spacetime near the event horizon (de la Cruz, Chase, and Israel 1970; Price 1972; Thorne 1973). Many problems of current interest in black-hole astrophysics cannot be posed, at least not without great danger, in the frozen-star viewpoint, because that viewpoint has difiiculty producing unambiguous boundary conditions (e.g., on electromagnetic fields) at the event horizon. These “dangerous" problems typically involve astrophysical phenomena in

which most of the interesting processes are occurring fairly far from the event horizon but in which the horizon is linked to those processes in a significant way (e.g., by electromagnetic fields connecting the black hole and an astrophysical plasma). [The most important discovery about black holes since the mid-19705, the Blandford-Znajek (1977) process for electromagnetic extraction of a hole’s rotational energy, is one such problem] In such problems it is inadequate simply to replace the black hole, in pictures and in calculations, by a “surface of no return". At the same time

these problems do not require the mathematical details of the full blackhole viewpoint, and such details would distract attention from the more

relevant physics. The need to study with ease such problems has led, in the past eight years, to the development of a new viewpoint (or paradigm) for black-hole physics — the “membrane paradigm“. This book is a pedagogical intro-

duction to that paradigm. The membrane paradigm is mathematically equivalent to the standard, full, general relativistic theory of black holes, so far as all physics outside the horizon is concerned. It adopts a frozen—star-like view of physics outside the horizon, but it contains within itself a simple prescription

for ignoring “irrelevant” near-horizon details in astrophysical problems. More specifically, in this viewpoint particles and fields very near the horizon possess a highly complex, frozen, “boundary-layer" structure, which

is essentially a relic history of the black hole’s past. This complex boun— dary layer has no influence on the present or future evolution of particles and fields above the boundary layer, in a way the membrane viewpoint

“stretches" the horizon to cover up the boundary layer and then imposes Simple and elegant, membrane-like boundary conditions on the stretched horizon. This sweeping away of irrelevancies entails small (and in practice negligible) errors, but it results in a formalism and viewpoint remarkably POWCrful for astrophysical studies. Moreover, it acquires a special

6

Introduction

elegance when one recognizes (Zurek and Thome 1985) that the entropy of a black hole is the logarithm of the total number of quantummechanically distinct configurations that could exist in the covered-up boundary layer. The membrane viewpoint was motivated by several startling results proved in the 19705 using the black-hole viewpoint: (i) Hawking‘s (1974, l975, 1976) proof that a stationary black hole radiates as though it were a black body with a finite surface temperature, and his realization (following a suggestion by Bekenstein l972a,b, 1973) that if the hole is regarded as having an entropy proponional to its surface area, then the laws of blackhole mechanics can be married to the laws of thermodynamics; (ii) the discovery by Hawking and Hartle (1972) (see also Hartle 1973, 1974) that external gravitational fields can tidally deform the horizon of a black hole and that the motion of the deformation produces entropy just as if the horizon were viscous; (iii) the attribution of an effective charge density to the horizon of a black hole by Hanni and Ruflini (1973) and by Hajicek (1974), and the discovery by Hanni and Ruflini (1973) that when a black hole is placed in a static external electric field, that field polarizes the horizon’s effective charge distribution; (iv) the discovery by Znajek (1976, 1978) that when electric current is run through a black hole (with, e.g.,

positive charges flowing in at the poles and negative charges at the equator) the horizon behaves as though it had an electric surface resistivity of order 30 ohms. Motivated by these results, Damour (1978, 1979, 1982) rewrote the

equations governing the evolution of a general black-hole horizon in a form in which one could identify explicitly terms that “looked like" electric conductivity, shear and bulk viscosity, surface pressure, surface momentum, temperature, entropy, etc.; and Znajek (1978, 1981, 1984) independently developed a nearly equivalent but initially less complete version of the equations. Damour’s beautiful and elegant formalism made

it obvious that the membrane viewpoint had sufficient richness and formal justification to become an important tool for astrophysical research. Unfortunately, Damour’s formalism was, in a sense, incomplete. It viewed the horizon as a three-dimensional null surface residing in fourdimensional spacetime without connecting the surface to the physics of the external universe in which the horizon lives. For Damour’s formalism to be made into a tool for astrophysics, it had to be married to an equally appealing description of the external universe. Such a description was provided by the “3+1" formulation of general relativity. The 3+1 formulation chooses a preferred family of 3—dimensional, spacelike hypersurfaces in spacetime (surfaces of “constant time”) and treats them as though they were a single 3-dimensional space that evolves as time passes (“decomposition of 4-dimensional spacetime into 3dimensional space plus l-dimensional time"). The general relativistic

I. The Membrane Paradigm

7

physics of black holes, plasmas, and accretion disks takes place in this 3— dimensional space; and the relativistic laws of physics that govern them, written in 3-dimensional language, resemble the nonrelativistic laws to which astrophysicists are accustomed. Thus, the 3+1 formulation is well suited to can'ying physicists” nonrelativistic intuition about plasmas, hydrodynamics, and stellar dynamics into the arena of black holes and

general relativity. Although the 3+1 formulation of general relativity is not treated in standard relativity textbooks, it has been developed in great detail by many researchers and has played an important role in relativity research in recent decades —— for example, in numerical solutions of the Einstein

field equations (e.g., Smarr 1979; Piran 1982), in the quantization of general relativity (e.g., Wheeler 1968b; Hawking 1984), in analyses of laboratory experiments to test general relativity (e.g., Braginsky, Caves, and Thome 1977), and most recently in the full formulation of the membrane

paradigm for black holes. A historical overview of the 3+1 formalism is presented in Sec. 2.1 of Thome and Macdonald (1982). The maniage of the 3+1 formalism to Damour’s membrane-horizon formalism was carried out in two steps: first for electromagnetic aspects of black holes, by Thome and Macdonald (1982); then for gravitational and mechanical aspects, by Price and Thome (1986). As a first application of the resulting full membrane paradigm, Macdonald and Thome (1982) studied the magnetosphere of a black hole surrounded by a magnetized accretion disk, giving emphasis to the Blandford-Znajek (1977)

process for magnetic extraction of the hole’s rotational energy. That study gave precise form to a set of pictures and mental images proposed originally as analogies by Blandford (1979) and Znajek (1976, 1978) [see also Rufiini and Wilson (1975), Damour (1975)] -— pictures of magnetic fields

threading the hole and being torqued by the hole's rotation, and pictures of magnetic field lines acting as the wires of a gigantic electrical circuit that links a battery in the hole’s horizon (power source) to a plasmaaccelerating region (electrical load) high in the hole’s magnetosphere. The membrane viewpoint further led to the realization that, no matter how chaotic the magnetic field may be when anchored in the accretion disk, When deposited on the hole by accretion it will be “cleaned” by the horizon in a time on the order of the hole’s classical light-crossing time, a cleaning that is important for the successful operation of the BlandfordZnajek process. And the membrane viewpoint revealed a remarkable

feature of the cleaned field: it distributes itself over the horizon in such a way as to minimize the horizon’s ohmic dissipation. _ The full membrane viewpoint, as used in this research, is mathematically equivalent to the standard view of black holes everywhere outside thehorizon — that is. everywhere of relevance for astrophysics. But the membrane viewpoint loses its validity (indeed, even ceases to exist) inside

8

Introduction

the horizon. For example, an observer who falls through the horizon dis-

covers that the horizon is not really endowed with electric charge and current; it merely looks that way from outside. Despite this ephemeral nature of the membrane, the researcher who takes it seriously and believes in it (if only temporarily, while working on a specific research project) may be rewarded with powerful insights.

The realm of applicability of the membrane paradigm is problems in which the black hole at issue is changing slowly (compared with the travel time of light across it). The mathematics of the membrane viewpoint is

capable of, but generally inconvenient for, more highly dynamical situations. In a static spacetime we have a special time coordinate, one which exhibits the “time"-independence of the fields. We therefore have a natural distinction between space and time, a distinction that is useful for visualization and calculation, although it destroys the “unity of space and time” usually thought of as a sacred cow of relativity. ln problems with slowly changing black holes this natural distinction between space and time remains, along with its advantage: 3-dimensional pictures of membrane-surfaced holes with more or less familiar physical properties. It is in such problems, as we will see, that the highly complicated boundary-layer structure can be replaced by a membrane-like stretched horizon, and that physics near the stretched horizon can be described with ease and insight that are missing from the frozen-star picture. The original mearch articles on the membrane paradigm are written largely in the language of a professional relativist. They are not easy for nonrelativists to penetrate. If the membrane paradigm is to realize its full power, however, it must be made accessible to astrophysicists; and it may be of some interest and amusement as well to researchers in other branches of physics. It is primarily for such people that this book is intended. The professional relativist may find this book an attractive introduction to the subject but will likely desire a deeper understanding of the underpinnings of the membrane paradigm. For that one must turn to the original literature: Znajek (1978, 1981, 1984) and Damour (1978,

1979) (horizon as membrane); Thorne and Macdonald (1982) and Price and Thorne (I986) (marriage of horizon formalism with 3+1 formalism); Thorne and Macdonald (1982) and Macdonald (1983, 1984) (application

to black-hole magnetospheres); Macdonald and Suen (1985) and Suen, Price, and Redmount (1986) (application to a variety of model problems in black-hole physics); Thorne and Hartle (1985) (the torquing of a black hole by an external gravitational field); Zurek and Thorne (1985) and Frolov and Thorne (1986) (black-hole atmosphere, black-hole evaporation, black-hole entropy, and other quantum-field-theory issues). This book will build up the membrane paradigm step by step, using model problems to illustrate each major concept as it is introduced. The

I. The Membrane Paradigm

9

model problems will be drawn not only from the recent literature and the

unpublished work of the authors, but also from the literature of the 19605 and I970s that preceded the membrane paradigm — in which case the

problems will be translated into membrane language. Throughout, attention will be restricted to the slowly evolving (Schwarzschild, Kerr, or nearly Schwarzschild or Kerr) holes for which the membrane paradigm is most powerful; an analysis of fully dynamical holes from the membrane

viewpoint is given by Price and Thome (1986). Attention will also be restricted to the astrophysically realistic situation where the total electric

charge on a black hole is so small that it can be treated as a linear perturbation of an uncharged hole; the generalization to strongly charged (Reissner-Nordstrom or Kerr-Newman) black holes is straightforward, but

has not yet been canied out. We begin, in Chap. II, by introducing the 3+1 formalism for Schwanschild spacetime and in particular the 3+1 split of the laws of electrodynamics. We then discuss the difliculty inherent in the 3+l viewpoint for the description of processes very near the horizon: the “freezing” of motion at the horizon. This motivates the introduction of the concept of “stretching" the horizon, replacing the null horizon with a timelike physical membrane endowed with electrical, mechanical, and thermodynamic properties. The result is the membrane paradigm. The use and the usefulness of this membrane paradigm are then illustrated, in the last part of Chap. 1], with a set of electromagnetic model problems. A model problem is presented for a black hole threaded by vibrating magnetic field lines that are anchored in a perfectly conducting surrounding sphere; the complex and (for most purposes) irrelevant structure of the field near the true horizon is hidden beneath the stretched horizon. The surface charge of the stretched horizon is illustrated by a model problem in which external charged bodies induce charge separation on the horizon. The stretched horizon’s surface current and resistivity are illustrated by a problem in which the hole is inserted into an electric circuit.

The ohmic heating of the horizon in this circuit is used to illustrate black-hole entropy and its law of increase. The Lorentz force exerted on the stretched horizon's surface current by a horizon-threading magnetic field is illustrated by a model problem in which a (nearly) Schwarzschild hole acts as the rotor of an electric motor. The motion of electric and magnetic fields in the vicinity of a horizon and the currents and charges induced on the stretched horizon by that motion are illustrated by model problems in which electric charges are dragged parallel to the horizon. In Chap. 111 we turn attention to rotating (Kerr) black holes. The 3+1 decomposition is presented for such holes, and the gravitomagnetic

field (the gravitational analog of a magnetic field) resulting from that decomposition is studied in some detail. The nature of the gravitomag-

netrc (GM) field is illustrated. in the weak-gravity limit far from the

10

Introduction

horizon, by its magnetic-like role in the geodesic equation and by its role in causing gyroscopes to precess (“Lense-Thirring effect”; “dragging of inertial frames”). The 3+l split of electrodynamics and other laws of physics and the stretching of the horizon are presented for a rotating hole. Model problems are then used to clarify the manner in which the GM field produces electric fields from magnetic, and magnetic fields from electric. As an important example, it is shown that a rotating black hole threaded by a magnetic field develops an electrical voltage drop between its poles and its equator — in effect, the hole’s GM field interacts with its magnetic field to produce a “surface battery" in the horizon. The physics of this surface battery is elucidated by a model problem involving charge separation on the horizon of a hole that has no currents entering or leaving it and by a model problem in which the surface battery drives currents in wires that link the hole to an external circuit. Finally, the interaction of magnetic fields and the eddy currents that they induce in the horizon is illustrated by a model problem with a hole in an external magnetic field oblique to the hole’s rotation axis. The insights developed from these model problems are used in Chap. W as foundations for an overview of black-hole magnetospheres. Among the phenomena described are the driving of an accretion disk into the hole‘s equatorial plane by the joint action of the hole’s GM field and the disk’s viscosity (Bardeen-Petterson effect); the deposition of magnetic fields on the hole by the accretion disk; the cleaning of the deposited fields by the horizon; the transformation of the cleaned fields into a plasmaendowed, force-free magnetosphere; and the extraction of rotational energy from the hole by that magnetosphere (Blandford-Znajek process), including circuit-theory and torque-balance analyses of the energy transfer. In the first four chapters, gravitational effects of the hole on external fields and matter are the focus of attention. The next three chapters deal with the inverse problem: the gravitational effects of external fields and matter (and black holes!) on the hole of interest. We begin in Chap. V with a study of the effects on a hole due to distant sources of gravity. Key roles in this study are played by the “asymptotic rest frame” of the hole and by a 3+] split of the tidal gravitational field of external bodies. These concepts are used to motivate and express 3+1 equations for the forces

and torques exerted on a hole by distant bodies; it is then shown how to convert those forces and torques into concrete equations of motion and precession for a black hole interacting with a complex external system, e.g., with other compact bodies in a several-body or many-body system. In the remainder of Chap. V, model problems are used to illustrate four types of precession that a black hole can undergo: a torqued precession due to coupling of the hole’s quadrupole moment to the tidal gravitational fields of external bodies, a gravitomagnetic precession due to coupling of the hole’s GM field to the GM fields of external bodies, a procession due

I. The Membrane Paradigm

1]

to gravitational spin-orbit coupling as the hole orbits a companion, and a precession due to the space curvature of the companion it orbits. The spin-orbit and space—curvature processions together make up “geodetic precession”, which is shown to dominate over other forms of precession in

binary motions. However, for a hole interacting with a surrounding accretion disk the gravitomagnetic (Lense-Thirring) precession is dominant; and for a hole interacting with a surrounding elliptical swarm of stars (nuclear bulge of a galaxy) with negligible net angular momentum, only the torqued precession is nonzero. In Chap. VI we retum attention to the strong-field region of a black hole, this time for the study of gravitational perturbations that can induce significant changes in the horizon. We develop a membrane paradigm for

these gravitational perturbations similar to (but more complicated than) the electromagnetic membrane paradigm of Chaps II and III. The starting point is the3+1 split of perturbing tidal fields, the gravitational analogs of E and B fields The influence of these tidal fields on the hole is first described in terms of their effects on the kinematics of the stretched horizon. Using the viewpoint of Damour (1979, 1982), we then interpret the resulting kinematics by assigning mechanical surface properties (surface viscosity, surface stresses, and surface densities of mass and momen-

tum) to the stretched horizon. These concepts and their effects are illustrated in Chap. VII by model problems in which external matter distorts

the horizon and causes it to evolve. The first model problem involves the quasistatic lowering of mass points toward the horizon of a nonrotating hole, and leads to a general discussion of quasistationary horizon distortions. Rotating perturbations are illustrated with a model problem in

which mass points orbit just above the equator of a Kerr horizon. A discussion is presented of the Spindown of a rotating hole in a tidal field produced by distant sources, a problem that first showed the viscous nature of the horizon (Hawking and Hartle 1972, Hartle I973, I974) and helped to motivate the membrane viewpoint.

Black-hole processes in the strong-

field near-horizon region are then discussed; these include the gravitational extraction of rotational energy, superradiant scattering of gravitational waves, and quasinormal pulsations of the hole. Finally, horizon dynamics is further illustrated by model problems in which point particles move

radially near the horizon, both in free fall and with an imposed acceleration.

The book concludes with Chap. VIII, a membrane-formalism description of the thermal atmosphere that quantum field theory predicts a black hole should possess The physical laws governing the thermal atmosphere and its evolution are presented, with emphasis on the manner in which vacuum polarization renormalizes the atmospheric observations

of static observers to give the energy and momentum that couple to gravity and produce the evolution of the stretched horizon. It is shown that

12

Introduction

the entropy of the hole is equal to the renonnalized statistical mechanical entropy of its atmosphere, which in turn is equal to Boltzmann’s constant times the logarithm of the number of ways that the black hole could have been made. This interpretation of the hole’s entropy leads to a proof that the second law of thermodynamics for a system that includes a black hole is merely a special case of the standard, nonrelativistic second law. Several model problems illustrate the physics of black-hole thermal atmoSpheres. These include the evaporation of a hole into a perfect vacuum, the interaction of a hole's atmosphere with an external thermal bath, the

injection of gravitons into the atmosphere in the course of the tidally induced spindown of a hole and the resulting increase in the hole's statistical mechanical entropy, and the dumping of quanta into and mining of quanta out of a black-hole atmosphere. Throughout this book we use the notation and sign conventions of Misner, Thorne, and Wheeler (1973; cited henceforth as MTW), including setting Newton’s gravitation constant G and the speed of light 0 to unity (“geometrized units”; cf. Box 1.8 of MTW). Occasionally we restore G and c to an equation just to remind the reader of where they go. We also follow MTW (Sec. 8.4, Box 9.2) in denoting coordinate basis vectors by partial derivatives, eg, 2; E 6/de

To give certain equations a more

elegant appearance, we use MTW‘s dyadic notation for second-rank tensors, but we write in index notation all equations for which dyadic notation would be ambiguous.

II

Nonrotating and Slowly Rotating Holes

by Douglas A. Macdonald, Richard H. Price, Wai-Mo Suen, and Kip S. Thorne

A. The 3+1 Split of Schwarzschild Spacetime Consider a black hole formed by the gravitational collapse of a star

with negligible angular momentum J > 1 and that the change in position occurs very smoothly. Under these conditions we should expect radiative elfects to be small; we shall in fact ignore them. For t < 0 the E field is just that for_.a point charge stationary at x - 0, y =- 0, z - z_9. The lines of that E field are pictured in Fig. 53. [The fact that the E field is normal to the horizon follows front.the boundary conditions (2.24) in the Rindler geometry together with B - 0 for a

stationary charge distribution] We might expect that for t > A: the E field is just that due to a static point charge at x - 0, y - 0, z — zo+Az, as depicted in Fig. 5b. This will in fact be the case except near the horizon. In Fig. 6 the situation is pictured on a Minkowski diagram. The region with stipples is the region of spacetime in which the original static field exists. The crosshatched region is the future of the event (I - At, 2 - zo+Az, x - y - 0) at which time the charge has reached its new stationary position; in this region the E field should be static, corresponding to the new height. In the white region the field should correspond to the transition. Notice that any slice of constant I > At reaches from the crosshatched region of the new static field, back through the white transition region, and into the stippled region of the old static field. In fact it is easy to show (Sec. lIlF of Macdonald and Suen 1985) that for any t > 0 the portion of the slice which lies in the originally static region will be that for which 2 < Zoe-3'“. Thus, sufliciently near the horizon (i.e., at small

29 "CI. The Frozen Boundary Layer f—

Tl

,t=2At //t=At I

x/

I

//

.95 , 1’

x’

//’/ //

z,“ ‘-H Z=io

Fig. 6. Minkowski spacetime diagram for a charged particle that is at rest at Rindler height 2 - 20 for all r < 0 (T < 0), then moves smoothly to Rindler height 2 - zo+Az during Rindler time Ar, and then remains at rest for all I > At. [:1 the stippled region the electric field is that of a static charge at rest at z - 1“. In the cross hatched region it is the field of a static charge at rest at z - zo+A:. This diagram also illustrates subsection 2’s discussion of the true horizon 1/. the stretched horizon 2's, and the two events 3’ on I and .9” on IS that correspond

to each other.

enough 2) the field will always “remember“ its original value. [Of course it will also be true that the field remembers its original value at large enough 2 (cf. Fig. 6); but that aspect of the finite propagation speed of the change to distant regions is familiar and expected] The qualitative evolution of the field. then, will_look something like the progression shown in Fig 7. In this figure the E field on the t - 0, t - At, and t — 2A: slices (shown in Fig. 6) is sketched. On each slice the field structure below the dashed slice is the original static field; and on the t - 2At slice, that above the dotted line is the final static field. This sluggishness of near-horizon fields, i.e., the long delay in field

30

Nonrotating and Slowly Rotating Holes

all (a)

(b)

Fig. 7. The electric field lines, as seen in Rindler space, due to the charged particle whose motion is shown in Fig 6. Diagram (a) shows the initial static field at the moment, t - 0, when the particle first begins to move upward. Diagram (b) shows the field at the moment, t - At, when the particle reaches its new, final height; below the dashed line the field is unchanged. Diagram (c) shows the field at t - 2A1, i.e., a time Al alter the particle stops moving. Above the dotted line the field is the same as that of a static charge at the new height (Fig. 5b); below the

dashed line it is the same as for a static charge at the original height.

evolution near the horizon, will be further elucidated in Sec. D] by a numerical study of vibrating magnetic fields near the horizon. The sluggishness is an inescapable consequence of the t-slicing; it is due to the fact that all slices of constant t are anchored at the same fixed location (T,Z) =- (0,0) in Fig. 6 and thus retain near there, i.e., near r = 2M, “relic fields” from the distant past. This is the real sense in which things are frozen in the frozen-star viewpoint; but the freezing is confined to a thin

(small-z) layer, a layer that cannot have any significant causal influence on the external universe and thus is of no consequence for the “real" dynamics, such as for the interaction of the fields with distant plasmas.

To eliminate the astrophysically irrelevant complications of this frozen boundary layer without eliminating the real influence of the black hole, we adopt an approach [proposed by Thome and Macdonald (1982, Sec. 5.3) and developed by Macdonald and Suen (1985)] in which boundary conditions are imposed not right at the horizon, but instead at a timelike surface just outside (in a well-defined sense) the frozen boundary layer. The theory of this “stretched horizon” is sketched in the next section. 2. The stretched horizon and its membrane-like electrical properties

To get rid of the irrelevant details of the frozen boundary layer, we introduce a surrogate boundary outside the bulk of the boundary layer, i.e., at a - a” where a” is some small positive value of the lapse function. The replacement of the horizon by this “stretched horizon” involves identifying with each event ? on the true horizon a corresponding event 9’ on the stretched horizon, and requiring that the values of physical variables be (very nearly) the same at the corresponding events .9" and E as

rties IICZ. The Stretched Horizon and Its Electrical Prope measured by (physically reasonable) freely falling observers.

31

(ln the

Rindler approximation the identified events, 3’ on the true horizon 2? and .9” on the stretched horizon 1’5, are linked by an ingoing null ray with T+Z - constant; see Fig. 6.) In Sec. VIC we shall develop the theory of the stretching of the horizon with some care. For now it will suffice to state that the horizon gets stretched to a nonzero but small cm and that boundary conditions for all

external physics are imposed there. It would be aesthetically displeasing to use boundary conditions that

depend on the location chosen for the stretched horizon, since that location a - a” is somewhat arbitrary.

For this reason, we choose not to

express the electromagnetic b_oundary_. conditions in terms of the divergently large tangential fields E. and B l, but instead we utilize finite and urn-independent “horizon fields” defined by

EH 2 (an).-..., EH 5 (amp-.. .

(2.38)

These horizon fields live in (are tangent to) the two-dimensional stretched horizon. Together with the normal electric and magnetic fields,

5,, E in, 3,, s Err

(2.39)

(where F1‘ is the unit radial vector 5}), these fields completely characterize the electromagnetic field at the stretched horizon. In terms of these fields the electromagnetic boundary conditions at the stretched horizon take on

a form which can be read directly ofl‘ the asymptotic field relations (2.24), (2.25):

E”, B", E”, and EH all finite , in - fiXEH

,

EH - —fiXEH

(2.4021) .

(2.40b)

As discussed following Eq. (2.25), Eqs. (2.4%) can be thought of as “ingoing-wave” boundary conditions. 01' course, these boundary conditions include slight errors because of the stretching of the horizon — fractional errors typically of order «“3. By choosing a” sufliciently small, one can make the errors as small as one wishes. As first shown by Znajek (1978) and by Damour (1978), it is very useful to regard these boundary conditions as arising from physical properties of a fictitious membrane residing at the location of the stretched horizon. More specifically, it is useful to pretend that the stretched hortzon lS endowed with a surface density of electric charge a” [first introduced by Hanni and Rufiini (1973) and by Hajicek (1973, 1974)], so defined that it terminates the normal components of all electric fields that pierce the stretched horizon; see diagram (3) of Fig. 8: =

cha e

r unit area

0’” _ on srtgretgteled horizon

_ = (En/4"")stretched horizon '

(2-41)

32

Nonrotating and Slowly Rotating Holes

(a) Gauss‘s Law

(c) Ohm's Low

(d) Charge Conservation

Fig. 8. Electromagnetic properties of the stretched horizon. Diagram (a) depicts Gauss‘s law [Eq. (2.4!)!, by which the stretched horizon‘s surface charge density a” is defined; (b) depicts Ampere‘s law [Eq. (2.43)], by which its surface current

density ,7" is defined; (c) depicts Ohm‘s law for the stretched horizon [Eq. (2.44)]; and (d) depicts the law of charge conservation at the stretched horizon [Eq. (2.46)]. (Figure adapted from Thorne, 1986.)

Similarly, it is convenient to pretend that there exists a surface current on the stretched horizon [first introduced by Znajek (1978) and by Damour (1978)], so defined that it terminates any tangential magnetic field in accordance with Ampere’s law; see diagram (b) of Fig. 8. But in making this idea precise, we must be careful about the type of time used in defining the surface current. The proper time of a FIDO must not be used because it ticks at a pathologically slow rate at the stretched horizon, and its ticking rate even depends on just where we choose the stretched horizon to be. The only reasonable choice is a time that is independent of the stretched horizon’s location: universal time t. Thus, we define

-‘ _ char e crossin a unitl n h r unit )7” = univergsa] time tgin the stgetgcthetltl)ehorizon ‘

(2‘42)

Since this surface current is smaller by a factor a” - dr/dt than that which would be measured by a FIDO at the stretched horizon, Ampere’s law must relate it to 4:- times a tangential magnetic field that is smaller by

rties IIC2. The Stretched Horizon and Its Electrical Prope

33

f

0,, than that measured by the FIDO, i.e., to 41' times the “horizon figd” B!” — aHB]. Thus, Ampere’s law, which is actually the definition of ,7“, says

_

3,, 5 419mm.

(2.43)

See diagram (b) of Fig. 8. An intuitively powerful consequence of this definition of the horizon's surface current is _its ability to reinterpret the “ingoing-wave”

boundary condition By -EH> Er, “locally measured energy" E has not been conserved.

IIIB4. Global Laws for Energy and Angular Momentum

93

’—

To obtain a conserved energy we obviously must correct for the kinetic energy of fall; or, more generally, we must correct for all work done by or against gravity as lumps of energy fall or rise in the hole’s gravitational field. It is conventional (e.g., Sec. 25.3 of MTW) to make the corrections in Such a manner that the “conserved energy” is the amount of locally measured energy that a lump would have if it were moved, under its own power and with perfect efficiency, from its actual location to a point far from the hole (“radial infinity”). The energy thus corrected is sometimes called “energy-at-infinity” and denoted Ea, sometimes “gravitationally redshifted energy" or just “redshifted energy”. The energy-at-infinity for an uncharged test particle moving near a rotating hole, when expressed in 4-dimensional language, is minus the

scalar product of the particle’s 4-momentum p“ with the generator of time translations 13/61, E- - —p“(6/8t)u - -po; cf. Sec. 25.2, 25.3, and 33.5 of MTW. Translated into 3+1 language, Ea. is

E... - aE—E-p - 5.5+sz ,

(3.68)

where E is the particle’s locallymeasured energy, E-m(l— —"°‘)""’; p is its locally measured momentumuo =mV(l— 'Q—v)""; L. is its 2-component of

angular momentum;_and w ispthe angular velocity of the FIDO at the particle5 location, -)3 - to 8/64: - as. In (3.68) the first term a5 is familia_r_ from the case of a nonrotating hole (Sec. 1132). The second term

-§p — wL- accounts for the orbital motion (if/dz - —E of local FlDOs relative to static observers at infinity. One can verify from the equation of motion (3.20) for an uncharged test particle that E... - aE—B-p‘ is con-

served along the particle’s trajectory. This conservation law is actually a consequence of the time-independence of the hole‘s gravitational fields gij, m, 0:; cf. Secs. 25.2, 25.3, and 33.5 of MTW. For matter and fields with a (locally measured) energy density _¢., energy flux (- momentum density) S, and momentum flux (= stress) T, the density and flux of energy-at-infinity are ‘E- '= ae-§‘§= (zed-ML, ; S_E_ = a§-E‘f= a§+w§L ,

(3.69a) i.e., Si}; - aSi+wTi¢-aSi+wa'-¢;

(3.69b)

and these obey global and local conservation laws identical to those for the z-component of angular momentum and for electric charge:

d f eE_dV- f a(-§E_+eE_V)dA 7(1)

67(1)

dc _

_ _.

a d: .. %-5-v £_-—v(.155)

(3.70) (3.71)

94

Rapidly Rotating Holes

The local law (3.71) can be derived from the laws of local energy balance

and force balance [Eqs. (3.57’) and (3.59)]; the global law (3.70) follows from the local law by Gauss’s theorem. Both laws are consequences of the time-independence of the hole's gravitational fields. For a charged particle which interacts with the hole’s gravity and with surrounding electric and magnetic fields, one can (as in flat-spacetime electrodynamics) take two different views of the energy and angular momentum associated with the particle’s electromagnetic interactions with the surrounding fields. The first view, which is valid in general and which

we shall adopt, includes the interaction energy and angular momentum in the fields’ eg_ and q" [Eqs. (3.55), (3.62) and (3.65) with cross terms between particle E, B and surrounding E, B included]; and it attributes

to the particle itself the same E... and L. as for an uncharged particle [Eqs. (3.68) and (3_.63)]. The second view, valid only when the hole5 surrounding E and B fields are time-independent and axisymmetric, incorporates the electromagnetic interaction energy and angular momentum into the particle’s own 15.. and L:: E... - aE + wL: - qu ,

(3.72a)

Here q is the particle’s charge; and A0 and A6, are the components of the

electromagnetic 4-potential along the spacetime vectors 8/81 and E}. In effect, the first viewpoint computes E a and L: from the particle’s kinematic 4-momentum, and the second viewpoint computes them from

its generalized (canonical) 4-momentum; cf. Eqs. (33. 3|) of MTW. For further details, including the 3+1 split of the electromagnetic 4-potential and its relation to E and B, see Secs. 5.2 and 33 of Thorne and Macdonald (1982); also Sec. 33.5 of MTW. C. The Stretched Horizon of a Rotating Hole 1. Stretching the horizon As for a nonrotating hole, so also for a rotating one, we shall stretch the horizon so as to avoid having to keep track of the layer-upon-layer of past history compacted near the true horizon.

Rotation introduces no

significant new features into the stretching of the horizon; the guidelines for the amount of stretching are the same as in the nonrotating case (last two paragraphs of Sec. 11D 1). It is desirable (for technical reasons discussed by Price and Thorne 1986) to choose the stretched horizon in Kerr spacetime to be a surface of constant lapse function, a = (1" > 2M) Thus, only the horizon contributes: !

—o

—o

—o

EMF- f—a,,xB,-d1.

(3.106)

.9”

Here the integral is along the stretched horizon from point .9” to point 5 (cf. Fig. 25). There are three different ways to view the physical origin of this EMF Two of the ways are the viewpoints underlying our two derivations

[Eqs. (3.104) and (3 105)]: The curve is star-fixed but moves relative to the FlDOs and thus also relative to the B that they measure, and that motion produces the EMF; or the curve is attached to the FIDOs and thus orbits the hole differentially, thereby linking a time-changing magnetic flux, which induces the EMF. The third viewpoint is to ignore the issue of how the curve moves (its motion doesn’t really affect the EMF anyway!) and to think of the EMF asproduced by an interaction between the hole’s gravitomagnetic potential ,3 and the externally imposed magnetic field B. We prefer this viewpoint and shall adOpt it below. Note that this view is in nice accord with the difl'erential-equation version of Faraday’s law, Eq. (3.52), which shows the explicit coupling of the magnetic and gravitomagnetic fields that produces a curl of 015. We can carry this view one step further and think of the magnetic and gravitomagnetic fields as jointly producing a “surface battery” in the horizon of the hole — a battery whose EMF between two points .9” and J is given by Eq. (3.106). However, we must be a bit careful with this battery viewpoint. Only when the closed curve 8’ rises out of the horizon parallel to the magnetic field and extends to large distances does the EMF get produced solely in the horizon. For other types of closed curves the magnetic-gravitomagnetic coupling formula (3.104) or (3.105) remains valid, but the pure horizon formula (3.106) fails because pieces of the closed curve outside the horizon give nonzero contributions to the EMF. Let us now specialize to an explicit magnetic configuration: a uniform external magnetic field parallel to the hole’s rotation axis (Fig. 26). [The precise definition of “uniform field" will be given in Eq. (3.108) below. Until then all we shall really need is for the magnetic field to be axially symmetric and time-independent and to thread the horizon] We wish to determine the physical consequences of the “battery-like” EMF in the hole’s horizon. We shall do so with the help of Ampere‘s law (3.51)

[11D]. Hole Immersed in an External Magnetic Field

105

f”

i

VV

\mt

\

”——

(a)

Fig. 26. A rotating black hole immersed in a uniform, time-independent magnetic field parallel to its rotation axis. Ampere’s law, applied to curve 9" in diagram (a) and combined with the horizons Ampere and Ohm laws, reveals that there can be

no tangential electric field at the stretched horizon The charge separation and the nature lof the E field lines (shown dashed) are sketched in diagram (b) lei. Eq. (3. I09)

applied to the circle 8’1n the stretched horizon of Fig. 263. By axisymmetry, and taking the viewpoint that 8"IS star-fixed so v — —§/a, ZTD’HBHQ a .£03111-

3’ -

f ExE-dni f E-dZ+41f(a7—peE)-dz . 9*

d’ .1:

(3.107)

y

Ihe first term on the right vanishes because the gravitomagnetic potential )9 is parallel to the curve 8"; the second term vanishes because any elec; tric field E generated by interaction of the time-independent B and )3

fields must itself be time-independent; and the third term vanishes because there are no charges pe or_currt_e.nts fpresent in the space outside the hole. Thus, the integral of a3 - 3,, around the curve 8" finishes, which means that the tit-component (“toroidal component”) of 3;, must

vanish at the stretched horizon. But the vanishing of the toroidal B,“ implies [by Ampere’s law for the stretched horizon, Eq. (3.95)] that the 0component (“poloidal component”) of the horizon current I" must van-

ish, which in turn implies (by Ohm’s law) that the poloidal component of

106

Rapidly Rotating Holes

the horizon electric field EH; - «HE; must vanish.

The vanishing of the horizon’s poloidal EH might seem peculiar in light of the poloidal, “battery-like" EMF (3.106) in the stretched horizon. Something must be counteracting that EMF to make the tangential electric field vanish. The only thing that can do so is “charge separation" in the stretched horizon (Fig. 26b). When the hole was first immersed in the magnetic field, the “battery-like” EMF must have pushed positive charges in the horizon toward the equator and negative charges toward the pole, until the charge separation created sufficient electric field of its own to counteract the EMF and leave the horizon free of tangential electric field and current. From the nature of the resulting charge distribution (Fig. 26b), we expect an electric field with quadrupole-like angular distribution in the absolute space around the hole — a field that will account for the net EMF (3.105), (3.106) around closed curves 8" of the type shown in Fig. 25. These qualitative conclusions derived by simple physical arguments can be verified from Wald's (1974) exact analytic solution for a Kerr black hole immersed in a uniform magnetic field; cf. Thorne and Macdonald (I982). The uniform magnetic field is given analytically by

_

Ba

B=

«El—X e- “JAEfl cal ZZsinB

330?:-

06

’

Br

at r>>M;

(3.108)

and the electric field is

E=_.

—B02a2 p

{ 3(a2 )+ Msiznze (22-4a2Mr)— _2 _. Br 2

era: . _Mrsp ;nv (22-4 « as 2

a

'

2

L.2

—M

ofl lm sinfi—ZMCOSZM 1 “0529”; r" a +30 It?cosflsinatl+cos2 0)e,

at

a M ,

(3-109)

where X .=.. (Sin20/p2)(22-4a2Mr) and p, A, E, and a are defined by Eqs.

lllDl. Hole Immersed in an External Magnetic Field

107r

(3.5) and (3.6). This E field is pictured in Fig. 26b. The forms of the field lines are somewhat surprising: they have the expected quadrupolar angular dependence 3coszo — 1, but instead of coming up from the hole’5

equatorial region and swinging around and back down into the polar region with a l/r4 radial dependence as do the fields of a localized quadru-

pole in flat space, the fields here are directed almost precisely radially with a 1/1'2 dependence, and their small angular component [at 0(l/r4)]is one that brings fields lines in from infinity in the polar regions, swings them around, and takes them out to infinity in the equatorial region. This peculiar behavior1s easily understood _The E field of Fig. 26b has_two sources, one for V- E and the other for VX(aE) The source of

V--E is familiar from flat—space intuition: it is_the horizon5 surface charge density and it affects the behavior of the E field for small radii. The source of VX(aE) is unfamiliar. It is the gravitomagnetic--magnetic coupling 5’53 [Eqs. (3 52), (3. 54)]. This source dies out, at large radii, as 1/r3 (the magnetic field B is constant, the gravitomagnetic potential {3 dies out as l/rz, and the Lie derivative 3’ brings1n a factor of Ur), and it gen-

erates directly the large-r l/r2 electric field of Fig. 26b. The charge density on the stretched horizon of Wald‘s solution can be read off the small-a form of the radial electric field (3.109) (0” ='

n/41r - E;/41r):

Boar"(rH —M) rHsin‘0-2Mcoszli( l +c0520) a” =

(3 l 10)

411’

(rHZHIZeosZtJ)2

1t exhibits explicitly the charge separation shown in Fig. 26b (with «H > 0 in equatorial regions and a” < 0 in polar regions). The integral of a” over the horizon vanishes, so the total stretched-horizon charge is zero as

it should be. The total magnetic flux through the horizon north of the equator may be calculated from Eq. (3.108) to be 11: -= 41rBoM(rH-M). As the rotation of the hole increases, the magnetic flux is expelled from it, as noted by Bicak and Dvorak (1976, 1980). For a maximally rotating hole

(a - M), no magnetic flux threads the horizon. We shall discuss this peculiar behavior at the end of subsection 4 below. 2. Hole endowed with a net electric charge

Consider next a hole which is threaded by a time-independent electric field rat1_1_er than a magnetic field -— i.e., Fig. 25 with the B lines replaced by E lines. Since (for the present) we are dealing with scurceless

regions of space, pi. -= fs 0, there is a complete duality between electricity and magnetism: Ampere5 law when applied to the closed curve of Fig 25 must yield the same conclusion [Eqs. (3.103)—(3.106)] as Faraday’5 law did, but with B—°—E and E-'B.

Thus, there must be a nonvanishing

108

Rapidly Rotating Holes

“magnetic EMF” around the closed curve:

f aE'dT- ffixE-di‘. 3'

(3.111)

3’

This version of Ampere’s law holds not just for the specific curve of Fig. 25, but for any closed curve 8’ in a vacuum, time-independent electric field. When the curve has the specific form of Fig. 25 (with Bfi-E), or equally well of Fig. 27a, the only nonzero contribution to the integral is from the horizon, so 6

!

fast-ark fExE-dT-= 41rfrTX(—0HEH)'dT ‘B’

.9”

(3.112)

.9“

[cf. Eq. (3.106)]. Just as we can think heuristically of the EMF in Eq. (3.106) as due to a “battery" in the stretched horizon, so we can think heuristically of the magnetic integral in Eq. (3.112) as due to the‘ current loops” which a distant observer sees created by the rotational motion dx/dt - —,6H - ope/3:1: of the horizon’5 charge a”. And just as the “battery” viewpoint is dangerous because it applies on_y to curves of the form

in Fig. 25 and not to curves that cross over the B lines, so the “currentloop“ viewpoint is dangerous because it applies only to the same special type of curves (e.g., Fig. 27a). An alternative viewpoint which is always secure and which we preferIS that of gravitomagneticinduction: interaction

of the GM potential ,6 or field H = cr—lVfl with any time-

independent electric field produces the magnetic-field integral of Eq. (3.1 1 1), just as the GM interaction with a magnetic field creates the EMF of Eq. (3.105). Turn now to a specific configuration of electric fields threading the stretched horizon: that obtained by placing a nonzero net electric charge on the horizon (Fig. 27a). The above Ampere analysis tells us there must

be a nonzero magnetic integral [Eq. (1112)] around the closed curve 6‘. However, in this sourceless, time-independent configuration there cannot be any poloidal magnetic field Bf - “Bé at the stretched horizon. [This one can see by an application of Faraday‘s law analogous to that used in the electromagnetically dual situation of Fig. 26a; see the discussion following Eq. (1107).] Consequently, the magnetic integral must be associated with a dipolar magnetic field that emerges orthogonally from the stretched horizon and loops around in the manner of Fig. 27b. In accord with Eq. (3.112), we can think of this dipole field as produced by the current loops which a distant observer sees created by rotational motion of the horizon‘s charge. The hole’s rotation-induced dipole magnetic field couples to its GM field and influences the structure of the electric field. More specifically, that coupling induces a “battery-like” EMF in the stretched horizon [Eqs (3.105), (3,106)], which drives charge separation just as it did in the

IIIDZ. Hole Endowed with a Net Electric Charge

109

'___——

(o)

(b)

Fig. 27. An electrically charged. rotating black hole. Ampere’s law applied to the curve 9 in (a) reveals the necessity of a dipolar B field as shown in (b).

uniform-E case of Fig. 26. However, in this case that charge separation does not produce a net negative charge near the poles; rather, it merely

pushes some of the excess positive charge away from the poles toward the equator, thereby causing a” to be larger at the equator than at the poles. These semiquantitative conclusions can be verified from the exact analytic solution for the electromagnetic field of a charged, rotating hole. In the case we are considering, where the total (redshifted) energy in the electromagnetic field is negligible compared with the mass of the hole, i.e.,

where Q - (charge on hole) > M . Note that the GM-induced magnetic field is truly dipolar at large radii, both in its angular dependence and its radial dependence. This contrasts with the GM-induced electric field in the case of a hole immersed in a

uniform magnetic field [Fig 26 and Eq. (1109)]. Here the inductive

110

Rapidly Rotating Holes

scurce (35E with Ea: acQ/r2 and )3 cc Ma/r- ) dies out so fast at large radii

(a: l/rs) that it cannot influencethe induced dipolar (a: 1/r3) B field; there the inductive soarce (2’53 with B - constant and B a: Ma/rz) remained so strong at large radii that it was the dominant influence on the induced B field at large radii. The exact charge distribution on the stretched horizon of Fig. 27b can be read off the small-a form of the radial electric field [Eq. (11321)]: __Q (r),2-a 10520) —— .

3.11

6-H

41 (rH2+a2c0520)2

(

3e)

This shows explicitly the concentration of the charge toward the hole‘s equator; for example, if the hole is “maximally rotating” (a - M) the charge distribution (3.113c) is, 6"-

—9— ——5i“20 41er (2:211:20)2 '

3 114 )

( ‘

In this case the charge is so strongly concentrated toward the equator that the charge density at the poles has been reduced to zero. 3. Magnetized, rotating hole as a battery for an external circuit

Let us return to the model problem of subsection 1 and Fig. 26a: a rotating black hole immersed in a uniform external magnetic field. in subsection l the hole was surrounded by vacuum; there was no possibility for currents to flow into and out of the horizon, so the battery-like EMF produced charge separation (Fig. 26b) rather than flowing currents. Now

we shall hook the hole up to an external circuit of the type studied in Sec. llD4 (Fig. 16), but with the external battery replaced by an external resistive load with total resistance RL (Fig. 28). The black hole now plays the role of the battery; its magnetic-gravitomagnetic interaction produces an EMF around the circuit, i.e., around a closed curve extending from the hole‘s equator out along the perfectly conducting disk to one of the distant wires, up the wire and through the resistive load to the top of the conical conductor, down the conical conductor to its tip at 0 = 00 > 2M. The disk’s str_ucture is governed by the joint action of gravity E, gravitomagnetism H, spatial geometry gm, pressure forces, and viscous

forces. The dominant driving force is the radial part of E, which produces a gravitational acceleration relative to the star-fixed coordinates of magnitude

d2)?

_fl (4.3a)

dt‘2

r2 '

E. radial

By contrast, the dominant, spherical part of ng produces effects smaller by a factor ~M/r, i.e., effects of “post-Newtonian order"

d"? dt2

g,..spherical

_—-2 M v __ 1"

M,— [fil—l -, r r

(4.31:)

see Sec. 39.6 of MTW. If all other gravitational effects were spherically symmetric like these, the disk would lie in a single plane. The warping in Fig. 33 is produced by nonspherical gravity. Nonspherical gravitational effects in E - —§lna are due to the hole’s quadrupole moment Jj-k —-— M02 [Eqs. (3.25), (3.26)] and produce nonSpher-ical accelerations of order 2

n a“)?

air2

~ Ma2 _- la a 2 M , ,

(4.3c)

Enonspherical

which is 0(a2/r2) smaller than the spherical E. (Recall that a/M ~ 1 for a rapidly rotating hole.) Similarly, in g), nonspherical effects are smaller

by 0(a2M/r3) than spherical efl‘ects [Eqs. (3.5), (3.6c)] 2

3

a2

fl sit2

E M ghnonspherical

I

i"2

M i M

r

r2

7

M

(4.3d)

r

By contrast, the dominant gravitomagnetic (E, 5") effects are nonspherical and of magnitude

fl (2’:2

~ I7.nonsphenca7

VXHln—TV r

[V B.

Disk and Jet Alignment by Gravitomagnetic Forces

127

l."2

~ Milfl r3r

(4.3e)

[Eqs (3.32), (3.36)], which is larger by (M/a)(r/M)"r "2 than the nonspheri-

cal efl'ects of g‘ and larger by (M/a)(r/M)3"'2 than the nonSpherical effects of gjk. Thus, one can compute the disk’s warped structure to high accuracy by ignoring nonsphericities in gm and E, and by treating the gravitomagnetic efiects (4.3e) as weak, nonspherical perturbations of an otherwise Newtonian disk. In such an analysis one might worry about omitting the spherical, “post-Newtonian” [O(M/r)l corrections to 5,7 [Eq. (4.3a)] and

contributions of gjk [Eq. (4.3b)], since they are larger by (M/a)(r/M)”2 than the gravitomagnetic efi'ects we are keeping [Eq. (4.3e)]. However, these spherical corrections presumably have negligible influence on the disk’s warp, so we ignore them. The key equation governing the disks structure is Navier-Stokes. It can be derived, in the weak-gravity limit of our 3+1 formalism, by inserting into the local law of force balance (3 59') the Newtonian mass»energy density 6, mementum flux S, and stress TI" _.

6 - Pm 9

S :- va :

T’k - pmvrvk+pgf*—2na1k .

(4.4a)

Here n is the coeflicient of shear viscosity (bulk viscosity is negligible), and «’7‘ is the rate of shear of the gas. The velocity that appears here is that

measured by the FIDOs: vj - a"(dx-’/dl+fll'). The factor a is postNeyvtonian and spherical and thus can be dropped; but 6’, though small (lfll ~ Ma/rz), m_1_ist be kept because it is a nonspherical, gravitomagnetic term, so we keep [3 and write

vi - ui+51 ;

uJ‘ 2 dedt .

(4.4b)

0n the other hand, we need not keep 5 everywhere. In astrophysical disks

the viscous stresses —2nafl‘ are generally very small compared with pmw‘v“, so in the shear we can ignore 6; Le, we can compute the shear from the

standard Newtonian formula ajk == —;—(uj”‘+u“”)—-;-g-‘ku".l -

(4.4C)

By inserting the e, S“, and TJ" of (4.4a) into the law of force balance (3.59') and deleting effects of spherical gravity that are post-Newtonian in magnitude, we obtain 6p

.

J'

.

[Jfipmv"M v’+pm §L+vkv1.k] at

- ngj—pIJHZWjL)Ik+HJkPi-nvk+flk(9m V’).k -

128

Astrophysical Applications of Black-Hole Electrodynamics

By then combining this with the law of energy balance (3.57'), which reads to the desired accuracy _+(pmvk_)lkijmj- 0,

anti with H - 63, H - Vxfi, and (4.41:) for v, we obtain to first order in .6

PM

L411" 6! uml - Pm(g' +f’k1ukH')—P'j+(2m’k)1k-

(4-5)

This15 the standard, nonrelativistic Navier-Stokes equation augmented by the gravitational and gravitomagnetic forces pm(g+u)__ r, >> r") that gravity is Weak, space is nearly flat, and consequently the gravitational fields of the hole and external bodies superpose very nearly linearly. The asymptotic rest frame is illustrated in Fig. 41 for a black hole

(left side of diagram) in a binary system with another black hole of comparable mass. The spatial region of the left hole’s asymptotic rest frame is shown stippled; it extends from the “inner radius” r, >> 7'” to the “outer radius” r0 ) + owl/.92)

(5.54)

(e.g., MTW Chap. 39, especially Exercise 39.15). Here (I: is the system’s Newtonian gravitational potential.

VAS.

Equations of Motion and Precession for a Hole

165

__—

Although equations (5.54) lose their validity near neutron stars and near the horizons of holes [and are replaced by expressions such as (5.12)], nevertheless here, as in the exact discussion of a moving Schwaizschild hole [Eqs. (5.12)], we can regard each hole or neutron star

as moving along some specific trajectory through the global, isotropic, Cartesian coordinates x’:

x" — xflt) - (trajectory of hole or neutron star K) .

(5.55)

And here as in Eq. (5.12), when one is near enough to body K that its gravity dominates, iii—36K! < r0 r, >> MK, the lapse and metric will be given by (5.54) with

MK 4’ = '-

12-)?“

small effects 0

+ external bodie3 at r, > R3, the GM precession is also negligi-

ble. The dominant, geodetic precession is similar to the precession of the electron spin in a classical model of the hydrogen atom. Just as motion of the electron through the electric field_ E of the nucleus induces in the elec_t_ron5 rest frame a magnetic field Bmdmd =- —(d£/dt)XE and a torque EXBinduccd and a resulting precession of the electron Spin (“atomic spinorbit coupling”), similarly [cf. Eqs. (5 6l)—(5.63)] motion of a gyrosc0pe

throughg“i" produces1n the gyroscope’s rest frame a gravitomagnetic field

Hmdumd =- —(dE/dt)>> My, so also here, the hole’s torqued precession and its GM precession due to the body’s spin angular momentum are negligible compared with geodetic precession. But now the GM precession contains a contribution from the motion of the body _

4M V _. .. 4M . _ urn—ii, mews“2” we at r-EH. lx-Egl b

50M - 4/21?" — 2;:an , a, - «Mg/bah.

(5.98a)

[Eqs. (5.56b) and (5.64)], which is comparable in size or larger than the geodetic precession

5m = fisc + 550 - g mgr".

(5.981))

By adding these two processions and using the Keplerian relations (5.97), we obtain the following result for the total precession rate of the hole’s spin

a + _90M - ~2—Z— 3M +u ..K, n ‘Mffii’ M M — [reggcsgd . (5.99) on.” So long as the disparity in masses of the hole and companion body is not too great, the orbital angular momentum will greatly exceed the spin angular momenta of the hole and body, which means that gravitorgagnetic forces will not make the orbit precess significantly and 9g”, + 90M will remain essentially constant in time. The precession formula (5.99) was first derived, for weakly gravitating bodies with u 0; dotted curve when h < 0]. Similarly, waves with “x" polarimtion

[2?“ - 8”,; - —'.w'2 in] deform the ring as shown on the right [dashed curve when It,‘ > 0; dotted curve when h). < 0]. For further detail see, e.g., Thome (1982) or See. 35.6 of MTW.

For outgoing gravitational waves far from the hole 3&ut _ _ $33“ _ 4.9g“ - —'/zjz-$m ,

(6.233)

980m _ ‘33? _ _ 3’33! _ 44/252"! ,

(6.23b)

where

A $‘“(t-r :0,¢) hgut - ___,

hgut -

A Sum—r ;0a¢)

r

(6.23c)

r

For these outgoing waves in as computed from the approximate formula (6.20), vanishes. However, a more careful calculation from the exact definition (6.19) of \P and from a more precise expression than (6.23) for the waves gives a nonzero result, smaller than (6.22) in magnitude by a factor of order l/(wr)“, where w is the characteristic frequency of the waves: 4 wont ~

WX

l

..

W(h+

.. and

hx)]

-

(6'24)

Put another way, the “l/r”, “radiative" part of an outgoing gravitational wave has no influence on 1/; but the “l/r”’, “inductive" part of the wave

194

Gravitational Interaction of a Black Hole with Nearby Matter

does influence ‘1! — and from its influence on ‘1! one can reconstruct the full wave. The mechanics of the reconstruction are rather complex, in part because the angular (0,45) dependence of the l/rs part of the wave contained in \II is rather different from the angular dependence of the l/r,

radiative part of the wave. For details of the reconstruction procedure, which relies on the resolution of \II into normal modes with separated variables, see Teukolsky and Press (1974) or chapter 9 of Chandrasekhar

(1983). [Had our 1: been chosen from the other class of Teukolsky functions, it would have incorporated the l/r, radiative part of outgoing waves

but only a l/r5, inductive part of ingoing waves]

Near the horizon T: (p/JKxaf, + a.) and ya = (l/x/i)e""(§‘5+ié’¢), where k E -tan“(ac059/rH) ,

(6.25)

so our definition (6. l9) of ‘1! reduces to

1 1' NEH?“ — a?“ — 297,9) w = Eez +1199; — Q“ + 2 3%)] as r — r”.

(6.26)

Although this expression is proportional to the square of the lapse function a, which goes to zero at the horizon, I! does not go to zero there. As we shall see in Sec C2, jus__t as FlDOs near the horizon see ingoing—wavelike, tangential E a_nd

B fields with divergently large magnitudes,

E“ a: l/a, B" = —eAXE” a: l/a, so those FlDOs also see ingoing-wave--like, tangential E” and 9 fields with doubly divergent magnitudes, 3’55- —8’“ - 0, and modes of negative helicity are associated with 0., < 0. In the quantum-field-theory formalism of Chap. VIII (below), by contrast, a... is always > 0 and negative helicity states have angular dependences -ZS,,,,(aa,.,0) different from lem(aa..,0). For discussion see the first few pages of Sec. VIIIC4. Note that a, is the angular frequency of a mode as

measured physically by a FIDO “at infinity”, r >> r"; it is analogous to Ea, the energy that a particle would have if it were to climb to “infinity”

under its own power [“energy-at-infinity”; Secs. llBZ(c) and lIIB4]. In Chap. VIII, when we quantize gravitational waves near a black hole, each graViton associated with a mode with frequency a- will have energy-atinfinity E, - from. A very powerful but complicated computational formalism has been built around the Teukolsky function; see Chandrasekhar (1983) for details and see Sasaki and Nakamura (l982a,b) for a major additional contribution. We shall call this the “Teukolsky-Chandrasekhar formalism.” In some of the model problems of Chap. Vll we shall present results which were derived using the Teukolsky-Chandrasekhar formalism. Fortunately, one need not master that formalism in order to understand the model problems and in order to gain from them deep physical insight about the

196

Gravitational Interaction of a Black Hole with Nearby Matte.-

behaviors of black holes. And since the primary purpose of this book is to teach physical insight rather than to teach computational techniques, we shall not delve further into the Teukolsky—Chandrasekhar formalism. In essence, then, this book is complementary to that of Chandrasekhar

(1983). For physical insight one should study this book. For full compu. tational power one should study Chandrasekhar. There are three other widely used computational formalisms by

which, in special situations, one can determine the tidal fields 8’}; and 915,; due to perturbing energy densities 6, energy fluxes S, and stresses 7". We shall describe these very briefly. First. for arbitrary perturbations of a nonrotating hole there is a perturbation-theory formalism due to Regge and Wheeler (1957). The Sasaki-Nakamura (l982a,b) variant of the Teukolsky-Chandrasekhar formalism reduces to it in the special case of a nonrotating hole. We do not know of any good pedagogical introduction to the Regge—Wheeler formalism, but important contributions to it will be found in Edelstein and Vishveshwara (1970), Zerilli (1970), Moncrief (1974), Chandrasekhar and Detweiler (1975), and chapter 4 of Chandrasekhar (1983). Second. for stationary, axially symmetric perturbations of a nonratating hole one can use a formalism due to Weyl (1917, 1919). This formalism, by contrast with the others, is capable of treating strong, nonlinear perturbations as well as weak ones. For recent studies of black-hole perturbations using the Weyl formalism see Geroch and Hartle (1982); Suen, Price, and Redmount (1986); and Suen (1985).

Third, for arbitrary perturbations of a nonrotating hole with perturbing sources that are very near the horizon, one can adopt the Rindler approximation; i.e., one can approximate the near-horizon spacetime of the unperturbed hole as Minkowski spacetime. This was discussed for nonrotating holes in Sec. 1136 and will be discussed below (See. VICI) for rotating holes. In this approximation one can use an analog of the Lignard-Wiechert potential of electrodynamics to solve the_ equation

gh‘” - —l61rT“' for the “trace-reversed metric perturbation" h“. From 12’" one can then compute the Riemann tensor Rah, by standard tech-

niques; and one can then compute 8’]; and .917; from the Riemann tensor. For the relevant Lienard-Wiechert potential and examples of computations that use it see Suen, Price, and Redmount (1986). C. Structure and Evolution of the Stretched Horizon

We turn attention now to the stretched horizon of a dynamically perturbed black hole. As an aid to our study of the stretched horizon we

introduce new spatial coordinates in subsection 1 — coordinates which unlike the star-fixed r, 0, «t are attached to the rotating horizon and thus rotate relative to the distant stars. Then in subsections 2 and 3 we study

VIC- Structure and Evolution of the Stretched Horizon

197

the radial (i.e., a--dependent) behaviors, near the stretched horizon, of the

dynamically perturbed tidal fields 3",}, Q”, and 1P — (Teukolsky func. tion), and of the energy density, momentum density, and stress 1, SJ, and

Tr] —— finding that, as for the electric and magnetic fields, some of these quantities are divergently large By renormalizing_the divergent quantities

we obtain physical horizon fields, analogous to EH and B”, which will play key roles as driving forces for the evolution of the stretched horizon

ln subsections 4—7 we introduce the concept of fiducial particles, or “fiducions” which live on the stretched horizon and act as surrogates for

the generators of the true horizon; and we introduce and study the evolution of the relative expansion and shear of these fiducions. In subsection 8 we show that the stretched horizons fiducions can be regarded as form-

ing a 2-dimensional viscous fluid endowed with surface densities of mass E, and momentum II” and with a standard viscous-fluid stress tensor

SH. In subsection 9 we formulate the laws of conservation of energy and momentum for the stretched horizon —- laws which say that any energy and momentum carried into the stretched horizon by infalling matter or nongravitational fields get intercepted, held, and conserved by the fiducion fluid. In subsection 10 we discuss the (very small) amount of gauge dependence and slicing dependence contained in our horizon equations. In subsection ll we formulate the evolution laws for the total mass and angular momentum of the hole. Finally, in subsection 12 we summarize our membrane formalism for the stretched horizon.

l. Horizon-fixed spatial coordinates a, 3, 3 In the next twelve subsections, as we study the dynamics and evolution of the stretched horizon, it would be a nuisance (though not an

impossibility) to use as our coordinates for absolute space the star-fixed Boyer-Lindquist coordinates r, 0, and 4». These coordinates have the distracting property, near the stretched horizon, of rotating relative to abso— lute space and its FlDOs with angular velocity -9H; so we shall remove

that distraction by introducing a new angular coordinate 3 - 4:— IQHdt, which is very nearly at rest in absolute space. The Boyer-Lindquist radial coordinate r is also distracting because in terms of it various near-horizon

formulae look rather complicated. We shall remove this distraction by using as our radial coordinate not r but the lapse function a. By switching from r to a we introduce into the near-horizon spatial metric an unnecessarily distracting gag term. To get rid of this third distraction we

introduce a slightly modified 9 coordinate, called 3. The precise definitions of these new coordinates — accurate to within the usual fractional errors of order a2, which as usual we ignore — are PH20 2 a-B-fia,

4311 PH

(6.313)

198

Gravitational Interaction of a Black Hole with Nearby Matter

a — ¢ — fflfldt, =- L

l~azsin20/2Mr 1/2 = _—__H

MrH/(rH—M)

>3 A

a

(6.311;) ”2

_

(r m)

I/2

,

(6.31c)

where 0H2 - rHZ+a2coszo is the horizon value of the function p2 and where

9H - a/ZMrH

(6.32)

is the Kerr-metric angular velocity [cf. Eq. (3.42)] computed from the mass M and rotation parameter a of the “background” hole, about which we perturb. (Warning; the coordinates here denoted F and d are denoted

9' and 95' by PT, and the Boyer-Lindquist 0,¢ of this book are denoted 0", 4" by PT; see their appendix C. Since '0 difi‘ers from 0 only by a quantity of 0(a2), the distinction between 5 and 9 is unimportant for almost all purposes. In this chapter we shall maintain the distinction; in chapters 7 and 8 we shall drop it, replacing 3 by 0 on and near the stretched horizon.) ln terms of the coordinates a, 5, 3 the unperturbed metric of abso-

lute space [Eq. (3.6)] takes the following beautifully simple form (aside from fractional corrections of order 012): 2

dsz - % + mid? + wHZdaz.

(6.33)

H

Here g”, p”, and w” are the unperturbed Kerr surface gravity and metric functions p and 23, evaluated at r - r” (a - 0) with the slowly evolving values of M and a: TH‘M gH

-

2M,”

2 1

p]!

2

-= r

H

2

+ a c0523

(ZMT'H)2 7

2:! 2 - —— sinzfi . H

pHZ

(6.34)

Relative to the (01,3, 35) coordinates, the very slow motions of the nearhorizon FlDOs (both unperturbed and perturbed) have magnitudes

ld_a

~ Q

adzmo

d'nDo

~ fl dz

~ 0(a2) .

(6.35)

FIDO

In our studies, below, of the structure and evolution of the stretched horizon we shall never have need of or be interested in the tiny 0(a2)

corrections to the spatial metric (6.33) or the FIDO motions (6.35) — with one exception. The 00:?) motions of the FIDOs tangential to the

stretched horizon will be of some interest — and they are among the few gauge- and slicing-dependent quantities that will appear in the membrane formalism. They have the form

[an “'1 FIDO

- OJ '- 9H +

gauge- and slicing-dependent . 2 perturbations of 0(a )

VIC].

Horizon-Fixed Spatial Coordinates

199

’—

.- _

a

l + 2M(rH - M)

M(er—azcosz'a') a2 + (perturbations), (6.36a) 2 rHPH

au e- and slicin -de

ndent

‘13 pe 2-:— F100 - [g gperturbationsgof 0(a2)

I

(6.36b)

Note that in (d3/dt)npo, although the leading term and the perturbations

are both of 0(a2), the leading term—being a property of the unperturbed Kerr hole—is “finite”, while the gauge- and slicing-dependent perturbations are “infinitesimal”.

In the immediate vicinity of a point (a - a”, 3 - 30, 3 - 30) on the stretched horizon and over lateral distances small compared with r", we can introduce local Cartesian coordinates based on a, 0, 7:5:

a 1:3” dam :-

_

d‘o‘______9_92,

x J”

2 pH

at (-

0’

y = we — an), 2 — i. 8H

(6.37)

In terms of these coordinates the metric (6.33) and the lapse function take on the Rindler form

d52 — dx2 + dyz + dz2 ,

a - gHz ,

(6.38)

aside from fractional corrections of order a2, xz/rflz, yz/rHZ, and xy/rHZ. This, together with the fact that [aside from fractional errors of 0(a2)] the FlDOs are at rest in these coordinates, permits us to use the Rindler-

approximation formalism (Sec. 1186) when studying local physics in a tiny (size t' ,

(6.85)

where t’ is the time that the shell hits.

More generally, the teleological nature of the horizon dictates that the tidal-force equation and the focusing equation both be solved using a Green’s function G(t,t’) whose response precedes the driving force: (—d/dt+gH)G(l,t’) -= 6(t—t');

(6.86a)

G(t,t') — explgH(t—t')} for t < t', - 0 for t > t‘ .

(6.86b)

This teleological Green’s function is shown graphically in Fig. 50. terms of it, the solution of the tidal-force equation (6.80) is

«gunm- f agga'wmw'm' ,

In

(6.87)

v1 C6.

Teleological Boundary Conditions

215

#7

Gum

t’

t

Fig. 50. The teleological Green‘s function of Eq. (6.861)), which is used to integrate the tidal force equation and focusing equation; cf. Eqs. (6.87) and (6.88).

and the solution of the focusing equation (6.81) is

0”(z,3,25)- f(aflafib+81r.9’”)(,:_m)6(t,t')dt' .

(6.88)

There is no teleology involved in the solution of Eq. (6.82) for the evolution of the stretched horizon’s metric. That evolution, between time to and time t, is given by a straightforward integration of the right-hand side of (6.82):

b.5035) - 'YaonfiJ) + 2355; + W731: ,

(6.8921)

where 25, is the time-integrated shear I

250.35) a fattuzth'

(6.8%)

l.

and 8“ is the time-integrated expansion f

Mam) a fo"(z',fi,as)dz' .

(6.89c)

(.1

It is straightforward to verify that, for the case of a Schwarzschild hole perturbed by an incoming spherical shell of matter (Fig. 49), Eqs. (6.88) and (6.89) give a metric perturbation that describes a stretched horizon with circumference which evolves as

?-4wM+41M,exp[gH(t-t')l for t < t’ -4«(M+M,) for t > t'. This is in accord with the behavior shown in Fig. 49.

(6.90)

216

Gravitational Interaction of a Black Hole with Nearby Mattel*__

7. Caustics in the stretched horizon

As a means of getting insight into the focusing equation, let us relax momentarily the restriction to weakly perturbed black holes. More specifically, let us consider the evolution of the stretched horizon of a very

large black hole when a very compact object (a small black hole or, perhaps, a neutron star) falls into it. The compact object has such strong gravity that it holds null rays (i.e., photon world lines) rather tightly in its own grip; and as it falls through the true horizon, it deposits some of its tightly held rays onto the true horizon, where they become generators and live forever after. This process is depicted in Fig. 51 and is analyzed with great generality by Penrose (1968) and in Sec. 34.4 of MTW (which is based on Penrose).

As shown in Fig. 51a (with the suppressed spatial dimension mentally restored), shortly before the new fiducions are deposited on the stretched horizon they form a shrinking 2-dimensional bubble surrounding, or perhaps inside, the infalling body. As this bubble nears the stretched horizon (Figs. 51b and 51c), the bubble and the stretched hor-

izon reach out and touch each other at a point (“caustic”), through which the fiducions then begin to flow from bubble to stretched horizon (Figs. 51c and 51d). Finally, after all the fiducions have flowed onto the stretched horizon, the caustic disappears and the stretched horizon smooths itself out (Fig. 51c). As they flow through the caustic the new fiducions, though infinitely close together, are expanding away from each other and onto the stretched horizon at a finite relative speed; and, as a result, their expansion rate

a” - (l/AA)(dAA/dt) is infinite. From this it should be clear that an infinite expansion 0" is the sign that a caustic has formed and that new fiducions are attaching themselves to the stretched horizon. Let us turn, now, to the exact focusing equation (6.78b) in a slicing with constant 3” and study how it gives rise to an infinite expansion at a caustic. By setting y E (flu/2g“, we bring the homogeneous version (-7}: =- arfi’ - 0) of this exact equation into the form

[—1 +ghy who»2 -0, dt

(6.91)

which should be contrasted with the weak-perturbation homogeneous

equation (6.83). if at some time to before driving forces act y has the value ya, then the solution to this equation is

y — [1 + (y, l — new-014 . (6.92) If ya ZgH (i.e., y > 1), the exact focusing equation forces that patch to develop at

an earlier time a caustic or wustics with infinite 0”. Our perturbation approach requires that 0" t; drop a large number of zero-angular-momentum particles into the final hole (thereby altering the hole’s mass but leaving its angular momentum unchanged). The tidal effects of each particle, as it passes through the stretched horizon, will deform the horizon's fiducion fluid elements and correspondingly will deform its com0ving coordinates in a characteristic way that we shall study in Sec. VIIEl (Fig. 63). Carefully choose the distribution (i.e., mass per unit area) of the particles that fall through the stretched horizon (one function of two variables 3, 3) so as to deform the com0ving d-constant coordinate lines into equally

spaced lines orthogonal to E. These coordinate lines will then coincide with the Boyer-Lindquist lines of constant longitude, and correspondingly

the basis vector (ii/63),; will satisfy 6/63-5- |£l2, the horizon momentum

density fl” -Ilf,,§ will have as its a component IIf-l—‘iHE and the hole‘s

VlCll.

Evolution of Entropy, Angular Momentum and Mass

233

final angular momentum Jfinal will be given precisely by the integral (6.131). Although this second evolutionary stage has altered the hole‘s mass, it has left the angular momentum Jm, unchanged, and by the symmetry of the tidal deformation pattern for each infalling particle (Fig. 63) it has contributed nothing (at the relevant second order) to the time integral of (6.134). Consequently, Eq. (6.131) now correctly represents Jfiml, and Jam. — Jintial - [the change in (6,131)] is given, correct through second order in the perturbation amplitude, by the time integral of (6.134) over the first evolutionary stage and t. < l < :2. This validates our use of (6.134) to describe the time evolution of the hole’s angular momentum in the pedagogical situation of evolution from one Kerr state to another. In more general situations, i.e., in the midst of a nonaxisymmetric evolution (including that analyzed above or an evolution that does not go from precise Kerr to precise Kerr), the angular momentum is not precisely defined; so we are free, if we wish, to take (6.131) and (6.134) as our definition of it and of its evolution. For alternative derivations of (6.134) sec Sec. IV of Teukolsky and Pms (1974) and Prior ( i977a,b).

Equation (6.134) for the torque on the hole can be rewritten, using

7“,, _=. 73b+225+0flygb and aygb/aa - o and «5t - 3251/61, as ”L251, FIii—2,5}; TI

+ 3,,H dA .

(6 . 135)

71-1; (J

Notice that when if - 0 (purely tidal perturbations), this torque is second order in the perturbation amplitude; and, correspondingly, the physically significant parts of the viscous force —2nHa§”’..b and the Navier-Stokes equation (6113) will be second order. To evaluate all details of the Navier-Stokes equation to second order would require knowing the perturbing tidal fields 8’51, accurate to second order —— a task beyond the sc0pe of any of the model-problem calculations in this book. Fortunately, as Eq. (6.135) shows, that part of the Navier-Stokes equation which determines the total torque dJ/dt on the hole is fixed to second

order by a first-order knowledge of 2:1,, or of of!” or of 8’5. Turn, finally, to the evolution of the hole’s total mass M. By com-

bining equations (6.130’) for THdSH/dt and (6.134) for dI/dt with the hole’s first law of thermodynamics dM - THdSH + and], we obtain the following equation for the rate of increase of M:

dM

dt

d'SH

til

T" dt + 9” dz 67cm H —

ayab + .7” + 9H 354]“ .

Iii" [ at M Here we have used the relation [Eqs (6.69)]

(6.1363)

234

Gravitational Interaction of a Black Hole with Nearby Matte,

i

3‘ a»

= i

-9”; 33 M .

( 6.13 6b)

3‘ 33

12. Summary of the laws of evolution of the stretched horizon The evolution of the stretched horizon of a weakly perturbed black

hole is “driven” by the tidal gravitational field 8’5 and by the flux of energy 9‘” and momentum 95' carried into the stretched horizon by electromagnetic fields and matter. One can compute the tidal field by solving the Teukolsky equation (6.28) for the Teukolsky function ‘1!” at

the horizon and then reading 8’5 off ‘1'” [Eq. (6.54)]. Alternatively, one can compute 8’51, by evaluating with some other technique the tidal field

3’“, =- Caébfi measured by a FIDO at the stretched horizon (end of Sec. V132), and then by taking the transverse-traceless pan and renormalizing,

8’55, = 1:1”ng [Eq. (6.4%)]. Further, one can compute the flux Y" of energy by renonnalizing the energy density 6 or normal energy flux S"

measured by a FIDO at the stretched horizon, .7" =- (1,126 =- -aHZS" [Eq. (6.56a)l, and the flux 3’5" of momentum by renormalizing the momentum density S,, or momentum flux T"a measured by a FIDO at the

stretched horizon, 315' -aHSa - —aHT”,, [Eq. (6.56b)l. For an electromagnetic field

9“ - ~(E”x§”)-r7 = 5”?” - (ohmic dissipation rate) (6.137a) 3’” - aHFH + 35455,, - (Lorentz force per unit area) (6.137b) [Eqs. (3.99), (3,100)].

The evolution of the stretched horizon’s shear 051 is driven by the tidal field 875, through the tidal force equation (6.80), which must be solved using the teleological Green‘s function (6.86), (6.87). The evolu-

tion of the expansion 9” and surface mass density 2" - -(l/81r)0“r is driven by the square of the shear and by the energy flux 9” through the focusing equation (6.81), or equivalently through the law of horizon

energy conservation (6.112), which must also be solved using the teleological Green5 function. The evolution of the horizon’5 surface density of momentum II” is driven by the divergence of the shear and by the momentum flux 3’” through the horizon’s Navier—Stokes equation (6.1 13). The evolution of the horizon’5 metric yab is driven by the expansion 0" and shear a5}, through the metric perturbation equation (6.82). And, finally, the evolutions of the hole’s mass M, angular momentum J, and entropy S” are driven by the shear 05,, the metric 74b, and the

energy and angular momentum fluxes y” and 9’5" through Eqs. (6.136), (6.135), and (6.130).

VII

Model Problems for

Gravitationally Perturbed Black Holes

by Richard H. Price, Ian H. Redmount, Wai-Mo Suen, Kip S. Thorne,

Douglas A. Macdonald, and Ronald J. Crowley We turn, now, to a series of model problems which illustrate the membrane formalism for gravitational perturbations of a black hole. We shall begin in Sec. A (and in Sec. B2) with model problems for quasista-

tionary perturbations of the horizon — i.e., perturbations with dynamical time scales to satisfying ID >> 35'. In Secs. C and D we shall give examples of the astrophysically more typical case :0 —-— 3”". Finally, we shall turn in Sec. E to model problems involving dynamical time scales with ID > r” above the north and south poles of a nonrotating black hole of mass M. (b) An embedding diagram showing the prolate, quadrupolar deformation of the black hole‘s stretched horizon due to the tidal fields 83'". of the suspended masses.

hole symmetrically — one along the hole’s north polar axis (0-0), the other along the south polar axis (Our); see Figs. 52 and 53. Do the lower-

ing on a time scale to that is arbitrarily large — at least as large as

1,, >> gfi' - 4M so the hole can adjust quasistatically to the tidal gravitational fields of the masses and ropes. We wish to study the evolution of the hole’s horizon in response to the masses’ tidal fields. But before discussing the evolution, we shall pause to comment on the importance of the ropes and the use of two masses rather than one.

This model problem is the obvious analog of the point charge at rest above a Schwarzschild hole, studied in Sec. IIDZ (Figs. 13 and 14). In the point-charge problem there was no difficulty with ignoring the suSpension mechanism (rope or whatever), because that mechanism could in principle

be electrically neutral and have vacuum permittivity and thus could leave the electromagnetic field unaffected. Not so in the gravitational case. The suspension mechanism (ropes) must exert a force on the masses to keep them from falling freely; it must thus have internal stresses; and those stresses will gravitate. Moreover, the Einstein field equations, which

VIIA].

Compact Masses Lowered toward Nonrotating Hole

237

F

rn

ISm

(a)

(b)

Fig. 53. (a) Same as Fig. 52a but with the suspended masses very close to the stretched horizon. (b) Embedding diagram, as in Fig 52b, showing the tidally induced, pimple-like defamation of the hole’s north and south poles.

describe the generation of gravity by mass, momentum, and stress, contain within themselves the equations of motion and force balance of all gravitating material (see, e.g., Sec. 17.2 of MTW). If in computing the tidal forces on the hole we include the gravity of the suspended masses but ignore that of the stressed ropes, the resulting gravitational field will be incompatible with the law of force balance for the masses; that law expects

the masses to start falling freely, and when the masses fail to fall, mathematical analysis will start producing nonsense (as the authors discovered in early misguided calculations). Similar considerations require that two masses be suspended on opposite sides of the hole. If we were to suspend only one mass, the hole

would start to fall toward it and the configuration would cease to evolve quasistatically. The gravitational force of the second mass on the hole counterbalances that of the first, thereby permitting the hole to be in quasistatic equilibrium. ‘

Return, now, to the horizon’s evolution during the quasistatic lower-

mg of the masses. The masses' slowly changing tidal field 8’" produces a

238

Model Problems for Gravitationally Perturbed Black Hula

shear of the horizon ‘5‘” [tidal force equation (6.80)]; that shear then pro. duces a horizon expansion 0” [focusing equation (6.81) or, equivalently, law of energy conservation (6112)]; and the shear and expansion then produce a change in the horizon metric [metric evolution equation (6.82)]. It should be intuitively obvious — and indeed is true — that because the shear '6‘” describes a rate of change of the horizon shape in

response to the tidal force, the more slowly the masses are lowered, the smaller will be the shear; and, in fact, the shear will be inversely pl'Opor.

tional to the arbitrarily large time scale ID on which the lowering is per. formed. This, together with the tidal-force equation (6.80) (where the time derivative is negligible because l/tD > r"; and (iii) the special case of masses very near the horizon, i.e., at a proper distance 3," = 4Mam > r”, we obtain the limit 3,1,. ~ m/rm3; when they are very close to the horizon, at the point on the horizon just below them we obtain

instead 83'", ~ m/sm3; in both cases this corresponds to the Newtonian formula 83'," -- (mass)/(proper distance)3. (iv) The area average of 83'," over the horizon vanishes. This property is demanded by the fact that for any 2-dimensional surface with the topology of a 2-sphere (including our black-hole horizon) the surface integral of the scalar curvature must be equal to 81-; since the surface area is unperturbed, the area integral of the unperturbed scalar curvature must give 81r, leaving zero contribution for

the integral of a}, - —2 $1,":

~2§ stud/1 - —25( an."- an)“ - flag—9mm - 31—3: — o.

(7.12)

To aid in the visualization of the horizon’s tidal-field-induced scalar curvature, we show in Figs. 52 and 53 embedding diagrams for the horizon 2-geometry as computed by Hartle (1974) and by Redmount (1983). Note that when the masses are very far from the hole (Fig. 52), the

242

Model Problems for Gravitationally Perturbed Black Holes ‘

horizon is distorted into a spheroid which is slightly prolate [stronger tidal field and curvature near the poles than near the equator, of. Eq. (6.105.) which shows the longitudinal tidal force of a distant mass to be a stretch with twice the magnitude of the transverse tidal squeeze]. By contrast, when the masses are very near the horizon (Fig. 53), the distortion is

strongly localized at the poles. Moreover, in the limit m/sm3 >> 1/r32' the curvature due to the perturbation is far larger at the poles than the

unperturbed

curvature (9}, >> .93;

“pimple on

the embedding

diagram"). This does not by any means indicate a breakdown in the analysis, however; the analysis remains valid so long as m/sm > 5," away from the pole; the angle of the cone is given by the standard formula (7.13) with 3' - a”, which integrates to

0d” - (Bm/sm)"’ .

(7.13')

At polar angles 0 2 am the conical embedding surface smoothly asymptotes to the unperturbed spherical geometry — and even in the conical region it has somewhat more net scalar curvature (deviation from precise conicality) than the unperturbed horizon. We conclude this subsection with some remarks intended for experts in general relativity. Geroch and Hartle (1982) have developed a very general mathematical formalism to describe nonrotating, gravitationally distorted, static black holes and the surrounding matter that distorts them.

By contrast with the weak-perturbation analysis given here, the GerochHartle formalism can treat arbitrarily strong (but necessarily static) pertur— bations, even perturbations so unrealistically strong as to change the horizon topology from spherical to toroidal. In the Geroch-Hartle formalism there are two possible definitions of the mass of the hole. One definition corresponds to that used here: a mass [mm in their notation; their Eq. (4.8)] which, in terms of the hor-

izon area A, is (A/l61r)"’, and which, in quasistatic perturbation processes, is unchanged by the gravitational fields of surrounding matter. The other definition corresponds to that used in most of the relativistic literature, e.g., Bardeen, Carter, and Hawking (1973) and Carter (1973, 1979): a

VIlAl.

Compact Masses Lowered toward Nonrotating Hole

243

___——

mass (m in the Geroch-Hartle notation) which is defined via the “Kama: integral" at infinity and which is affected by the gravitational fields of surrounding matter. Our adoption of the former and less conventional definition is dictated by our quest for simplicity when treating fully dynamical situations: our simple prescription (6.136a) for the computa-

tion of dM/dt in dynamical situations produces M - min, in static, defonned-hole situations (which are obtained from unperturbed black holes via quasistatic processes, such as those described at the beginning of this section).

2. Quasistationary deformations of a rotating hole Tum attention, briefly, from a quasistatically deformed, nonrotating hole, to a quasistationarily deformed, rotating hole. For any deformation of the rotating horizon to be quasistationary, it must change, as seen by stretched-horizon FIDOs and fiducions, at a very slow rate compared with the hole’s surface gravity, i.e., the time scale of change on the horizon 10 must satisfy [/10 > gH", but also in marginally quasistationary situations, -gH'1. From this relation and Eqs. (7.14) we obtain order-of-

magnitude estimates of 05,, 0”,25, and 0”, all valid for ID 2 gfil:

05% ~ —1—l(22$;,) , a"

to —(_> gH" corresponds to B 0 is a vanishing rate of shear «fl, and a nonvanishing, time-independent net deformation in the tangential direction with ellipticity

VIIEl.

Radial Fall of Star into Nonrotating Hole

,

'

~3/9H

‘2/9'4

271

t

_|/gH

O

H -_ 31'. 2%? 299

@H

—3/gH

‘Z/QH

I

I

‘l/‘JH

O

t

. 62. The time dependence of the horizon tidal field ?, shear «5, expansion

, and metric perturbations 251, E (time-integrated shear) and O” E (timeintegrated expansion), produced by the radial fall of a compact star into the north

pole of a nonrotating black hole. The shear. expansion, and metn‘c nurbations grow on a timescale l/gH, anticipating the sharply peaked burst of which signals the passage of the star through the horizon at t - 0. The shear and expansion then die 05 on a timescale much shorter than 1/3", while the metric perturbations become constant. (Figure adapted from Suen, Price, and Redmount 1986.)

. diameter! e a {tangential _ l _ 1+2M _ 1 a, (2“ _ Eat) (radial diameter)

[+233

32:2“ .

(7.78)

Similarly, the expansion 0” leaves each bundle of fiducions at t > 0 with a slightly increased surface area: AA

1677:11412

74— - o” -

(7.79) 732

'

This model problem provides a good example of the jumping of new

fiducions onto the stretched horizon, as discussed in Sec. VIC7 and Fig.

272

Model Problems for Gravitationally Perturbed Black HOIes

000 O 0 0000900 OOfl OQQO 00%0 £00 GOO GOO

QOwQOO Fig. 63. Distortions of the horizon produced by a compact radially infalling star. lnitial circles of fiducions are distorted into ellipses as the horizon deformsm anticipation of the arrival of the tidal shock wave Haccompanyin the plunge of the

star. The nature of the distortions illustrates 25 > 0 and E

< 0 as derivedIn

the text

51 above. The criterion for new fiducions to jump on is that 0” grow as large as 2g” (which then forces it to have grown, slightly earlier, to infinity). Our perturbation-theory formalism starts to break down if ever

this regime, 0,, ~ 2g” is reached, but the formalism is valid up to near that regime and thus can give us an order-of-magnitude criterion for the jumping-on of new fiducions. The expansion 0”, as described by Eq. (7.74), attains its largest value

at time t- —gfi'ln2, shortly before the tidal pulse hits; and its value divided by 2g” then is

-64 mZM2 4

[3”— 23H

max

(7.30)

U

If this quantity exceeds ~l at or - R E (radius of star), then sometime shortly before the plunge of the star fiducions necessarily will jump onto

the stretched horizon in anticipation of the arrival of strong tidal fields: f— R S- 2 2mM

for new fiducions to be pushed onto the stretched horizon by the infalling star

(7.31)

vflEl.

Radial Fall of Star into Nonrotating Hole

273

#—

correspondingly, the criterion for the validity of our perturbation analysis is the following:

4 2 R >> 64(W)

for perturbation analysis to be valid everywhere outside the star,

574 >> 64(mM)2

for perturbation analysis to be valid at a: .

(7.82)

In the star’s interior Our analysis must be modified by the inclusion of an inflowing material energy flux .7 in the focusing equation and by a

modification of (reduction in the strength 01) the tidal field 3’5, which drives the shear. The details of this are left as an exercise for the reader. This stellar-plunge problem provides a good example of the gauge changes (infinitesimal changes of spatial coordinates) which one may wish to perform, after a perturbation turns off, in order to return the spatial coordinates to the standard Boyer-Lindquist or Schwarzschild form; see

Sec. VlClO. Consider, initially, only the first-order (shear-induced) contributions to the perturbed metric. The full metric (7.75), after the pertur-

bations have ceased, then has the form

ail - rH2( 1 +2zg)d92+ r,.,2(1+22‘.;:(,,)sinzad.i2 - rH2 1—

32mM 202 dflzhw2 I'H

HL ’g‘gMIawz at 9 [1113413)

’

(7.98)

701) I: ygb+225+ofl73b 9 (799)

2

_52(nli+znfsm?2 3_m €ng

2353 _ —2£3-

fort < tmaxW)

m

(7 100) "-2771 WII2

sm(l+f)_

“Hill-In _2—m(l—+D for! > tmax(w)

VIIE2.

Particle Radially Accelerated above Nonrotating Hole

279

,_f

4 4—"12L in.

Sm2(1+f)2

o”-

2

s

5m (H-f)2

e EMI (2e gulm_ egut )

w

4

2

w

for: < Imam)

2

(7.101)

s,"

At any time I >> g,“ the shear and expansion vanish within the annulus, i.e., for v < cram-Aer, and fall off with increasing 1:: outside

the annulus, a > army-Aer. The time-integrated shear 25 corresponds, as in the stellar plunge problem, to a tangential elongation and radial compression of an initial circle of fiducions. At fixed a: this distortion grows as the expanding annulus approaches, then when hit by the

annulus‘s pulse of 8’" field it suddenly stops growing and becomes static afier the annulus passes. Both inside and outside the annulus, despite the nonzero 25, the horizon geometry is intrinsically flat at first order (25order): a simple coordinate change like that of Eq. (7.84) converts the first-order metric to standard flat form at 221 < awn—Aw and U > wmax+Aax

This

is

not

so,

however,

in

the

annulus

itself,

lax—arm! 5 Av - Sm. There the horizon geometry has a genuine, physical, first-order distortion — a distortion not embodied in the above equa-

tions, which rely on the 6-function approximation.

Since 05 vanishes inside the annulus, the fiducion fluid’s viscous dissipation takes place only outside the annulus. There the dissipation rate as given by Eq. (6.130) is

d

—-M

M - T” S”

dt

ZnHagafibZrwa

dt - a... 2

_ __.L. L". 16(l+,02

2

2

i"— 6.23": _ fi fl es”!

M

arm“

16 M

(7.102) ’

with the result that the hole's total increase in mass up to time t is l

AM

_.. M dt dt

M

4M e EH1

(7.103)

Since the analysis presented is valid only in the Rindler approximation, i.e., for a; 0

Same as above

(1 — IRKI2)n{g 2 0 (1 — IRKI2)n{§.E£2 IRKInfh> >0 (r — IRKI2)n{f, > 0

Reversed from ab0ve

nfiEK 2 0

0< IRKI2
0

—9HLK/a s E" < 0 IRKI2 > 1

Range of locally measured energies deep inside atmosphere

r15, > 0

0

SR

Range of redshifted frequencies

Values of azimuthal quantum number

Characteristic

Table l. Superradiant (SR) boson modes contrasted with nonsuperradiant (NSR) boson modes

293

Thermal Atmosphere of a Black Hole

5. Interaction of the atmosphere with the extemal universe We are now in a position to derive formulae for the rates of increase of the mass and angular momentum of a black hole’s atmosphere due to interaction with the external universe, and as a byproduct of our deriva. tion we shall elucidate our choice of conventions for the number of quanta in SR modes. Our derivation and elucidation will make use of a

“box” placed around the hole, which consists of the region between spher. ical Surfaces of radii r. >> r” and r2 >> r. (Fig. 68). We shall quantize all fields inside this box in the usual elementary way, using periodic boundary conditions. Inside the box, where gravity is negligible, we can divide the modes of all fields into two classes, I modes whose radial propagation is outward and 1 modes whose radial propagation is inward. We shall presume that the external universe is bombarding the hole

with quanta, and we shall denote by (nfir) the mean number of quanta that impinge on the hole in the mode (K,IN). Since the 1 modes can be populated only by [N quanta in the ingoing channel (never by UP quanta in any channel), the mean number of quanta in a 1 mode is

("£9 - ("1’50 -

(8.13a)

The I modes of the box get populated in two ways: by reflected IN quanta and by evaporating UP quanta; thus, the mean number of I quanta is

M) - 1RK|2(nIS/) + (1 — laxiszg .

(8.13b)

[The sign of the second factor 1/2 in Eq. (8.120) above is dictated by the demand that when the zero-point contributions (factors 1/2) are included in the IN and UP modes, they produce the correct l/2 quantum in the I modes:

(M) - IR’92((n.’$4)+1/2)+(l-IRKIZXfimH/Z) - {nf}+l/2 .

(8.13c)

If we had reversed the sign of the second [/2 in (8.12c), thereby obtaining n - —n°]d rather than n - -(nold+l), the zero-point contributions would not have worked out right in Eq. (8.13c).] Since the quanta in the 1 modes empty out of the box and into the external universe in a time At - (r2 - r.)/vK

(8.14)

(where v“ is the radial velocity of a quantum in state X when it is far from the hole), and since and the quanta in the 1 mode empty out of the box and into the hole’s atmosphere in the same time At, and since each

quantum carries a redshifted energy E5 and angular momentum L", the total rates at which mass and angular momentum are injected into the atmosphere are

VIIIBS.

Interaction of Atmosphere with External Universe

299

Fig. 68 A box in the form of a thick spherical shell surrounds a black hole and is used for quantization of fields. in the box one can characterize a mode by its direction of radial propagation, l for outward propagation and l for inward Propagatlon.

‘L—“f - 3'7 §( — 725ml: ,

(8.15a)

% - i gun—om“

- figl — IRK|2)({n}fv} — Him".

(3.15m

Since, far from the hole, the quanta propagate with radial speed v", the mode-K" field far from the hole has spacetime dependencec

field ~ r-'SJMm,.(o)eimte-i'- _ "mp-5 '(8'24b) K Again, in the Minkowski vacuum (precisely thermal FIDO-measured state), this vanishes. [Note that here and throughout this chapter we treat gravitons (gravitational-wave perturbations) on the same footing as other particles;

304

Thermal Atmosphere of a Black Hole

in particular, we attribute to them a stress-energy tensor T‘”.

For a

rigorous justification of this in the classical (nonquantum) regime see

Isaacson (1968) or the review in Secs. 35.13—35.15 of MTWJ 7. Renormalized angular momentum and redshifted energy In a black-hole atmosphere

The membrane fonnalism’s fiducial observers, very close to the horizon of a stationary, rotating black hole, are physically indistinguishable from the Rindler FlDOs of flat spacetime. This fact shows up clearly in the Rindler approximation to the metric, lapse function, and shift function near a rotating horizon (Sec. VIC 1). Within a distance 2 of the horizon, the metric, lapse, and shift are those of the Rindler spacetime aside from fractional corrections of order a:2 ~ (gHz)2. So long as one is not interested in such corrections, one can treat the hole’s FIDOs as though they were Rindler observers in flat spacetime. This is true not only classically, but also -— so far as calculations have yet revealed — quantum mechanically. In particular, calculations thus far (Candelas 1980; Sciama, Candelas, and Deutsch 1981; Frolov and

Thorne 1986) are compatible with the presumption that in the (nearly) thermal atmosphere very near a blackhole horizon, as in flat spacetime, the dominant parts ofi' Tm are equal to the stress-energy that the HDOs would compute from their own physical measurements, minus a vacuum

polarization correction equal to that of a precisely thermal atmosphere — an atmosphere that comoves with the near-horizon FIDOs and thus has angular velocity 9”, and that has locally measured temperature T related to the FIDO acceleration a by T - hill/21kg - TH/a. Here by “dom-

inant parts” is meant those parts of < T“) which, in the FIDO reference frame, diverge as the horizon is approached. The dominant parts typically

are the energy density (a) = (Too), the inward radial flux of energy —LS’ — —(T°’) and the radial stress {T"), which are all equal to each other and are divergent as l/al, and the density and flux of lateral

momentum (3‘?)=- ( T90), SW - ( T”), < T"), and-fTM (and of angular momentum < T¢0) - aH’T‘U (T,> - UH< TM ), which typically are divergent as 1/01. (The divergences are associated, of course, with the near-horizon redshift of FIDO proper time, dr/dt — a > r”, the FIDOs are essentially inertial and wavelengths are small com red with the radius of curvature of spacetime, so all components of {T55 can be computed from FIDO measurements without renonnaliza— tion; such computations will reveal outflowing energy due to evaporation and inflowing energy due to accretion. For further details see, e.g., Frolov and Zel’nikov (1985), Frolov and Theme (1986), and the many earlier references cited therein. In studies of the properties of a black hole‘s atmosphere at heights 2 > I and 2; a + 9H> IO‘7g in response to its emission of quanta. Figure 71 shows Page’s results for the spectrum (energy per unit time per unit redshifted frequency) of quanta of various types emitted by a

nonrotating black hole of mass M >> 10'7g.

These spectra are the

integrand of Eq. (8.48a) for spin 5 - 1/2, 1, 2, summed over helicity h and angular quantum numbers I, m, and multiplied by a factor 6 for s - V: to account for the 6 independent species of neutrinos and antineutrinos. Notice, as expected, that at frequencies on g g” - l/4M, where most of the energy is emitted, the higher the spin of the field the more the emission is suppressed. Notice also that at high frequencies the emission is

316

Thermal Atmosphere of a Black Hole

io-2

1 O

3

ion 3:;

T-

5

gr? I0"

2 "6

'0

IO‘3 U 89

2|; 1

A 3° 10'5

2 Q

5% v

l0"

Fig. 71 The spectrum of quanta of various types emitted by a nonrotating black hole of mass M >> 10%. (Based on calculations by Page, [9762! except the neu-

trino and total curves. which are extrapolated from Page‘s calculations which assumed 4 independent neutrino species to the presently known 6 neutrino species.) (Figure adapted from Page, 1976a.)

that of a black body with surface area 271rM2 and with temperature equal to the Hawking temperature TH (dashed line). This is because at high frequencies the quanta can be described by geometric optics, and in the geometric optics limit (“point particles" of zero rest mass) the hole presents a capture cross section of 27-er2 to the external universe (MTW

Exercise 25.25). By integrating the spectra of Fig. 71 over all frequencies, Page obtains for the rate of evaporative loss of mass from the hole W dt

h CMZ

,

(8.49)

where C - 2.830>> 10'7g

(8.50)

(a value extrapolated from Page’s original calculations, based on 4 neutrino species, to the presently known 6 neutrino species). In all, 86.7 percent of the radiated energy is canied off by neutrinos (1/3 of it as electron neutrinos and antineutrinos; l/3 as muon neutrinos and antineutrinos;

VIIIC 1.

Evaporation of Hole into Perfect Vacuum

1/3 as tau neutrinos and antineutrinos).

317

Because they have so much

greater difficulty escaping than neutrinos, photons carry only 11.9 percent

of the power, and gravitons only 1.4 percent.

When the hole’s mass lies in the range 5X]0”g rH,

(8.77a)

Iorl

dN

(aw-m9”)

Marc]

1

I ml 2

31m

"1’

' rH(DSlm)2

- :t sign(o..—mQH)

at r.. 0) , (8.82b)

l — Ill/"ml2 - —|T,,,,,_|2

for SR modes (am-m9}, < 0).

(8.82c)

This is just the law of conservation of quanta. From this law we see that

the SR modes have reflection probabilities lRl2 greater than unity, and the NSR modes have IRI2 less than unity, in accord with the discussion in Sec. B4 above.

The values of the reflection probabilities, IR,,,,,_I2, which play such an important role in the quantum theory of the hole’s atmosphere, can be derived by solving the radial wave equation (8.67) with potential (868), subject to the boundary conditions (8.80) or (8.81). The potential V(r)

contains, through the angular eigenvalue OElm [equal to 1([+l) for a nonrotating hole], the effects of centrifugal forces on the wave’s propagation; and it also contains, through the hole's mass M (which appears in A) and its rotation parameter a, the effects of spacetime curvature. The qualitative influences of these efi'ects were discussed in Sec. B1 above. Analytic calculations based on the radial wave equation (8.67), the

potential (8.68), and the boundary conditions (8.80) or (8.81) give for the scalar-wave reflection probability at low frequencies (Starobinsky and Chuiilov 1973) 2

_

1

2

(l—s)!(l+s)l

'R-"'"°" a 2[ (21)!(21+ 1):! Zl+l X 0.,

_

m9”

3;; AH 2 =70...

AHgH a}.

2w for

5-0,

I II

I +

0,.

,, -1 l-m-O.

_

2 m9”

n3" (8.83)

This formula is written in a form valid for any boson field (5 - 0 for

330

Thermal Atmosphere of a Black Hole

scalar waves, 5 -= l for electromagnetic waves, 3 -= 2 for gravitational

waves). The strong suppression of transmission for waves with 6.. > r" ,

(8.873)

332

Therm] Atmosphere of a Black Hole

Vida—"19””?! (455; 1115.5)!” 1 ‘

“ISM“!dm’a)er(m3+»)e-i(a-—m 01:)(14 r.)

‘lmhaz

at r... >rH , (8.883)

Iorl __d_N

a2

dAdt

.

.

2

i 4«h(a-—mnfl) 'Efi’hE“ S

2

- :I: sign(a.,-m93% at r, > 7H u

w—

9

ifllL‘éhM at r.

r si m-mfl tfl—H) 271

at r...