Applied general relativity 9783030196721, 9783030196738

Table of contents :
Preface......Page 6
Contents......Page 11
List of Acronyms......Page 17
List of Symbols......Page 19
1 Introduction......Page 21
1.1 Time and Reference Systems......Page 22
1.2 Space......Page 27
1.3 Astrometry......Page 29
1.4 Celestial Mechanics......Page 31
1.5 Relativistic Astrophysics and Cosmology......Page 33
2.1 Space-Time Manifold and Fields......Page 34
2.2 Coordinates, Differentials and Tensors......Page 35
2.3 Tensor Algebra......Page 38
2.4 The Lie-Derivative......Page 39
2.5 The Covariant Derivative......Page 40
2.6 Geodesics......Page 44
2.7 Curvature- and Ricci Tensor......Page 45
2.8 The Metric Tensor......Page 49
2.9 Metric Connections......Page 51
2.9.1 Riemann Tensor and Its Symmetries......Page 55
2.10 The Levi-Civita Symbol and Tensor......Page 57
2.11 Symmetric Spaces......Page 59
2.11.1 Maximally Symmetric Spaces......Page 60
2.11.2 Maximally Symmetric 3-Spaces......Page 61
2.12 GRTensor......Page 62
3.1 Newtonian Theory of Gravity......Page 65
3.2.1 The Galilean Group......Page 66
3.2.2 Weak Equivalence Principle and Newtonian Theory of Gravity......Page 67
3.3.1 Spherical Multipole-Moments......Page 71
3.3.2 Spherical Mass-Moments of an Oblate Spheroid......Page 77
3.3.3 STF-Tensors......Page 80
3.3.4 Cartesian Multipole-Moments......Page 86
3.4.1 Newtonian Tidal Moments......Page 91
3.4.2 The l=2 Tidal Potential for External Point-Masses......Page 93
3.5 Translational Equations of Motion......Page 95
3.6 Rotational Equations of Motion......Page 97
3.6.1.1 The Torque as Lie-Derivative of UE......Page 98
3.7.1 Integrals of Motion......Page 100
3.7.2 Orbital Equation; Kepler's First and Third Law......Page 105
3.7.3 Classification of the Conic Sections......Page 107
3.7.4 Kepler's Equation......Page 109
3.7.5 Fourier-Analysis in the Elliptical Orbit......Page 112
3.7.6 The Elliptical Kepler Orbit in Space......Page 113
3.7.6.1 Calculation of x and from the Orbital Elements......Page 114
3.7.6.2 Calculation of Orbital Elements from x and......Page 117
3.8.1 Variation of Constants......Page 118
3.8.2.1 Vectorial Elements in the Kepler-Problem......Page 119
3.8.2.2 Perturbation Theory with Vectorial Elements......Page 121
4.1 Relativity......Page 132
4.2.1 Maxwell's Equations......Page 133
4.3 The Minkowskian Metric, Lorentz-Transformation......Page 137
4.3.1 Addition of Velocities......Page 144
4.3.2 Thomas Precession......Page 145
4.3.3 General Coordinate Transformations and a Derivation of the Lorentz-Transformation......Page 149
4.4 The EM-Field of a Moving Point Charge......Page 152
4.5.1.1 The Uniformly Moving Point Charge......Page 156
4.5.1.2 The Harmonic Hertzian Dipole......Page 158
4.5.2.1 The Front Velocity......Page 162
4.5.2.2 The Group-Velocity......Page 167
4.6 Energy and Momentum......Page 169
5.1 General Relativity......Page 173
5.2 Einstein's Equivalence Principle......Page 174
5.3 The Motion of Test Bodies......Page 177
5.4 Einstein's Theory of Gravity......Page 178
5.5 The Problem of Observables......Page 180
5.5.2 The Spectroscopic Observable......Page 181
5.5.3 The Astrometric Observable......Page 183
5.6 Tetrads and Tetrad Induced Coordinates......Page 185
5.7 Proper Reference Systems of Accelerated Observers......Page 190
5.8.1 The Landau-Lifshitz Field Equations......Page 195
5.8.2 Harmonic Gauge......Page 197
6.1 Minkowskian Space-Time......Page 201
6.2 Stationary Space-Times......Page 203
6.2.1 Stationary Axially Symmetric Space-Times......Page 208
6.2.2 The Hartle-Thorne Metric......Page 214
6.2.3 Static Axially Symmetric Space-Times......Page 215
6.2.3.1 The Schwarzschild Metric......Page 218
6.2.3.2 The Erez-Rosen Metric......Page 220
6.2.3.3 The Quevedo-Mashhoon M-Q-S Metric......Page 222
6.2.4 Spherically Symmetric Space-Time......Page 225
6.2.4.1 Harmonic Coordinates......Page 226
6.3.1 Boyer-Lindquist Coordinates......Page 229
6.4.1 The Cosmological Principle......Page 231
6.4.2 Robertson-Walker Metric......Page 233
6.4.3 De Sitter Space......Page 237
6.4.4 Schwarzschild: De Sitter Solution......Page 239
6.5.1 Geroch-Hansen Moments......Page 240
6.5.2 Thorne Moments......Page 242
6.5.3 The FHP Theorem......Page 246
7.1 The Post-Newtonian Expansion......Page 250
7.2 The General Form of the Metric......Page 251
7.3 Field Equations and the Gauge Problem......Page 256
7.4 The External Post-Newtonian Field of a Body......Page 261
7.5 The Multi-Polar, Post-Minkowskian (MPM) Formalism......Page 268
7.6 Several Expansions......Page 269
7.7 First Post-Minkowskian Approximation......Page 271
7.8 The MPM Algorithm......Page 287
7.8.1.1 The Inner Metric......Page 289
7.8.1.2 The External Metric......Page 291
7.8.1.3 Formal Matching in the Overlap Region......Page 296
7.8.1.4 Matching Equations for V and Vi......Page 297
7.8.2 The MPM Iteration Scheme......Page 298
7.8.2.1 The General External Metric......Page 299
8 First Applications of the PN-Formalism......Page 304
8.1 Equipotential Surfaces and Relativistic Geoid......Page 305
8.1.1 Post-Newtonian Equipotential Surfaces......Page 306
8.2.1 Synchronization of Nearby Clocks......Page 308
8.2.2 Rates of Clocks in the Earth's Vicinity......Page 309
8.2.3 Synchronization of Clocks in the Vicinity of the Earth......Page 311
8.2.4 Coordinate Time Synchronization......Page 312
8.2.5 The Relation Between Coordinate and Proper Time......Page 313
8.2.5.1 Clock's on the Earth's Surface......Page 314
8.2.6.1 TWSTFT......Page 315
8.2.7 TAI, TT and UTC......Page 319
8.3 Barycentric Timescales TCB, Teph, TDB......Page 321
8.4 Fairhead–Bretagnon Series......Page 323
8.5 Light-Rays in the PN-Field of a Single Body......Page 324
8.5.2 The Astrometric Observable......Page 329
8.5.3 The Gravitational Time Delay......Page 331
8.6 The PN Motion of a Torque-Free Gyroscope......Page 333
8.7 Geodesic Motion in the PN-Schwarzschild Field......Page 338
8.8.1 Post-Newtonian Schwarzschild Effects......Page 344
8.8.2 The Lense-Thirring Effect......Page 347
9.1 The Problem of Celestial Mechanics......Page 351
9.2 Transformation Between Global and Local Systems......Page 352
9.3 Split of Local Potentials, Multipole-Moments......Page 357
9.4 Local Harmonic Proper Coordinates......Page 362
9.5 The Standard xμ→Xα Transformation......Page 366
9.6.1 Post-Newtonian Tidal Moments......Page 368
9.7 BCRS and the Expansion of the Universe......Page 378
10 The Gravitational N-Body Problem......Page 380
10.1 Local Evolution Equations......Page 381
10.2 The Translational Motion......Page 383
10.2.1 The LD-EIH Lagrangian......Page 387
10.2.2 Laws of Motion......Page 388
10.2.3 Equations of Motion......Page 390
10.3 The PN Two-Body Problem......Page 393
10.3.1 The Brumberg Representation......Page 395
10.3.2 The Wagoner-Will Representation......Page 397
10.3.3 The Damour-Deruelle Representation......Page 400
10.4.1 Landau-Lifshitz and Fock Spin......Page 405
10.4.2 The PN-Spin in the N Body Problem......Page 407
10.5.1 Angular Velocity......Page 410
10.5.2 Rigidly Rotating Multipoles......Page 411
11.1 Historical Remarks......Page 413
11.2 Light-Rays for 1PN Stationary Multipoles......Page 418
11.2.1 The Shapiro Time Delay......Page 424
11.2.1.2 The Quadrupole Time Delay......Page 425
11.2.2 The Time Transfer Function......Page 426
11.2.3 The TTF for a Body Slowly Moving with Constant Velocity......Page 427
11.3 Light-Rays to Post-Minkowskian Order......Page 428
11.3.1 The Shapiro Time Delay......Page 433
11.4 The Klioner-Formalism......Page 435
11.4.1 Relativistic Aberration......Page 436
11.4.3 Parallax......Page 437
11.4.4 Proper Motion and Radial Velocity......Page 438
12.1 Pulsar Timing......Page 442
12.1.1 Pulsar Timing Arrays......Page 453
12.2 GNSS......Page 454
12.2.1 Global Positioning System......Page 455
12.2.1.1 Space Segment......Page 456
12.2.1.4 GPS Signals, Code and Carrier......Page 457
12.2.2 GLONASS......Page 459
12.2.3 GALILEO......Page 460
12.3.1 Satellite Laser Ranging......Page 461
12.3.2 Lunar Laser Ranging......Page 464
12.4 VLBI......Page 473
12.4.1 The Gravitational Time Delay......Page 476
12.4.1.1 Mass-Monopoles to pN Order at Rest......Page 477
12.4.1.2 Mass-Monopoles to pN Order in Motion......Page 478
12.4.1.3 The Influence of Mass-Quadrupole Moments......Page 479
12.4.2.1 Barycentric Baselines......Page 480
12.4.2.2 Geocentric Baselines......Page 481
12.4.3 Radio Sources at Finite Distance......Page 484
12.5 Doppler Measurements......Page 489
12.6.1 Passive Sagnac Interferometers......Page 491
12.6.1.1 Active Sagnac Interferometers......Page 493
12.7 Astrometry......Page 499
12.7.1 Hipparcos......Page 501
12.7.2 The Astrometric Project Gaia......Page 504
13.1 Legendre-Polynomials......Page 508
13.1.1 Ql(x) for x ≥1......Page 509
13.2 Relations for STF-Tensors......Page 510
13.3 Differential Geometry: Formulas......Page 512
13.4 Spherically Symmetric Metric......Page 513
13.5 Spherically Symmetric Static Metric......Page 514
13.6 The Kerr Metric: Geometry......Page 515
13.7.1 Multipole-Moments Derived from ξ-Moments......Page 521
13.7.2 Multipole-Moments Derived from Spherical Weyl-Moments......Page 523
13.8 Weyl-Moments as Functions of Mass Multipole-Moments......Page 525
Bibliography......Page 528
Index......Page 545

Citation preview

Astronomy and Astrophysics Library

Michael H. Soffel Wen-Biao Han

Applied General Relativity Theory and Applications in Astronomy, Celestial Mechanics and Metrology

Series Editors:

Martin A. Barstow, University of Leicester, Leicester, UK Andreas Burkert, University Observatory Munich, Munich, Germany Athena Coustenis, Paris-Meudon Observatory, Meudon, France Roberto Gilmozzi, European Southern Observatory (ESO), Garching, Germany Georges Meynet, Geneva Observatory, Versoix, Switzerland Shin Mineshige, Department of Astronomy, Sakyo-ku, Japan Ian Robson, The UK Astronomy Technology Centre, Edinburgh, UK Peter Schneider, Argelander-Institut für Astronomie, Bonn, Germany Virginia Trimble, University of California, Irvine, CA, USA Derek Ward-Thompson, University of Central Lancashire, Preston, UK

More information about this series at http://www.springer.com/series/848

Michael H. Soffel • Wen-Biao Han

Applied General Relativity Theory and Applications in Astronomy, Celestial Mechanics and Metrology

123

Michael H. Soffel Institute of planetary geodesy Lohrmann-Observatory Dresden, Germany

Wen-Biao Han Shanghai Astronomical Observatory Chinese Academy of Sciences Shanghai, China

ISSN 0941-7834 ISSN 2196-9698 (electronic) Astronomy and Astrophysics Library ISBN 978-3-030-19672-1 ISBN 978-3-030-19673-8 (eBook) https://doi.org/10.1007/978-3-030-19673-8 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: ‘Space-Time curvature’ (2019) by M. Soffel and W.-B. Han with the St. Marien (Rostock) astronomical clock overlaid (photograph by M.L. Preis). This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

At present, there is a vast number of textbooks on Einstein’s theory of gravity (general relativity, GR) that are available for different kinds of readers and at different levels of technical complexity. There is a series of classical treatments, e.g., Eddington (1922), Tolman (1934), Bergmann (1942), Weyl (1950), Pauli (1958), Fock (1959), Synge (1960), Adler et al. (1965), Fokker (1965), Rindler (1969), Carmeli et al. (1970), Landau and Lifshitz (1971), Weinberg (1972), Møller (1972), Misner et al. (1973), and Hawking and Ellis (1973), but also many more modern books, like Geroch (1978), Wald (1984), Schutz (1985), Woodhouse (2007) or Carroll (2013) to name just a few. In addition, there are the Living Reviews in Relativity, such as e.g. Will (2006) or Blanchet (2014), that can be downloaded from the web for free. Many books deal with tests of GR; the standard reference is Will (1993, 2006), but only a few deal with specific applications, e.g. in the important field of metrology. In the field of ‘Applied General Relativity’ it is essentially the books by Soffel (1989) and Kopeikin et al. (2011) where the reader can learn how general relativistic effects enter such fields as the realization of time scales, practical clock synchronization, satellite- and lunar laser ranging or very long baseline interferometry. Now, the first of these books is completely obsolete, whereas the second one is not really a textbook, written in a homogenous style where the reader should be able to understand the arguments step-by-step. In some sense, this book presents an improvement, extension and actualization of my old Springer book (Soffel 1989). This is especially true with respect to the selection of subjects treated in this book: the main emphasis lies on relativity in astrometry, celestial mechanics and metrology, thus on certain aspects of applied science. We have borrowed heavily from that book; some parts that we think are still up to date were taken almost literally (we have also borrowed several parts from Soffel and Langhans 2013). This book is clearly not a textbook on all aspects of Einstein’s theory of gravity (‘general relativity’, GR). Though some aspects related with exact solutions of the Einstein field equations are treated, the physics of objects with strong gravitational fields, like black holes, neutron stars or white dwarfs or gravitational waves, will not be discussed here. v

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In another sense, this is a completely new book. After Soffel (1989) came out, both, the theoretical relativistic formalisms and the observational techniques, have drastically been improved so that large parts of Soffel (1989) became obsolete. A good example for theoretical improvements is the Brumberg-Kopeikin DamourSoffel-Xu (BK-DSX) formalism for relativistic celestial mechanics. For the first time in history, a new formalism for treating the relativistic celestial mechanics of systems of N arbitrarily composed and shaped, weakly self-gravitating, rotating, deformable bodies was introduced. This formalism is aimed at yielding a complete description, at the first post-Newtonian approximation level, of (1) the global dynamics of such N -body systems (‘external problem’), (2) the local gravitational structure of each body (‘internal problem’), and (3) the way the external and the internal problems fit together (‘theory of reference systems’) (Damour et al. 1991; DSX-I). This BK-DSX formalism is based on the first post-Newtonian approximation of Einstein’s theory of gravity, and an extension to higher orders will be difficult. Nevertheless, it is sufficient for many applications at the present level of accuracies. The multipolar post-Minkowskian (MPM) formalism that has been worked out by Blanchet, Damour and Iyer (see Blanchet 2014 for an overview) is another example for that. Though important papers on that subject date back to the second half of the 1980s, it has only been in recent years that the MPM formalism has been worked out completely. The MPM formalism is able to describe the gravitational field of weak-field sources inside of some compact region basically to all orders of GM/(c2 R) in a single coordinate system and has been employed very successfully to the emission of gravitational waves from binary systems. Also the character of the book is very different from Soffel (1989). For example, a lot of work has been spend on didactical aspects, like a large number of (partially solved) exercises have been included. The title of the book, Applied General Relativity (AGR), points to two aspects: applied science on the one side and theoretical framework of GR on the other side. It is not difficult to realize that these two aspects usually are represented by two different expert groups. It is one goal of the book to illustrate to one of these groups the discipline of the other. The field of AGR has advanced to a multidisciplinary stage so that both groups should fertilize each other. Chapter 2 deals with the language of relativity: differential geometry, in which the reasons for that will be discussed later. This treatment is fairly standard. For more details, the reader is referred to the standard literature (e.g. Beyer and Gostiaux 1988; Pressley 2010; Bär 2011; Kobayashi and Nomizu 2014). Chapter 3 introduces Newtonian celestial mechanics. It starts with the Weak Equivalence Principle (universality of free-fall) and Newton’s theory of gravity. Of special interest is the exterior gravitational field of some matter distribution (body) and its description with multipole moments. Here, in addition to the usual spherical moments that are based upon an expansion of the exterior Newtonian potentials in terms of spherical harmonics, the expansion in terms of Cartesian symmetric and trace-free (STF) tensors is introduced. Whereas the spherical moments are very well known, e.g., for geodesists under the name of ‘potential coefficients’,

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this is still not the case for the STF moments. For experts in relativity, they play a crucial role, e.g. because Lorentz transformations usually are formulated in Cartesian coordinates. As was, e.g., nicely demonstrated by Hartmann et al. (1994) the use of STF moments can be employed very efficiently, like for the derivation of translational and rotational equations of motion in the N -body problem or for a representation of the tidal potential in Newtonian celestial mechanics. The Newtonian two-body problem of two mass monopoles moving under their mutual gravitational attractive forces is treated exhaustively since it serves as basis for a description of the relativistic (post-Newtonian) two-body problem. The usual classical celestial mechanical first-order perturbation theory is outlined. There are special topics in classical celestial mechanics that are of interest for the central subject of the book, e.g. the anomalous perihelion shift of planetary orbits due to the action of other planets, which can be treated with such a first-order perturbation theory. Later, it will also be used in connection with relativistic dynamical problems such as planetary or satellite motion. Chapter 4 is devoted to Maxwell’s theory of electromagnetism as an introduction to the field of relativity. Here, a metric tensor is introduced from a physical point of view for the first time, and the Lorentz transformation is derived. The electromagnetic field (Liénard-Wiechert potentials) of a moving point charge is discussed in some detail. The problem of the ‘speed of propagation’ in electromagnetism is exhaustively treated here. The attentive reader will ask for a reason for this. First, this problem is generally not understood very well, though all details can be found in the literature. One often hears that relativity means nothing moves faster than light in vacuum, which is absolutely not the case. Though causality is always assured by the use of retarded potentials, physics is full of superluminal speeds as explained in the text. To many readers, it might not be clear which velocity is restricted to the vacuum speed of light if the propagation of an electromagnetic wave through a dielectric medium is concerned. Another common error is that the use of retarded potentials implies retarded propagating action with the vacuum speed of light in all cases. In any case, this part sheds light on the problem of the ‘speed of gravity’ that has been the subject of many controversial discussions in the past. Chapter 5 introduces Einstein’s theory of gravity. The field equations are derived, and the problem of coordinates, the gauge problem in GR, is discussed. Observables have to be coordinate-independent quantities (as measured objects, they cannot depend upon a certain coordinate system that a theoretician employs for his calculations) and, therefore, have to be described as scalars. This implies that observers and parts of the measuring devices (e.g. in form of tetrad-vectors) have to be introduced explicitly into the formalism. For AGR, there are only three types of observables that play a central role besides proper time (a time that might be read-off some idealized clock): (1) the ranging observable as measurable time interval between emission and reception of some electromagnetic signal (as in laser ranging) (2) the spectroscopic observable presenting measurable frequencies of some incident light ray and (3) the astrometric observable as the angle between two incident light rays as seen by some observer.

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Chapter 5 also deals with the Landau-Lifshitz formulation of Einstein’s theory which presents the basis for the MPM formalism, a perturbation theory that formally can be extended to any order of corresponding small parameters. The LandauLifshitz formulation chooses the harmonic gauge from the beginning and works with √ the ‘gothic metric’ gμν = −gg μν instead of the usual one, g μν (g = det gμν ). Chapter 6 presents some exact solution of Einstein’s field equations in the vacuum that might be of some use for the field of AGR. I am not convinced that from a methodological point of view, it is preferable to start from some exact solutions before one deals with approximations: Einstein’s theory of gravity might be violated at some level of accuracy; to deal with exact models, a real problem has to be over-simplified and so on. Nevertheless, exact models sometimes might be of help how to extend a certain approximative framework to higher orders. In this chapter also, cosmologically relevant space times are introduced in relation to the following question: How does the global expansion of the universe influence the gravitational physics in the solar system? Chapter 6 also introduces multipole moments for stationary gravitational fields obeying the vacuum field equations and poses the properties of asymptotic flatness. Due to the definitions of Geroch (1970) and Hansen (1974), such field moments can be defined rigorously. Later, they will be related with body moments as integrals over the energy-momentum tensor of the body. Thorne in 1981 has introduced field moments differently by the structure of the metric tensor in special coordinates systems (asymptotically Cartesian and mass-centred coordinates) that were later shown to be equivalent to the Geroch-Hansen moments (Gürsel 1983). Chapter 7 introduces the post-Newtonian approximation of GRT as slowmotion, weak field approximation and the multipolar post-Minkowskian formalism. A canonical form of the metric tensor is discussed in the first post-Newtonian (PN) approximation, where the gravitational field is entirely described with two potentials only: a scalar potential w that generalizes the Newtonian gravitational potential U and a vector potential w that describes gravito-magnetic-type gravity resulting from matter currents (moving or rotating masses). The corresponding field equations are very similar to Maxwell’s equations of electromagnetism. This chapter also deals with the exterior field of a body to first PN-order, where post-Newtonian multipole moments (Blanchet-Damour moments) come into play, and the last part is devoted to the multipolar post-Minkowskian (MPM) formalism. First applications, which do not require a formalism for the gravitational N -body problem, are discussed in Chap. 8 on the basis of the first PN framework. Discussed are the gravitational field of the Earth, equipotential surfaces, clocks and time scales (TCG, TAI, TT, UTC) in the vicinity of the Earth and in the barycentric system (TCB, TDB), clock synchronization, the gravitational light deflection and time delay in the field of a single central body, the PN motion of torque-free gyroscopes and the satellite motion in the field of a rotating Earth to PN-order. Chapter 9 is devoted to the BK-DSX framework of relativistic celestial mechanics which is based upon a total of N + 1 different coordinate systems in the gravitational N-body problem: one global system with coordinates (ct, x) that, neglecting all matter outside the system of N bodies and assuming asymptotic

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flatness, extends to (spatial) infinity and is used to describe the overall motion of the N bodies and N local systems and one for each body A with coordinates (cTA , XA ) that is co-moving with body A and used for a description of physics (e.g., geophysics) in the local A-system. For many applications, the BK-DSX formalism is the best one, providing highest accuracy at present. Chapter 10 deals with the post-Newtonian gravitational N -body problem. Laws and equations of motion for the translational and rotational cases are discussed here. For a system of Pure mass monopoles, the famous Lorentz-Droste Einstein-InfeldHofmann equations (in harmonic gauge) for translational motion are derived. They form the basis for any modern high-precision numerical ephemeris, such as one from the DE, INPOP or EPM series (see e.g. Soffel and Langhans 2013). Chapter 11 is devoted to relativistic astrometry while Chap. 12 to relativistic metrology, where many techniques like pulsar timing, navigation by means of GNNS, satellite and lunar laser ranging (SLR and LLR), very long baseline interferometry (VLBI), etc. are theoretically described in a consistent relativistic framework. Of course, there are other books that deal with Applied General Relativity in one way or another. We would like to mention especially the book by Kopeikin et al. (2011), where many of the subjects are in common with those of this book. A comparison, however, reveals that these books are of very different character. It is the hope that this book fills the obvious gap in the field of ‘Applied General Relativity’. It is a textbook with a clear red thread, containing many exercises with solutions in most cases. We hope that we have not forgotten a field of application that really is of practical interest. It is a pleasure for us to thank all those people who have contributed to this book in one way or another: Andreas Bauch, Francisco Frutos, Franz Hofmann, Enrico Gerlach, Sergei Klioner, Sergei Kopeikin, Jürgen Müller, Gerhard Schäfer, Maximilian Schanner, Harald Schuh, Irina Tupikova and Sven Zschocke, to name just a few. Clearly, for all the mistakes, only the authors are responsible. Dresden, Germany Shanghai, China April 2019

Michael H. Soffel Wen-Biao Han

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Time and Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Relativistic Astrophysics and Cosmology. . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 7 9 11 13

2

Elements of Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Space-Time Manifold and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Coordinates, Differentials and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Symmetrization and Antisymmetrization . . . . . . . . . . . . . . . . . 2.3 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Lie-Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Curvature- and Ricci Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Metric Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Riemann Tensor and Its Symmetries . . . . . . . . . . . . . . . . . . . . . . 2.10 The Levi-Civita Symbol and Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1 Maximally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.2 Maximally Symmetric 3-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 GRTensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 16 19 19 20 21 25 26 30 32 36 38 40 41 42 43

3

Newtonian Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Newtonian Theory of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Newtonian Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 The Galilean Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Weak Equivalence Principle and Newtonian Theory of Gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 48 48 49

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3.3

3.4

3.5 3.6

3.7

3.8

4

5

Gravitational Field of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.1 Spherical Multipole-Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Spherical Mass-Moments of an Oblate Spheroid . . . . . . . . . 59 3.3.3 STF-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.4 Cartesian Multipole-Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 The Tidal Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.1 Newtonian Tidal Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.2 The l = 2 Tidal Potential for External Point-Masses . . . . . 75 Translational Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Rotational Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6.1 The Torque Resulting from an External Mass-Monopole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 The Newtonian 2-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.7.1 Integrals of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.7.2 Orbital Equation; Kepler’s First and Third Law . . . . . . . . . . 87 3.7.3 Classification of the Conic Sections . . . . . . . . . . . . . . . . . . . . . . . 89 3.7.4 Kepler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.7.5 Fourier-Analysis in the Elliptical Orbit . . . . . . . . . . . . . . . . . . . . 94 3.7.6 The Elliptical Kepler Orbit in Space . . . . . . . . . . . . . . . . . . . . . . . 95 Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.8.1 Variation of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.8.2 Perturbation Equations, Derived from Vectorial Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Electrodynamics and Special Theory of Relativity . . . . . . . . . . . . . . . . . 4.2.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Minkowskian Metric, Lorentz-Transformation . . . . . . . . . . . . . . . . 4.3.1 Addition of Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 General Coordinate Transformations and a Derivation of the Lorentz-Transformation. . . . . . . . . . . . . . . . . 4.4 The EM-Field of a Moving Point Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 The Speed of Propagation in Electromagnetism . . . . . . . . . . . . . . . . . . . . 4.5.1 The Vacuum Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Propagation in a Uniform Dielectric Medium. . . . . . . . . . . . . 4.6 Energy and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 116 116 120 127 128

Einstein’s Theory of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Einstein’s Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Motion of Test Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Einstein’s Theory of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 157 158 161 162

132 135 139 139 145 152

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5.5

xiii

The Problem of Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The Ranging Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 The Spectroscopic Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 The Astrometric Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tetrads and Tetrad Induced Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proper Reference Systems of Accelerated Observers . . . . . . . . . . . . . . The Landau-Lifshitz Formulation of GR. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 The Landau-Lifshitz Field Equations . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Harmonic Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 165 165 167 169 174 179 179 181

6

Exact Solutions—Field Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Minkowskian Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Stationary Space-Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Stationary Axially Symmetric Space-Times . . . . . . . . . . . . . . 6.2.2 The Hartle-Thorne Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Static Axially Symmetric Space-Times . . . . . . . . . . . . . . . . . . . 6.2.4 Spherically Symmetric Space-Time . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Kerr Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Boyer-Lindquist Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Cosmologically Relevant Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Cosmological Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Robertson-Walker Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 De Sitter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Schwarzschild: De Sitter Solution . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Field Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Geroch-Hansen Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Thorne Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 The FHP Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185 185 187 192 198 199 209 213 213 215 215 217 221 223 224 224 226 230

7

The Post-Newtonian and MPM Formalisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 The Post-Newtonian Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The General Form of the Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Field Equations and the Gauge Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 The External Post-Newtonian Field of a Body . . . . . . . . . . . . . . . . . . . . . 7.5 The Multi-Polar, Post-Minkowskian (MPM) Formalism . . . . . . . . . . . 7.6 Several Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 First Post-Minkowskian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8 The MPM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 The First PN Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.2 The MPM Iteration Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

235 235 236 241 246 253 254 256 272 274 283

8

First Applications of the PN-Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.1 Equipotential Surfaces and Relativistic Geoid . . . . . . . . . . . . . . . . . . . . . . 290 8.1.1 Post-Newtonian Equipotential Surfaces . . . . . . . . . . . . . . . . . . . 291

5.6 5.7 5.8

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8.2

The Problem of Time in the Vicinity of the Earth . . . . . . . . . . . . . . . . . . 8.2.1 Synchronization of Nearby Clocks . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Rates of Clocks in the Earth’s Vicinity . . . . . . . . . . . . . . . . . . . . 8.2.3 Synchronization of Clocks in the Vicinity of the Earth . . . 8.2.4 Coordinate Time Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 The Relation Between Coordinate and Proper Time . . . . . . 8.2.6 Clock Comparisons: Practical Aspects . . . . . . . . . . . . . . . . . . . . 8.2.7 TAI, TT and UTC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Barycentric Timescales TCB, Teph , TDB . . . . . . . . . . . . . . . . . . . . . . . . . . . Fairhead–Bretagnon Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Light-Rays in the PN-Field of a Single Body . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 The Celestial Sphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 The Astrometric Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 The Gravitational Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The PN Motion of a Torque-Free Gyroscope . . . . . . . . . . . . . . . . . . . . . . . Geodesic Motion in the PN-Schwarzschild Field. . . . . . . . . . . . . . . . . . . Celestial Mechanical Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Post-Newtonian Schwarzschild Effects. . . . . . . . . . . . . . . . . . . . 8.8.2 The Lense-Thirring Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 293 294 296 297 298 300 304 306 308 309 314 314 316 318 323 329 329 332

9

Astronomical Reference Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 The Problem of Celestial Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Transformation Between Global and Local Systems . . . . . . . . . . . . . . . 9.3 Split of Local Potentials, Multipole-Moments . . . . . . . . . . . . . . . . . . . . . . 9.4 Local Harmonic Proper Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 The Standard x μ → Xα Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 The Description of Tidal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Post-Newtonian Tidal Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 BCRS and the Expansion of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . .

337 337 338 343 348 352 354 354 364

10

The Gravitational N-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Local Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 The Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 The LD-EIH Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Laws of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 The PN Two-Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 The Brumberg Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 The Wagoner-Will Representation . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Damour-Deruelle Representation . . . . . . . . . . . . . . . . . . . . . 10.4 The Rotational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Landau-Lifshitz and Fock Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The PN-Spin in the N Body Problem . . . . . . . . . . . . . . . . . . . . . . 10.5 Rigidly Rotating Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Rigidly Rotating Multipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367 368 370 374 375 377 380 382 384 387 392 392 394 397 397 398

8.3 8.4 8.5

8.6 8.7 8.8

Contents

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Light-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Light-Rays for 1PN Stationary Multipoles. . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 The Shapiro Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 The Time Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 The TTF for a Body Slowly Moving with Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Light-Rays to Post-Minkowskian Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 The Shapiro Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 The Klioner-Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Relativistic Aberration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.2 Gravitational Light Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.3 Parallax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.4 Proper Motion and Radial Velocity . . . . . . . . . . . . . . . . . . . . . . . .

401 401 406 412 414

12

Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Pulsar Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 Pulsar Timing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 GNSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Global Positioning System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.2 GLONASS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 GALILEO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.4 BEIDOU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 SLR–LLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Satellite Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 Lunar Laser Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 VLBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 The Gravitational Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 The Geometrical Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.3 Radio Sources at Finite Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Doppler Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Passive Sagnac Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.1 Hipparcos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7.2 The Astrometric Project Gaia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

431 431 442 443 444 448 449 450 450 450 453 462 465 469 473 478 480 480 488 490 493

13

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Legendre-Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Ql (x) for x ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Relations for STF-Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Differential Geometry: Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Spherically Symmetric Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Spherically Symmetric Static Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 The Kerr Metric: Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497 497 498 499 501 502 503 504

415 416 421 423 424 425 425 426

xvi

Contents

13.7

13.8

Relations Concerning Multipole-Moments . . . . . . . . . . . . . . . . . . . . . . . . . 13.7.1 Multipole-Moments Derived from ξ -Moments . . . . . . . . . . . 13.7.2 Multipole-Moments Derived from Spherical Weyl-Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weyl-Moments as Functions of Mass Multipole-Moments . . . . . . . .

510 510 512 514

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

List of Acronyms

AC ADM adv ALGOS APOLLO BCRS BD BIPM can CMO DM EAL EFE EIH EM FP GCRS GD GLONASS GNSS GPS ILRS INRIM IPTA ISM JD LAGEOS LLR LL LNE-SYRTE LT

Astrographic Catalog Arnowitt-Deser-Misner Advanced Algorithm Involved in the Determination of TAI Apache Point Observatory Lunar Laser-ranging Operation Barycentric Celestial Reference System Blanchet-Damour Bureau International des Poids et Mesures Canonical Calculated Minus Observed Dispersion Measure Echelle Atomic Libre, a free time scale Einstein Field Equations Einstein-Infeld-Hoffmann Electromagnetic Finite Part Geocentric Celestial Reference System Geodetic Deviation Russian Satellite Navigation System Global Navigation Satellite Systems Global Positioning System International Laser Ranging Service Istituto Nazionale di Ricerca Metrologica (Torino, Italy) International Pulsar Timing Array Interstellar Medium Julian Date Laser Geodynamics Satellite Lunar Laser Ranging Landau-Lifshitz Laboratoire national de métrologie et d’essais (Paris, France) Lense-Thirring xvii

xviii

MPM NANOGrav NIST NPL OCA PN PPS PPTA PRN PTA PTB ret RF SATRE SLR STF TAI TCB TCG TIC TIC TS TTF TT TWSTFT UT1 UTC VLBI

List of Acronyms

Multipolar Post-Minkowski North American Nanohertz Observatory for Gravitational Waves National Institute of Standards and Technology (Boulder, Colorado, USA) National Physical Laboratory (Middlesex, GB) Observatoire de la Côte d’Azur Post-Newtonian Pulse Per Second Parkes Pulsar Timing Array Pseudo Random Noise Pulsar Timing Array Physikalisch-Technische Bundesanstalt (Braunschweig, Germany) Retarded Radio Frequency Satellite Time and Ranging Equipment Satellite Laser Ranging Symmetric and Trace-Free International Atomic Time Barycentric Coordinate Time Terrestrial Coordinate Time Tetrad-Induced Coordinates Time Interval Counter Time Scale Time Transfer Function Terrestrial Time Two-Way Satellite Time and Frequency Transfer Phase Angle of Earth’s Rotation Coordinated Universal Time Very Long Baseline Interferometry

List of Symbols

List of symbols used in the text (acronyms can be found at the end of the book) t T xμ dx μ μν Tαβ Lv μ νλ μ A;ν R ρ αμν Rμν Gμν gμν ds g |M| [μ1 μ2 . . . μn ]: μ1 μ2 ...μn U Utidal (Clm , Slm ) Jl , Jl TˆL Mˆ L SˆL GL HL f E

Physical time coordinate, especially TCB Time coordinate, especially T = TCG Coordinates Coordinate differential Tensor Lie derivative in the direction of v Affine connection, usually Christoffel symbol Covariant derivative of Aμ with respect to x ν Curvature tensor Ricci tensor Einstein tensor Metric tensor Infinitesimal distance between neighbouring points in a manifold Determinant of gμν : det gμν Determinant of matrix M Levi-Civita symbol Levi-Civita tensor Newtonian gravitational potential Newtonian tidal potential Potential coefficients, spherical mass moments Spherical mass moments for axial symmetry and their dimensionless counterpart STF part of TL STF mass moments STF spin moments STF (gravito-electric) tidal moments STF (gravito-magnetic) tidal moments True anomaly Eccentric anomaly xix

xx

S, T , W M E, B jμ F αβ Aμ kμ uμ Rai T μν μ e(α)

(a)(b) μ zobs ωμ w E tmixed or t± gμν αβ tLL B , G , C δ(x) f (n) (x)

i , e w, wi kn , hn fp Xα

μν  ∂i , ∂i⊥ tobs mI , mG ηN Pl (x), Ql (x)

List of Symbols

Radial, transverse and normal component of the perturbing function Mean anomaly EM field vectors EM current density EM field tensor EM vector potential Wave vector, tangent to some light ray Four-velocity Rotation matrix Energy-momentum tensor Tetrad (field) Ricci rotation coefficients World-line of observer Twist four vector Twist three vector Complex Ernst potential (1/2)(tret + tadv ) Gothic metric tensor Landau-Lifshitz pseudotensor Expansion parameters Dirac delta function (distribution) nth derivative of f with respect to x Inner and exterior zone Potentials of the PN-metric Love numbers Plasma frequency Local coordinates, usually geocentric coordinates Landau-Lifshitz complex Partial derivative with respect to x i parallel and perpendicular to some three vector n Time of observation Inertial and gravitational mass Nordtvedt parameter Legendre functions of the first and second kind

Chapter 1

Introduction

In 1905, Albert Einstein published four papers that changed modern science fundamentally. Among them there was an article on the electrodynamics of moving bodies that laid the foundation of Special Relativity Theory (SRT) (Einstein 1905). About 10 years later, Einstein revolutionized Newton’s theory of gravitation and formulated a space-time geometrized picture of the gravitational interaction: Newton’s gravitational force was replaced by the curvature of space-time (Einstein 1915). Actually Einstein’s theory of gravity (General Relativity) was an applied science form the very beginning when in 1915 Einstein derived the anomalous perihelion advance of Mercury, thus solving the most important problem of celestial mechanics of the nineteenth century (Roseveare 1982; Pais 1992). In 1859, Urbain Le Verrier was the first to report that the slow precession of Mercury’s orbit around the Sun could not be completely explained by Newtonian mechanics. For the first time in history he was able to compute the perturbations of the known planets onto Mercury’s perihelion advance: 153.6 from Jupiter, 277.8 from Venus, 90.0 from Earth, 7.3 from Saturn and 2.5 from Mars (in arc s/century). Together with the general precession of the classical astronomical reference system at epoch 1900 of 5025 .6/cen. the calculations yield a total value of 5557 .0/cen., whereas the observed value amounts to 5599 .7/cen. (Will 1993). So the anomalous perihelion advance of Mercury is about 43 /cen., that was immediately derived from Einstein’s General Relativity without further obscure assumptions such as, e.g., the existence of a new planet, Vulcan, near the Sun. In addition to that General Relativity was able to explain the deflection of light by the gravitational field of the Sun that amounts to 1.75 for a light-ray that just grazes the limb of the Sun. Since the light deflection angle decreases like 1/r with increasing distance from the Sun, for light rays incident at about 90◦ from the Sun the angle of light deflection still amounts to 4 mas (milliarcseconds). In Einstein’s theory of gravity light-rays move along geodesics (“shortest curves between two points”) in curved space-time and the curvature of space deflects an © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_1

1

2

1 Introduction

observed stellar image outwards from the Sun. Historically, the light deflection in the gravitational field of the Sun had been first detected by the British expeditions to Sobral (Brazil) and Principe (Gulf of Guinea) taking photographic pictures of the solar vicinity during the solar eclipse on the 29th May, 1919. These measurements, though provided with large errors, confirmed the predictions from GRT and helped to make Einstein famous (e.g., Weinberg 1972; Soffel 1989; Will 1993). Already in 1916, de Sitter (1917) derived a relativistic precession of the lunar orbit about the Earth when moving in the gravitational field of the Sun, now called the de Sitter effect or geodetic precession. In 1918, Lense and Thirring (Lense 1918; Thirring 1918, 1921; Lense and Thirring 1918; Mashhoon et al. 1984) predicted the precession of some satellite orbit about a central gravitating body due to its relativistic gravito-magnetic field induced by its rotational motion. Despite of these early ‘applications’ General Relativity for a long time was mainly understood as some esoteric, mathematically oriented discipline far from ‘real applications’ in science and technology.

1.1 Time and Reference Systems This situation, however, changed with the development of clocks with high accuracy and stability. A clock in principle is a frequency generator or oscillator with stochastic properties. Its accuracy describes the capability to realize the SI-second, whereas stability refers to the fluctuations around some averaged clock-frequency. Figure 1.1 shows the increase in accuracy of mechanical and atomic clock until the year 2000, when stabilities of order 10−15 for cesium fountains were achieved. In 1927 the first quartz clock was built by Marrison and Horton (1928) at Bell Telephone Laboratories in Canada, whereas the first practical cesium atomic frequency standard was built at the National Physical Laboratory in England in 1955 by Louis Essen and Jack Parry (Fig. 1.2). Though cesium clocks especially in the form of fountains achieve a remarkable level of stability, they are clearly limited. A stability of order 10−15 can be achieved only after averaging over a day which presents a problem for real time applications. Now, the achievable stability of an atomic clock is related with the clock frequency that for cesium clocks lies in the microwave region at 9.2 GHz. Meanwhile optical clocks are built with clock frequencies of about 1015 Hz, where comparable stabilities can be achieved already after a second (Soffel and Langhans 2013). Figure 1.3 shows the evolution of fractional frequency uncertainties for atomic clocks based on microwave and optical transitions. Likely in the near future optical clocks will be superior to conventional atomic clocks. For more details on optical clocks the reader is referred e.g., to Ludlow et al. (2015) and references cited therein. Meanwhile fractional frequency uncertainties in the 10−18 region have been reported, e.g., for the 27 Al+ single ion clock at NIST (Chou et al. 2010) and the 87 Sr optical lattice clock at JILA (Bloom et al. 2014) (from Poli et al. 2013). Such a stable clock would be off by less than a

1.1 Time and Reference Systems

3

Fig. 1.1 Achieved accuracies of mechanical and atomic clocks until the year 2000 (Image credit: Bauch 2018)

second after the age of our universe (about 14 billions years or 4 × 1017 s). This is an amazing progress indeed that will change our world in a variety of ways. Already in 1905 Einstein showed that a clock moving with  speed v with respect to some observer appears to be slowed down by a factor of 1 − (v/c)2 . Note, that for two clocks moving with constant speed with respect to each other the situation is completely symmetrical for the two clocks. If, however, this symmetry is broken, e.g., if any of the two clocks undergoes phases of acceleration than they will display different times if brought together. In General Relativity it is the curvature of space-time that describes the gravitational interaction. The curvature in the time is difficult to visualize; it is related with the gravitational redshift: gravitational redshift if some monochromatic electromagnetic signal passes through a gravitational field in the direction of a smaller Newtonian potential U , then its frequency will be red-shiftet. As consequence the natural frequencies of two clocks, f1 and f2 , located at x1 and x2 respectively, are related by f2 /f1 = 1 + [U (x2 ) − U (x1 )]/c2 so that clocks in a gravitational field are slowed down. To illustrate this point, let us consider a GPS satellite that moves at 3874 m/s with respect  to the geocenter. According to SRT, GPS clocks are slowed by a factor of 1 − (v/c)2 1 + 8.3 × 10−11 with respect to some inertial geocentric observer. This time dilation would cause a time error of 7.2 μs per day, and this should be compared with the accuracy of the GPS system time of some tens of nanoseconds (Müller et al. 2008). The contribution from the space-time curvature is even larger: 1 − U/c2 1 + 5.28 × 10−10 , where U is the difference between

4

1 Introduction

Fig. 1.2 The world’s first caesium-beam atomic clock represented a revolutionary advance in timekeeping. It was designed by Louis Essen (right) and fellow physicist, Jack Parry (left), and was constructed at the National Physical Laboratory (NPL) in 1955. Although it was not the first apparatus to use atoms for timekeeping, it was the first to keep time better than any other clock in existence, including both pendulum and quartz clocks (Image credit: National Physical Laboratory, UK)

Fig. 1.3 Evolution of fractional frequency uncertainties of atomic frequency standards based on microwave (Cs clocks) and optical transitions (Reproduced with kind permission of Società Italiana di Fisica from Poli et al. 2013)

1.1 Time and Reference Systems

5

the gravitational potential between the Earth’s surface and satellite altitude. As the sign of the potential part is opposite to the sign of the velocity part, one obtains a total of +4.45 × 10−10 , i.e., GPS satellite clocks run faster by about 38 μs per day than the receiver clocks on Earth (Müller et al. 2008). Because the observed time indicated by an atomic clock (proper time τ as idealization) depends upon the clock’s velocity and gravitational potential at the clock’s location one has the problem to define useful timescales, i.e., time coordinates valid in a certain part of space and time, and to relate them to the readings of a certain set of atomic clocks. In general, a time coordinate is only one part of a space-time coordinate system such as the Geocentric Celestial Reference System (GCRS) with coordinates (T , X), usually chosen with origin in the geocenter. The coordinate T of the GCRS, called Geocentric Coordinate Time TCG, Geocentric Coordinate Time is the basic timescale for physics in the immediate environment of the Earth (especially for geodesy and geophysics) (Soffel et al. 2003). TCG is realized indirectly from atomic clocks indicating proper time τ due to the relation dτ/d(TCG) 1 − (U + v2 /2)/c2 . For an Earth-bound clock co-rotating with the Earth this relation reads dτ/d(TCG) 1−Ug /c2 , where Ug is the gravity potential, including both gravitational and rotational potentials. Thus, on an equipotential surface Ug = const. the rates of atomic clocks are the same, whereas for a height difference of R the relative difference in frequency f/f (GM/c2 R) × (R/R), where R is the radius of the Earth and (GM/c2 R) = 7 × 10−10 . Hence, for R = 1 km the corresponding rates differ by about 10−13 and clocks with 10−18 stability should be able to measure height difference of less than 1 cm. This offers the possibility to determine small differences if the gravity potential by comparing the readings of optical clocks at different locations and to bridge the gap between the global scale geometry of the geoid (determined by dedicated satellite such as GOCE) and the local one. To this end glass-fiber networks will be employed to compare the rates of different clocks possibly 1000 km apart. Such a comparison of optical clocks related with extremely accurate time and frequency measurements, will have a variety of applications, e.g., in fundamental physics (e.g., Kozlov et al. 2018), where one searches for temporal changes of fundamental constants such as the fine-structure constant α or the electron to proton mass ratio. In oceanography one is interested in the sea surface topography relative to the geoid to deduce ocean currents that might be influenced by climate changes. Practically the time scale TCG is realized by International Atomic Time (TAI) International Atomic Time or Terrestrial Time (TT). Originally TT was defined in IAU Resolution A4 (1991) as: “a time scale differing from the Geocentric Coordinate time (TCG) by a constant rate, the unit of measurement of TT being chosen so that it agrees with the SI (Système international d’unités) on the geoid”. Some shortcomings appeared in this definition of TT when considering accuracies below 10−17 because of the uncertainty in Ug on the geoid and the realization of the geoid (Müller et al. 2008; Petit 2003). For this reason, the IAU decided to fix the TT-TCG relation by d(TT)/d(TCG) = 1 − LG with a defining constant

6

1 Introduction

LG = 6.969290134 × 10−10 ensuring continuity with the current best estimate for Ug = 62,636,856 (m/s)2 at the geoid (Groten 2000). Terrestrial Time TT is realized from international atomic time TAI via its relation TT = TAI + 32.184 s . TAI is derived form the readings of more than 400 clocks in various laboratories around the globe. These readings, after a reduction to some ‘quasi-geoid’ (compatible with TT), first generate some free time scale called Echelle Atomique Libre (EAL). From the EAL, the frequency of TAI is finally tuned by a few number of primary frequency standards (PFS). Finally, the time scale UTC, coordinated universal time, differs from TAI by a certain number of leap seconds to ensure that the difference between UTC and a certain Earth’s rotation angle, UT1, is always smaller than 0.9 s. This has the advantage that over a very long time span, UTC stays related with the Sun via the UT1 which has advantages for ordinary life. Without the introduction of leap seconds, UTC–UT1 would show a secular drift mainly due to the secular decrease of the Earth’s rotation rate resulting from tidal effects. The various zonal times (national times) usually differ from UTC by an integer number of hours according to a division of the Earth’s surface into time zones. TCG, TT, TAI and UTC are geocentric time scales that should be used for the description of physics in the vicinity of the Earth in some suitably chosen reference system the is co-moving with the Earth: they are geocentric time scales. Especially TCG is the time coordinate of the Geocentric Celestial Reference System (GCRS), whose origin is usually chosen to agree with the geocenter. For other purposes, e.g., for planetary ephemerides or interplanetary spacecraft navigation, barycentric time scales should be used. Barycentric coordinate time, TCB, is the time coordinate of the Barycentric Celestial Reference System (BCRS), with origin in the barycenter of the solar system. The realization of TCB is achieved with its relation to TCG:   d(TCG)/d(TCB) 1 − c−2 Uext (zE ) + (1/2)v2E . Some of the best solar system ephemerides, the DE ephemerides of the Jet Propulsion Laboratory (JPL), do not employ TCB as basic time variable (Soffel and Langhans 2013). The original idea was to use a time scale Teph that differs from TT practically only by periodic terms which cannot be realized with ultimate precision due to arbitrarily long periods in the motion of the solar system. For that reason another barycentric time scale called TDB has been defined by TDB = TCB − LB × (JDTCB − T0 ) × 86,400 s + TDB0 with T0 = 1.550519768 × 10−8 , TDB0 = −6.55 × 10−5 s. JD is Julian Date and the value for LB = 1.550519768 × 10−8 was chosen so to minimize the linear drift between TDB and TT for the ephemeris DE405. The planetary ephemerides developed by Paris Observatory called INPOP (Fienga et al. 2009), is 4-dimensional, i.e., in addition to positions and velocities of solar system bodies it provides TT-TCD values since INPOP08. The establishment of time scales is not possible without the procedures of clock synchronization and time dissemination. Nowadays clock synchronization is accomplished by means of Global Navigation Satellite Systems (GNSS) or TwoWay Satellite and Frequency Transfer (TWSTFT). Using carrier phase TWSTFT

1.2 Space

7

measurements (TWCP) for frequency transfer accuracies of order a few picoseconds at 1 s integration time and a transfer stability of order 10−14 at 100 s have been reported for intercontinental distances (e.g., Schäfer et al. 1999). For very long baselines up to 9000 km short-term stabilities of order 10−13 (at 1 s) have been obtained (e.g., Fujieda et al. 2014). Since relativistic ‘effects’ are of order 7 × 10−10 they clearly have to be taken into account. Since the Einstein procedure of clock synchronization (two clocks with constant distance are synchronized by some central device, located exactly in the middle between the two clocks, that emits electromagnetic signals to the two clocks simultaneously. The two clocks can then be synchronized from the arrival times of the signals) is not possible on the rotating Earth (due to the Sagnac effect in time) clock synchronization is usually a coordinate time synchronization using the relation between proper time and TCG: two clocks showing proper times τ1 and τ2 are synchronous if their corresponding TCG-values agree. Note, that the Sagnac effect can amount to hundreds of nanoseconds; a GPS timing error of one nanosecond can lead to a navigational error of 30 cm (Ashby 2004).

1.2 Space Geodetic or astronomical measurements of ‘space’ and thus the realization of spatial coordinates, usually involve electromagnetic signals being emitted by some observer, reflected by some device so that at least some photons return to the observer and the propagation time interval between emission and reception of the signal is measured. Measurement of ‘space’ then is based upon measurements of time. For short distances the geodesist uses tachymeters for local measurements of ‘space’. Of tremendous importance is the establishment of the spatial parts of geodetic-astronomical reference frames. For the Earth it is the International Terrestrial Reference System, the ITRS, with the ITRF as practical realization. The ITRS is geocentric with a center of mass referring to the whole Earth including oceans and atmosphere. Its spatial orientation was initially given by a BIH orientation at 1984.0 and the time evolution of the orientation is involving a plate tectonic model via a ‘no-net-rotation’ condition. Since 1988 the ITRS was presented in form of 13 realizations (ITRF89–ITRF2014), distributed by the International Earth Rotation and Reference Systems Service (IERS; formerly called: International Earth Rotation Service). Each realization estimates the geocentric coordinates and velocities of a set of stations observed by the Global Positioning System GPS, SLR (Satellite Laser Ranging), LLR (Lunar Laser Ranging), DORIS (Doppler Orbitography by Radiopositioning Integrated on Satellite) and VLBI (Very Long Baseline Interferometry) thus carefully mapping the complicated motion of the Earth’s crust. Relativity enters in many places, in metrology, satellite dynamics, dynamics of the solar system, signal propagation etc. that will be discussed in the main part of the book.

8

1 Introduction

The International Celestial Reference System (ICRS) presently is the standard celestial quasi-inertial (i.e., with respect to rotations) reference system that extends far into the universe to the most distant objects in space. Its realization, the ICRF, is in the form of catalogues of extragalactic radio sources (mainly quasars), whose positions and structure images are obtained with VLBI. The original ICRF1 was adopted by the IAU in 1998, the update ICRF2 in 2009 and the ICRF3 in 2018. The ICRF3 contains positions of 4536 extragalactic sources out of which 303 have been identified as defining sources. Presently global distances on the Earth’s surface can be determined via VLBI with accuracy of a few mm; now, an accuracy of one millimeter corresponds to a light travel time of about 3 ps and in a corresponding geodetic VLBI model all terms down to the order 0.3 ps should be taken into account (Heinkelmann and Schuh 2010). At this level of accuracy a large number of relativistic ‘effects’ have to be taken into account, such as those resulting from the mass-monopole and quadrupole of solar system bodies to post-Newtonian order, post-post Newtonian terms related with the solar mass, velocity effects etc. The ITRS might be directly related with the GCRS, the ICRS with the BCRS and the corresponding coordinates (t, x) (BCRS) and (T , X) (GCRS) are related by complicated space-time transformation that will extensively be discussed in this book. The applied aspects of relativity in the modern age can nicely be seen when one considers the motion of the ITRS with respect to the GCRS, that classically is split by introducing some intermediate system into an astronomically dominated part (precession-nutation, length of day (LOD) variations) and a geophysically dominated part (polar motion). This motion is described by Earth orientation parameters (EOP) whose dynamics results from the physics of the various subsystems of the Earth (atmosphere, ocean, hydrosphere, cryosphere, elastic mantle, fluid outer core, solid inner core) and their complex interactions, including the tidal effects from the Sun, Moon and planets (Fig. 1.4). From this important information e.g., for global climate variations and even for anthropogenically induced environmental changes can be derived. Figure 1.5 clearly demonstrates that El Niño events can be seen in VLBI data on LOD variations. Periodically, El Niño appears every 3– 7 years. The Spanish word El Niño means child but actually means Christ child, since this phenomenon usually appears around Christmas time. With the beginning of EN, the water in front of the coast of Peru becomes warmer and fish food from the cold water disappears. El Niño begins with a weakening of the trade winds who can even change their direction. Usually the trade winds blow very violently. After an EN event during one to 2 years again normal pressure gradients built up again and the trade winds blow normal again in the tropics of the Pacific. These oscillations of air pressure are also called Southern Oscillation, linked with El Niño this is called: El Niño SOUTHERN OSCILLATION (ENSO). It is not only the economically significant El Niño events that one can derive from VLBI data on EOPs, but also affects as mean sea level rise, melting of ice masses at the polar caps or glaciers. Most parts of your complicated system Earth can be monitored with modern geodetic techniques that for reaching utmost precision require models where relativity should be considered.

1.3 Astrometry

9

Fig. 1.4 Components and influences in the system Earth. Direct effect are indicated by solid lines, influences and effects in relation with the deformation of the Earth by dashed lines (from: Schuh 2003, modified)

1.3 Astrometry Astrometry, the discipline to measure stellar positions and velocities, has made a tremendous progress with the space astrometric missions Hipparcos and Gaia. The ESA satellite mission Hipparcos (1989–1993) measured the positions of about 120,000 stars with the precision of about 1 mas (milli-arcsec), the Gaia mission (satellite launch: 19.12.2013) reaches incredible accuracies depending on stellar magnitudes: about 4 μas for very bright stars. This corresponds roughly to the appearance of a one EUR coin on the Moon as seen from the Earth! Up to now Gaia data sets have been released containing information on 1.7 billion stars, quasars, asteroids and galaxies. This includes precise measures of distance and motion across the sky, brightness and colours for 1.3 billion stars (our Milky Way has about 100 billion stars), radial velocities for 7 million stars, stellar parameters for some 100 million stars, variability over time for 550,000 stars, and accurate orbital data for 14,000 asteroids. It should be clear that at the μas level of accuracy quite a refined relativistic model has to be used for the data analysis. In practice it is the relativistic Klioner-model called Gaia Relativity Model (GREM), extensively described in this book, that is used for the Gaia mission. The astrometric, photometric and spectral

10

1 Introduction

Fig. 1.5 Time series of daily LOD variations (black), atmospheric angular momentum values (blue) around three extreme El Niño events. The dashed red line presents the (scaled) monthly Niño 3.4 index, referring to Sea Surface Temperature variations in the Niño 3.4 region (5◦ N–5◦ S, 120–170◦ W). The ticks on the x-axis indicate the first day of each month (from: Lambert et al. 2017)

data from the Gaia mission will clearly revolutionize all parts of astrophysics. It will provide new insights into – – – – – – – –

the origin and evolution of our Milky Way galaxy, stellar physics, stellar multiple systems, the field of Exo-planets, solar system bodies and their dynamics, the realization of astronomical reference frames, fundamental physics, quasars and distant galaxies.

For example, in the field of fundamental physics tests of Special and General Relativity will be performed and hopefully the Gaia data will shed some new light on the problem of dark matter. For more details the reader is referred to the literature (see the websites of ESA and the publications related with the Gaia mission).

1.4 Celestial Mechanics

11

1.4 Celestial Mechanics We have already mentioned the relativistic perihelion advance of Mercury’s orbit due to the relativistic mass monopole of the Sun of order 42. 98/cen.; this relativistic orbital precession amounts to 8.63 for Venus, 3.84 for the Earth, 1.35 for Mars, 0.06 for Jupiter and 0.01 for Saturn (all in  /cen.). Even for the motion of certain asteroids this relativistic perihelion precession should be taken into account. E.g., this orbital motion amounts to 0. 101/y for 1566 Icarus, 0. 043/y for 2062 Aten and 0. 101/y for 3200 Phaeton (Shadid-Saless and Yeomans 1994). In modern solar system ephemerides also the relativistic effects arising from the gravitational fields of the planets should be taken into account. For that reason the post-Newtonian equations of motion for a whole set of mass-monopoles (‘point masses’), the Einstein-Infeld-Hoffmann (EIH) equations are the basis of the three state-of-the-art solar system ephemerides: the American one, DE (Development Ephemeris; JPL); the Russian one, EPM (Ephemerides of Planets and the Moon; IPA, St. Petersburg); and the French one, INPOP (Intégration Numérique Planétaire de l’Observatoire de Paris) (Soffel and Langhans 2013 and references quoted therein). With respect to the motion of artificial Earth satellites we have to keep in mind that the gravitational radius of the Earth RG = GME /c2 is about 0.44 cm. This implies that for a model of satellite motion relativistic terms become important for high precision orbit determination which is e.g., possible for the LAGEOS, LAGEOS II, and LARES. If an orbit cannot be determined with cm accuracy or better relativistic effects might be absorbed in the orbital parameters (Soffel and Frutos 2016). Figure 1.6 shows a variety of accelerations of some Earth satellite in km/s2 as function of the orbits semi-major axis in km. The red curves labelled by ‘Moon’,‘Sun’, ‘Venus’ and ‘Jupiter’ refer to the (Newtonian) tidal accelerations. The contributions from the various zonal harmonics of the Earth are indicated by the green curves with the corresponding index. The three dominant relativistic accelerations in the GCRS are: (1) the contribution from the post-Newtonian spherical field of the Earth (the Schwarzschild acceleration) (dotted blue curve, labelled rel. Monopole), (2) the Lense-Thirring acceleration due to the gravitomagnetic field of the rotating Earth (dotted red curve) and (3) the relativistic acceleration due to the oblateness of the Earth (dotted blue curve, labelled rel. Quadrupole). In addition to that since the GCRS is NOT (locally) inertial we face a relativistic Coriolis force related with geodesic precession. This leads to ˙ GP of satellite orbits; for the LAGEOS orbit this is an additional nodal drift

of order 17.60 mas per year. For the LAGEOS orbit we also included estimates of the direct solar radiation pressure (), (e.g. Anselmo et al. 1983), the Earth albedo (), infrared pressure (∇) and the atmospheric drag (maximal and minimal values) (Rubincam 1982). The orders of magnitude imply that for a measurement of relativistic effects (maybe apart from the Schwarzschild term) by means of SLR data from a single satellite, the even zonal harmonics of the Earth have to be known with extreme precision. E.g., the secular nodal drift of the LAGEOS orbit due the LenseThirring effect of the rotating Earth is of order 2 × 10−5 per revolution, roughly

12

1 Introduction

Fig. 1.6 Various accelerations of an Earth’s satellite as function of the orbit’s semi-major axis. Other orbital parameters: e = 0.004, I = 110◦ , = 197◦ , ω = 72◦ . The vertical line is for the semi-major axis of LAGEOS, a = 12,270 km. The various curves and symbols are explained in the text (from: Soffel 1989 modified by Schanner 2019)

comparable with the effect from the l = 12 multipole. Ciufolini and colleagues (Ciufolini 1986a,b; Ciufolini and Pavlis 2004; Ciufolini et al. 2010, 2016; see also Iorio 2009a,b), however, succeeded to measure the Lense-Thirring effect with the orbital data of LAGEOS, LAGEOS II and LARES with a precision of a few percent.

1.5 Relativistic Astrophysics and Cosmology

13

1.5 Relativistic Astrophysics and Cosmology For many researchers there are additional fields of Applied General Relativity of great importance, especially the fields of relativistic astrophysics and cosmology. The physics of compact objects (white dwarfs, neutron stars, black holes), of active galactic nuclei, quasars, dark matter, dark energy, the structure and evolution of the universe on all scales are very exciting and up to date topics of general interest. These topics have been treated exhaustively in the literature (e.g., Weinberg 1972; Rees et al. 1974; Kolb and Turner 1994; Zeldovich and Novikov 1997; Börner 2003; Dodelson 2003; Hyong 2006; Maggiore 2007; Demia´nski 2008; Weinberg 2008; Belusevic 2008; Giacconi and Ruffini 2009; Straumann 2012; Liddle 2015; Böhmer 2016; Ryden 2016; Maggiore 2018 to name just a few) and are not included in this book.

Chapter 2

Elements of Differential Geometry

2.1 Space-Time Manifold and Fields Physics deals with the behavior of certain objects (particles, bodies, fields, etc.) in 3-space in course of time. Formally, time and space can be combined to a 4dimensional space-time, though the two entities, space and time, are always clearly distinguished. Usually 4-dimensional space-time is described by the mathematical picture of a manifold, a 4-dimensional space that locally looks like the Euclidean R4 . Clearly this manifold picture is an idealization since arbitrarily small distances in space or time cannot be measured in principle. Moreover, because of quantum effects, one expects the manifold picture to break down for distances of order the Planck-length lP = (Gh/c ¯ 3 )1/2 1, 6 × 10−35 m or smaller. Presently, it is unclear how to describe gravitational physics at such small length scales. Both the electromagnetic as well as the gravitational interaction are described with fields. Such fields are usually defined over a space-time manifold and characterize geometrical, i.e., coordinate independent physical objects (though below we will use coordinate components to describe them). Fundamental laws of physics, that are obeyed by such fields, are differential relations between the fundamental geometrical objects and, therefore, can also be formulated in a coordinate independent manner; they are formulated in the language of differential geometry. Using geometrical fields, like tensor fields, and coordinate independent laws of physics is often described as ‘principle of covariance’, though it basically is no ‘principle’, but a language form. E.g., in Chap. 3 on Newtonian Celestial Mechanics it is shown how Newton’s theory of gravity can be formulated in a covariant way. Of course one way nature shows us lies in the selection of fields that one must use to describe the physical reality. The so-called equivalence principle implies that the gravitational interaction can be described with a single metric field. If, however, the equivalence principle breaks down at some level, then additional fields might play a role in the description of the gravitational interaction. © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_2

15

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2 Elements of Differential Geometry

2.2 Coordinates, Differentials and Tensors Mathematically a manifold (Fig. 2.1, an example of a two-dimensional manifold) is a triple (M, {Uα , α }), where M is a set of points, {Uα } a collection of open sets in M with M = ∪α Uα and α are differentiable functions Uα → Rn . For each point p ∈ M, there is at least one Uα with p ∈ Uα ; then α defines a local coordinate system in a surrounding Uα of p. In a certain region R of an N -dimensional manifold M the various points of R are described by coordinates x μ : R → Rn , x μ = (x 1 , x 2 , . . . , x n ). Example 2.1 An elementary example is the 2-dimensional Euclidean space that can be described by Cartesian coordinates (x 1 , x 2 ) = (x, y) or by polar coordinates with x  1 = r, x  2 = φ. The relation between these two sets of coordinates reads  x = r cos φ ; r = x2 + y2 (2.2.1) y = r sin φ ; φ = arctan y/x . Example 2.2 As an another example we study the 3-dimensional Euclidean space. It can be described by Cartesian coordinates x μ = (x, y, z) . Alternatively, we might use spherical coordinates (Fig. 2.2) x ν = (r, θ, φ) .

Fig. 2.1 The surface of a torus as a two-dimensional manifold

2.2 Coordinates, Differentials and Tensors

17

z

Fig. 2.2 Spherical coordinates in the Euclidean R3

r y x

The relation between two such sets of coordinates is a coordinate transformation x μ → x ν with  r = x 2 + y 2 + z2  x2 + y2 (2.2.2) θ = arctan z y φ = arctan . x The inverse transformation reads x = r sin θ cos φ y = r sin θ sin φ

(2.2.3)

z = r cos θ . Objects dx μ are called: coordinate differentials. Coordinate differentials transform under coordinate transformations according to the chain-rule. E.g., for the 3-dimensional Euclidean space in Cartesian and spherical coordinates we have: dx =

   3   ∂x ∂x ν dx dx ν , ≡ ∂x ν ∂x ν ν=1

=

∂x ∂x ∂x dr + dθ + dφ ∂r ∂θ ∂φ

= (sin θ cos φ) dr + (r cos θ cos φ) dθ − (r sin θ sin φ) dφ dy =

∂y ∂y ∂y dr + dθ + dφ ∂r ∂θ ∂φ

= (sin θ sin φ) dr + (r cos θ sin φ) dθ + (r sin θ cos φ) dφ ,

(2.2.4)

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2 Elements of Differential Geometry

dz =

∂z ∂z ∂z dr + dθ + dφ ∂r ∂θ ∂φ

= cos θ dr − r sin θ dθ . Here the Einstein’s summation convention was employed: over every pair of indices, one contravariant upper index and one covariant lower index, a summation is employed automatically even when the summation-symbol is dropped. Generally we write  dx = μ

∂x μ ∂x ν



dx ν .

(2.2.5)

The matrix (∂x μ /∂x ν ) is called the (inverse) Jacobi-matrix. Objects that transform like coordinate differentials dx μ are called contravariant vectors:  ν  ∂x Aμ . Aμ → Aν : Aν = (2.2.6) ∂x μ μ ...μ

n 1 Quantities Tν1 ...ν m are called n-fold contravariant, m-fold covariant tensors if under a coordinate transformation they transform according to

...λm T λσ11 ...σ n

 =

∂x λ1 ∂x μ1







∂x λn ··· ∂x μn

∂x ν1 ∂x σ1



∂x νm ∂x σm

 1 ...μn . Tνμ1 ...ν m

(2.2.7)

One also says that such a tensor is of rank m + n (the total number of tensor indices). By this definition a contravariant vector is a one-fold contravariant tensor. A covariant vector Aμ is a one-fold covariant tensor that transforms according to Aν =



∂x μ ∂x ν

 Aμ .

(2.2.8)

From the transformation rules it is clear that a set of tensors where each contravariant index has a corresponding covariant one and it is summed over all indices like in μν

Tαβ Aμ Bναβ is a scalar, i.e., a coordinate independent object. Exercise 2.1 Proof the last statement by considering general coordinate transformations.

2.3 Tensor Algebra

19

2.2.1 Symmetrization and Antisymmetrization Let Aμν and Aμνλ be arbitrary tensors. The components of the new completely symmetrized tensors are distinguishes by round brackets (e.g., Misner et al. 1973; Exercise 3.12) and written as 1 (Aμν + Aνμ ) 2 1 ≡ (Aμνλ + Aνλμ + Aλμν + Aνμλ + Aμλν + Aλνμ ) . 3!

A(μν) ≡ A(μνλ)

(2.2.9)

Similarly, the corresponding completely antisymmetrized tensors are write with square brackets in the form 1 (Aμν − Aνμ ) (2.2.10) 2 1 ≡ (Aμνλ + Aνλμ + Aλμν − Aνμλ − Aμλν − Aλνμ ) . 3!

A[μν] ≡ A[μνλ)]

Exercise 2.2 Show that Tμνλ = A[μνλ] with an arbitrary tensor Aμνλ changes sign if the indices of any pair are interchanged, e.g., Tμνλ = −Tμλν .

2.3 Tensor Algebra There are several rules that follow from the definition of tensors: (1) The sum of two tensors of the same kind is again a tensor; (2) Multiplication of a tensor with a real number is again a tensor; γ ···γ ···αn (3) If Tβα11···β and Sδ11···δpo are tensors then also m α ···α γ ···γ

γ ···γ

···αn Gβ11 ···βmn δ11 ···δop = Tβα11···β · Sδ11···δpo ; m ···αn (4) If Tβα11···β is a tensor then also (summation over σ ) m α ···α

σ

n−1 Tβ11···βm−1 σ .

This contraction process lowers the number of covariant- and contravariant indices by one respectively. In the same way any other pair of indices (one covariant and one contravariant) can be contracted.

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2 Elements of Differential Geometry

2.4 The Lie-Derivative A mapping φ : M → N between two manifolds, M and N is called differentiable (db) if for some pair of charts {Um , m } of M and {Un , n } of N the mapping n ◦ φ ◦ −1 m from real numbers (coordinates for M) to real numbers (coordinates for N ) is differentiable. A mapping φ : M → N is said to be a diffeomorphism, if φ is bijective (i.e., the inverse φ −1 is well defined) and both, φ and φ −1 are differentiable. Let v be a db vector-field on M. Then there is a curve γ (λ) through each point p ∈ M such that γ (0) = p and whose tangent vector at q = γ (λ) is just v(q). In coordinates, let γ (λ) = x μ (λ) and v μ the coordinate components of v then the integral curve x μ (λ) of v through p is uniquely determined by the first order differential equation dx μ = v μ (x ν (λ)) . dλ

(2.4.1)

By means of their integral curves a db vector-field v induces a family of diffeomorphisms φλv in the neighborhood U (p) by taking each point of U a parameter distance λ along the integral curve of v, called the flow of v. It is useful to describe the flow of v by a mapping of the form x μ → x  ≡ x μ + v μ (x) μ

(2.4.2)

where | |  1. A scalar-field φ then transforms, to first order in as φ(x  ) = φ(x μ + v μ (x)) = φ(x) + Lv φ(x) ,

(2.4.3)

where Lv φ ≡ v μ φ,μ

(2.4.4)

is the Lie-derivative of φ in the direction of v and the commy denotes the partial derivative, v μ φ,μ ≡ v μ (∂φ/∂x μ ). Next we consider a covariant vector-field with components uμ . We first take u at the point q with coordinates x  and then transform it back to the point p with uμ (x) =

∂x  ρ ρ uρ (x  ) = uμ (x  ) + uρ v,μ = uμ (x) + [Lv u]μ ∂x μ

(2.4.5)

where [Lv u]μ = lim

→0

uμ (x) − uμ (x)

ρ = v ρ uμ,ρ + uρ v,μ

(2.4.6)

2.5 The Covariant Derivative

21

is the Lie-derivative of u in the direction of v. Similarly,  (x) = gμν (x) + [Lv g]μν gμν

(2.4.7)

κ κ + gμκ v,ν . [Lv g]μν = gμν,κ v κ + gκν v,μ

(2.4.8)

with

For a general r-fold contravariant, s-fold covariant tensor with components μ1 ...μr the Lie-derivative is given by Tν1 ...ν s μ1 ...μr κ κ...μr μ1 1 ...μr (Lv T )μ ν1 ...νs = Tν1 ...νs ,κ v − Tν1 ...νs v,κ − . . . (all upper indices) μ1 ...μr κ + Tκ...ν v,ν1 + . . . (all lower indices) . s

(2.4.9)

2.5 The Covariant Derivative In the following we will assume the scalar-, vector- or tensor-fields to be differentiable over a certain part of the underlying manifold M. Let ϕ be a scalar-field over M. Then ∂ ϕ ≡ ϕ,μ = Bμ ∂x μ is a covariant tensor-field, since a coordinate transformation x μ → x  ν leads to  μ  μ ∂ ∂ ∂ ∂x ∂x Bμ . Bν =  ν ϕ  =  ν ϕ = ϕ = ∂x ∂x ∂x  ν ∂x μ ∂x  ν The partial derivative acting on a scalar-field yields a tensor-field. This is, however, not the case if the partial derivative is applied to vector- and tensor-fields of higher ranks. E.g., for a contravariant vector-field we get dA = d ν



∂x  ν ∂x μ



 2 ν   ν  ∂ x ∂x σ μ Aμ = dx dAμ , A + ∂x μ ∂x σ ∂x μ

i.e., only under linear transformations dAμ transforms as a tensor. We now will introduce a derivative for tensor-fields that leads to new tensors. To this end we consider again a contravariant differentiable vector field Aμ along some curve γ (λ) (Fig. 2.3). First we will consider such a vector-field in some Euclidean space with Cartesian coordinates. Let us first consider a vector-field with constant components in these coordinates, i.e., dAμ (x ν ) = 0. Now we switch to

22

2 Elements of Differential Geometry

Aμ (xν (λ + dλ)) =

Aμ (xν (λ))

Aμ (xν + dxν ) λ + dλ

λ γ(λ) Fig. 2.3 A contravariant vector field along some curve γ (λ)

new coordinates x  ν , where 

ν

A =

∂x  ν ∂x μ

 Aμ

will no longer be constant in general. Along γ (λ) A ν will vary according to dA ν = dλ



∂ 2xν ∂x μ ∂x σ



dx σ μ A + dλ



∂x  ν ∂x μ



dAμ = dλ



∂ 2xν ∂x μ ∂x σ

This we will write in the form  2 ν   σ   ρ   μ  ∂x dx ∂ x ∂x dA ν τ A = dλ ∂x μ ∂x σ ∂x  ρ dλ ∂x  τ =

ν −ρτ

dx  ρ  τ A dλ



dx σ dλ

 Aμ .

(2.5.1)

with ν =− ρτ

∂ 2 x  ν ∂x α ∂x β . ∂x α ∂x β ∂x  ρ ∂x  τ

(2.5.2)

That means that infinitesimal changes of Aμ along the curve γ (λ) are bi-linear in ν is valid only for our special case. For arbitrary dx ρ and Aμ itself. This form of ρτ vector-fields we now write ν dx ρ Aτ δAν = −ρτ

(2.5.3)

ν . We will consider (2.5.3) as a rule for a parallel with at first arbitrary coefficients ρτ ν μ displacement of A from x (λ) to the neighboring point x μ (λ + dλ) = x μ + dx μ , ν = Aν + δAν (Fig. 2.4). i.e., Aν → A

2.5 The Covariant Derivative

23

Fig. 2.4 Infinitesimal parallel displacement of a vector Aν from x μ to x μ + dx μ resulting in the ν (x μ + dx μ ) vector A

A˜ν

Aν

γ(λ)

Aν λ + dλ xμ + dxμ

λ xμ

ν are called affine connections iff Definition The quantities ρτ

ν ) DAν (Aν − A = lim λ→0 Dλ λ λ+λ

(2.5.4)

is a tensor. In that case DAν /Dλ is called the covariant derivative of Aν along γ (λ). We have ν ν |x+dx = Aν (x μ ) + dAν − (Aν (x μ ) − ρτ DAν = Aν − A Aτ dx ρ ) ν = dAν + ρτ Aτ dx ρ

and we see that for our special case above (Euclidean space, constant vectorcomponents in Cartesian coordinates) the covariant derivative of Aν vanishes, if ν are given by (2.5.2). the affine connections ρτ ν be affine Next we come to the transformation rule for affine connections. Let ρτ  μ + dx μ ), i.e., connections, then DAν transforms as a vector and also A(x

 ∂x  ν μ (x ν + dx ν ) |x+dx A ∂x μ

  ν   2  ν  ∂ x ∂x σ μ (x μ + dx μ ) + dx A = ∂x μ ∂x μ ∂x σ x

 ν (x  ν + dx  ν ) = A



or A −   ρτ dx  A = ν

ν

ρ

τ



∂x  ν ∂x μ



 +

∂ 2xν ∂x μ ∂x σ



μ dx σ (Aμ − αβ dx α Aβ ) (2.5.5)

at the place defined by x μ . Therefore,   ν ∂ 2xν ν ρ τ μ ∂x dx  A = − αβ μ − α β dx α Aβ . −ρτ ∂x ∂x ∂x

24

2 Elements of Differential Geometry

From  dx A = α

β

∂x α ∂x  ρ



∂x β ∂x  τ



dx  A ρ

τ

we finally get the transformation rule for affine connections in the form 

ν = ρτ

∂x  ν ∂x α ∂x β μ ∂ 2 x  ν ∂x α ∂x β  − ∂x μ ∂x  ρ ∂x  τ αβ ∂x α ∂x β ∂x  ρ ∂x  τ

(2.5.6)

μ

that reduces to (2.5.2) for the case αβ = 0. One writes ν DAν = dAν + ρτ Aτ dx ρ = Aν;ρ dx ρ

(2.5.7)

ν Aτ Aν;ρ ≡ Aν,ρ + ρτ

(2.5.8)

where

is called the covariant derivative of Aν with respect to x ρ . It generalizes the partial derivative, Aν,ρ ≡

∂Aν . ∂x ρ

Let Aν and Bμ be differentiable vector-fields. Then Aν Bν is a scalar-field and, therefore, ν dx ρ Aτ )Bν + Aν (δBν ) δ(Aν Bν ) = 0 = (δAν )Bν + Aν (δBν ) = (−ρτ

or ν τ dx ρ Aτ Bν = (ρν dx ρ Bτ )Aν . Aν (δBν ) = ρτ

From this we derive τ dx ρ Bτ δBν = +ρν

and DBν = Bν;ρ dx ρ ,

(2.5.9)

where τ Bτ Bν;ρ ≡ Bν,ρ − ρν

(2.5.10)

2.6 Geodesics

25

is the covariant derivative of Bν with respect to x ρ . In a similar way the covariant derivative of an arbitrary tensor-field is defined: μ ...μ

r 1 1 ...μr +  μ1 T κ...μr + · · · +  μr T μ1 ...κ Tν1 ...ν = Tνμ1 ...ν ρκ ν1 ...νs ρκ ν1 ...νs s ,ρ s ;ρ

κ κ − ρν T μ1 ...μr − · · · − ρν T μ1 ...μr . s ν1 ...κ 1 κ···νs

(2.5.11)

2.6 Geodesics Let γ (λ) be some curve in M. The covariant derivative of some contravariant vector-field Aν along γ is given by DAν ν ) uρ = (Aν,ρ + ρτ Dλ where uρ ≡

dx ρ dλ

(2.6.1)

is the tangent vector-field to the curve γ (λ). Definition The vector-field Aν is called parallel along γ (λ) if dAν DAν ν ρ τ =0= + ρτ u A . Dλ dλ

(2.6.2)

A curve γ (λ) is called a geodesic if Duν = h(λ)uν Dλ

(2.6.3)

with uν ≡ dx ν /dλ. Hence, a geodesic is a curve where the tangent vectors are parallel to themselves. The equation for a geodesic, therefore, reads ρ τ d 2xν dx ν ν dx dx = h(λ) . + ρτ 2 dλ dλ dλ dλ

(2.6.4)

In general one can eliminate the right hand side of this equation by a suitable choice of the curve-parameter. Such a parameter is called affine. With respect to an affine parameter κ the geodetic-equation takes the form ρ τ d 2xν ν dx dx = 0. +  ρτ dκ dκ dκ 2

(2.6.5)

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2 Elements of Differential Geometry

2.7 Curvature- and Ricci Tensor The curvature of a manifold is usually introduced by means of parallel transport of a vector around a closed curve. Such a situation is depicted in Fig. 2.5, where we start at the north pole of a sphere and consider the tangent vector t1 to some meridian running through the pole and the equator. Since a meridian is a geodesic (see Example 2.2 below) parallel displacement of t1 leads to the tangent vector e1 in the point E1 of the equator. Parallel transport along the equator to E2 leads to e2 and finally going back to the pole by parallel displacement along the meridian running through E2 we end up with a vector t2 that differs from t1 because of the curvature of the sphere. For convenience let us consider the parallel transport of a covariant vector Aα along some curve x μ (λ). The changes of the coordinate components of Aα are then determined by (2.6.2)   dx μ dx μ β = Aα,μ − αμ Aβ dλ dλ μ dx dAα β − αμ = Aβ dλ dλ

0 = Aα;μ

or

Aα =

β αμ Aβ

dx μ dλ . dλ

(2.7.1)

We now assume the curve γ given by x μ (λ) to be closed, i.e., x μ (λ0 ) = x μ (λ1 ) for some suitably chosen value of λ1 , given λ0 . One might then consider γ as the edge of some two-dimensional surface S, and divide S into small cells bounded by little closed curves γn (Weinberg 1972). From Fig. 2.6 it becomes clear that Aα [γ ] =



n Aα [γn ] .

n

Fig. 2.5 Parallel transport of a vector around a closed curve leads to a different vector at the initial position due to curvature of the manifold

t1 P t2

E1 e1

E2 e2

2.7 Curvature- and Ricci Tensor

27

A A1

A2

Fig. 2.6 Integration around a closed curve as edge of some surface A is given by a sum of integrations over partitions A1 and A2 of A

In other words it is sufficient to consider the parallel transport of Aα around some sufficiently small surface. Let Xσ ≡ x σ (λ0 ), e.g., the starting point of our route and x σ ≡ x σ (λ). We now will compute the change in Aα to second order in x σ − Xσ β and since the integral over dx μ is of first order we need the changes of αμ and Aβ σ σ only to first order. To first order in x − X we get β β (x) = αμ (X) + (x σ − Xσ ) αμ

∂  β (X) . ∂Xσ αμ

(2.7.2)

Similarly, dropping all terms of second order in x − X one has from (2.7.1) β (X)Aβ (λ0 )(x μ (λ) − Xμ ) . Aα (λ) = Aα (λ0 ) + αμ

(2.7.3)

Inserting the last two relations into (2.7.1) one obtains an expression valid to second order in x − X: λ

∂ β  (X) ∂Xσ αμ λ0   dx μ ρ dλ × Aβ (λ0 ) + βσ (X)Aρ (λ0 )(x σ (λ) − Xσ ) dλ

λ μ dx β dλ Aα (λ0 ) + αμ (X)Aβ (λ0 ) λ0 dλ

λ μ ∂ ρ ρ β σ σ dx + dλ .  (X) +  (X) (X) A (λ ) (x (λ) − X ) ρ 0 αμ αμ βσ ∂Xσ dλ λ0

Aα (λ) Aα (λ0 ) +

β αμ (X) + (x σ (λ) − Xσ )

For a parallel transport around some small closed curve with x μ (λ0 ) = x μ (λ1 ) obviously

λ1

λ0

dx μ dλ = 0 dλ

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2 Elements of Differential Geometry

and the total change of Aα is of second order in x − X proportional to the surface element spanned by the small curve γn : Aα ≡ Aα (λ1 ) − Aα (λ0 )

 ∂ ρ ρ β =  (X) + βσ (X)αμ (X) Aρ (λ0 ) x σ dx μ . ∂Xσ αμ

(2.7.4)

The integral appearing on the right hand side of this equation can be interpreted as surface element S σ μ enclosed by x μ (λ) which is antisymmetric in σ and μ since  S

σμ

=

x dx = σ

μ

λ0

 =−

λ1

d σ μ (x x )dλ − dλ

λ1

λ0

xμ

dx σ dλ dλ

x μ dx σ = −S μσ .

For that reason we can replace the coefficient of S σ μ on the right hand side of (2.7.4) by its antisymmetric part and write Aα =

1 ρ R αμσ Aρ S μσ , 2

(2.7.5)

where ρ

ρ

ρ ρ β β R ρ αμσ = ασ,μ − αμ,σ + βμ ασ − βσ αμ

(2.7.6)

is called the curvature tensor. From its definition we see that the curvature tensor is antisymmetric in the last pair of indices: R μ νλσ = −R μ νσ λ .

(2.7.7)

Exercise 2.3 Proof by direct calculation that for any vector fields Aρ and Aρ the following relations holds ρ Aα . Aρ ;σ μ − Aρ ;μσ = +Rαμσ

(2.7.8)

α Aα . Aρ;σ μ − Aρ;μσ = −Rρμσ

(2.7.9)

and

Proof of (2.7.8): We have: Aρ ;σ = Aρ ,σ + σρν Aν ≡ Tσρ ,

2.7 Curvature- and Ricci Tensor

29

ρ

where Tσ is a tensor. Therefore; ρ

ρ ρ α + μα Tσα − μσ Tαρ Aρ ;σ μ = Tσ ;μ = Tσ,μ

(2.7.10)

ρ ρ α = (Aρ ,σ + Aν σρν ),μ + (Aα ,σ + Aν σαν )μα − (Aρ ,α + Aν αν )μσ .

Using (2.7.10) relation (2.7.8) can be shown directly. The second Bianchi identities are fundamental relations for the curvature tensor. They read: R μ νλσ ;κ + R μ νκλ;σ + R μ νσ κ;λ = 0 .

(2.7.11)

The proof of the second Bianchi identities (2.7.11) is especially simple if special coordinates are introduced at some arbitrary point P0 of the manifold: Riemann normal coordinates (see e.g., Sect. 11.6 in Misner et al. 1973) that in a 4-dimensional space-time are related with local inertial coordinates of some freely falling observer. μ In such coordinates the affine connections νλ vanish at P0 , i.e., in Riemann normal coordinates μ

νλ (P0 ) = 0

(2.7.12)

and derivatives thereof are given by the components of the curvature tensor (more about normal coordinates are presented in Sect. 5.6 where the manifold is assumed to have a metric and the set of basis vectors at a certain point of the manifold is chosen as orthonormal tetrad. The corresponding coordinates will then be called ‘tetrad-induced’). In Exercise 2.4 one proofs that the second Bianchi identities are true at some point P0 in Riemann normal coordinates. But, since the left hand side of theses identities is a tensor the relations are true in any suitable coordinate system and at every point of the manifold. Exercise 2.4 Proof by direct calculation the (second) Bianchi-identity by using Riemann normal coordinates at some point P0 . Proof Using (2.7.12) the components of the curvature tensor are given by: μ

μ

R μ νλσ = νσ,λ − νλ,σ

(2.7.13)

so that μ

μ

μ

μ

μ

μ

R μ νλσ ;κ = R μ νλσ,κ = νσ,λκ − νλ,σ κ R μ νκλ;σ = R μ νκλ,σ = νλ,κσ − νκ,λσ R μ νσ κ;λ = R μ νσ κ,λ = νκ,σ λ − νσ,κλ . Summing up the left hand sides, therefore, gives zero.

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2 Elements of Differential Geometry

An object that will play an important role in the following is the Ricci tensor. The Ricci tensor Rμν is defined by σ σ κ σ σ κ Rμν ≡ R σ μσ ν = νμ ,σ − σ μ ,ν + νμ σ κ − κν μσ .

(2.7.14)

2.8 The Metric Tensor The geometry of a manifold is locally described by the metric tensor gμν . Consider two points P1 and P2 in the Euclidean 3-space. The distance s between the two points, according to the Pythagorean theorem, is given by (Fig. 2.7): (s)2 = (x)2 + (y)2 + (z)2 with x = x2 − x1 etc. Infinitesimally we write this as ds 2 = dx 2 + dy 2 + dz2

(2.8.1)

ds 2 = gμν dx μ dx ν .

(2.8.2)

or generally

gμν is called the metric tensor. Since ds 2 has to be a coordinate independent object gμν is a two-fold covariant tensor. In our example gμν is given by ⎛

gμν

Fig. 2.7 The distance between two points P1 and P2 in Euclidean 3-space

1 =⎝ 0 0

0 1 0

⎞ 0 0 ⎠ ≡ δμν . 1

Δs P1



P2

Δz

2

(Δx)

+ (Δy

)2

2.8 The Metric Tensor

31

Since ds 2 should be independent of the coordinates used and the transformation rule (2.2.5) for differentials the metric tensor gμν transforms according to  gμν =

∂x α ∂x β gαβ . ∂x μ ∂x ν

(2.8.3)

Since we know how the differentials dx, dy and dz in (2.8.1) transform into dr, dθ and dφ (relations (2.2.4)) we find for our Euclidean 3-space ds 2 = dx 2 + dy 2 + dz2 = dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) .

(2.8.4)

Thus, with respect to x 1 = r,

x 2 = θ,

x3 = φ

the nonvanishing components of the metric tensor read: g11 = 1 ,

g22 = gθθ = r 2 ,

g33 = gφφ = r 2 sin2 θ .

(2.8.5)

So, in general the metric tensor depends upon coordinates. The inverse metric tensor is denoted by g μν , i.e.,  g

μν

gνλ =

μ δλ

≡ δμλ =

1 if μ = λ 0 otherwise.

(2.8.6)

For our Euclidean 3-space ⎛

g μν

1 =⎝ 0 0

0 1 0

⎞ 0 0 ⎠ 1

(2.8.7)

in Cartesian coordinates and ⎛

g μν

1 =⎝ 0 0

0 r −2 0

⎞ 0 ⎠, 0 −2 −2 r sin θ

(2.8.8)

in spherical coordinates, i.e., the non-vanishing components read g 11 = 1,

g 22 = r −2 ,

g 33 = r −2 sin−2 θ .

(2.8.9)

Definition (Raising and Lowering of Indices) If a manifold is endowed with a metric tensor (and its inverse), we can associate with each contravariant tensor-index

32

2 Elements of Differential Geometry

a corresponding covariant one and vise versa according to Aμ = g μν Aν ;

Bμ = gμν B ν ;

T μσρκτ = g μν Tν σρκτ

etc.

Such process behind such mappings is called the raising and lowering of indices. The metric tensor defines a scalar-product (A, B) of two vectors Aμ and B ν according to (A, B) ≡ gμν Aμ B ν .

(2.8.10)

If gμν is positive definite, i.e., if (A, B)|p ∈ R+ , ∀p ∈ M, A, B = 0 , gμν is called a Riemannian metric and (M, g) a Riemannian space. If (A, B) can also attain negative values one speaks of a pseudo-Riemannian metric and pseudoRiemannian space.

2.9 Metric Connections σ is called metric, if the covariant derivative of the metric An affine connection μν tensor vanishes, i.e., σ σ gσ ν − νλ gσ μ = 0 . gμν;λ = gμν,λ − μλ

(2.9.1)

Metric connections conserve the scalar product of two vector-fields that are both parallel along some curve γ (λ). Let Aμ and B ν be vector-fields along γ , i.e., DAμ DB ν = = 0. Dλ Dλ The variation of (A, B) along γ is then given by Dgμν μ ν dx σ μ ν D d [gμν Aμ B ν ] = [gμν Aμ B ν ] = A B = gμν;σ A B dλ Dλ Dλ dλ and vanishes for metric connections. Especially the norm of a vector field parallel along γ is constant for a metric connection. Such a norm can be associated with natural constants such as the rest-mass of elementary particles etc. For that reason in the following we will always assume the connection to be metric if relativity comes into play.

2.9 Metric Connections

33

Theorem 2.1 On a pseudo-Riemannian manifold (M, g) there is only one affine connection that is metric, i.e., condition (2.9.1) fixes the connection uniquely which in this case is given by the Christoffel-symbols μ

νλ =

1 μσ g (gσ ν,λ + gσ λ,ν − gνλ,σ ) . 2

(2.9.2)

Proof From (2.9.1) we get σ σ 0 = gμν;λ = gμν,λ − gσ ν μλ − gσ μ νλ ≡ gμν,λ − ν|μλ − μ|νλ

and therefore gμν,λ = +ν|μλ + μ|νλ gλμ,ν = +μ|λν + λ|μν −gνλ,μ = −λ|νμ − ν|λμ . Adding these three equations leads us to μ|νλ =

1 (gμν,λ + gμλ,ν − gνλ,μ ) . 2

Raising the first index of μ|νλ then leads us to the Christoffel-symbols from (2.9.2). Lemma 2.1 For metric connections the geodesics are curves of maximal (minimal) length between two points P1 and P2 . Consider two fixed points P1 and P2 and all kinds of curves x μ (λ), where λ is some curve parameter, that connect these two points. If the connection is metric the geodesic connecting these two points will obey a relation of the form

P2

δ P1



dx μ dx ν gμν dλ dλ

 dλ = 0 .

(2.9.3) μ

This relation can be understood in the following way: let xγ (λ) be the desired geodesic. One then considers small variations of that curve, i.e., curves of the form μ xγ (λ) + δx μ (λ) with fixed endpoints: δx μ (P1 ) = δx μ (P2 ) = 0 . With the rule δA(x μ ) =

∂A δx μ ≡ A,μ δx μ ∂x μ

(2.9.4)

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2 Elements of Differential Geometry

(i.e., the comma denotes a partial derivative) for any differentiable function A(x μ ) and the notation x˙ μ ≡

df dx μ , (f )˙ ≡ dλ dλ

we get

0=

=

=

  (δgμν x˙ μ x˙ ν + 2gμν x˙ μ δ x˙ ν dλ   gμν,ρ δx ρ x˙ μ x˙ ν + 2gμρ x˙ μ (δx ρ )˙ dλ   gμν,ρ x˙ μ x˙ ν − (2gμρ x˙ μ )˙ δx ρ dλ



1 gμρ x¨ μ + g˙ μρ x˙ μ − gμν,ρ x˙ μ x˙ ν δx ρ dλ 2

1 μ ν μ μ ν gμρ x¨ + gμρ,ν x˙ x˙ − gμν,ρ x˙ x˙ δx ρ dλ = −2 2

 μ ν 1 μ gμρ x¨ + gμρ,ν + gνρ,μ − gμν,ρ x˙ x˙ δx ρ dλ = −2 2

1 = −2 gαρ x¨ α + g ασ (gμσ,ν + gνσ,μ − gμν,σ ) x˙ μ x˙ ν δx ρ dλ . 2

= −2

In the first line only the chain rule was used, in the second the variation of gμν was performed according to the rule (2.9.4), the 3rd line involved an integration by parts, in the 4th line a factor of 2 was taken in front of the integral and a dotderivative was written out, the dot-derivative of gμν was written out in the 5th line, the middle-term in the 5th line was written symmetrically with respect to the indices μ and ν in the 6th line, and finally a factor gαρ was taken out of the bracket in the last line. I.e., the equation for the curve with extremal length reads α μ ν x˙ x˙ = 0 , x¨ α + μν

(2.9.5)

α are just the Christoffel-symbols from (2.9.2). where μν

As an example we compute the Christoffel symbols for the Euclidean 3-space in spherical coordinates x 1 = r, x 2 = θ, x 3 = φ .

2.9 Metric Connections

35

The components of the metric tensor are given by (2.8.5) and (2.8.9). From this we find e.g., 2 33 =

1 22 g (g32,3 + g23,3 − g33,2 ) = − sin θ cos θ . 2

Similarly one gets for the non-vanishing Christoffel symbols: 1 22 = −r; 2 2 = 12 = 21

1 ; r

1 33 = −r sin2 θ 2 = − sin θ cos θ 33

3 3 = 32 = cot θ ; 23

3 3 31 = 13 =

(2.9.6)

1 . r

Example 2.2 As another example we consider the geodesics on a unit sphere (Fig. 2.8). As coordinates we choose usual spherical coordinates x 1 = θ,

x2 = φ

with length element ds 2 = dθ 2 + sin2 θ dφ 2 . For the Christoffel symbols we get from (2.9.6) 1 = − sin θ cos θ ; 22

2 2 12 = 21 = cot θ .

For the geodesic equation one finds e.g., 1 θ¨ + μν x˙ μ x˙ ν = 0

Fig. 2.8 Geodesics on the unit sphere running through the poles

(2.9.7)

36

2 Elements of Differential Geometry

and the indices μ and ν both have to take the value 2. Therefore, θ¨ − sin θ cos θ φ˙ 2 = 0 .

(2.9.8)

φ¨ + 2 cot θ φ˙ θ˙ = 0 .

(2.9.9)

Similarly one finds

E.g., the geodesics through the poles θ+ = 0, θ− = π are given by the meridians φ = const.,

θ =λ

λ ∈ [0, π ] .

2.9.1 Riemann Tensor and Its Symmetries If the curvature tensor results from the Christoffel-symbols of some metric tensor it is called Riemann tensor . The Riemann tensor has the following symmetries: Rμνλσ = −Rνμλσ Rμνλσ = −Rμνσ λ

(2.9.10)

Rμνλσ = +Rλσ μν . First Bianchi-identities (cyclic identities): R μ νλσ + R μ σ νλ + R μ λσ ν = 0.

(2.9.11)

The second Bianchi-identities, relation (2.7.11) R μ νλσ ;κ + R μ νκλ;σ + R μ νσ κ;λ ≡

1 μ R ν[λσ ;κ] = 0 3

(2.9.12)

have already been discussed in Exercise 2.4. Exercise 2.5 Proof that the first Bianchi identities can be written in the form R μ [νλσ ] = 0 .

(2.9.13)

Proof Using the definition of R μ [νλσ ] and the relation (2.7.7) we get R μ [νλσ ] =

1 μ (R νλσ + R μ λσ ν + R μ σ νλ ) = 0 . 3

Exercise 2.6 Proof the symmetries (2.9.10) of the Riemann tensor.

(2.9.14)

2.9 Metric Connections

37

Exercise 2.7 Proof the first Bianchi-identity (2.9.11) using (2.9.14) and the condiμ μ tion νλ = λν . Proof μ

μ

μ

μ

μ

μ

μ

α μ α 3R μ [νλσ ] = νσ,λ − νλ,σ + αλ νσ − ασ νλ μ α μ α λν,σ − λσ,ν + ασ λν − αν λσ μ

μ α σ λ,ν − σ ν,λ + αν σ λ − αλ σαν μ

μ

μ

μ

μ = (λν − νλ ),σ + (νσ − σμν ),λ + (σ λ − λσ ),ν μ

α α μ α α μ α + αλ (νσ − νσ ) + ασ (λν − νλ ) + αν (σαλ − λσ )

= 0. The Einstein tensor Gμν is defined by 1 Gμν ≡ Rμν − gμν R , 2

(2.9.15)

R ≡ g μν Rμν

(2.9.16)

where

is the curvature scalar. Exercise 2.8 Proof that the Einstein tensor is divergenceless, i.e. Gνμ;ν = 0 .

(2.9.17)

Show that (2.9.17) is equivalent to the second Bianchi identity (2.7.11). Exercise 2.9 Calculate the Riemann curvature tensor, the Ricci tensor, and the curvature scalar for a 2-sphere of radius a. Solution The metric is given by (2.8.5) with dr 2 = 0, r = a and the Christoffel symbols can be taken from (2.9.6). Direct calculation gives R 1 212 = R θ φθφ = =

1 ∂ 1 ∂22 1 α 1 α − 21 + α1 22 − α2 21 ∂θ ∂φ

1 ∂22 1 2 − 22 21 = sin2 θ − cos2 θ + sin θ cos θ cot θ ∂θ

hence R 1 212 = sin2 θ .

(2.9.18)

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2 Elements of Differential Geometry

Similarly we find R 2 121 = 1 .

(2.9.19)

From this we find the components of the Ricci tensor R11 = 1,

R22 = sin2 θ .

(2.9.20)

Since g 11 = a −2 and g 22 = a −2 sin−2 θ the curvature scalar reads R=

2 . a2

(2.9.21)

Often a useful measure for the local curvature is the Kretschmann-scalar K. It is defined by K ≡ Rμνλσ R μνλσ .

(2.9.22)

2.10 The Levi-Civita Symbol and Tensor The Levi-Civita symbol [μ1 μ2 . . . μn ] is defined to be completely antisymmetric, i.e., it changes sign when two indices are exchanged and [01 . . . (n − 1)] = +1

[12 . . . n] = +1 .

or

(2.10.1)

E.g., [0132] = −1, [321] = −1. In three dimensions one usually writes [ij k] ≡ ij k . The Levi-Civita symbol is especially useful when dealing with determinants. Let μ Mν be some n × n matrix, |M| ≡ Det(M) , then [ν1 ν2 . . . νn ]|M| = Mνμ11 · · · Mνμnn [μ1 . . . μn ] .

(2.10.2)

Think of [μ1 . . . μn ] as being the coordinates of some geometrical object in flat Rn in some Cartesian coordinate system x μ . We then consider a coordinate transformation x μ → x  ν and put  Mνμ

≡

∂x μ ∂x  ν

 .

2.10 The Levi-Civita Symbol and Tensor

39

Then from (2.10.2) we get  μ   μn  ∂x ∂x 1 ∂x · · · [μ1 . . . μn ] , [ν1 . . . νn ]  = ν1  ∂x ∂x ∂x  νn

(2.10.3)

where |∂x/∂x  | is the Jacobian of the transformation x  → x. We get   μ   μn  ∂x ∂x 1 ∂x [ν1 . . . νn ] = · · · [μ1 . . . μn ] , ν1  ∂x ∂x ∂x  νn

(2.10.4)

since ∂x ∂x  −1 = ∂x  ∂x .

(2.10.5)

 ∂x J ≡ ∂x

(2.10.6)

Let

be the Jacobian of the transformation x → x  , then a quantity that transforms as  Tν1 ...νn = J m

∂x μ1 ∂x  ν1



 ···

∂x μn ∂x  νn

 Tμ1 ...μn

(2.10.7)

is called a tensor-density of weight m. From (2.10.4) we see that the Levi-Civita symbol can be considered as a tensor-density of weight +1. As an example consider the volume element d n x that transforms as  ∂x n d x, d n x  = (2.10.8) ∂x hence it is a scalar density of weight +1. Let us consider the determinant of the gμν , g ≡ Det(gμν ) < 0 in a 4-dimensional spacetime manifold. From  = gμν

∂x ρ ∂x σ gρσ ∂x  μ ∂x  ν

we get ∂x 2 g  =  g . ∂x

(2.10.9)

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2 Elements of Differential Geometry

√ Thus g is a scalar density of weight −2 and we can use powers of −g to convert √ each tensor density into a real tensor. E.g., −g d n x is an invariant volume element. We define the Levi-Civita tensor generally by √ μ1 ...μn ≡ + −g [μ1 . . . μn ] , .

(2.10.10)

This implies that in a 4-dimensional spacetime √ 0123 = + −g ;

(2.10.11)

(Note, that Weinberg (1972) is using a different sign convention). We can then raise the indices with the inverse metric tensor sgn(g) [μ1 . . . μn ] . μ1 ...μn = g μ1 ν1 · · · g μn νn ν1 ...νn = √ |g|

(2.10.12)

In a 4-dimensional spacetime 1 μ1 ...μ4 = − √ μ ...μ . −g 1 4

(2.10.13)

2.11 Symmetric Spaces A space is called symmetric if the metric tensor is form-invariant under the flow of  (x) = g (x) or a vector-field ξ . For Eq. (2.4.7) form-invariance implies that gμν μν [Lξ g]μν = 0 .

(2.11.1)

The last equation can be re-written in the form 0 = gμν,κ (g κσ ξσ ) + gκν (g κσ ξσ ),μ + gμκ (g κσ ξσ ),ν κσ μσ = ξμ,ν + ξν,μ + g κσ gμν,κ ξσ + gκν g,μ ξσ + gμκ g,ν ξσ

= ξμ,ν + ξν,μ + ξ κ (gμν,κ − gκν,μ − gκμ,ν ) σ = ξμ,ν + ξν,μ − 2ξσ μν

or ξμ;ν + ξν;μ = 0 .

(2.11.2)

A vector-field ξ satisfying this Killing equation is called Killing vector-field. Each Killing vector-field describes a symmetry of the manifold. In that case the Liederivative of the metric tensor with respect to the Killing vector-field vanishes.

2.11 Symmetric Spaces

41

As an example we consider 3-dimensional Euclidean space with Cartesian coordinates where the Killing equation takes the form ξi,j + ξj,i = 0 .

(2.11.3)

One solution to this equation reads: ξ i = const. that describes an infinitesimal translation of the form x → x + ξ . A second solution reads ξi = rij x j

(2.11.4)

with rij = −rj i that describes an infinitesimal rotation about x = 0.

2.11.1 Maximally Symmetric Spaces (n)

A set of ξμ , n = 1, . . . , N of N Killing vector fields is said to be linearly dependent if there are constants cn such that 

cn ξμ(n) = 0 ,

(2.11.5)

n

otherwise the ξμ(n) ’s are said to be linearly independent. A space of dimension N is called maximally symmetric if it has N (N + 1)/2 independent Killing vector fields. Lemma 2.2 An N-dimensional space has at most N (N + 1)/2 independent Killing vector fields. Proof The definition of the Riemann curvature tensor and the first Bianchi-identity (2.9.11) imply that (e.g., Weinberg 1972; (13.1.8)) ξσ ;ρμ − ξσ ;μρ + ξμ;σρ − ξμ;ρσ + ξρ;μσ − ξρ;σ μ = 0

(2.11.6)

for any vector field ξμ . From the Killing equation, ξμ;ν = −ξν;μ one infers that ξρ;μσ = −ξμ;ρσ etc. so that for a Killing vector field ξσ ;ρμ − ξσ ;μρ − ξμ;ρσ = 0

(2.11.7)

λ ξλ . ξμ;ρσ = Rσρμ

(2.11.8)

and using relation (2.7.9),

Similarly all higher covariant derivatives of ξμ at some point X can be derived from ξλ and ξλ;ν at X. Therefore, ξλ (X) and ξλ;ν (X) completely determine ξλ (x) if x is in a certain neighborhood of X. In some N-dimensional space ξλ has N independent

42

2 Elements of Differential Geometry

components and ξλ;ν , because of the Killing-equation, has N (N − 1)/2 independent quantities which gives a total of maximally N (N + 1)/2 independent Killing vector fields. Lemma 2.3 If an N -dimensional space is maximally symmetric, then Rμν = (N − 1)kgμν Rμνλσ = k(gσ ν gμλ − gλν gμσ )

(2.11.9) (2.11.10)

where the curvature constant k is defined by R = Rσσ ≡ N (N − 1)k .

(2.11.11)

The proof is left as an exercise (see e.g. Weinberg 1972). The curvature constant k has the dimension 1/length2 . As we have seen the curvature scalar for a two-sphere of radius r0 is given by R = 2/r02 so that k = r0−2 .

2.11.2 Maximally Symmetric 3-Spaces A 3-space is maximally symmetric if it has a total of six independent Killing vector fields. Clearly the flat Euclidean 3-space is maximally symmetric, but there are more 3-spaces with maximal symmetry. It is clear that such a 3-space is spherically symmetric so that in suitable spherical coordinates r, θ, φ the metric tensor takes the form ds 2 = e2β(r) dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) .

(2.11.12)

The non-vanishing components of the Ricci-tensor read (β  (r) ≡ dβ/dr): Rrr =

2  β (r) r

Rθθ = e−2β(r) (rβ  (r) − 1) + 1 Rφφ = Rθθ sin2 θ .

(2.11.13)

Using condition (2.11.9) for maximal symmetry, Rij = 2kgij , one obtains e.g., 2  β (r) = e2β(r) r

2.12 GRTensor

43

and thus 1 β(r) = − ln(1 − kr 2 ) . 2

(2.11.14)

Thus in suitable coordinates the metric for a maximally symmetric 3-space takes the form ds32 =

dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) . 1 − kr 2

(2.11.15)

The case k = 0 is the flat Euclidean 3-space, if k > 0 the space has positive curvature describing a spherical closed space, if k < 0 the 3-space is called open.

2.12 GRTensor For the treatment of problems related with differential geometry the employment of a Computer Algebra System (CAS) such as REDUCE, MATHEMATICA or Maple is useful. A very efficient tool for dealing with differential geometrical objects is a package called GRTensor. Ir was developed by Peter Musgrave, Denis Pollney and Kayll Lake from the Queen’s University at Kingston, Ontario in Canada. Originally GRTensor was a standard Maple package. GRTensorII version 1.50 was developed in 1994–1999 and updated by GRTensorIII to work efficiently also with new version of Maple (meanwhile it is also available for MATHEMATICA). GRTensorIII software is freely available with documentation and examples from https://github.com/grtensor/grtensor. If one has downloaded the GRTensorIII Maple package one can start with a new Maple file where you have to define the link to the library of GRTensorIII: libname := libname, “libpath”: where ‘libpath’ could read, e.g.: D:\\Maple\\grtensor. If one wants to have excess to the library of metrics one also has to define the corresponding link: grOptionMetricPath := “metricpath”: where ‘metricpath’ could read D:\\Maple\\grtensor \\metrics. As an illustration let us compute differential geometrical objects such as the μ Ricci tensor Rμν , the curvature tensor Rμνλσ , the Christoffel symbols νρ and the Kretschmann-scalar K for the Schwarzschild metric discussed in Chap. 6 in standard coordinates (t, r, θ, φ). To this end we first produce an ASCII file with the name ‘SchwarzSelf.mpl’ that reads:

44

2 Elements of Differential Geometry Ndim_ := 4: X1_ := r: X2_ := theta: X3_ := phi: X4_ := t: complex_ := {}: g11_ := 1/(1-2*m/r): g22_ := r^2: g33_ := sin(theta)^2*r^2: g44_ := -(1 - 2*m/r): Info_:= `Schwarzschild-metric`:

A program that does the job could read: > > > >

>

##################################################### # The Schwarzschild metric ##################################################### restart:

>

##################################################### # define the path to the grtensor library ##################################################### libname := libname, "D:\\Maple\\grtensor":

>

with(grtensor);

>

##################################################### # define the path to the library of metrics ##################################################### grOptionMetricPath := "D:\\Maple\\grtensor\\metrics":

> >

> > >

> > > > >

> > > >

> >

##################################################### #load your private file for the Schwarzschild metric # in standard coordinates (r,theta,phi,t) ##################################################### qload( SchwarzSelf ):

##################################################### # display the components of g_ab ##################################################### grdisplay(g(dn,dn)):

##################################################### # calculate and display the components of R_ab

2.12 GRTensor > > >

> > > >

> > >

> > > > > >

##################################################### grcalc( R(dn,dn)): grdisplay( R(dn,dn) ):

##################################################### # calculate and display the components of R_abcd grcalc( R(dn,dn,dn,dn) ): grdisplay( R(dn,dn,dn,dn) ):

##################################################### # display the components of Chr_ab^c grdisplay( Chr(dn,dn,up) ): ##################################################### # calculate and display the Kretschmann scalar K ##################################################### grcalc(RiemSq): grdisplay(RiemSq): #####################################################

45

Chapter 3

Newtonian Celestial Mechanics

3.1 Newtonian Theory of Gravity Newton’s theory of gravity is based upon absolute time and space (the Newtonian space-time). According to Newton’s Philosophiae Naturalis Principia Mathematica (originally published in 1687 in Latin), absolute time and space respectively are independent aspects of objective reality: Absolute, true and mathematical time, of itself, and from its own nature flows equably without regard to anything external, and by another name is called duration: relative, apparent and common time, is some sensible and external measure of duration by the means of motion, which is commonly used instead of true time . . .

According to Newton, absolute time exists independently of any perceiver and progresses at a consistent pace throughout the universe. Also, space in the Newtonian framework has absolute character, in the sense that it regulates the inertial forces that appear if some observer is accelerated (or rotates) with respect to Newton’s absolute space, and that cannot be understood as arising from some kind of interaction with the direct physical neighbourhood (in this sense Newton’s theory is ‘non-relativistic’ but nevertheless can be formulated in a covariant manner). The absolute aspects of the Newtonian space-time lead to the globally determined bundle of inertial frames, where inertial forces are absent, and to the symmetries defined by the Galilean group. The Newtonian gravitational field equation relates the matter density ρ as source to the curvature tensor of the Newtonian space-time. Basically, the curvature tensor describes the tidal forces, i.e., relative accelerations of neighbouring ‘particles’ are the outcome of gradiometric measurements. A convenient way to describe this curvature tensor is by means of a Newtonian potential U (t, x), where t refers to the time- and x to the space coordinates of a point in the Newtonian space-time manifold. If one is interested in the problem of celestial mechanics, then in Newton’s theory the form of U outside of a body is of special interest. The definition of a body © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_3

47

48

3 Newtonian Celestial Mechanics

here presents no problem: a certain space-time region B , where ρ has compact support. Note, that in a non-liner theory of gravity (such as Einstein’s theory), where gravitational fields also act as field generating sources, the definition of a body presents a real problem. For celestial mechanical problems the potential U outside a body, that determines the global equations of motion is usually expanded in terms of multipole moments. Such expansions usually converge outside some coordinate sphere of radius RB that completely contains the body under consideration. There are different multipole expansions of the external potential of a body, the most common one being the expansion of Uext in terms of (scalar) spherical harmonics Ylm . An equivalent multipole expansion employs Cartesian Symmetric and TraceFree (STF) tensors, i.e, mathematical objects Tk1 ...kl with l different Cartesian indices running over three Cartesian values (1, 2, 3), symmetric in all l indices and completely trace-free, i.e., if two indexes are set equal and summed over all three components, the object vanishes. It turns out, that the set of (scalar) spherical harmonics is equivalent to the set nˆ L , where nL = nk1 · · · nkl , nk = x k /r (x k denotes the three Cartesian coordinates (x, y, z) and r 2 = x 2 + y 2 + z2 ) and the hat indicates that all traces have to be removed from nL . The multipole expansion in terms of STF-tensors is especially useful for the derivation of equations of motion and in relativistic theories, where Lorentz-transformations (usually formulated in terms of Cartesian coordinates) play an important role. Below we illustrate various aspects of Newtonian celestial mechanics that will be useful for later applications related with General Relativity. These parts are fairly standard; less common might be the derivation of perturbation equations by means of the (perturbed) integrals of motion of the Keplerian two-body problem. This part should also serve as a bridge between readers that are more familiar with Newtonian celestial mechanics and those familiar with certain aspects of relativity.

3.2 The Newtonian Space-Time 3.2.1 The Galilean Group In Newton’s theory there exists an absolute time coordinate t that is determined uniquely up to linear transformations (origin and unit) t → at + b ;

a ∈ R+ , b ∈ R .

and preferred Cartesian inertial coordinates x with the following property: consider a closed system of N ‘particles’ (bodies i = 1, . . . , N ) interacting via a 2-body force of the form Fij = xij · fij (|xij |) ;

xij = xi − xj ,

3.2 The Newtonian Space-Time

49

obeying the law “actio = reactio”: Fij = −Fj i . then the equations of motion read: mi x¨ i =



Fij .

(1 ≤ i ≤ N )

(3.2.1)

j =i

Since (t, x i ) are inertial coordinates no inertial forces appear in the dynamical equation of motion. If xi (t) represents a solution of (3.2.1) then also xi (±t + b) ≡ R xi (t) + vt + d , where v and d are constants and R is a constant rotation matrix. Hence, if (t, x) and (t  , x ) are inertial coordinates they are related by a Galilean transformation t = t + b x = R x + vt + d

(3.2.2)

or simply for Rij = δij , v = vex , d = b = 0 t = t x  = x + vt .

(3.2.3)

3.2.2 Weak Equivalence Principle and Newtonian Theory of Gravity Let us write the Newtonian law for the free-fall of test bodies in some external gravitational field produced by some mass M in the form mI x¨ = −

GMmG x ≡ mG g . r2 r

(3.2.4)

Here, mI denotes the inertial mass of the test body, mG its (passive) gravitational mass and g is the gravitational acceleration. Weak Equivalence Principle The mG /mI -ratio is identical for all test bodies, independent of their shape and composition. This implies, that we can take mI = mG

(3.2.5)

50

3 Newtonian Celestial Mechanics

and therefore we can cancel the two masses of the test body in (3.2.4). In this way we are led to the Law of Galileo: In the Newtonian space-time there exists a preferred class of reference frames with Galilean coordinates x μ = (x 0 , x 1 , x 2 , x 3 ) = (t, x)

(3.2.6)

in which the dynamical law for free-fall takes the form x¨ = ∇U.

(3.2.7)

Here, the gravitational potential U is a coordinate and frame dependent function. We can now write this as ν σ d 2xi i dx dx = 0; +  νσ dλ dλ dλ2

d 2t =0 dλ2

(3.2.8)

with i 00 =−

∂U ≡ U,i ∂x i

(3.2.9)

μ

and all other quantities νσ = 0 in our Galilean, i.e. Cartesian and inertial coordinate system. This is the surprising consequence of the weak equivalence principle or the universality of free fall: the equation for free-fall can be understood as geodesic equation in some affine space-time! From (3.2.8) we see that t = aλ + b with constants a and b; therefore, we can choose the affine parameter λ as the absolute Newtonian time t. Note, that (3.2.8) is written in covariant form, i.e., it is valid for arbitrary coordinates. Now, in Newton’s theory one would not like to change the universal time coordinate but one often needs the equation in curvilinear (e.g., spherical) or rotating coordinates. If the dot stands for d/dλ then for arbitrary spatial coordinates x j the free-fall equation (3.2.8) reads: i j i x˙ + 00 = 0. x¨ i + ji k x˙ j x˙ k + 20j

(3.2.10)

This equation now also contains the terms necessary in curvilinear coordinates and the inertial forces. E.g. i = −U,i + [ × ( × x)]i 00

(3.2.11)

also contains the Centrifugal forces and i x˙ j = ( × x˙ )i 0j

(3.2.12)

3.2 The Newtonian Space-Time

51

the Coriolis force. Here,  is the angular velocity of the coordinates with respect to inertial coordinates (e.g., Galilean coordinates). Exercise 3.1 From (3.2.10) compute the equation of motion of a freely falling test particle in some potential U in spherical non-rotating coordinates x i = (r, θ, φ). i = 0 and  i = −U . It is then Solution Since our coordinates are inertial 0j ,i 00 clear that we only have to compute the quantities ji k which are the Christoffel symbols for the Euclidean 3-space in spherical coordinates given by (2.9.6). The equations for a test mass (e.g., satellite) in spherical coordinates therefore read:

r¨ − r θ˙ 2 − r sin2 θ φ˙ 2 − U,r = 0 2 θ¨ + r˙ θ˙ − sin θ cos θ φ˙ 2 − U,θ = 0 r 2 φ¨ + 2 cot θ θ˙ φ˙ + r˙ φ˙ − U,φ = 0 . r

(3.2.13)

For a central spherical monopole field, U = μ/r, we can take the fixed orbital plane e.g., by θ = π/2, so that (3.2.13b) is satisfied. Then we obtain the usual equations for the Newtonian Kepler problem r¨ − r φ˙ 2 +

μ =0 r2

2 φ¨ + r˙ φ˙ = 0, r

(3.2.14)

where (3.2.14b) leads to the angular momentum integral in the form r 2 φ˙ = const.

(3.2.15)

Now, each set of affine connections has an associated curvature tensor. In Newton’s theory this curvature tensor is not a Riemann curvature tensor because the Newtonian space-time does not have a (non-degenerate) space-time metric; here, space is flat and the metric is the Euclidean one and this Euclidean space-metric is completely independent of the time-metric. In Galilean coordinates the Newtonian curvature tensor reads: i κ i R i 0j 0 = 00,j − ji 0,0 + ji κ 00 − 0κ jκ0 i = 00,j = −U,ij .

(3.2.16)

The physical meaning of the curvature tensor is that it describes the tidal forces: “curvature tensor”

=

“tidal force tensor”.

52

3 Newtonian Celestial Mechanics

Fig. 3.1 A bundle of neighboring freely-falling test particles

n

σ1

σ2

σ3

To see this we consider a bundle of freely-falling test particles. Each of these particles we give a number σ , so σ1 is the first particle, σ2 the second and so on. We think of having infinitely many of such particles such that we can differentiate with respect to the number σ (see Fig. 3.1). Then from x¨ − ∇U = 0 we get ∂ ∂σ



d 2xi ∂U − i ∂x dt 2

= 0.

If we now write ∂ ∂x j ∂ ∂ = ≡ nj j , j ∂σ ∂σ ∂x ∂x

(3.2.17)

where nj can be understood as a vector connecting two neighboring test particles for constant times, we get d 2 ni ∂ 2U j d 2 ni − n = + R i 0j 0 nj = 0. ∂x i ∂x j dt 2 dt 2

(3.2.18)

This is the Newtonian Jacobi equation or geodesic deviation equation. It describes the relative motion of two neighboring test particles under the influence of the gravitational field, which is given by the difference of the gravitational forces acting upon the two particles (this explains the occurrence of the second derivatives of the Newtonian potential). We now come to the field equation. In Newton’s theory of gravity this is the Poisson equation: U = −4π Gρ. Here,  is the Laplacian =

∂2 ∂2 ∂2 + 2+ 2, 2 ∂x ∂y ∂z

(3.2.19)

3.3 Gravitational Field of a Body

53

G is the Newtonian gravitational constant and ρ is the (gravitational) mass density which generates the gravitational field. With (3.2.16) we can write the field equations in the form R i 0i0 = −U = +4π Gρ or R00 = 4π Gρ

(3.2.20)

R00 = −U .

(3.2.21)

with

The meaning of this form of the field equation is clear: gravitational mass density ρ produces curvature of space-time. In Newton’s theory the curvature is in the direction of time; space in Newton’s theory is Euclidean, i.e., flat.

3.3 Gravitational Field of a Body For a single body (E) we will impose the boundary condition lim UE (t, x) = 0

(3.3.1)

|x|→∞

for the Newtonian potential U . The Poisson equation (3.2.19) then implies

d 3x

UE (t, x) = G E

ρ(t, x ) ≡G |x − x |

E

dM . |x − x |

(3.3.2)

3.3.1 Spherical Multipole-Moments Outside of the matter distribution where ρ = 0 the potential U satisfies the Laplace equation U (t, x) = 0

(3.3.3)

and we want to expand the right hand side of (3.3.2) in terms of multipole moments. To this end we might employ an expansion in terms of spherical harmonics (phase convention as in Condon and Shortley (1953) or Jackson (1975); especially in the geodetic literature the phase convention of the spherical harmonics often differs by

54

3 Newtonian Celestial Mechanics

a factor of (−1)m ) Ylm (θ, φ) = Nlm Plm (cos θ )eimφ

(m ≥ 0) .

(3.3.4)

Here, Nlm are normalization constants  Nlm ≡

2l + 1 (l − m)! , 4π (l + m)!

(3.3.5)

Plm (cos θ ) are associated Legendre functions: Plm (x) = (−1)m (1 − x 2 )m/2

dm Pl (x) , dx m

(3.3.6)

where Pl (x) are ordinary Legendre polynomials defined by Rodrigues’ formula Pl (x) =

1 dl 2 (x − 1)l . 2l l! dx l

(3.3.7)

Then the associated Legendre polynomials can be written in the form Plm (x) =

l+m (−1)m 2 m/2 d (1 − x ) (x 2 − 1)l . 2l l! dx l+m

(3.3.8)

Exercise 3.2 Show that: Pl (x) =

[l/2] (2l − 2k)! 1  x l−2k , (−1)k l 2 k!(l − k)!(l − 2k)!

(3.3.9)

k=0

where [l/2] ≡ kmax equals l/2 if l is even and equals (l − 1)/2 if l is odd. Thus the associated Legendre functions are finite polynomials in x = cos θ in our case. Solution From the binomial formula (x − 1) = 2

l

l    l k=0

k

x 2(l−k) (−1)k

and (2l − 2k)! l−2k d l 2(l−k) x x = (2l − 2k)(2l − ak − 1) · · · (l − 2k + 1)x l−2k = dx l (l − 2k)!

3.3 Gravitational Field of a Body

55

we get l   dl 2 dl  l l (x − 1) = (−1)k x 2(l−k) k dx l dx l k=0

=

k max  k=0

=

k max

l k

 (−1)k

(−1)k

k=0

(2l − 2k)! l−2k x (l − 2k)!

(2l − 2k)! l−2k l! x . k!(l − k)! (l − 2k)!

in agreement with (3.3.9). Exercise 3.3 Use (3.3.9) to show that for m ≥ 0 dm Pl (x) = dx m

[(l−m)/2] 

a lmk x l−m−2k

(3.3.10)

(2l − 2k)! (−1)k . l 2 k!(l − k)! (l − m − 2k)!

(3.3.11)

k=0

with a lmk ≡

Using (3.3.10) we see that Ylm (θ, φ) for m ≥ 0 can be written in the form Ylm (θ, φ) = (−1) Nlm (e sin θ ) m

iφ

m

[(l−m)/2] 

a lmk (cos θ )l−m−2k .

(3.3.12)

k=0

For negative values of m ∗ (θ, φ) Yl,−m (θ, φ) = (−1)m Ylm

(3.3.13)

which is then true for all values of m. Taking all possible values of (l, m) they form a complete set of orthonormal functions over the unit sphere; the normalization and orthogonality condition takes the form



2π

dφ 0

0

π

sin θ dθ Yl∗ m (θ, φ)Ylm (θ, φ) = δll  δmm .

(3.3.14)

Moreover, one finds l  m=−l

|Ylm (θ, φ)|2 =

2l + 1 4π

(3.3.15)

56

3 Newtonian Celestial Mechanics

and l ∞  l  r< 1 1 = 4π Y ∗ (θ  , φ  )Ylm (θ, φ) . l+1 lm |x − x | 2l + 1 r>

(3.3.16)

l=0 m=−l

Here, r< (r> ) is the smaller (larger) value of |x| and |x |. For a few small l values explicit expressions for Ylm (θ, φ) are given by: l=0

1 Y00 = √ 4π 

l=1

Y11 = − 

3 cos θ 4π  3 sin θ e−iφ Y1,−1 = + 8π  1 15 Y22 = sin2 θ e2iφ 4 2π  15 Y21 = − sin θ cos θ eiφ 8π    5 3 1 cos2 θ − Y20 = 4π 2 2  15 sin θ cos θ e−iφ Y2,−1 = + 8π  1 15 sin2 θ e−2iφ Y2,−2 = 4 2π  1 35 sin3 θ e3iφ Y33 = − 4 4π  1 105 2 sin θ cos θ e2iφ Y32 = 4 2π  1 21 sin θ (5 cos2 θ − 1)eiφ Y31 = − 4 4π    7 5 3 3 cos θ − cos θ Y30 = 4π 2 2 Y10 =

l=2

l=3

3 sin θ eiφ 8π

3.3 Gravitational Field of a Body

57

Y3,−1 Y3,−2 Y3,−3

 1 21 =+ sin θ (5 cos2 θ − 1)e−iφ 4 4π  1 105 2 sin θ cos θ e−2iφ = 4 2π  1 35 sin3 θ e−3iφ =+ 4 4π

Some selected spherical harmonics are shown in Fig. 3.2. Inserting expression (3.3.16) into (3.3.2) we get outside the matter distribution for which r> = |x|, r< = |x | UE (t, x) = G

 l,m

≡G

d 3x

E

l ∞   l=0 m=−l

4π ρ(t, x ) l ∗   Ylm (θ, φ) r Ylm (θ , φ ) 2l + 1 r l+1

Mlm

Ylm (θ, φ) . r l+1

(3.3.17)

Here Mlm are the complex, multipole moments of our matter √ spherical mass √ distribution. Since Y00 = 1/ 4π , M00 = 4π M, where M is the mass of our body. We will assume that such an expansion converges outside a coordinate sphere B that completely encompasses the matter distribution. Often different mass multipole

Fig. 3.2 Graphics of selected spherical harmonics: Y00 , Y10 , Y20 (upper part) and (Y21 ), Y30 , (Y31 ) (lower part) (Image credit: P. Wormer, file licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)

58

3 Newtonian Celestial Mechanics

moments are used in practise. Let l 

Mlm Ylm (θ, φ) =

m=−l

l 

Plm (cos θ ) [Clm cos mφ + Slm sin mφ]

m=0

(3.3.18) so that with real potential coefficients Clm and Slm , ∞

UE (t, x) =

G 1 Plm (cos θ )[Clm cos mφ + Slm sin mφ] . r rl l

(3.3.19)

l=0 m=0

Frequently in the geodetic literature one employs dimensionless potential coeffi∗ and S ∗ : cients, Clm lm l   ∞ GM   R l ∗ ∗ UE (t, x) = Plm (cos θ )[Clm cos mφ + Slm sin mφ] r r

(3.3.20)

l=0 m=0

with ∗ Clm = (MR l )Clm ∗ Slm = (MR l )Slm

(3.3.21)

and R is some suitably chosen radius of the central body E. Then (Clm , Slm ) or ∗ , S ∗ ) are real multipole moments that are also called potential coefficients. (Clm lm They are related with our complex mass multipole moments by Clm = Nlm (2 − δm0 )[Mlm ] Slm = −2Nlm (1 − δm0 )[Mlm ] .

(3.3.22)

For a body with axial symmetry m = 0 in (3.3.20) so that (assuming the mass dipole term with l = 1 to vanish)    l ∞  R GM Jl Pl (cos θ ) 1− UE (t, x) = r r

(3.3.23)

l=2

∗. where Jl = −Cl0 The usual spherical mass multipole-moments (potential coefficients) might be replaced by Cartesian mass multipole-moments. These quantities are symmetric and trace-free (STF) Cartesian tensors that will be studied in Sect. 3.3.3.

3.3 Gravitational Field of a Body

59

3.3.2 Spherical Mass-Moments of an Oblate Spheroid The Newtonian gravitational potential U (t, x) of some matter distribution with density ρ in spherical coordinates r, θ, φ is given by



r  dr  sin θ  dθ  dφ  2

U (t, x) = G

ρ . |x − x |

We will now assume the matter distribution to be axially symmetric and static, so that outside the body

∞

U (x) = 2π G

r  dr  2

0

π

sin θ  dθ 

0

ρ(r  , θ  ) , < |x − x | >

where the brackets indicate an average over the angle φ  . From (3.3.16) and the expression for the spherical harmonics it is clear that for the axially symmetric case only the m = 0 term contributes so that ∞   1  r l 1 = Pl (cos θ )Pl (cos θ  ) . < |x − x | > r r

(3.3.24)

l=0

Therefore,

π  1 R(θ  ) l+2  U (x) = 2π G dr sin θ  dθ  r  ρ(r  , θ  )Pl (cos θ  )Pl (cos θ ) , r l+1 0 0 l

where R(θ ) defines the outer boundary of our body. Assuming ρ(r  , θ  ) = ρ0 = const., we can write U (x) =



Jl

l

Pl (cos θ ) r l+1

(3.3.25)

with

R(θ  )

Jl = 2π Gρ0

dr 

0

π

sin θ  dθ  r 

l+2

Pl (cos θ  ) .

0

Integration over r  leads to 2π Gρ0 Jl = l+3

π 0

sin θ  dθ  R l+3 (θ  )Pl (cos θ  )

60

3 Newtonian Celestial Mechanics

and writing z = cos θ  2π Gρ0 Jl = l+3

+1

−1

dzPl (z)R(z)l+3 .

(3.3.26)

We will now assume that our body has the shape of an oblate spheroid, given in Cartesian coordinates by 1=

z2 x2 + y2 + , a2 c2

(3.3.27)

where a(c) is the semi-major (semi-minor) axis of the rotational ellipsoid. For our oblate body a > c. From this one gets 1=

R 2 sin2 θ R 2 cos2 θ + 2 a c2

or a R(z) =  1/2 1 + αz2

(3.3.28)

with α≡

a 2 − c2 . c2

Since R(z) = R(−z) all odd multipole moments J2n+1 vanish. For the even massmoments, l = 2n, one gets J2n =

3GMc2n 4π Gρ0 a 2n+3 (−α)n (1 + α)−n−1/2 = (−1)n αn . (2n + 1)(2n + 3) (2n + 1)(2n + 3) (3.3.29)

Here we have used (Magnus et al. 1981)

+1 −1

dz

P2n (z) 2 (−α)n (1 + α)−n−1/2 , = 2 n+3/2 2n + 1 (1 + αz ) M=

4π ρ0 a 3 (1 + α)−1/2 3

is the mass of the spheroidal body (with volume V = (4π/3)a 2 c)), and  c 2 α = α. 1+α a

(3.3.30)

(3.3.31)

3.3 Gravitational Field of a Body

61

Introducing the dimensionless mass multipole moments Jl by Jl = Jl (GMa l )

(3.3.32)

we get J2n = (−1)n

3 2n , (2n + 1)(2n + 3)

(3.3.33)

where the ellipticity is defined by 2 = 1 −

c2 . a2

This result (3.3.33) can be found e.g., in Antonov et al. (1988) or Pohanka (2011). Exercise 3.4 Use the formula  ∞  1 1  r< l Pl (cos φ) , = |x − x | r> r>

(3.3.34)

l=0

where r< (r> ) is the smaller (larger) value of |x| and |x |, and φ is the angle between x and x , as shown in Fig. 3.3, to derive the potential of a ring of total mass M and radius a inside the ring in the ring’s plane. Also compute the gravitational acceleration of a unit mass that is located at such a point. Solution The Newtonian potential U is given by

dm U =G |x − x | with dm = (M/2π ) dφ. Using (3.3.34) we get: U=

∞ GM   r l Il , a a

(3.3.35)

l=0

z

Fig. 3.3 Geometry in relation (3.3.34)

x r φ

x r y

x

62

3 Newtonian Celestial Mechanics

where Il ≡

1 2π

2π

Pl (cos φ)dφ . 0

Now, Il = 0 for odd values of l and

I2k =

2 P2k (0)

(2k)! = 2k 2 (k!)2

2 .

(3.3.36)

Therefore, the Newtonian gravitational potential of a ring in the ring’s plane inside of the ring is given by: U=

∞  r 2k GM  2 P2k (0) . a a

(3.3.37)

k=0

The acceleration induced by the ring is then given by (n = x/r) aring =

∞  r 2k−1 GM  ∂U 2 ·n= 2 (2k)P2k (0) n. ∂r a a

(3.3.38)

k=1

3.3.3 STF-Tensors A Cartesian l-tensor is a set of real or complex numbers Ti1 i2 ...il with l different indices i1 to il , each taking the values 1, 2, 3 or equivalently (x, y, z). A Cartesian 1-tensor thus is a three-component vector Ta with a = 1, 2, 3 = x, y, z. A Cartesian 2-tensor is a 3 × 3 matrix Tij with i, j = 1, 2, 3. For the sake of compactness often a set of l Cartesian indices is abbreviated by a multi-index, e.g., L ≡ i1 i2 . . . il etc. Usually Einstein’s summation convention is assumed, i.e., if some index appears twice a summation over that index is implied automatically, e.g., AL BL ≡ Ai1 i2 ...il Bi1 i2 ...il ≡

3 

Ai1 i2 ...il Bi1 i2 ...il .

(3.3.39)

ij =1

Given a Cartesian tensor TL , we denote its symmetric part by parentheses T(L) ≡ T(i1 ...il ) ≡

1 Ti ...i , l! σ σ (1) σ (l)

(3.3.40)

where σ runs over all l! permutations of (12 . . . l). If TL is a Cartesian l-tensor; each quantity where we put two arbitrary indices equal with a subsequent summation is

3.3 Gravitational Field of a Body

63

called a trace of TL . If every trace of TL vanishes it is called trace-free. Of great importance are symmetric and trace-free (STF) Cartesian tensors. The STF-part of TL is denoted indifferently by TˆL ≡ T ≡ T . The explicit expression of the STF part reads (Pirani 1964; Thorne 1980) TˆL =

[l/2] 

akl δ(i1 i2 . . . δi2k−1 i2k Si2k+1 ...il )a1 a1 ...ak ak

(3.3.41)

k=0

where SL = T(L) , akl =

(−1)k (2l − 2k − 1)!! l! , (2l − 1)!! (l − 2k)!(2k)!!

(3.3.42)

[l/2] denoting the integer part of l/2, i.e., the largest integer equal or smaller than l/2. For instance, 1 Tˆij = T(ij ) − δij Taa 3  1 δij T(kaa) + δj k T(iaa) + δki T(j aa) . Tˆij k = T(ij k) − 5 One has for every positive integer l l! = l · (l − 1) · (l − 2) · · · 2 · 1 ;

l!! = l · (l − 2) · (l − 4) · · · (1 or 2) .

(3.3.43)

and (2l)!! = 2 l! ; l

(2l + 1)! ; (2l + 1)!! = 2l l!

  l! l . = k k!(l − k)!

(3.3.44)

Exercise 3.5 Proof relation (3.3.41) with (3.3.42) for STF-tensors and show that the coefficients akl from (3.3.42) can also be written in the form akl

     −1 l l 2l = (−1) . k 2k 2k k

(3.3.45)

Proof A proof of this exercise can be found in Pirani (1964) If we denote by ej (j = 1, 2, 3) the Cartesian basic vectors (ejk = δjk ), it can be verified that a basis of the (2l + 1)-dimensional vector space of STF-l tensors can be constructed out of the STF parts of the one-fold tensorial products E+ ⊗ · · · ⊗

64

3 Newtonian Celestial Mechanics

E+ ⊗ E0 ⊗ · · · ⊗ E0 , where ⎛ ⎞ 1 E+ ≡ e1 + ie2 = ⎝ i ⎠ , 0

⎛ ⎞ 0 E0 ≡ e3 = ⎝ 0 ⎠ 1

(3.3.46)

(with i 2 = −1) and their complex conjugates. More precisely such a basis is YˆLlm with −l ≤ m ≤ +l where, for m ≥ 0, lm , YˆLlm = Alm E

(3.3.47)

+ 0 0 ELlm = E+ i1 . . . Eim Eim+1 . . . Eil ,

(3.3.48)

with

and

Alm

2l + 1 = (−1) (2l − 1)!! 4π(l − m)!(l + m)!

1/2

m

.

(3.3.49)

Explicitly one has (e.g., (2.12) of Thorne (1980); see Exercise (3.6)): YˆLlm = Yˆklm = (−1)m Nlm 1 ...kl × δk3m+1

[(l−m)/2]  j =0

    1 2 1 2 · · · δ × a lmj δ(k + iδ + iδ k k (k m m 1 1

aj a 1 . . . δk3l−2j (δkal−2j δ a1 ) · · · (δkl−1 δklj) ) +1 kl−2j +2

(3.3.50) where a lmj is given by (3.3.11). For m < 0 we have YˆLlm = (−1)m (YˆLl,−m )∗ .

(3.3.51)

The orthonormality condition is such that (2l + 1)!!  . YˆLlm (YˆLlm )∗ = δmm 4π l!

(3.3.52)

Many important relations for STF-tensors are known (e.g., Blanchet and Damour 1986; Thorne 1980). Some of them are listed in Appendix. The importance of STF-tensors and the basic tensors YˆLlm results from their relations with the usual scalar spherical harmonics (Ylm ). The basic relation between YˆLlm and these spherical harmonics is obtained from (x = r sin θ cos φ , y =

3.3 Gravitational Field of a Body

r sin θ sin φ ,

65

z = r cos θ )

nx + iny =

x y + i = sin θ eiφ ; r r

nz =

z = cos θ . r

(3.3.53)

It reads: Ylm = YˆLlm nL = YˆLlm nˆ L ,

(3.3.54)

x i1 . . . x il rl

(3.3.55)

where nL = ni1 ...il =

Using the orthogonality relation (3.3.52) the inverse relation nˆ L =

l  4π l! ∗ YˆLlm Ylm (2l + 1)!!

(3.3.56)

m=−l

can be derived. Exercise 3.6 Using relation (3.3.12) to derive expression (3.3.54) Ylm = YˆLlm nL . Proof Since nx + iny = eiφ sin θ and nz = cos θ in the sum for YˆLlm nL we can replace the m terms of the form δ 1 + iδ 2 by eiφ sin θ each and the l − m − 2j terms of form δ 3 by cos θ . The j terms of the form δ a δ a that are related with the traces of nL should be replaced by 1 because of na na = 1, so that we end up with relation (3.3.12) for Ylm (θ, φ). Let us illustrate relation√(3.3.54) for l = 0, 1, 2. For l = 0 the normalization gives the single number 1/ 4π . For l = 1 we have explicitly 

Yˆj11

3 + E =− 8π

 Yˆj10

=

3 0 E 4π

 Yˆj1,−1

=+

3 − E 8π

(3.3.57)

with ⎛

⎞ 1 E− = (E+ )∗ = ⎝ −i ⎠ . 0

(3.3.58)

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3 Newtonian Celestial Mechanics

It is easy to see that Y1m = Yˆj1m x j /r. For l = 2 we have ⎛

⎞ 1 i 0 5 ⎝ 3 =+ i −1 0 ⎠ 2 24π 0 0 0 ⎛ ⎞  001 5 ⎝ 3 =− 00 i⎠ 2 24π 1i 0 ⎛ ⎞  −1 0 0 1 5 ⎝ =+ 0 −1 0 ⎠ 2 4π 0 0 2 ⎛ ⎞  0 0 1 5 ⎝ 3 =+ 0 0 −i ⎠ 2 24π 1 −i 0 ⎛ ⎞  1 −i 0 5 ⎝ 3 =+ −i −1 0 ⎠ 2 24π 0 0 0 

Yˆj22k

Yˆj21k

Yˆj20k

Yˆj2,−1 k

Yˆj2,−2 k

(3.3.59)

j k 2 and Y2m = Yˆj2m k (x x /r ) is easily verified.

. Use this Exercise 3.7 Use relation (3.3.50) to get expression (3.3.59) for Yˆk20 1 k2 result to show that Y20 = Yˆ 20 nk1 nk2 . k1 k2

Proof We have Yˆk20 = N20 (a 200 δk31 δk32 + a 201 δks1 δks2 ) , 1 k2 where N20 =

√ 5/(4π ), a 200 = 3/2 and a 201 = −1/2. Therefore, 1 = N20 (3δk31 δk32 − δks1 δks2 ) Yˆk20 1 k2 2

in agreement with (3.3.59). Furthermore, Yˆk20 nk1 nk2 = (1/2)N20 (3nz nz − 1) = 1 k2 Y20 . Suppose, TˆL is some Cartesian STF l-tensor. Then it can always be written in the form TˆL =

l  m=−l

Tlm YˆLlm ,

(3.3.60)

3.3 Gravitational Field of a Body

67

where, because of the normalization relation (3.3.52), the (2l + 1) numbers Tlm are given by Tlm =

4π l! ˆ ˆ lm ∗ TL ( Y L ) . (2l + 1)!!

(3.3.61)

Generally, Tl,−m = (−)m (Tlm )∗ .

(3.3.62)

For l = 1 one finds  T11 = −  T10 = +

4π (T1 − iT2 ) 6 4π T3 3

and for l = 2 1√ 30π (Tˆ11 − Tˆ22 − 2i Tˆ12 ) 15 2√ =− 30π (Tˆ13 − Tˆ23 ) 15 2√ =+ 5π Tˆ33 . 5

T22 = + T21 T20

Another useful formula reads     1 1 = Nlm a lm0 (∂x + i∂y )m ∂zl−m YˆLlm ∂L r r from which the well-known Maxwell relation   1 Ylm lm ˆ Y L ∂L = (−1)l (2l − 1)!! l+1 r r

(3.3.63)

(3.3.64)

can be derived. Explicitly one gets (see e.g., Hobson 1955)  l+m

(−1)

1 2l + 1 (∂x + i∂y )m ∂zl−m 4π (l + m)!(l − m)!

  1 Ylm = l+1 . r r

(3.3.65)

By applying (3.3.63) twice and making use of (3.3.64), one obtains jk YˆLlm ∂L YˆJ ∂J

  Yl+j,m+k 1 = (−1)l+j (2l − 1)!!(2j − 1)!!γjlm k r r l+j +1

(3.3.66)

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3 Newtonian Celestial Mechanics

with  γjlm k ≡

2j + 1 (p + q)!(p − q)! 2l + 1 (l + m)!(l − m)! (j + k)!(j − k)! (2p + 1)4π

(3.3.67)

where p ≡ l + j, q ≡ m + k.

3.3.4 Cartesian Multipole-Moments We will now introduce an equivalent Cartesian (i.e., non spherical) multipole moment expansion of U (t, x) outside a coordinate sphere B of some body where ρ = 0. Using the Taylor-expansion  (−1)l 1 = x i1 · · · x il ∂i1 ···il  |x − x | l! l≥0

  1 r

(3.3.68)

where r ≡ |x|, we can write the Newtonian potential in the form

ρ |x − x |    (−1)l 1 . =G d 3 x  ρ  x i1 · · · x il ∂i1 ···il l! r

U (t, x) = G

d 3x

(3.3.69)

l≥0

Let ϕi1 ...il

    1 1 ∂l ≡ i . ≡ ∂i1 ···il i l 1 r ∂x · · · x r

Then because of   1  =0 r

(3.3.70)

the symmetric Cartesian (every index takes the values x, y, z) tensor ϕi1 ...il is tracefree, i.e., ϕi1 j ...j il = 0 etc. ϕi1 ...il is a Cartesian STF-tensor. Let   1 . (3.3.71) ϕL = ϕˆL = ∂L r

3.3 Gravitational Field of a Body

69

Then ϕL = (−1)l (2l − 1)!!

nˆ L . r l+1

(3.3.72)

The proof of this important relation is by induction: for l = 0 we get ϕ = 1/r in agreement with (3.3.72). For l = 1 we get ϕi = ∂i (1/r) = −x i /r 3 = −ni /r 2 also in accordance with (3.3.72). We now assume this relation to be valid for l. Then (tt: trace-terms) ϕL+1

 

1 xˆ L l = ∂i (−1) (2l − 1)!! 2l+1 = ∂Li r r   1 + tt = (−1)l (2l − 1)!!x L ∂i r 2l+1 xLxi + tt r 2l+3 nˆ L+1 = (−1)l+1 (2(l + 1) − 1)!! 2l+3 + tt r = (−1)l+1 (2(l + 1) − 1)!!

as was to be shown. With that result we can write the Cartesian multipole expansion of U in the form U (t, x) = G

 (2l − 1)!! l!

l≥0

ML

nˆ L . r l+1

(3.3.73)

Using the obvious fact that Aˆ L Bˆ L = AL Bˆ L = Aˆ L BL

(3.3.74)

we have ML = Mˆ L =

d 3 x  ρ  xˆ L .

(3.3.75)

This Cartesian multipole expansion of U is equivalent to the expansion in terms of spherical harmonics. Clearly the mass multipole moments, ML in the Cartesian language, depend upon the origin of the Cartesian coordinate system x. Let MLx be the Cartesian mass moments with respect to some coordinates x; let x = y + d with y being a new set of Cartesian coordinates whose origin differs by the constant vector d from that of the x-system. Then, from the definition of ML one obtains: MLx =

 K≤L

y

d .

(3.3.76)

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3 Newtonian Celestial Mechanics

Thus, associated with a single body there is a unique Cartesian inertial system that is mass-centered where the mass dipole moment vanishes. Using relations (3.3.17) and (3.3.73) the correspondence between the Cartesian moments ML and the potential coefficients (Clm , Slm ) or the spherical massmoments Mlm can be found. Using also (3.3.13) and (3.3.51) one finds that our Cartesian and spherical mass multipole moments, Mˆ L and Mlm are related by Mlm =

4π (Yˆ lm )∗ Mˆ L . 2l + 1 L

(3.3.77)

 The inverse relation is obtained by projecting with YˆLlm , summing over m and using the orthogonality relation (3.3.52):

Mˆ L =

+l  l! Mlm YˆLlm . (2l − 1)!!

(3.3.78)

m=−l

Exercise 3.8 Derive the relations between the Cartesian and spherical quadrupole mass-moments (l = 2) explicitly simply by expressing the spherical harmonics in terms of Cartesian coordinates. Solution We start with U (2) (x) =

G [P20 C20 + P21 (C21 cos φ + S21 sin φ) + P22 (C22 cos 2φ + S22 sin 2φ)] . r3

Since (x = r sin θ cos φ, y = r sin θ sin φ, z = r cos φ), r 2 P20 = (2z2 − x 2 − y 2 )/2 r 2 P21 cos φ = 3xz r 2 P21 sin φ = 3yz r 2 P22 cos 2φ = 3(x 2 − y 2 ) r 2 P22 sin 2φ = 6xy , we get U (2) (x) =

G [C20 (2z2 − x 2 − y 2 )/2 + 3C21 xz + 3S21 yz + 3C22 (x 2 − y 2 ) r5 + 6S22 xy] . (3.3.79)

In the STF-language we have U (2) (x) =

3G Mij x i x j , 2 r5

(3.3.80)

3.3 Gravitational Field of a Body

71

where a complete summation over i and j over 1, 2, 3 is assumed. Considering, e.g., the xz-term, one has 3

G G C21 xz = 3 5 M13 xz 5 r r

so that M13 = C21 . In this way one finds 1 M11 = +2C22 − C20 3 1 M22 = −2C22 − C20 3 2 M33 = C20 3 M12 = 2S22 M13 = C21 M23 = S21 .

(3.3.81)

Because of symmetry and the tracelessness, i.e., M11 + M22 + M33 = 0, there are five independent components of Mij . The inverse relations read 1 C20 = M33 − (M11 + M22 ) 2 C21 = M13 C22 =

1 (M11 − M22 ) 4

(3.3.82)

S21 = M23 S22 =

1 M12 . 2

Exercise 3.9 Derive the relations between ML and the potential coefficients Clm and Slm for l = 3 by using relations (3.3.22) Solution From (3.3.47) we get ij3mk = A3m · E ij k = A3m · (E(ij k) − 1 · (δij E(kaa) + δki E(j aa) + δj k E(iaa) )) Y 5

72

3 Newtonian Celestial Mechanics

that reads explicitly for m = 0, 1, 2, 3 ij30k = 15 Y 12



7 π

  1 3 3 3 3 3 3 δ i δj δk − δij δk + δki δj + δj k δi 5



   1 7 (δi1 + iδi2 )δj3 δk3 + (δk1 + iδk2 )δi3 δj3 + (δj1 + iδj2 )δk3 δi3 − 12π 3  1  δij (δk1 + iδk2 ) + δki (δj1 + iδj2 ) + δj k (δi1 + iδi2 ) − 15

ij31k = −15 Y 4

ij32k Y



7 1 1 (δi + iδi2 )(δj1 + iδj2 )δk3 + (δk1 + iδik )(δi1 + iδi2 )δk3 30π 3  + (δj1 + iδj2 )(δk1 + iδk2 )δi3

15 = 4

ij32k = −15 Y 4



 7  1 (δi + iδi2 )(δj1 + iδj2 )(δk1 + iδk2 ) 180π

with (3.3.78) the Cartesian STF-multipole moments can be computed : ML =

+l l   l! 2l! Llm = Llm ) Mlm Y (Mlm Y (2l − 1)!! (2l − 1)!! m=−l

=

m=0

3 12  L3m ) (M3m Y 15 m=0

so that 3 M111 = − C31 + 6C33 5

1 M112 = − S31 + 6S33 5

1 M122 = − C31 − 6C33 5

3 M222 = − S31 − 6S33 5

2 M113 = − C30 + 2C32 5

2 M223 = − C30 − 2C32 5

M123 = +2S32 .

3.4 The Tidal Potential

73

3.4 The Tidal Potential 3.4.1 Newtonian Tidal Moments For the description of tidal forces the equivalence principle is of special relevance. This principle implies that in suitable local coordinates (t, X) the gravitational action of external bodies can be described by some tidal potential Utidal of the form Utidal (X) = U ext (zE + X) − U ext (zE ) −

d 2 zE · X. dt 2

(3.4.1)

For an observer in free-fall, d 2 zi = U,iext dt 2 so that the linear term in the effective potential vanishes and an expansion in terms of co-moving spatial coordinates starts with quadratic terms. Deviations from such free-fall behavior of some astronomical body results either from non-gravitational forces acting on the body or from couplings to the external gravitational field resulting from the non-spherical components of the body’s own gravitational field. In practise it is common to expand U eff in a tidal-series, i.e., a Taylor-series in positive powers of X, U eff (X) = Gi Xi +

1 1 Gij Xi Xj + · · · + Gi1 ...il Xi1 · · · Xil + · · · 2! l!

(3.4.2)

where the Newtonian tidal-moments felt by body E read Gi (t) ≡ ∂i U ext (zE ) − Gi1 ...il (t) ≡ ∂i1 ...il U

ext

d 2 zEi , dt 2

(3.4.3)

(l ≥ 2) .

(zE )

Because of the Laplace equation, UA = 0 outside body A, the tidal moments Gi1 ...il are automatically symmetric and trace-free (L ≡ i1 . . . il ) GL ≡ STFL [∂L U ext (zE )]

(l ≥ 2) .

(3.4.4)

Thus, with the tidal moments of (3.4.3) Utidal can be written as Utidal

∞  1 = GL XL . l! l=1

with the Newtonian tidal-moments GL which are STF-tensors.

(3.4.5)

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3 Newtonian Celestial Mechanics

Assuming zE to coincide with the geocenter (center of mass) and inserting the expression (see (3.5.5) below) ⎡ ⎤   ∞ ∞  i j   d 2 zE 1 (−1) E ⎣ ⎦, ME =G ML MJA ∂iLJ l!j ! rEA dt 2 A=E

(3.4.6)

l=0 j =0

i and r where ∂iE ≡ ∂/∂zE EA ≡ |zE − zA |, the tidal potential can be written in the form: ⎡ ∞ ∞    (−1)j J L E 1 ⎣G MA X ∂LJ Utidal (t, X) = l!j ! rEA A=E

l=2 j =0

⎤ ∞ ∞ G   (−1)j L J i E 1 ⎦ M MA X ∂iLJ − . ME l!j ! rEA

(3.4.7)

l=1 j =0

The second term on the right hand side of (3.4.7) (the geodesic deviation term) results from the fact that due to higher multipole couplings the Earth (E) is not freely falling. Expression (3.4.7) is the most general form of the tidal potential since all mass multipole moments of the bodies are taken into account. Replacing the Cartesian multipole moments by their spherical counterparts one gets the spherical representation of the tidal potential (X given by (R, , ), zE − zA given by (rEA , θEA , φEA )) Utidal (t, X) = ⎡ j ∞  l  ∞    Yl+j,m+k (θEA , φEA ) 4π l A ∗ ⎣G (−1)l γjlm k R Mj k Ylm ( , ) l+j +1 2l + 1 r A=E

l=2 m=−l j =0 k=−j

EA

⎤ j ∞ l ∞ G     l lm E A i E Yl+j,m+k (θEA , φEA ) ⎦ . − (−1) γj k Mlm Mj k X ∂i l+j +1 ME r l=1 m=−l j =0 k=−j

EA

Here,  ∇

1 r p+1



 Ypq

=−

1 2r p+2

⎞ ⎛ α Y − α− Yp+1,q+1 2p + 1 ⎝ + p+1,q−1 iα+ Yp+1,q−1 + iα− Yp+1,q+1 ⎠ 2p + 3 2α0 Yp+1,q (3.4.8)

with α± ≡

 (p ∓ q + 2)(p ∓ q + 1) ,

α0 ≡



(p + q + 1)(p − q + 1) .

(3.4.9)

3.4 The Tidal Potential

75

Exercise 3.10 Proof that expression (3.4.8) for the tidal potential in spherical coordinates follows from the Cartesian expression (3.4.7) (Hartmann et al. 1994).

3.4.2 The l = 2 Tidal Potential for External Point-Masses The tidal potential in the vicinity of the Earth’s surface is dominated by the l = 2 term. E.g., for the Moon as external body, the l = 3 term is smaller by a factor of 6400 km R 0.016 . d 400,000 km In the following we will concentrate on the l = 2 tidal potential raised by one external point mass A. For l > 1 generally we have (dA = |XA |)  GL = ∂L Uext (xE ) = GMA ∂L

1 |x − xA |

 = (2l − 1)!! E

GMA ˆ NL (XA ) . dAl+1 (3.4.10)

Hence, Gij = 3

GMA dA5

  1 i j XA XA − δij X2A 3

(3.4.11)

and (2) Utidal



1 3 GMA 1 2 2 i j 2 (X · XA ) − X XA . = Gij X X = 2 2 dA5 3

(3.4.12)

(2)

If ψ denotes the angle between X and XA (Fig. 3.4), Utidal can also be written as (2)

Utidal (t, X) =

Fig. 3.4 Geometry in the problem of tidal forces

GMA R 2 P2 (cos ψ) . dA dA2

(3.4.13)

Tide raising body A

Observer

X

ψ

XA

76

3 Newtonian Celestial Mechanics

This, however, follows immediately from UA (X) =

 ∞  G MA  R l Pl (cos ψ) . dA dA

(3.4.14)

l=0

It also follows from (3.4.8) with 1 lm γ00 =√ , 4π so that (2) GMA Utidal

R2 dA3



4π 5

  2

∗ Y2m (θ, φ)Y2m (θA , φA ) .

(3.4.15)

m=−2

If the spatial coordinates refer to the principal axes, θ and φ can be interpreted as co-latitude and longitude of the observer with radial coordinate R, the corresponding quantities with a label A refer to geocentric spherical coordinates of the tide raising body A. We have P2 (cos ψ) = P2 (cos θ ) P2 (cos θA ) 1 + P21 (cos θ ) P21 (cos θA ) cos (φ − φA ) 3 1 + P22 (cos θ ) P22 (cos θA ) cos [2 (φ − φA )] . 12

(3.4.16)

Thus, Utidal (t, X) V20 + V21 + V22

(3.4.17)

with V2i = c2i (t)P2i (cos θ )

 2 R a

(i = 0, 1, 2)

(3.4.18)

with c20 (t) =

GMA

a2 dA3

P20 (cos θA ) ,

c21 (t) =

a2 1 GMA 3 P21 (cos θA ) cos(φ − φA ) , 3 dA

c22 (t) =

a2 1 GMA 3 P22 (cos θA ) cos[2(φ − φA )] . 12 dA

(3.4.19)

(3.4.20)

3.5 Translational Equations of Motion

77

Now, φ (e.g., the observer’s longitude) is constant for an earthbound observer, but φA changes due to the rotation of the Earth. Since the inertial motion of Moon and Sun about the Earth is ‘slow’, φA has almost a diurnal period. For that reason V20 describes the long-periodic tides, V21 the diurnal tides and V22 the semi-diurnal tides. Concentrating on V21 , relevant for nutation, we have V21 =

R2 1 GMA 3 P21 (cos θ )P21 (cos θA ) cos(φ − φA ) . 3 dA

Since cos(φ − φA ) = cos φ cos φA + sin φ sin φA , V21 can be written in the form V21 = − =3

GMA dA3

GMA dA5

P21 (cos θA ) [XZ · cos φA + Y Z sin φA ] [(XA ZA ) · XZ + (YA ZA ) · Y Z] .

(3.4.21)

3.5 Translational Equations of Motion Considering the gravitational N -body problem in the accelerated E-frame the evolution equations for the motion of matter read: ∂  i ∂ρ + ρVE = 0 , j ∂t ∂XE  ∂(ρVEi ) ∂  i j ∂ ij + ρV =ρ V + t UEeff . E E j ∂t ∂XEi ∂XE

(3.5.1)

Here, VE = v i − dzEi /dt is the velocity with respect to the local frame and t ij denotes the 3 × 3 material stress tensor. Using these local equations of motion one finds d E M (t) = 0 , dt

∞ E  ∂Utidal d2 E 3 M (t) = d Xρ = Mˆ LE GEiL . dt 2 i ∂XEi E

(3.5.2)

l=0

Note, that the self-potential U E does not contribute to the right hand side of the second equation of (3.5.2) (‘action and reaction principle’). Let us now assume the origin of the local E-system to coincide with the center of mass of body E defined

78

3 Newtonian Celestial Mechanics

by the vanishing of the mass dipole moment MiE = 0 .

(3.5.3)

Then, the global equation of motion for zEi (t) can be obtained from the equilibrium condition (’d’Alembert’s principle’, see D’Alembert 1743) d2 E M = 0. dt 2 i

(3.5.4)

Using (3.5.2) and expression (3.4.3) for GEi , Eq. (3.5.4) leads to: ∞

−ME GEi = ME

1 d 2 zEi E M E GE − ∂i Uext (zE ) = 2 l! L iL dt l=1

or, writing GEiL explicitly ⎡ ⎤   ∞ j   d 2 zEi 1 (−1) E ⎣ ⎦. MLE MJA ∂iLJ ME 2 = G l!j ! rEA dt A=E

(3.5.5)

l,j =0

Here again ∂iE ≡ ∂/∂zEi etc. and rEA = |zE − zA |. It is interesting to note that the force acting on body E can be written as the gradient of a two-body interaction potential, i.e., ME

 d 2 zEi = ∇zE UEA 2 dt

(3.5.6)

A=E

with UEA

∞  (−1)j E A E 1 ML MJ ∂LJ . =G l!j ! |zE − zA |

(3.5.7)

l,j =0

This condensed formula contains all multipole-multipole couplings. Using (3.3.66) and (3.3.78) this expression is easily converted to the representation with spherical harmonics (e.g., Gleixner 1982; Ilk 1983; Hartmann et al. 1994) UEA = G

+j +l  ∞  ∞   l=0 m=−l j =0 k=−l

E Mlm MjAk (−1)l γjlm k

Yl+j,m+k (θEA , φEA ) l+j +1

rEA

where zE − zA = ˆ (rEA , θEA , φEA ) and γjlm k is given by (3.3.67).

,

(3.5.8)

3.6 Rotational Equations of Motion

79

3.6 Rotational Equations of Motion In the local E-frame we define the spin vector (intrinsic total angular momentum vector) of body E by (dropping the index E if possible)

Si ≡ iab

d 3 XXa V b .

(3.6.1)

E

Using the local evolution equations (3.5.1) we obtain dSi Di ≡ = iab dt

d 3 XρXa E

∂Utidal . ∂Xb

(3.6.2)

Also here the self-potential UE does not contribute. We then get: Di =

 A=E

=





∞  1 E M GbL iab l! aL



l=0

⎡ ⎣ iab

A=E

⎤   ∞  1 ⎦ (−1)j E A E MaL MJ ∂bLJ . l!j ! rEA

(3.6.3)

l,j =0

The conversion to the spherical representation is tedious (see Hartmann et al. 1994); one obtains (Yp,q = Yp,q (θEA , φEA )): D= G

+j ∞  +l  ∞  

⎛ E A (−1)l γjlm k Mlm Mj k

A=E l=0 m=−l j =0 k=−j

1 l+j +1

rEA

⎞ −iαYp,q−1 − iβYp,q+1 ⎝ αYp,q−1 − βYp,q+1 ⎠ −imYp,q (3.6.4)

where l+m α≡ 2



p−q +1 , p+q

l−m β≡ 2



p+q +1 p−q

and p =l+j,

q = m+k.

(3.6.5)

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3 Newtonian Celestial Mechanics

3.6.1 The Torque Resulting from an External Mass-Monopole We will now concentrate on the torque

 1 3 a L d XρX X GbL Di = iab l! E l≥0

induced by a single external mass-monopole MA . Since 1 lm γ00 =√ 4π A = and M00

√

4π MA , the torque is given by: D = GMA

 Mlm (−LYlm )|A R l+1

(3.6.6)

l,m

with i Lx Ylm = + (a+ Y+ + a− Y− ) 2 1 Ly Ylm = + (a+ Y+ − a− Y− ) 2 Lz Ylm = imYlm ,

(3.6.7)

where a± = [(l ± m + 1)(l ∓ m)]1/2 and Y± = Yl,m±1 . 3.6.1.1

The Torque as Lie-Derivative of UE

Now, L can be understood as an operator, that has an interesting physical meaning. For an external point-mass the external potential is given by (RA = rEA ) Uext =

GMA RA

3.6 Rotational Equations of Motion

81

and ∂c Uext = −GMA

c Xc − XA . |X − XA |3

It is easy to see that the term with Xc does not contribute to the torque. Therefore, Da =

c abc GMA XA

ρXb d X |X − XA |3 E 3

.

Since

d 3 X

∂b UE (X) = −G E

ρ b (Xb − X ) |X − X |3

the torque resulting from a tide generating point-mass MA can be written in the form c ∂b U E | A . Da = abc MA XAE

(3.6.8)

This equation can be interpreted in the following way: Epot ≡ −MA UE (xA )

(3.6.9)

is the potential energy of the point-mass MA in the gravitational field of body E and the torque (3.6.8) is given in geocentric coordinates X by Da = La Epot ,

(3.6.10)

where La = abc Xb

∂ ∂Xc

(3.6.11)

is the Lie-derivative with respect to an infinitesimal rotation about the corresponding axis. In quantum-mechanics −iL presents the usual angular-momentum operator (h¯ = 1); for that reason many relations involving L can be found in any textbook about quantum-mechanics. Using UE (R, θ, φ) = G

l ∞   l=0 m=−l

Mlm

Ylm (θ, φ) , R l+1

(3.6.12)

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3 Newtonian Celestial Mechanics

where (R, θ, φ) are the polar coordinates of X (X = R sin θ cos φ, Y R sin θ sin φ, Z = R cos θ ), the torque can be written in the form of (3.6.6): D = GMA

=

 Mlm (−LYlm )|A , R l+1 l,m

since the derivatives of R do not contribute to the torque. The following relations for the angular momentum operator in spherical coordinates are well known: Lx = − sin φ ∂θ − cot θ cos φ ∂φ Ly = + cos φ ∂θ − cot θ sin φ ∂φ

(3.6.13)

Lz = +∂φ Defining   L± ≡ ie±iφ ±∂θ + i cot θ ∂φ

(3.6.14)

L± Ylm = i{(l ± m + 1)(l ∓ m)}1/2 Yl,m±1

(3.6.15)

one finds

and, in this way, we recover our old formula (3.6.8) for the torque.

3.7 The Newtonian 2-Body Problem 3.7.1 Integrals of Motion Let us consider two celestial bodies as mass-monopoles with masses m1 and m2 , that move solely due to their mutual gravitational attractions. Let r1 and r2 be the two position vectors in some suitably chosen inertial coordinate system. From (3.5.5) we get the two dynamical equations r1 − r2 |r1 − r2 |3 r2 − r1 m2 r¨ 2 = −G m1 m2 . |r2 − r1 |3 m1 r¨ 1 = −G m2 m1

(3.7.1)

These equations imply the existence of 12 scalar constants of integration for the two initial positions and the two initial velocities. If one subtracts the second equation

3.7 The Newtonian 2-Body Problem

83

of (3.7.1) multiplied by m2 from the first one multiplied by m1 one obtains r¨ 2 − r¨ 1 = −G(m1 + m2 )

r2 − r1 , |r2 − r1 |3

which we would like to write as r¨ = −GM

r . r3

(3.7.2)

Here, r ≡ r2 − r1 is the relative vector from m1 to m2 and M ≡ m1 + m2 is the total mass. If we add the two equations of (3.7.1) and integrate we obtain m1 r˙ 1 + m2 r˙ 2 = a and m1 r1 + m2 r2 = at + b. Here, a and b present six integration constants. The left hand side of the last equation we write in the form m1 r1 + m2 r2 = MrS , where rS ≡

m1 r1 + m2 r2 M

(3.7.3)

denotes the center-of-mass vector of the system. From this we find MrS = at + b ,

(3.7.4)

i.e., the center-of-mass moves linearly and uniformly in our Cartesian inertial coordinate system. We now want to derive equations for the relative motion with respect to the center-of-mass of the system. We have (m1 + m2 )r1 − (m1 r1 + m2 r2 ) = −m2 (r2 − r1 )

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3 Newtonian Celestial Mechanics

or r1 − rS = −

m2 r. M

r2 − rS = +

m1 r. M

In an analogous way we find

From this we get r¨ 1 − r¨ S = −

r m2 r1 − rS r¨ = Gm2 3 = −GM M r r3

and finally r¨ 1 − r¨ S = −GM

 m 3 r − r 2 1 S . M |r1 − rS |3

(3.7.5)

r¨ 2 − r¨ S = −GM

 m 3 r − r 1 2 S . M |r2 − rS |3

(3.7.6)

Similarly we obtain

After the center-of-mass law we will derive the law of (specific) angular momentum conservation which is valid for any kind of central force with r¨ ∝ r, implying that r × r¨ = 0, or r × r˙ = C ,

(3.7.7)

where C are three further integration constants. Similarly the following equations hold: (r1 − rS ) × (˙r1 − r˙ S ) = C1 ;

(r2 − rS ) × (˙r2 − r˙ S ) = C2 .

The quantities C1 and C2 , however, are not new independent constants, but can be derived from C. E.g., M 2 C1 = (Mr1 − MrS ) × (M r˙ 1 − M r˙ S ) = (Mr1 − m1 r1 − m2 r2 ) × (M r˙ 1 − m1 r˙ 1 − m2 r˙ 2 ) = m22 (r1 − r2 ) × (˙r1 − r˙ 2 ) = m22 (r × r˙ ) = m22 C ,

(3.7.8)

3.7 The Newtonian 2-Body Problem

85

i.e.,  m 2 2

C1 =

M

C

and, similarly, C2 =

 m 2 1

M

C.

The law of (specific) angular momentum conservation (3.7.7) is equivalent to Kepler’s second law, the area rule. The (oriented) area that is swept out by the relative vector r during a short interval of time t is given by F =

1 1 r(t) × r(t + t) C · t, 2 2

i.e., a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. We now come to the energy conservation. From r2 = r 2 we get r · r˙ = r r˙ and, therefore, r˙ · r¨ = −

GM GM r · r˙ = − 2 r˙ , r3 r

or 1 d 2 d r˙ = GM 2 dt dt

  1 . r

This relation can be integrated and with U (r) =

GM r

we get the conservation law for the (specific) energy in the form 1 2 r˙ − U (r) = h. 2

(3.7.9)

Here, the integration constant h has the dimension of energy per mass (specific energy). For the motion relative to the center-of-mass we get by r → r1 − rS and

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3 Newtonian Celestial Mechanics

M → M(m2 /M)3 first  m 3 1 1 2 (˙r1 − r˙ S )2 − GM = h1 2 M |r1 − rS | and correspondingly  m 3 1 1 1 (˙r2 − r˙ S )2 − GM = h2 . 2 M |r2 − rS | From this we find  m 3 M 1 2 2 ˙ ˙ ˙ r r (M r − m − m ) − GM 1 1 1 2 2 M m2 r 2M 2 2 2   m2 2 Gm2 m2 2 = = r˙ − h, 2 Mr M 2M

h1 =

(3.7.10) or h1 =

 m 2 2

M

h

(3.7.11)

h.

(3.7.12)

and, similarly, h2 =

 m 2 1

M

So far we found a total of ten scalar integrals of motion: the six center-of-mass constants (a, b), the three components of the specific angular momentum vector, C, and the specific energy h . The remaining two integration constants can be found in the following way: We first multiply Eq. (3.7.2) with C = r × r˙ : r¨ × C = −GM

r × (r × r˙ ) . r3

With a × (b × c) = (a · c)b − (a · b)c we get 

r · r˙ r˙ r− 3 r r   d r . = GM dt r

d (˙r × C) = −GM dt



 = GM

 r˙ r˙ − 2r r r

3.7 The Newtonian 2-Body Problem

87

This leads us to the Laplace-integral in the form r  r˙ × C = GM + eP . r

(3.7.13)

Here, P is a unit vector (|P| = 1). Since P lies in the orbital plane perpendicular to C, e and P present the last two integration constants. The vector L = GMe P is called Runge-Lenz vector. From the Laplace-integral (3.7.13) we can write the Runge-Lenz vector in the form   GM GM L=v×C− r = v2 − r − (r · v)v . (3.7.14) r r

3.7.2 Orbital Equation; Kepler’s First and Third Law The orbital equation can be derived from the Laplace-integral (3.7.13). Since (a × b) · c = (c × a) · b we get   P·r C 2 = C · C = (r × r˙ ) · C = (˙r × C) · r = GMr 1 + e . r We will denote the angle between P and r by f , the true anomaly. With P·

r = cos f r

we get the orbital equation in the form r=

p 1 + e cos f

(3.7.15)

with p=

C2 . GM

This is Kepler’s first law: the orbit of every planet is an ellipse with the Sun at one of the two foci (Fig. 3.5). The point of closest approach of the two bodies, the pericenter, is given by f = 0 , implying that the Runge-Lenz vector or our vector P points towards the pericenter. The value of the numerical eccentricity e of the conic section indicates if the orbit is an ellipse (e < 1), parabola (e = 1) or hyperbola (e > 1). In case of an ellipse p = a(1 − e2 ) , where a denotes the semi-major axis of the ellipse.

(3.7.16)

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3 Newtonian Celestial Mechanics

y

P b

f x

a·e

a √ Fig. 3.5 Geometry of the Keplerian elliptical orbit. a: semi-major axis; b = a 1 − e2 (semiminor axis); e: eccentricity; F : one of the two foci of the ellipse; f : true anomaly

For the elliptical orbit, according to the area rule with Aellipse = π a 2 (1 − e2 )1/2 C = 2F /T = 2π a 2 (1 − e2 )1/2 /T , where T denotes the orbital period. From C 2 = GMa(1 − e2 ) we get 2π T = n

 with n =

GM a3

1/2 ,

implying Kepler’s third law in the form GM = n2 a 3 .

(3.7.17)

Note, that for two planets moving about the Sun this law in correct form reads M + m 1 = M + m 2



a1 a2

3 

T2 T1

2 .

(3.7.18)

Only if we neglect the planetary masses with respect to the solar mass the cubes of the semi-major axes are proportional to the squares of the orbital periods. Finally we would like to remark that for the motion relative to the center-of-mass |r1 − rS | =

p1 ; 1 + e cos(f1 − f10 )

|r2 − rS | =

p2 1 + e cos(f2 − f20 )

with p1 =

m  2

M

p;

p2 =

m  1

M

p.

3.7 The Newtonian 2-Body Problem

89

3.7.3 Classification of the Conic Sections A classification of the conic sections can be managed by means of the energyintegral 1 2 GM r˙ − = h. 2 r With r = r er ;

r˙ = r˙ er + r e˙ r = r˙ er + r f˙ ef

we get r˙ 2 + (r f˙)2 − 2

GM = 2h; r

Furthermore, C = |r × r˙ | = |rer × (˙r er + r f˙ef )| = r 2 f˙. With r = p/(1 + e cos f ) and p = C 2 /(GM) we get for e < 1: r˙ 2 =

C2 2GM − 2 + 2h = r r

=

C 2 (GM)2 2(GM)2 (1 + e cos f ) − (1 + e cos f )2 + 2h C2 C4

=

(GM)2 (1 − e2 cos2 f ) + 2h . C2

On the other hand r˙ =

pe sin f e μe e sin f, f˙ = r 2 f˙ sin f = C sin f = 2 p p C (1 + e cos f )

with μ = GM. A comparison reveals r˙ 2 =

μ2 μ2 e2 2 2 2 (1 − e + e sin f ) + 2h = sin2 f C2 C2

or h=−

1μ 1 μ2 (1 − e2 ) = − . 2 2C 2a

(3.7.19)

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3 Newtonian Celestial Mechanics

Fig. 3.6 Geometry of conic sections: circle, ellipse, parabola and hyperbola (from: https://www. onlinemathlearning.com/conic-parabolas.html)

This implies that for an elliptical orbit with e < 1 the specific energy h is negative; the orbit is bound. h is determined by the semi-major axis of the relative motion. Figure 3.6 shows the different conic sections presenting possible orbits in the Keplerian two-body problem. For the case of hyperbolic orbits with e > 1 one finds h = +μ/(2|a|) > 0 and the specific energy h just vanishes, h = 0, for parabolic orbits with e = 1. If we write the energy conservation in the elliptical orbit in the form   2 1 2μ + 2h = μ − , v = r r a 2

we see that the velocity in a circular orbit, e = 0, is simply  v=

GM . a

3.7 The Newtonian 2-Body Problem

91

3.7.4 Kepler’s Equation After we have found the form of the 2-body orbit we now turn to the time dependence in the elliptical orbit. To this end it is useful to introduce Cartesian coordinates with origin in the center of the orbital ellipse. Let b = a(1 − e2 )1/2 be the semi-minor axis of the ellipse. We then write (x, y) = (a cos E, b sin E), where the angle E is called eccentric anomaly. From Fig. 3.7 we see that r cos f = a(cos E − e) r sin f = a(1 − e2 )1/2 sin E .

(3.7.20)

From this we get r = (r 2 cos2 f + r 2 sin2 v)1/2 = a(cos2 E − 2e cos E + e2 + (1 − e2 ) sin2 E)1/2 = a(1 − 2e cos E + e2 cos2 E)1/2 or r = a(1 − e cos E).

(3.7.21)

This result, using r = a(1 − e2 )/(1 + e cos f ), leads to an expression for cos f : cos f =

cos E − e . 1 − e cos E

(3.7.22)

Fig. 3.7 True and eccentric anomaly in the elliptical Keplerian orbit

P a b

E C

f

F

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3 Newtonian Celestial Mechanics

From this we get √ 1 − e2 sin E sin f = 1 − e cos E

(3.7.23)

and analogous one finds e + cos f cos E = ; 1 + e cos f

√ sin E =

1 − e2 sin f . 1 + e cos f

(3.7.24)

Later we will need an expression for df/dE. To this end we will first differentiate the expression for sin f , relation (3.7.23), with respect to E:   df cos E − e df d sin f = cos f = dE dE 1 − e cos E dE √ √ 1 − e2 [(1 − e cos E) cos E − e sin2 E] 1 − e2 (cos E − e) = = 2 (1 − e cos E) (1 − e cos E)2 and obtain √ 1 − e2 df = . dE 1 − e cos E

(3.7.25)

Similarly one finds √ dE 1 − e2 = . df 1 + e cos f For practical calculations another relation between f and E is useful: tan2

1 − cos f 1 − e cos E − cos E + e f = = 2 1 + cos f 1 − e cos E + cos E − e   1+e E (1 + e)(1 − cos E) = tan2 = (1 − e)(1 + cos E) 1−e 2

or f tan = 2



E 1+e tan . 1−e 2

(3.7.26)

3.7 The Newtonian 2-Body Problem

93

The time dependence in the elliptical orbit is finally obtained from the area rule. From r 2 f˙ = C one derives

f

C(t − t0 ) =

f0

= a2

r 2 (v)dv =



E

 a 2 (1 − e cos E)2

E0

df dE

 dE

1 − e2 (1 − e cos E) dE

 E  = a 2 1 − e2 E − e sin E , E0

or, using C 2 = μa(1 − e2 )  E √ μa(t − t0 ) = a 2 E − e sin E . E0

If we divide both sides by a 2 we get a factor of (μ/a 3 )1/2 = n on the left hand side. Again, n is the mean motion as it appears in Kepler’s third law, n2 a 3 = GM . This, finally, leads us to Kepler’s equation M = E − e sin E ,

(3.7.27)

M = n(t − T )

(3.7.28)

where the angle M,

is called mean anomaly. For a circular orbit true and mean anomaly are equal. For the elliptical case we image to have a further fictitious body that moves along a circular orbit with the same mean motion, n, as our celestial body. This fictitious body only serves for the calculation of the time dependence in the orbit. The quantity T indicates the time of perigee passage, where M = E = f = 0 and real and fictitious body meet after each revolution. The time dependence in the elliptical orbit then follows from Eq. (3.7.28): for any instance of time we first compute the mean anomaly M. Kepler’s equation (3.7.27) can then be solved for the eccentric anomaly E and relation (3.7.26) leads to the true anomaly f that appears in the orbital equation (3.7.15). Given the mean anomaly M, Kepler’s equation presents a transcendental equation for the eccentric anomaly E. Many different methods to solve Kepler’s equation can be found in the literature. One possibility, for small value of e, is a solution by iteration. Let E0 be a zeroth order approximation to E, then we can derive a correction E0 from Kepler’s equation: M = E − e sin E = E0 + E0 − e sin(E0 + E0 ) = E0 − e sin E0 + (1 − e cos E0 ) E0 + . . .

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3 Newtonian Celestial Mechanics

i.e., E0

M − M0 ; 1 − e cos E0

M0 = E0 − e sin E0

It is clear that this procedure can be iterated; experience shows that for e < 0.2 such an iteration usually converges.

3.7.5 Fourier-Analysis in the Elliptical Orbit Another possibility to solve Kepler’s equation is to expand E − M = e sin E as an odd function of E (and hence also of M) in a Fourier-series of the form e sin E = 2

∞ 

bs sin(sM)

s=1

with

1 (e sin E)d(cos sM) sπ 0 π

π

1 1 cos(sM)dE − cos(sM)dM = cos(sM)ed(sin E) = sπ sπ 0 0

π 1 1 cos[s(E − e sin E)] dE = Js (se) , = sπ 0 s

bs =

1 π

π

(e sin E) sin(sM)dM = −

where Js (z) are Bessel-functions (of the first kind). For small values of e these Bessel-functions admit a Taylor-series expansion of the form 2J1 (1e) = e −

e3 + O(e5 ) 8

2J2 (2e) = e2 −

e4 + O(e6 ) 3

2J3 (3e) =

9e3 + O(e5 ) 8

2J4 (4e) =

4e4 + O(e6 ). 3

3.7 The Newtonian 2-Body Problem

95

The first terms result from the asymptotic behavior of Js (z). For fixed values of s (positive integers) and z → 0 one gets: 1  z s . s! 2

Js (z) ∼ Therefore, e sin E =

∞  2 s=1

s

Js (se) sin(sM)

and therefore, E=M+

∞  2 s=1

s

Js (se) sin(sM) .

(3.7.29)

Inserting the Taylor-series expansion for the Bessel-functions we obtain    1 2 1 4 1 3 e − e sin 2M E = M + e − e sin M + 8 2 6 

+

3 3 1 e sin 3M + e4 sin 4M + O(e5 ). 8 3

(3.7.30)

Also the other variables in the elliptical motion can be expressed in terms of corresponding Fourier-series. E.g., a differentiation of Kepler’s equation with respect to t leads to: n = (1 − e cos E)

dE  r  dE = · , dt dt a

i.e., ∞

 a dE = /n = 1 + 2 Js (se) cos(sM). r dt

(3.7.31)

s=1

3.7.6 The Elliptical Kepler Orbit in Space So far we have characterized the position of one celestial body with respect to the other by elements a, e and M, quantities defined in the space-fixed orbital plane. If an arbitrary x-axis is chosen in the orbital plane then we also need the angle between this x-axis and the direction towards the pericenter. This argument of the pericenter is usually denoted by ω. For practical applications it is useful, however, to employ

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3 Newtonian Celestial Mechanics

P

a f Ω

ω

I

Fig. 3.8 The elliptic Kepler orbit in space

a different reference plane. In satellite theory usually one uses a suitably defined equatorial plane as reference plane for orbital elements. This situation is depicted in Fig. 3.8. The reference plane, e.g., the celestial equator at a certain epoch, is the x − y-plane of our fundamental Newtonian inertial reference system. The astronomical x-axis might be defined by the vernal equinox, i.e., the intersection of the reference plane with the (mean) ecliptic of the epoch. The line of intersection between reference and orbital plane is called line of nodes. It defines two points in the Keplerian orbit: the ascending and the descending node. If the body crosses the reference plane from negative (positive) to positive (negative) z-values it goes through the ascending (descending) node. In Fig. 3.8 we see further elements I and that define the orientation of the orbital plane in space. I describes the inclination of the orbital plane with respect to the reference plane. The argument of the ascending node, , describes the angle between the x-axis of our fundamental reference system and the direction towards the ascending node. The argument of the pericenter, ω is reckoned from the ascending node. Altogether, the Keplerian orbit in space can be characterized by six orbital elements (a, e, ω, , I, T ), equivalent to the Cartesian vectors of position and velocity for a given instance of time.

3.7.6.1

Calculation of x and x˙ from the Orbital Elements

Let the Cartesian coordinates of our inertial reference system be denoted by (x, y, z). Let us introduce a second (right-handed) Cartesian coordinate system (X, Y, Z) with the properties: the X − Y -plane agrees with the orbital plane and

3.7 The Newtonian 2-Body Problem

97

the X-axis points towards the pericenter. The Z-axis points perpendicular to the orbital plane in the direction of the angular momentum vector. The coordinates of our celestial body are then given by X = r cos f = a(cos E − e);

Y = r sin f = a(1 − e2 )1/2 sin E;

Z = 0. (3.7.32)

Using E˙ = (a/r)n, that follows from Kepler’s equation, dM/dt = n = (1 − ˙ one finds for the velocity e cos E)E, X˙ = −a E˙ sin E = −(a 2 n/r) sin E;

Y˙ = (a 2 n/r)(1 − e2 )1/2 cos E;

Z˙ = 0 . (3.7.33)

Now, two right-handed Cartesian coordinate systems with the same origin are related by a rotation matrix, in our case x = RxX · X .

(3.7.34)

To get from X to x we first have to rotate about the Z-axis by an angle −ω (a rotation with a positive rotation angle is counter-clockwise if we look from positive values on the rotation axis onto the plane where the coordinate are rotated) which transforms the X-axis into the line of nodes, a second rotation about this new X axis by an angle −I so that the new X − Y  -plane agrees with the fundamental x − y-plane, plus a third rotation about the z-axis by an angle − . Thus, RxX = R3 (− ) · R1 (−I ) · R3 (−ω)

(3.7.35)

with ⎛

⎞ 1 0 0 R1 (θ ) = ⎝0 cos θ sin θ ⎠ 0 − sin θ cos θ

(3.7.36)

and ⎛

⎞ cos θ sin θ 0 R3 (θ ) = ⎝− sin θ cos θ 0⎠ . 0 0 1

(3.7.37)

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3 Newtonian Celestial Mechanics

One finds: RxX = cos cos ω − sin sin ω cos I RxY = − cos sin ω − sin cos ω cos I RxZ = sin sin I RyX = sin cos ω + cos sin ω cos I RyY = − sin sin ω + cos cos ω cos I RyZ = − cos sin I RzX = sin ω sin I RzY = cos ω sin I RzZ = cos I .

(3.7.38)

˙ and Since the same rotation matrix applies for the corresponding velocity vectors, X x˙ , we can express x and x˙ by means of the orbital elements. E.g., one gets x = RxX X + RxY Y + RxZ Z = [cos cos ω − sin sin ω cos I ] r cos f + [− cos sin ω − sin cos ω cos I ] r sin f = r [cos (cos ω cos f − sin ω sin f ) − sin (sin ω cos f + cos ω sin f ) cos I ] = r[cos cos(ω + f ) − sin sin(ω + f ) cos I ].

In this way for the position vector one finds: x = r[cos cos u − sin sin u cos I ] y = r[sin cos u + cos sin u cos I ] z = r sin u sin I , where u≡ω+f . For any given instance of time r can be obtained from r=

p , 1 + e cos f

(3.7.39)

3.7 The Newtonian 2-Body Problem

99

where the true anomaly results from the Kepler equation and relation (3.7.26) between f and E.

3.7.6.2

Calculation of Orbital Elements from x and x˙

The specific angular momentum vector C = x × x˙ has Cartesian components Cx = y z˙ − zy˙ Cy = zx˙ − x z˙

(3.7.40)

Cz = x y˙ − y x˙ , from which we can calculate the quantity p (μ = GM): C2 . μ

p=

(3.7.41)

With r 2 = x2 one obtains the semi-major axis a from  v2 = x˙ 2 = μ

2 1 − r a

 .

(3.7.42)

The inclination I is obtained from I = arccos[Cz /C] .

(3.7.43)

We then compute the Runge-Lenz or eccentricity vector   GM x − (x · v)v L = v2 − r

(3.7.44)

from which we get the eccentricity by e=

|L| GM

(3.7.45)

and the true anomaly f by # f =

L·x arccos |L||x| L·x 2π − arccos |L||x|

for x · v ≥ 0 otherwise.

(3.7.46)

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3 Newtonian Celestial Mechanics

The eccentric anomaly E is then given by ⎡

⎤ tan f2 ⎦. E = 2 arctan ⎣  1+e 1−e

(3.7.47)

The longitude of the ascending node and the argument of the pericenter ω are then obtained by means of the nodal vector n pointing towards the ascending node: ⎛

⎞ −Cy n = ez × C = ⎝ +Cx ⎠ . 0

(3.7.48)

One has #

=

nx arccos |n| nx 2π − arccos |n|

for ny ≥ 0 for ny < 0

(3.7.49)

for Lz ≥ 0 for Lz < 0 .

(3.7.50)

and # ω=

L·n arccos |L||n| L·n 2π − arccos |L||n|

Finally the mean anomaly M is obtained from Kepler’s equation M = E − e sin E.

3.8 Perturbation Theory 3.8.1 Variation of Constants Let us now consider a perturbed Keplerian problem, where the equation of motion takes the form r¨ = ∇(U0 + R)

(3.8.1)

with U0 =

GM . r

Here, the quantity R is called perturbing function and we will assume that |R|  |U0 |.

3.8 Perturbation Theory

101

Without the perturbing function R we face our Kepler-problem that is completely integrable with r = r(α, t);

r˙ = r˙ (α, t) .

(3.8.2)

Here α stands for the complete set of orbital elements α = (a, e, I, , ω, T ). We now think of getting a solution of (3.8.1) by considering the orbital elements α as time dependent quantities. In this case one speaks of variation of constants. We can then write ∂r ∂r dαi ∂r ∂r dr ˙ = + ≡ + · α. dt ∂t ∂αi dt ∂t ∂α If, for each instance of time, we want the velocity in the perturbed orbit to agree with the corresponding velocity of the instantaneous Keplerian orbit, we can require an osculation condition of the form ∂r · α = 0. ∂α

(3.8.3)

In that case the perturbed orbit is described with osculating orbital elements. For each instance of time the elements α(t) describe an osculating ellipse that yields the position and velocity in the actual perturbed orbit.

3.8.2 Perturbation Equations, Derived from Vectorial Elements 3.8.2.1

Vectorial Elements in the Kepler-Problem

In the Kepler problem we had r¨ = −

GM r r2 r

and for each instance of time t the solution is given by the six time independent orbital parameters a, e, I, , ω and M0 = −nT . The first five of these orbital elements are determined by the vectors C, the specific angular momentum vector, and f, the Runge-Lenz vector, with C = r × r˙ ;

  GM r − (r · r˙ )˙r. f = r˙ 2 − r

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3 Newtonian Celestial Mechanics

Fig. 3.9 Various unit vectors in the Keplerian orbit

Q m

k

P

Ω

perigee

l I

We now introduce a system of three orthonormal unit vectors, l, m, k, such that: – l lies in the nodal line of the orbit pointing towards the ascending node, – m lies in the orbital plane perpendicularly to l and – k in the direction of C, perpendicular to the orbital plane. These unit vectors (Fig. 3.9) are given by ⎛

⎞ cos

l = ⎝ sin ⎠ ; 0

⎛

⎞ − cos I sin

m = ⎝ cos I cos ⎠ ; sin I

⎛

⎞ sin I sin

k = ⎝ − sin I cos ⎠ ; cos I (3.8.4)

The position vector can then be written as r = r [l cos(ω + f ) + m sin(ω + f )] . In the orbital plane we can rotate the two unit vectors l and m by the angle ω such that one of them, P, points towards the pericenter P = l cos ω + m sin ω;

Q = −l sin ω + m cos ω .

(3.8.5)

The two vectorial elements C and f can then be written as (p = a(1 − e2 )): C = (GMp)1/2 k;

f = GMe P .

(3.8.6)

For the Keplerian orbit we have: r = r(P cos f + Q sin f )   GM 1/2 r˙ = [−P sin f + Q(cos f + e)] . p

(3.8.7) (3.8.8)

3.8 Perturbation Theory

3.8.2.2

103

Perturbation Theory with Vectorial Elements

With a perturbing acceleration F the dynamical equation now reads r¨ = −

GM r + F. r2 r

(3.8.9)

We now image a solution of this equation to be given by a set of osculating elements. Five of these six elements can be derived from the vectorial elements k, P and Q. The time dependence of a, e, I, and ω is then determined by the time dependence of these three vectorial elements. E.g., ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ sin I sin

cos

cos I sin

d ⎝ d d

dI ⎝ sin ⎠ + ⎝ − cos I cos ⎠ , k= − sin I cos ⎠ = sin I dt dt dt dt cos I 0 − sin I or d d

dI k = sin I l− m. dt dt dt

(3.8.10)

Similarly one finds    dω d

dI d

+ cos I Q + sin ω − cos ω sin I k dt dt dt dt     dω d

dI d

dQ =− + cos I P + cos ω + sin ω sin I k. dt dt dt dt dt dP = dt



(3.8.11) (3.8.12)

From these relations we can derive the temporal variations of the two vectorial elements C and f. From   d 1 GM dp d C= k + GMp k dt 2 p dt dt we get  d 1 C= dt 2

   dI GM dp d

k + GMp sin I l− m . p dt dt dt

From df de d = GM P + GMe P dt dt dt

(3.8.13)

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3 Newtonian Celestial Mechanics

it follows that de df = GM P+GMe dt dt



  

dω d

dI d

+ cos I Q + sin ω − cos ω sin I k . dt dt dt dt (3.8.14)

On the other hand there are relations between the Runge-Lenz vector f and the perturbing acceleration F. From C = r × r˙ ;

d C = r × r¨ dt

one sees that d C=r×F dt

(3.8.15)

Due to  f = r˙ × (r × r˙ ) −

GM r

 r

one finds that d d f = r¨ × (r × r˙ ) + r˙ × (r × r¨ ) − dt dt



GM r



r .

The last term cancels with the corresponding term in the unperturbed equation so that we obtain df = F × (r × r˙ ) + r˙ × (r × F) dt = 2(˙r · F) − (r · F)˙r − (r · r˙ )F.

(3.8.16)

A scalar multiplication of dC/dt with k, l, m and of df/dt with P and Q yields five independent perturbation equations (Fig. 3.10). Let us derive two of them in detail. From GM = n2 a 3 we get 

GMp =



 n2 a 4 (1 − e2 ) = na 2 1 − e2 .

Using (3.8.13) we get  ˙ · l = na 2 1 − e2 sin I d = (r × F) · l. C dt

3.8 Perturbation Theory

105

Fig. 3.10 Three orthogonal components of the perturbing acceleration F: S = F · eS (radial-part), T = F · eT (transverse-part in the orbital plane) and W = F · eW (normal-part, orthogonal to the orbital plane); eS = r/r, eT = k × eS and eW = k

eT

eW

eS

Now, r = r [l cos(ω + f ) + m sin(ω + f )] , so that the right hand side is equal to r sin(ω + f )(m × F) · l = r sin(ω + f )(l × m) · F = r sin(ω + f ) W with W ≡ k · F. From this we finally obtain d

r sin(ω + f ) W. = √ dt na 2 1 − e2 sin I ˙ If we project the C-equation onto m, we obtain  dI ˙ · m = (r × F) · m =C −na 2 1 − e2 dt = r cos(ω + f ) (l × F) · m = r cos(ω + f ) (m × l) · F = −r cos(ω + f )W . Hence, dI r cos(ω + f ) W. = √ dt na 2 1 − e2

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3 Newtonian Celestial Mechanics

We define S≡

1 (r · F) , r

T ≡

1 (k × r) · F , r

W ≡ k · F.

(3.8.17)

S is the radial component of F, T the transverse component perpendicular to the radial direction in the orbital plane. W is called the normal component of F. Since r = r(P cos f + Q sin f ) we have S = (P · F) cos f + (Q · F) sin f T = (Q · F) cos f − (P · F) sin f . In this way five (out of a total number of six) perturbation equations can be derived:  2 p da = √ Se sin f + T dt r n 1 − e2 √ de 1 − e2 = [S sin f + T (cos f + cos E)] dt na r cos(ω + f ) dI = W √ dt na 2 1 − e2

(3.8.18)

r sin(ω + f ) d

= W √ dt na 2 1 − e2 sin I √  

d

r 1 − e2 dω = − cos I + −S cos f + T 1 + sin f . dt dt nae p These equations have to be augmented with an additional one for M0 or T . This last equation will be derived from the osculation condition for the velocity dr/dt. From r = r(P cos f + Q sin f ) we obtain d df ˙ sin f ) . r = r˙ (P cos f + Q sin f ) + r(−P sin f + Q cos f ) + r(P˙ cos f + Q dt dt ˙ Q ˙ and r˙ = (GM/p)1/2 e sin f we obtain: Inserting P,  r˙ =

GM e sin f (P cos f + Q sin f ) p + r(−P sin f + Q cos f )

df dt

3.8 Perturbation Theory

107

   

dω d

d

dI + cos I − cos ω sin I +r Q + k sin ω cos f dt dt dt dt   

 d

dI d

dω + cos I + k cos ω + sin ω sin I sin f . + r −P dt dt dt dt The osculation condition then implies that the right hand side of this equation, according to (3.8.8), should be equal to  GM [−P sin f + Q(cos f + e)] . p The terms proportional to Q yield df na 2  = 2 1 − e2 − dt r



 dω d

. + cos I dt dt

Correspondingly, for the eccentric anomaly one finds dE na r = − dt r a(1 − e2 )1/2



dω d

sin f de + cos I + dt dt 1 − e2 dt

 .

Using Kepler’s equation, M = E − e sin E, we get dM dE de dE r dE de = − sin E − e cos E = − sin E . dt dt dt dt a dt dt In the Kepler problem we had M = M0 + n(t − T ) , that, in case of osculating elements, leads to dM0 dn dM =n+ + · (t − T ). dt dt dt This implies, that we would in the time derivative of the mean anomaly we would face terms proportional to time t. To avoid such terms in the presence of perturbations, one defines the mean anomaly via

M = M0 +

t

n dt . t0

Then, dM dM0 r dE de =n+ = − sin E . dt dt a dt dt

(3.8.19)

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3 Newtonian Celestial Mechanics

In this equation we can finally insert the expressions for the time derivatives of E and e. In this way the last perturbation equation can be derived. It reads:    dω d

2r dM0 = − 1 − e2 + cos I −S 2 . dt dt dt na

(3.8.20)

Equations (3.8.18) and (3.8.20) are the usual celestial mechanical perturbation equation in STW-form (or Gauss-form). So far these equations are exact, i.e., it was not assumed that the perturbation is small compared to the Keplerian acceleration. Instead of the argument of perigee ω and the mean anomaly M one often uses the longitude of perihelion,  , and the mean longitude at the epoch, , given by  =ω+

(3.8.21)

and

M + =

n dt + .

(3.8.22)

 

r 1 − e2 −S cos f + T 1 + sin f nae p

(3.8.23)

These quantities obey the relations d d 2 I =2 sin + dt dt 2

√

and   d e2 d  2rS d 2 I 2 = +2 − 2 . 1 − e sin √ 2 dt dt dt 2 na 1+ 1−e

(3.8.24)

To derive another form of the perturbation equations that was first derived by Lagrange, we assume that the perturbing acceleration can be derived from a potential R = R(t, r), the perturbing function, F=

∂R . ∂r

Let α be some orbital element. Then, ∂R ∂R ∂r ∂r = · =F· . ∂α ∂r ∂α ∂α From r = a(1 − e cos E) one finds ∂r/∂a = r/a and, therefore, ∂r r = , ∂a a

(3.8.25)

3.8 Perturbation Theory

109

leading to ∂R ∂r 1 r =F· = F · r = S. ∂a ∂a a a

(3.8.26)

Similarly one finds  

r ∂R = a − cos f S + sin f 1 + T ∂e p ∂R = rW sin(ω + f ) ∂I ∂R = rT ∂ω

(3.8.27)

∂R = rT cos I − r cos(ω + f ) sin I W ∂

 ∂R p  a e sin f S + T . = √ ∂M0 r 1 − e2 Inserting these relations into the STW-form of the perturbation equations (3.8.18) and (3.8.20), we obtain the Lagrange-equations in the form: 2 ∂R da =− 2 dt n a ∂T $  1 1 ∂R de ∂R =− 2 + (1 − e2 ) 1 − e2 dt ∂ω n ∂T na e  $ ∂R 1 ∂R dI cos I = − √ dt ∂ω ∂

na 2 1 − e2 sin I √ dω cot I 1 − e2 ∂R ∂R = − √ 2 dt na e ∂e na 2 1 − e2 ∂I d

∂R 1 = √ 2 2 dt na 1 − e sin I ∂I 1 − e2 ∂R 2 ∂R dT = 2 2 + 2 . dt n a e ∂e n a ∂a Exercise 3.11 Derive the Lagrange perturbation equations. Solution From d

r sin(ω + f ) W = √ dt na 2 1 − e2 sin I

(3.8.28)

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3 Newtonian Celestial Mechanics

and rW sin(ω + f ) =

∂R ∂I

we get the d /dt equation. Furthermore, dI r cos(ω + f ) sin I = W √ dt na 2 1 − e2 sin I



 ∂R + r cos I T = (na 2 1 − e2 sin I )−1 − ∂



 ∂R ∂R 2 −1 2 − + cos I . = (na 1 − e sin I ) ∂

∂ω

The last Lagrange-equation for the time of passage through the pericenter T , can also be written in the form dM 1 − e2 ∂R 2 ∂R =n− − . dt na ∂a na 2 e ∂e

(3.8.29)

To employ the Lagrange form of the perturbation equation the perturbing acceleration requires a scalar potential, R, that has to be expressed in terms of the orbital elements R = R(a, e, I, , ω, T ) . Exercise 3.12 Calculate the secular Newtonian perihelion precession of Mercury’s orbit due to the gravitational action of an outer planet with mass M  and semi-major axis a  . To this end smear the mass M  along a ring of radius a  . Consider the orbit of Mercury to lie in the ring’s plane. Employ Gauss’ perturbation equation for ω (Mercury) by considering only lowest order terms in the eccentricity e of Mercury’s orbit. Solution From Exercise 3.4 we know that the ring produces a radially outward acceleration of form aring =

∞ GM    r 2k−1 2kP2k (0)2 · n ≡ S · n . a a2 k=1

Gauss’ perturbation equation for the argument of perihelion ω reads ω˙ = −

S cos f , nae

3.8 Perturbation Theory

111

where we neglected e2 -terms in the nominator. Using GM = n2 a 3 (M: solar mass) the average precession velocity per revolution of Mercury reads: (ω) ˙ rev = −

∞  r 2k−1  a 2k+1 M n  (2k)P2k (0)2 < cos f >  , M e a a

(3.8.30)

k=1

where 1 < Q >≡ 2π

2π

Q dM . 0

Since  a 2 df = + O(e2 ) dM r we get
= 2π

2π

 r 2k+1 a

0

e cos f df = − (2k + 1) . 2

The average perihelion precession (radians) per revolution therefore reads:  (ω)rev = π

M M

 ∞

ηk

 a 2k+1

k=1

(3.8.31)

a

with 2 (0) . ηk = 2k(2k + 1)P2k

One has η1 =

3 2

η2 =

45 16

η3 =

525 128

η4 =

11025 2048

η5 =

218295 . 32768

Mercury makes 414.9 revolutions per century so to get the result in arcsec/century one has to multiply the precession per revolution, (3.8.31), by a factor of  414.9

 360 3600 . 2π

For the contributions of the various planets to Mercury’s secular perihelion precession see Table 3.1. Modern values are from Will (1993).

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3 Newtonian Celestial Mechanics

Table 3.1 Perihelion precession of Mercury’s orbit due to outer planets in arcsec./century

Pert. planet Modern value LeVerrier Value from (3.8.31) Venus 277.8 280.6 280.9 Earth 90.0 83.6 95.0 Mars 2.5 2.6 2.3 Jupiter 153.6 152.6 159.9 Saturn 7.3 7.2 7.7

Exercise 3.13 Compute the precession of Mercury’s longitude of perihelion  =

+ ω due to the oblateness J2 of the Sun by means of Lagrange’s perturbation equations. To this end consider a non-vanishing small inclination of Mercury’s orbit with respect to the solar equator. Solution The relevant perturbation equations read: dω = dt

√ 1 − e2 ∂R ∂R cot I − √ 2 2 na 2 e ∂e na 1 − e ∂I

∂R 1 d

= . √ dt na 2 1 − e2 sin I ∂I From (3.3.20) we find a perturbing function of the form R=−

GM J2 r

 2   R 3 1 cos2 θ − , r 2 2

(3.8.32)

where cos θ = z/r = sin I sin u (u = ω + f ). Inserting R into the corresponding perturbation equation we get (p = a(1 − e2 )) 3 dω = J2 dt 2

 2   R 5 2 n 2 − sin I p 2

or (ω)rev

 2   R 5 2 2 − sin I = 3π J2 p 2

(3.8.33)

for the drift per revolution. Neglecting the inclination term we get: (ω)rev

 2 R = 6π J2 . p

(3.8.34)

Similarly for the secular drift of the node one obtains: d

3 = − J2 dt 2

 2 R n cos I , p

(3.8.35)

3.8 Perturbation Theory

113

or ( )rev = −3π J2

 2 R cos I . p

(3.8.36)

Neglecting the inclination, we end up with a secular precession of Mercury’s longitude of perihelion  of (e.g., Will 1993) ( )rev = 3π J2

 2 R . p

(3.8.37)

Inserting numbers J2 = 2 × 10−7 , R = 6.96 × 108 m, a = 5.79 × 1010 m and e = 0.2 we obtain a drift for  of  = 1.3 × 105 J2 = 2.6 × 10−2 arc-seconds per century.

Chapter 4

Relativity

4.1 Relativity Already in 1864 Maxwell (1864, 1865) published his fundamental equations of electromagnetism that contain a central natural constant: the vacuum speed of light c. Later, it was found by experiments that the vacuum speed of light velocity c obeys a principle of constancy. This principle of the constancy of the speed of light in vacuum has a harmless part, as well as a critical one. The harmless part says that c is independent of light frequency, amplitude and polarization, as well as the speed of light-source. The critical part, however, says that the vacuum speed of light is also independent upon the speed of the observer which was first tested in the famous experiment by Michelson and Morley (1887) (for modern tests see e.g., Antonini et al. 2005; Eisele et al. 2009; Haugan and Will 1987; Herrmann et al. 2005, 2009; Müller et al. 2003, 2007; Stanwix et al. 2005, 2006; Wolf et al. 2003, 2004). Because of this constancy of the vacuum speed of light, the absolute character of Newtonian space-time had to be abandoned and the Galilean group that relates different inertial systems in the absence of gravity has to be replaced by the Lorentz (Poincaré) group. A space-time with this symmetry is called ‘relativistic’. In the absence of gravitational fields the physical structure is called ‘Special Relativity’. Einstein’s theory of gravity is called ‘General Relativity’, though it is not more relativistic than ‘Special Relativity’ and both theories will be formulated covariantly (of course accelerated observers can be discussed in the framework of ‘Special Relativity’).

© Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_4

115

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4 Relativity

4.2 Electrodynamics and Special Theory of Relativity 4.2.1 Maxwell’s Equations We will discuss the theory of electromagnetism first in the absence of gravity. In that case a set of preferred inertial coordinates t and x i can be introduced in which dynamical equations of motion take a particularly simple form. Introducing four time-space coordinates of the same dimension (length), x μ = (x 0 , x 1 , x 2 , x 3 ) = (ct, x i ), we will call them Minkowskian coordinates in the following. The charge density ρ(t, x) and the current density j(t, x) act as sources of the electromagnetic field. Since j represents the flow of moving charges it is related with ρ by a continuity equation expressing the law of charge conservation of the form ∂ρ +∇ ·j=0 ∂t

(4.2.1)

in Minkowskian coordinates. Introducing a contravariant charge-current vector j μ in a four-dimensional Minkowskian space-time by j μ = (cρ, j)

(4.2.2)

the continuity equation (4.2.1) can be written simply as j μ ,μ = 0

(4.2.3)

since j μ ,μ =

∂j i ∂j μ ∂j 0 ∂ρ + + ∇ · j = 0. = = ∂x μ ∂ct ∂x i ∂t

The electromagnetic field in vacuum is described by the electric field strength E(t, x) and the magnetic field strength B(t, x). These are related with the field sources by the inhomogeneous Maxwell equations ∇ ·E = ∇ ×B =

ρ 0

(4.2.4)

1 ∂E + μ0 j . c2 ∂t

(4.2.5)

Here, 0 = 8.8542 × 10−12 C2 N−1 m−2

(4.2.6)

4.2 Electrodynamics and Special Theory of Relativity

117

is the electric permittivity of free space and μ0 = 4π 10−7 NA−2

(4.2.7)

is the magnetic permeability of free space. These two constants are related with the vacuum speed of light by c2 =

1 . 0 μ0

(4.2.8)

The first Maxwell equation (4.2.4) is Coulomb’s law. For a point charge Q located at the origin of our coordinate system we have

(∇ · E) d 3 x = V

Q = 0

E · df = 4π r 2 E ∂V

where Gauß’s theorem was used, so that E(t, x) =

1 Q x 4π 0 r 2 r

(4.2.9)

(Coulomb’s law). With the Coulomb potential (t, x) =

1 Q 4π 0 r

(4.2.10)

the electric field strength vector E is obtained from E = −∇ .

(4.2.11)

The second Maxwell equation (4.2.5) describes Ampère’s law The remaining two homogenous Maxwell equations read: ∇ ·B = 0 ∇ ×E+

∂B = 0. ∂t

(4.2.12) (4.2.13)

The third equation (4.2.12) states the absence of free magnetic monopoles whereas the last one describes Faraday’s law. It is convenient to introduce the electromagnetic field strength tensor ⎞ 0 Ex /c Ey /c Ez /c ⎜ −Ex /c 0 Bz −By ⎟ ⎟. =⎜ ⎝ −Ey /c −Bz 0 Bx ⎠ −Ez /c By −Bx 0 ⎛

F αβ

(4.2.14)

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4 Relativity

Note that F 0i = c−1 Ei ;

F ij = ij k Bk .

(4.2.15)

Here ij k = +1 if (ij k) is an even permutation of (123) it is −1 for an odd permutation of (123) and zero otherwise. The inhomogeneous Maxwell equations can then be written as F αβ ,β = μ0 j α .

(4.2.16)

Exercise 4.1 Proof that (4.2.16) is equivalent to the two inhomogeneous Maxwell equations (4.2.4) and (4.2.5). Solution If we put α = 0 we get F 00 ,0 + F 0i ,i =

∂ 0i 1 F = ∇ · E = μ0 cρ . ∂x i c

Since c2 = ( 0 μ0 )−1 Maxwell’s equation (4.2.4) is recovered. For α = i one finds Eq. (4.2.5) F i0 ,0 + F ij ,j =

∂ 1 ∂ i0 ∂ 1 ∂ F + j F ij = − 2 Ei + ij k j Bk c ∂t ∂x ∂x c ∂t

= μ0 j i . The inhomogeneous Maxwell equations (4.2.12) and (4.2.13) can be written in the form Fαβ,γ + Fβγ ,α + Fγ α,β = 0 ,

(4.2.17)

where ⎞ 0 −Ex /c −Ey /c −Ez /c ⎜ Ex /c 0 Bz −By ⎟ ⎟. =⎜ ⎝ Ey /c −Bz 0 Bx ⎠ Ez /c By −Bx 0 ⎛

Fαβ

(4.2.18)

Note that Fαβ is obtained from F αβ by changing the sign of E. Exercise 4.2 Proof that (4.2.17) is equivalent to the two homogenous Maxwell equations (4.2.12) and (4.2.13). Proof Taking α = 1, β = 2, γ = 3 we get F12,3 + F23,1 + F31,2 =

∂ ∂ ∂ Bz + Bx + By = ∇ · B = 0 . ∂z ∂x ∂y

4.2 Electrodynamics and Special Theory of Relativity

119

If we set one index to zero, e.g. α = 0, one finds e.g. for β = 1, γ = 2 F01,2 + F12,0 + F20,1 = −

1 ∂ 1 ∂ 1 ∂ Ex + Bz + Ey , c ∂y c ∂t c ∂x

i.e., the z-component of (4.2.13). Introducing an electromagnetic potential vector Aμ by Fμν = Aν,μ − Aμ,ν

(4.2.19)

Aμ = (−A0 , Ai ) ≡ (/c, A)

(4.2.20)

and

we find e.g., Ex = cF10 = c(A0,1 − A1,0 ) = −

∂ ∂  − Ax ∂x ∂t

and Bz = F12 = A2,1 − A1,2 =

∂ ∂ Ay − Ax . ∂x ∂y

Generally one has E=−

∂A − ∇ , ∂t

B = ∇ × A.

(4.2.21)

Note that from the last relations the components of the potential vector Aμ are not determined uniquely; instead one can impose certain gauge conditions that fix Aμ . One useful gauge condition is the Lorentz-gauge Aα ,α =

1 ∂  + ∇ · A = 0. c2 ∂t

(4.2.22)

With the Lorentz-gauge the inhomogeneous Maxwell equations take the form   1 ∂2 − 2 2 +  Aα = −μ0 j α . c ∂t

(4.2.23)

Here  is the usual Laplacian =

∂2 ∂2 ∂2 + + . ∂x 2 ∂y 2 ∂z2

(4.2.24)

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4 Relativity

Exercise 4.3 Show that the last statement is true. First write (4.2.4) and (4.2.5) with the potentials  and A. Then use the Lorentz-gauge condition to derive the wave equation (4.2.23). A special solution of the inhomogeneous wave equation (4.2.23) is given by the retarded potential Aαret (t, x)

μ0 = 4π

j α (tret , x ) 3  d x , |x − x |

(4.2.25)

|x − x | c

(4.2.26)

where tret ≡ t −

is called the retarded time. We finally come to the equation of motion for a sufficiently small test charge q. In a Minkowskian coordinate system it reads   dp 1 =q E+ v×B . dt c

(4.2.27)

Here p is the momentum of the particle with charge q and the right hand side is known as the Lorentz-force.

4.3 The Minkowskian Metric, Lorentz-Transformation It is quite obvious that the 3-dimensional space in SRT is the Euclidean R3 . There are several equivalent ways to introduce a 4-dimensional space-time metric tensor gμν in Minkowski space. We first consider the differential operator appearing in the wave-equation (4.2.23): ≡−

1 ∂2 + c2 ∂t 2

(4.3.1)

∂ ∂ , ∂x μ ∂x ν

(4.3.2)

that can be written as  = ημν where ημν = diag(−1, +1, +1, +1) ≡ ημν

(4.3.3)

4.3 The Minkowskian Metric, Lorentz-Transformation

121

is a good candidate for a 4-dimensional metric tensor. Let us consider the vacuum wave equation Aα = 0

(4.3.4)

in Minkowskian coordinates. Obviously plane waves Aα (t, x) = Aα0 exp(ikμ x μ ) ≡ Aα0 exp(i )

(4.3.5)

are solutions of the wave equation provided −k0 k0 + ki ki = ημν kμ kν = 0 = ημν k μ k ν where k μ = (−k0 , ki ) .

(4.3.6)

If we consider ημν as components of a metric tensor g μν (μ, ν = 0, 1, 2, 3) in Minkowskian coordinates then the wave vectors kμ or k μ are of zero length; they are called null-vectors. Integral curves of kμ are light-rays that run perpendicular μ to the surfaces of constant phase . Let us consider a certain light-ray xγ (λ). The curve parameter is assumed to have the dimension of length, e.g., λ = ct. In a certain point its tangent vector is given by μ

kμ ∝

dxγ dλ

and we will choose the constant such that μ

kμ =

dxγ 1 f0 · c dλ

(4.3.7)

(the wave vector) has the dimension of an inverse length and f0 is a constant frequency (Fig. 4.1). Fig. 4.1 Surfaces of constant phase, light rays and wave-vectors

k k

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4 Relativity

From μ

0 = ημν k μ k ν = ημν

μ ημν dxγ dxγν dxγ dxγν = dλ dλ dλ dλ

(4.3.8)

the identification of ημν with a metric tensor becomes clear. In other words: in the 4dimensional Minkowskian manifold one introduces a metric tensor gμν that reduces to ημν in Minkowskian coordinates. From a physical point of view this metric tensor has several fundamental properties. As we have already seen: Metric Property 1 Light-rays are curves of zero length (null-curves). In Minkowskian coordinates this simply means that 0 = ds 2 = −c2 dt 2 + dx2 , i.e., the speed of light c takes a constant value in every Minkowskian coordinate system. This constancy of the speed of light has profound consequences for our understanding and measurement of time. Consider some primitive version of an idealized light-clock consisting of two mirrors in vacuum (see Fig. 4.2) with constant separation L and some light signal bouncing to and fro between the two mirrors. At first the two mirrors are considered to be at rest. In that case the time needed for the signal to travel from one mirror to the other and back is given by τ =

2L . c

In the right part of Fig. 4.2 the same situation is depicted with the two mirrors moving in the direction of their extensions with constant speed v. Also in this case the observer measures the same propagation velocity for the light pulse. According

Fig. 4.2 (a) the light-clock at rest. Here a light impulse is reflected to and fro between two mirrors of constant separation L; (b) the moving light-clock. Here the two mirrors move with constant speed v with respect to some observer along the mirror’s extensions

4.3 The Minkowskian Metric, Lorentz-Transformation

123

to the Pythagorean theorem we now get  L + 2

v t 2

2

 =

c t 2

2

from which we derive t = 

τ 1 − v 2 /c2

.

(4.3.9)

In words: a moving clock appears to go slower. Metric Property 2 If x μ (λ) is the worldline of an idealized clock then the proper time interval dτ as indicated by the clock is related with the metric tensor via: dτ 2 = −

1 2 ds c2

(4.3.10)

where the length element ds refers to two neighbouring points on x μ (λ). Let us describe the clock’s worldline in Minkowskian coordinates x μ . Then   v2 ds 2 = −c2 dt 2 + dx2 = −c2 dt 2 1 − 2 c or −

1 2 ds = dt 2 (1 − v 2 /c2 ) c2

in accordance with relation (4.3.9). Exercise 4.4 Compute the metric tensor gμν with a Minkowskian time coordinate t and spatial spherical coordinates r, θ, φ. Solution It is clear that we only have to take the metric of Euclidean R3 from Eq. (2.8.5) and add the time part of the metric, i.e., ds 2 = −c2 dt 2 + dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) .

(4.3.11)

For a massive particle or body proper time τ is a natural quantity to parametrize its worldline γ : x μ (τ ). The tangent vector onto γ uμ ≡

dt dx μ = (c, v i ) dτ dτ

(4.3.12)

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4 Relativity

with vi =

dx i dt

(4.3.13)

is called 4-velocity of the body. In Minkowskian coordinates we have ηαβ uα uβ = ηαβ

dx μ dx ν ds 2 = −c2 . = dτ dτ dτ 2

(4.3.14)

The introduction of a 4-dimensional metric tensor gμν has many advantages. E.g., one can relate co- and contravariant components of a tensor as in Aμ = gμν Aν ;

T α βγ = gβσ T ασ γ

(4.3.15)

etc. Furthermore, all equations of physics can be written in a coordinate independent manner, i.e., in covariant form, by using covariant derivatives. E.g., Maxwell’s equations in any coordinate system take the form 4π α j c + Fβγ ;α + Fγ α;β = 0

F αβ ;β =

(4.3.16)

Fαβ;γ

(4.3.17)

and the continuity equation can be written as j μ ;μ = 0 .

(4.3.18)

Let us come to the force equation for a small test charge q. Let us define f μ ≡ F μν uν

(4.3.19)

with uν = gνσ uσ . In Minkowskian coordinates where gμν = ημν we have uν = (−c, v i ). The spatial part of f μ then reads in such coordinates f i = F i0 u0 + F ij uj = cE i + ij k Bk v j or   1 f=c E+ v×B c

(4.3.20)

4.3 The Minkowskian Metric, Lorentz-Transformation

125

proportional to the Lorentz-force from Eq. (4.2.27). Let pμ = m uμ = (p0 , p)

(4.3.21)

be the 4-momentum of a body of mass m and aμ ≡

Duμ = uμ ;ν uν dτ

(4.3.22)

its 4-acceleration. Then the covariant version of the force equation (4.2.27) reads m aμ =

q μν F uν . c

(4.3.23)

Finally the metric tensor leads to a generalization of the Galilean group. We have already used the fact that the metric tensor gμν takes the same form in every set of inertial (Minkowskian) coordinates, i.e., gμν = ημν = diag(−1, +1, +1, +1) .

(4.3.24)

We can use this condition of metric invariance to derive the transformation rules between two such inertial coordinate systems x μ and x ν . Assuming first the two x-axes to be aligned and v is the constant velocity of the x  -system with respect to the x-system, then one finds a space-time coordinate transformation of the form (y  = y, z = z)  vx  t = γ t − 2 c x  = γ (x − vt)

(4.3.25)

with −1/2  v2 γ ≡ 1− 2 . c

(4.3.26)

Obviously this restricted Lorentz-transformation reduces to the Galileantransformation (3.2.3) for c → ∞. It is not difficult to see that the Lorentz-transformation (4.3.25) leaves the metric tensor invariant. From  v  dt  = γ dt − 2 dx c dx  = γ (dx − v dt)

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4 Relativity

we get ds 2 = ημν dx μ dx ν = −c2 dt 2 + dx 2 + dy 2 + dz2   2v v2 = −c2 γ 2 dt 2 − 2 dx dt + 4 dx 2 c c   +γ 2 dx 2 − 2v dx dt + v 2 dt 2 + dy 2 + dz2 = −c2 dt 2 + dx 2 + dy 2 + dz2 = ημν dx μ dx ν . More generally the (inhomogeneous) transformations) can be written in the form

Lorentz-transformations

ν μ x μ = μ νx +a

(Poincaré-

(4.3.27)

with 00 = γ vi 0i = i0 = −γ c

(γ − 1) Rjk , ij = δik + v i vk v2

(4.3.28)

where Rjk is a constant rotation matrix and a μ describes translations in time and space. Note, that for most cases we will use aligned coordinates, i.e., Rjk = δkj . Then the x → x  transformation reads:  v · x t = γ t − 2 c (4.3.29) (x · v) x = x + (γ − 1) 2 v − γ vt . v For some calculations the relation (β i ≡ v i /c) v i vk γ2 i β βk (γ − 1) = 2 1+γ v

(4.3.30)

is useful. The inverse transformation, x  → x, reads:   v · x  t =γ t + 2 c (x · v) x = x + (γ − 1) v + γ vt  . v2 

(4.3.31)

4.3 The Minkowskian Metric, Lorentz-Transformation

127

Since −1/2  v2 1 v2 3 v4 γ = 1− 2 =1+ + + O(c−6 ) 2 c2 8 c4 c

(4.3.32)

we get a post-Galilean transformation (Chandrasekhar and Contopoulos 1967) of the form     1 v2 v · x 1 v2 3 v4 t − 1 + t = 1 + + + O(c−5 ) (4.3.33) 2 c2 8 c4 2 c2 c2   1 1 v2 vt + 2 (v · x)v + O(c−4 ) . (4.3.34) x = x − 1 + 2 2c 2c

4.3.1 Addition of Velocities We again consider two inertial systems, x and X , where we write Xα = (cT , X) instead of x  α . Let v be the constant velocity of the origin of the X -system with respect to the x -system. We then consider a particle that moves with constant velocity w in the X -system, i.e., its trajectory reads: XP = wT . At first we consider w to be parallel to v and orient the two spatial coordinates such that both velocities point in x-direction. In the X -system the trajectory of the particle is then given by XP = wT and a one-dimensional Lorentz-transformation t = γ (T + vX/c2 ), x = γ (X + vT ) leads to xP = γ T (w + v) and t = γ T (1 + vw/c2 ), so that for ux = xP /t we get v+w . 1 + vw/c2

ux =

So for w being parallel to v we have a rule for the addition of velocities of the form u=

v+w 1 + βv · βw

(4.3.35)

with βv ≡

v ; c

βw ≡

w . c

Next we consider the case that w is orthogonal to v in the sense that formally v · w = 0 and the v · X-terms vanish in the general Lorentz-transformation (4.3.31).

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4 Relativity

Following the same argument but now with xP = wT + γ vT we get: u = v + γ −1 w⊥ .

(4.3.36)

Combing results (4.3.35) and (4.3.36) we get the general relativistic law for the addition of velocities u=

v + w + γ −1 w⊥ . 1 + βv · βw

(4.3.37)

w = w + w⊥

(4.3.38)

Exercise 4.5 Use the relations

with w =

(v · w)v ; v2

w⊥ = w − w

(4.3.39)

to show that (4.3.37) can be written in the form (γ = (1 − v 2 /c2 )−1/2 ): 

 1 γ −1 u= γ w+ 1+ v . β ·β 1 + βv · βw 1+γ v w

(4.3.40)

4.3.2 Thomas Precession We now consider three inertial systems: x μ , x  μ and x  μ , where the origin of x  moves with constant velocity v with respect to x and the origin of x  moves with constant velocity w in x  . From the last section we know that the origin of x  moves with velocity u (given by (4.3.37)) in x . We now consider the two Lorentz-boosts to go from x μ to x  μ : ν κ μ κ x  = μ ν (w)κ (v)x ≡ Tκ x μ

(4.3.41)

μ

with ν (v) being given by (4.3.28) in the form 00 = γ , 0i = −γβi , i0 = −γβ i , ij = δji + μ

μ

γ2 i β βj . 1+γ

ν (w) and ν (u) are of the same form with γ and β being replaced by γw , βw or μ γu , βu respectively and ask, if this is in agreement with the single boost κ (u).

4.3 The Minkowskian Metric, Lorentz-Transformation

129

We first consider the simple case that w = v, where βu ≡ u/c = 2β/(1 + β 2 ) μ μ with β = v/c, one finds (γ = (1 − β 2 )−1/2 , ν = ν (v)) T00 = 0σ σ0 =

1 + β2 = 00 (u) 1 − β2

Ti0 = 0σ σi = −2γ 2 βi = 0i (u)

(4.3.42)

T0i = iσ σ0 = −2γ 2 β i = i0 (u) Tji = iσ σj = δji + 2γ 2 β i βj = ij (u) . μ

μ

This agreement between Tν and ν (u) is true for all w and v which are parallel to each other. μ

μ

Exercise 4.6 Show that if w and v are parallel to each other, then Tν = ν (u). Let us now consider the case where v and w are not parallel to each other; for simplicity we take v = (v, 0, 0) and w = (0, w, 0), so that v and w are orthogonal to each other, i.e., v · w = 0. For this special case ui = vδ1i + γv−1 wδ2i and u2 = 1 − (γv γw )−2 . c2

(4.3.43)

γu = (1 − βu2 )−1/2 = γv γw ,

(4.3.44)

βu2 ≡ From the last equation we se that

where γv ≡ (1 − βv2 )−1/2 , βv ≡ v/c and γw ≡ (1 − βw2 )−1/2 , βw ≡ w/c. We get T00 = γu Ti0 = −γu βv δi1 − γw βw δi2 T0i = −γv βv δ1i − γu βw δ2i

(4.3.45)

Tji = δji + γu βv βw δj1 δ2i + (γv − 1) δ1i δj1 + (γw − 1) δ2i δj2 and 00 (u) = γu 0i (u) = −γu βv δi1 − γw βw δi2 (4.3.46) i0 (u) = −γu βv δ1i − γw βw δ2i  γu − 1  2 i 1 −2 2 i 2 −1 i 2 i 1 β ij (u) = δji + δ δ + γ β δ δ + γ β β (δ δ + δ δ ) . v w v 1 j v w 2 j v 1 j 2 j βu2

130

4 Relativity

From this we see that Tσ0 = 0σ (u) , but Tσi = iσ (u). It turns out that Tσi = Rji jσ (u)

(4.3.47)

where Rji is a 3 × 3-rotation matrix. For our example with v = (v, 0, 0) and w = (0, w, 0) it is clear that this matrix describes a rotation about the unchanged z-axis, i.e., ⎛

⎞ cos α sin α 0 Rji (α) = ⎝− sin α cos α 0⎠ . 0 0 1

(4.3.48)

Let us consider (4.3.47) with σ = 0, leading to the two equations j

T01 = Rj1 (α)0 (u) = + cos α10 (u) + sin α20 (u) j

T02 = Rj2 (α)0 (u) = − sin α10 (u) + cos α20 (u)

(4.3.49)

from which we get cos α =

γv γu βv2 + γu γw βw2 . γu2 βv2 + γw2 βw2

(4.3.50)

Some re-writing yields 1 + cos α =

(1 + γv + γw + γu )2 . (1 + γv )(1 + γw )(1 + γu )

(4.3.51)

Exercise 4.7 Assuming v = (v, 0, 0) and w = (0, w, 0) compute sin α and cos α. Then check relation (4.3.47) for σ = j numerically for βv = 0.777555 and βw = 0.643354. Solution One finds that cos α = 0.941260, sin α = −0.337682, so that e.g., for σ = 1: T11 = 1.590293, 11 = 1.847779, 21 = 0.441087, i.e., T11 = cos α11 + sin α21 up to rounding errors. For arbitrary velocities v and w one finds: Tαi = Rji jα (u) ;

Tα0 = 0α (u) .

(4.3.52)

4.3 The Minkowskian Metric, Lorentz-Transformation

131

Here, Rji is a 3×3-rotation matrix with Rji Rjk = δik ; it is called the Thomas rotation matrix (Thomas 1926). This important fact is often expressed symbolically by: boost ◦ boost = rotation ◦ boost . Many authors have derived expressions for the rotation matrix appearing in (4.3.52), e.g., Salingeros (1986) or Sexl and Urbantke (2001). However, the most compact and exact form was given by Klioner (2008): Rji = δji + A

v i vj w i vj v i wj w i wj + B + C + D c2 c2 c2 c2

(4.3.53)

with (1 − γw )γv2 (1 + γv )(1 + γu )   γ u − γv γw γv γw 1+2 B= 1 + γu (1 + γv )(1 + γw ) γv γw C=− 1 + γu

A=

D=

γw2 (1 − γv ) . (1 + γw )(1 + γu )

Here, γw = (1 − w 2 /c2 )−1/2 etc. If w is parallel to v so that w = av, then Rji = δji +

 v i vj  2 A + aB + aC + a D = δji , c2

(4.3.54)

since the expression in the bracket vanishes. The angle α of Thomas-rotation can be derived from the trace of Rji using the standard formula 1 + 2 cos α = Rss

(4.3.55)

which leads to expression (4.3.51) above. From the above it is clear that if v and w are parallel then α = 0. From (4.3.53) we can derive many results from the literature. E.g., Rji = δji +

2γv [i v β β + O(βw2 ) . 1 + γv w j ]

(4.3.56)

132

4 Relativity

Here, A[i Bj ] ≡ 12 (Ai Bj − Aj Bi ). Defining δβ ≡ β u − β v this equation can be re-written as (Klioner 2008) Rji = δji +

2γ 2 δβ [i βj ] + O(|δβ|2 ) , 1+γ

(4.3.57)

where β = β v and γ = γv . This expression that can be found, e.g., in Jackson (1975) and Møller (1972). Exercise 4.8 Take the general expression (4.3.53) and analyse the situation where v = (v, 0, 0) and w = (0, w, 0). Especially show that R11 = R22 , or Aβ 2 = Dβw2 and R21 = −R12 or B = −C. Derive expression (4.3.51) from R11 = cos α = 1 + Aβ 2 . For accelerated motion the accelerated frame has a local inertial frame at every instant of time with the consequence that Thomas-rotation leads to a precession of some ‘inertial axis’ in space. If we consider an accelerated torque-free gyroscope then it will precess around kinematically non-rotating axes with Thomas-precession frequency: T = −

1 c2



 γ2 v × a, γ +1

(4.3.58)

with v and a being the velocity and acceleration of the gyro. Thomas precession (Thomas 1926) plays a role in atomic physics because of the electron spin. Here the Thomas precession leads to an interaction energy of ET = S · T that can be rewritten to take a form similar to that of the usual spin orbit coupling resulting from the magnetic dipole intercation of the electron spin (Jackson 1975; Soffel 1989): ET = S · T = −

1 1 dV , (S · L) r dr 2m2e c2

(4.3.59)

where L denotes the electronic angular momentum and V the Coulomb potential of the nucleus. Thomas precession therefore reduces the spin orbit interaction energy by a factor of two.

4.3.3 General Coordinate Transformations and a Derivation of the Lorentz-Transformation We will now derive the Lorentz-transformation in a constructive manner. Let us consider two coordinate systems x μ = (ct, x i ) and Xα = (cT , Xa ) with corresponding metric tensors gμν and Gαβ and gμν = Gαβ = diag(−1, +1, +1, +1) .

(4.3.60)

4.3 The Minkowskian Metric, Lorentz-Transformation

133

Let us write the transformation Xα → x μ in the general form x μ (Xα ) = zμ (T ) + eaμ (T )Xa + ξ μ (T , Xa ) ,

(4.3.61)

where ξ μ is at least quadratic in Xa . Let Aμ α =

∂x μ ∂Xα

(4.3.62)

be the Jacobi-matrix of this transformation. Because of (4.3.60) the transformation rule for the metric tensor leads to (summation over the index i) ν 0 0 i i Gαβ = Aμ α Aβ gμν = −Aα Aβ + Aα Aβ

(4.3.63)

or explicitly − 1 = −A00 A00 + Ai0 Ai0 0 = −A00 A0a + Ai0 Aia δab =

−A0a A0b

+ Aia Aib

(4.3.64) .

Generally we have μ

1 d ea a 1 d μ X + ξ c dT c dT ∂ μ μ Aμ ξ , a = ea (T ) + ∂Xa μ

μ

A0 = e0 (T ) +

(4.3.65)

where e00 (T )

dt 1 dz0 = ≡ c dT dT Xa =0

(4.3.66)

vi 1 dzi = e00 (T ) . c dT c

(4.3.67)

e0i (T ) ≡ Here,

vi ≡

dzi dt

(4.3.68)

is the coordinate velocity of the origin of the Xα -system as seen in x μ coordinates. For the transformation between two inertial systems with (4.3.60) we assume that ξ μ (T , Xa ) = 0 ,

d μ e (T ) = 0 dT a

(4.3.69)

134

4 Relativity

and show that the matching conditions (4.3.64) can indeed be satisfied. With these assumptions, that will be discussed later, they read: − 1 = −e00 e00 + e0i e0i

(4.3.70)

0 = −e00 ea0 + e0i eai

(4.3.71)

δab =

−ea0 eb0

+ eai ebi

.

(4.3.72)

Inserting e0i = e00 v i /c into (4.3.70) we get e00

 −1/2 v2 =γ = 1− 2 c

(4.3.73)

and, therefore, vi . c

(4.3.74)

vi i e c a

(4.3.75)

e0i = γ From (4.3.71) and (4.3.72) we obtain ea0 = and

γ −1 j Ra , = δ +v v v2

eai

ij

i j

(4.3.76)

j

where Ra is a constant rotation matrix with j

j

Ra Rb = δab .

(4.3.77)

Exercise 4.9 Proof by direct calculation that eai from (4.3.76) solves the matching equation (4.3.72). Finally, inserting expression (4.3.76) for eai into (4.3.75) one finds ea0 = γ

vi i R . c a

(4.3.78)

A comparison with (4.3.27) shows that for Rai = δia this transformation agrees with the Lorentz-transformation above.

4.4 The EM-Field of a Moving Point Charge

135

4.4 The EM-Field of a Moving Point Charge Let us consider a point charge q with arbitrary world-line Lq . For its description it is useful to introduce the Dirac delta function δ(x) . In the mathematical language it is a distribution (e.g., Lighthill 1958) with the following properties:

δ(x − a) = 0 +∞

−∞

for

x = a

f (x)δ(x − a) dx = f (a) δ(f (x)) =

 i

1 δ(x − xi ) , df dx (xi )

(4.4.1) (4.4.2) (4.4.3)

where f (x) is assumed to have only simple zeros, located at x = xi . The 3dimensional delta function δ 3 (x) is defined by δ 3 (x) = δ(x) δ(y) δ(z) .

(4.4.4)

The current density of our point charge can then be written as j μ = (cρ; j) = quμ (t)δ 3 (x − z(t))

(4.4.5)

if z(t) parametrizes the wold-line of q. We now want to solve the corresponding wave equation (at several places we write x for a 4-dimensional point in space-time with coordinates (t, x)) Aμ (x) = −μ0 j μ (x) . Let us define a Green’s function D(x, x  ) such that D(x − x  ) = −δ 4 (x − x  ) = −δ(ct − ct  )δ 3 (x − x ) .

(4.4.6)

One of such Green’s functions, the retarded Green’s function DR (x − x  ), satisfying (4.4.6), is given by (see e.g., Poisson and Will 2014, Box 6.5) DR (x − x  ) =

1 0 0

(x 0 − x  )δ(x 0 − x  − |x − x |) 4π |x − x |

(4.4.7)

where 

(x ) = 0

1 x0 > 0 0 otherwise .

(4.4.8)

is the Heaviside step-function. Note, that the appearance of (x 0 − x  0 ) ensures causality, i.e., a current density at x cannot influence physics in the past of x.

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4 Relativity

The retarded solution of the wave equation can therefore be written as

μ

AR (x) = μ0

d 4 x  DR (x − x  )j μ (x  )

(4.4.9)

since μ AR (x)

= μ0

  d 4 x  DR (x − x  ) j μ (x  )

= −μ0

(4.4.10) 4  4





d x δ (x − x )j (x ) = −μ0 j (x) . μ

μ

For our moving point charge we obtain for x 0 > x  0 μ0 q μ AR (t, x) = 4π c μ0 q = 4π



d 4x

 δ t − t +

|x−x | c  |x − x |

 uμ (t  )δ 3 (x − z(t  ))



μ  |x − z(t  )|  u (t )  dt δ t −t + . |x − z(t  )| c

(4.4.11)

Let r(t) ≡ x − z(t) ;

r(t) ≡ |r(t)| ;

n(t) ≡ r(t)/r(t) .

Then, using

g(x)δ[f (x) − α] dx =

g(x) df/dx

(4.4.12) f (x)=α

with f (t  ) ≡ t  + r(t  )/c and df ≡ κ = 1 − n(t  ) · β(t  ) dt  we obtain μ

AR (t, x) =

μ

u μ0 q . 4π κr R

(4.4.13)

These are the Liénard-Wiechert potentials for a point-charge q. β(t) is the instantaneous velocity of the point charge divided by c: β = z˙ /c. The index ‘R’ refers to an

4.4 The EM-Field of a Moving Point Charge

137

event eR on Lq with proper time τR such that x 0 = ct > ct (τR )

(4.4.14)

x 0 − z0 (τR ) = |x − z(τR )| ,

(4.4.15)

and

i.e., eR = eret is given by the intersection of Lq with the backward light-cone through the event (ct, x) (i.e., the space-time point where the potentials and fields are to be evaluated) (Fig. 4.3). Since κr|tR = [r − r · β]tR = [(x 0 − z0 ) − (x − z) · β]tR = ρ(tR )

(4.4.16)

with β

ρ ≡ |ηαβ (x α − zα )e0 |

(4.4.17)

μ

and e0 ≡ dzμ /d(ct) = uμ /c, we can write the Liénard-Wiechert potentials in the Lorentz-invariant form μ

u μ0 μ q A (t, x) = (4.4.18) 4π ρ R or 1 q  4π 0 κr R μ0  qv  A(t, x) = . 4π κr R

(t, x) =

(4.4.19)

Fig. 4.3 Backward light-cone and the retarded event eR

e at at (t, x) or (T, X) Backward light-cone eret LQ

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4 Relativity

Exercise 4.10 Expand the expression (4.4.17) in terms of 1/c and show that up to terms of order 1/c4   1 1 ρ(tR , x) = r 1 + (β · n)2 + 2 a · r . (4.4.20) 2 2c From the Liénard-Wiechert potentials, the E- and B-fields can be derived (a derivation can be found e.g., in Jackson 1975):

' ( q (n − β)(1 − β 2 ) q  n E(t, x) = × (n − β) × β˙ + 3 2 3 4π 0 4π 0 κ r κ r R R (4.4.21) and B(t, x) =

1 (n × E) . c2

(4.4.22)

It is interesting to note, that the Liénard-Wiechert potentials depend only on position and velocity of the charge at retarded time and NOT upon its acceleration. However, since the E- and B-fields are obtained by differentiation of the potentials for them the acceleration of q enters explicitly. Looking at (4.4.21) we see that the fields are given by two terms: the first term is independent of acceleration and falls off as r −2 ; this is the static field carried by the charge. The second term depends linearly on the acceleration of the charge. It falls off as 1/r and both, E- and B-field, are transverse to the radius vector r. This is a typical radiation field that is dominant in the far zone from the charge. Exercise 4.11 Consider two inertial system, IX and Ix , where the origin z of IX (moving system) moves with constant speed in Ix so that z = vt (Fig. 4.4). Consider some event with coordinates T , X in IX and t, x in Ix . Let TR = T − |X|/c be the retarded time in the moving system. By means of Lorentz-transformations show that the corresponding retarded time tR can be expressed in the form tR = t − γ 2

 1/2 v·r 2 2 2 r − γ − (β × r) , c2

(4.4.23)

with r(t) ≡ x − z(t). Fig. 4.4 Two inertial systems and one event

Ix

IX (t, x) (T , X) z(t) = vt

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139

Solution A Lorentz-transformation X=r+

v(v · r) (γ − 1) c2

leads to X2 = r 2 + γ 2 (β · r)2 , so that |X| = (r 2 + γ 2 (β · r)2 )1/2 = γ (r 2 (1 − β 2 ) + (β · r)2 )1/2 = γ (r 2 − (β × r)2 )1/2 . Therefore, tR = γ TR = γ T − γ 2 (r 2 − (β × r)2 )1/2 . A Lorentz-transformations yields T = γt − γ

v·x c2

and we get the result (4.4.23) with x = r(t) + vt.

4.5 The Speed of Propagation in Electromagnetism The form of the Liénard-Wiechert potentials might suggest that the electromagnetic interaction in vacuum ‘propagates with the vacuum speed of light’. This point, however, gave rise to a lot of confusion in the literature and that is the main motivation for this section (another one being the ‘speed of gravity’ problem). One thing, however, should be clear from the beginning: causality is automatically assured by the choice of the retarded Green’s function. So the real question concerning the problem of ‘propagation’ in electromagnetism is: what is possible under the condition that causality is not violated.

4.5.1 The Vacuum Case 4.5.1.1

The Uniformly Moving Point Charge

Let us first consider a point charge q moving with constant velocity in some inertial system I with coordinates (ct, x). According to (4.4.21) the electric field at some

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4 Relativity

field point is given by E(t, x) =

q 4π 0



n−β γ 2κ 3r 2

.

(4.5.1)

R

Let r(t) be a vector pointing from the charge at position βct to the field point x: r(t) ≡ x − βct. Then, r = |r|, n = r/r and κ = (r − r · β)/r so that r(tR ) − r(tR )β q . 2 4π 0 γ (r(tR ) − r(tR ) · β)3

E(t, x) =

(4.5.2)

Now, tR is determined from the equation tR = t −

|x − βctR | c

so that r(tR ) = c(t − tR ) .

(4.5.3)

r(tR ) − r(tR )β = r(t)

(4.5.4)

Thus

and E(t, x) =

r(t) q . 4π 0 γ 2 (r(tR ) − r(tR ) · β)3

(4.5.5)

From (4.5.4) we also infer that r(tR ) − r(tR ) · β = r(t)[(1 − β 2 )ρ − β cos φ] ,

(4.5.6)

where ρ ≡ r(tR )/r(t) and r(t) · β = r(t)β cos φ. From (4.5.4) one also finds that (1 − β 2 )ρ − β cos φ = (1 − β 2 sin2 φ)1/2

(4.5.7)

so that E(t, x) =

q r(t) ·k 4π 0 r 3 (t)

(4.5.8)

with k=

1 γ 2 (1 − β 2 sin2 φ)3/2

.

(4.5.9)

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141

Fig. 4.5 A Hertzian dipole, driven by some AC

So apart from the factor k that equals to one if we neglect β 2 -terms and results from a Lorentz-transformation from an inertial system I  , co-moving with the charge to our reference system I, the result is just the static electric field of a point charge at rest. To talk about some ‘speed of propagation’ is obviously meaningless for this situation (Fig. 4.5).

4.5.1.2

The Harmonic Hertzian Dipole

Let us consider two charges of equal magnitude but opposite sign separated by a small distance that oscillate with a period ω such that j(t, x) = j(x)e−iωt ,

ρ(t, x) = ρ(x)e−iωt .

(4.5.10)

Such an equation should be understood that on the right hand side the real part should be taken. Such a Hertzian dipole is depicted in Fig. 4.5, excited by some AC. In the Lorentz-gauge the vector potential is then given by A(t, x) =

μ0 4π

d 3x

j(x )e−iωtR . |x − x |

(4.5.11)

Since 

e−iωtR = e−i(t−|x−x |/c) we have μ0 A(t, x) = 4π



ei(k|x−x |−ωt) d x j(x ) |x − x | 3 



(4.5.12)

with k ≡ ω/c. Let λ≡

2π k

(4.5.13)

be much larger than p, the dimension of the electric dipole. Then the spatial region with p  r  λ is called the near (static) zone, p  r ∼ λ the intermediate

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4 Relativity

(induction) zone and p  λ  r the far (radiation) zone. In the far zone, |x |  |x| and |x − x | r − n · x , where n ≡ x/r, r ≡ |x|. Then, μ0 ei lim A(t, x) = kr→∞ 4π r

d 3 x  j(x )e−ikn·x



∞ μ0 ei  (−ik)m = d 3 x  j(x )(n · x )m 4π r m!

(4.5.14)

m=0

where  ≡ kr − ωt .

(4.5.15)

For a very small value of p we keep only the m = 0 term so that μ0 ei A(t, x) 4π r

d 3 x  j(x ) .

(4.5.16)

Under this assumption this expression is also valid in the near zone. Using the continuity equation in the form iωρ = ∇ · j(x)

(4.5.17)

we get

d 3 x  j(x ) = −

d 3 x  x (∇  · j(x )) = −iωp0 ,

where

p0 ≡

d 3 x  ρ  x

(4.5.18)

is the constant electric dipole moment of the two charges. Thus, A(t, x) = −

μ0 ei iωp0 4π r

(4.5.19)

which implies, that the vector potential is always oriented parallel to the dipole moment and takes the form of a spherical wave propagating away from the origin.

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143

Fig. 4.6 Spherical coordinates and corresponding base vectors

z er

eφ

x θ φ

eθ y

x

Taking the orientation of p0 in z-direction the vector potential in spherical coordinates takes the form Ar = A cos θ ;

Aθ = −A sin θ ;

Aφ = 0

(4.5.20)

with A=−

ei μ0 iωp0 . 4π r

(4.5.21)

The B-field is obtained from the vector potential A by B = ∇ × A: B(t, x) =

μ0 2 ei k c(n × p0 ) 4π r

 1+

i kr

 (4.5.22)

or in spherical coordinates (see Fig. 4.6) Br = B θ = 0 ;

Bφ = −

μ0 2 ei k cp0 sin θ 4π r

  i 1+ . kr

(4.5.23)

Finally, the E-field is obtained from the Ampère-Maxwell equation E=i

c2 ∇ × B, ω

(4.5.24)



 1 ei 1 ik i 2 E(t, x) = k (n × p0 ) × n + [3n(n · p0 ) − p0 ] 3 − 2 e 4π 0 r r r (4.5.25)

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4 Relativity

or in spherical coordinates   i i ei Er = − 2kp0 cos θ 1 + 4π 0 kr r 2  i  e 1 1 2 i − 2 2 Eθ = − k p0 sin θ 1 + 4π 0 kr r k r

(4.5.26)

Eφ = 0 . In the near zone the dominating fields are lim E(t, x) =

kr→0

1 e−iωt [3n(n · p0 ) − p0 ] 3 4π 0 r

lim B(t, x) = i

kr→0

μ0 e−iωt kc(n × p0 ) 2 . 4π r

(4.5.27) (4.5.28)

The E-field, that dominates over the magnetic field very close to the dipole is just the static electric dipole field, apart from the oscillatory part. In the far zone, lim E(t, x) = cB × n

(4.5.29)

kr→∞

lim B(t, x) =

kr→∞

μ0 2 ei k c(n × p0 ) 4π r

(4.5.30)

which is a typical radiation field. The Poynting-vector, describing the electromagnetic energy-flux density S=

1 (E × B) μ0

(4.5.31)

takes the form 

 cos 2  cos  sin  − + k 2 p02 [(3 cos2 θ − 1)n − 2 cos θ pˆ0 ] r2 kr 3   )  cos2  − sin2  k 1 × + cos  sin  − 5 r4 r3 kr

c S= 16π 2 0

#

k 4 p02 sin2 θ n

(4.5.32) where pˆ 0 ≡ p0 /|p0 |. Note, that this expression is also valid in the near zone. Thus the energy flux shows a very complex behavior especially in the near zone. When, however, we consider a time average over one full period with < cos2  > = < sin2  > =

1 ; 2

< sin  cos  > = 0

4.5 The Speed of Propagation in Electromagnetism

145

the simple result reads 1 = 32π 2 0



k 4 p02 sin2 θ r2

 nc .

(4.5.33)

This expression is not only valid in the far-zone but also in the near-zone. In the far zone =

1 Enc , 2

(4.5.34)

where E=

0 2 1 2 E + B 2 2μ0

(4.5.35)

is the electromagnetic energy-density.

4.5.2 Propagation in a Uniform Dielectric Medium 4.5.2.1

The Front Velocity

A dielectric medium tends to be polarized in the presence of an electric field. If an electromagnetic wave propagates through such a medium electrons will be accelerated and add their contribution to the incident wave. To account for these polarization effects an electric displacement field D is introduced with D = 0 E ,

(4.5.36)

where is the relative permittivity of the (isotropic) medium. It is related with the susceptibility χ (ω) by (ω) = 1 + χ (ω) .

(4.5.37)

Often the permittivity is derived from a Lorentz-model where electrons with charge e and mass me are harmonically coupled to protons with characteristic oscillation frequency ω0 so that (ω) = 1 +

ωp2 ω02 − ω2 − ig ωω0

,

(4.5.38)

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4 Relativity

where g a dimensionless damping constant and  Ne e2 0 m e

ωp =

(4.5.39)

is the plasma frequency with Ne being the electron number density. The wave equations in such a medium take the form   ∂2 − 2 2 + E=0 c ∂t

(4.5.40)

∂B . ∇ ×E=− ∂t Plane wave solutions of (4.5.40) can be written as E = E0 ei(k·x−ωt) ,

B = B0 ei(k·x−ωt)

(4.5.41)

with B0 = (k × E0 )/ω

(4.5.42)

and vp =

c ω = , k n

(4.5.43)

where n=

√

(4.5.44)

is the medium’s index of refraction. vp is the phase velocity, i.e., the velocity for  ≡ k · x − ωt = const. For the Lorentz-model (4.5.38) the index of refraction is a function of ω: the medium is √ dispersive. Figure 4.7 shows the real part of and the phase velocity with n = ( ) for the Lorentz-model (4.5.38) (ω0 = 1, ωp = 0.1, g = 0.1). One sees that in the vicinity of the resonance at ω = ω0 the phase velocity becomes larger than the vacuum speed of light. We now consider a one-dimensional problem (Fitzpatrick 2015): a dispersive uniform medium extends from x = 0 to x = +∞. An incident wave of frequency ω∗ coming from negative x-values is supposed to have an amplitude  f (t) =

0 for t < 0 sin(ω∗ t) for t ≥ 0

(4.5.45)

4.5 The Speed of Propagation in Electromagnetism

0

147

w

Fig. 4.7 Relative permittivity (Real part; brown curve) and phase velocity (blue curve) in units of c in the Lorentz-model

at x = 0 and we ask how the wave propagates to a point x > 0 in the medium. As shown e.g., in Fitzpatrick (2015) the amplitude for x ≥ 0 can be written as

dω 1 f (t, x) = ei(kx−ωt) (4.5.46)  2π ω − ω∗ C+ where the integration contour in the complex ω-plane C+ = {ω|ωI ≡ (ω) = z ∈ R+ }

(4.5.47)

extends from ωR ≡ (ω) = +∞ to −∞. For t < 0 we can choose z → +∞ so that the integrand has a term exp(ωI t) that vanishes exponentially so that f (t, x) = 0 as it has to be. Exercise 4.12 Show that for x ≥ 0 the wave amplitude obeying (4.5.45) is given by expression (4.5.46). The proof is given in Fitzpatrick (2015). Let s ≡ t − x/c. For s < 0 (or v > c) we can again choose z → ∞ so that in the Lorentz-model +1/2 * ωp2 ω ω ω k= n= −→ 1+ 2 2 c c c ω0 − ω − ig ωω0

for |ω| → ∞ .

(4.5.48)

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4 Relativity

In that case i(kx − ωt) = −iω(t − x/c) = −iωs has a large negative real part so that f (t, x) = 0. Thus the wave-front cannot propagate through the dispersive medium with velocity greater than c. We now consider s > 0 (v < c) but very small. Starting from

dω ω∗ ei([k−ω/c]x−ωs) 2 2π C+ ω − ω∗2

f (t, x) =

(4.5.49)

that is equivalent to (4.5.46) for s being sufficiently small we can deform C+ into a large semi-circle of radius R in the upper ω−plane plus two segments of the real axis as shown in Fig. 4.8. Because of the denominator ω2 −ω∗2 the integrand tends to zero as 1/ω2 for large |ω| on the real ω-axis. Adding the integration along the dotted curve in Fig. 4.8 along which the integrand vanishes exponentially for s > 0 and large values of R we get ω∗ f (t, x) = 2π

 ei([k−ω/c]x−ωs) S

ω2

dω . − ω∗2

(4.5.50)

For |ω| → ∞ one has ⎞ ⎛ ωp2 ωp2 ω⎝ ω k− → 1 − 2 − 1⎠ − c c 2cω ω so that ω∗ f (t, x) = 2π

 S

e[i(ξ/ω−ωs)]

dω ω2

(4.5.51)

Fig. 4.8 Integration contour for the Sommerfeld-precursor

(ω)

R

(ω) S

4.5 The Speed of Propagation in Electromagnetism

149

with ωp2

ξ≡

2c

x.

(4.5.52)

We now parametrize the integration circle by writing  ω=

ξ iu e , s

(4.5.53)

(0 ≤ u ≤ 2π ) so that dω =i ω2



s −iu e du ξ

and  2π √ s e−2i ξ s cos u e−iu du ξ 0  2π √ ω∗ s =i e−2i ξ s cos u cos u du . 2π ξ 0

ω∗ f (t, x) = i 2π

(4.5.54)

The last line follows from

2π

cosn (u) sin(u) du = 0

0

which results from cos(π ± x) = − cos(x) and sin(π ± x) = ∓ sin(x). Now, the Bessel function of first order (Fig. 4.9) is given by J1 (z) = −

i 2π

2π

eiz cos θ cos θ dθ

(4.5.55)

 s J1 (2 ξ s) ξ

(4.5.56)

0

so that  f (t, x) = ω∗

since J1 (−z) = −J1 (z). Equation (4.5.56) describes the behavior of the Sommerfeld-precursor. Its amplitude is extremely small compared to that of the √ incident wave since |fS | ∼ ω∗ s/ξ = ω∗ /ω  1. Since ξ is proportional to x the amplitude of the Sommerfeld-precursor decreases like 1/x with increasing value of x. The initial period of oscillation is determined by the first maximum of J1 where s ∼ 1/ξ , hence its oscillation frequency is extremely high and independent of ω∗ .

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4 Relativity

0

Fig. 4.9 The Bessel function J1 (z)

4.5.2.2

The Group-Velocity

We now consider a one-dimensional wave-packet with amplitude

f (t, x) =

+∞

−∞

dk F (k)ei(kx−ωt)

(4.5.57)

and first assume the wave packet to be almost monochromatic with dominant wave vector k0 . To perform the integral ω has to be considered implicitly a function of k. Substituting a linear relation of the form ω(k) ω0 + (k − k0 )vgr

(4.5.58)

with  vgr ≡

dω dk

 (4.5.59) 0

leads to

f (t, x) = ei(k0 x−ω0 t)

+∞ −∞

dk F (k)ei(k−k0 )(x−vgr t) .

(4.5.60)

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151

The first factor describes a perfect monochromatic wave with wave-vector k0 with peaks and troughs moving with phase velocity vph = ω0 /k0 under the envelope of the wave-packet. The second term implies that the envelope propagates with the group-velocity vgr . For a general wave-packet with large bandwidth partial waves with different frequencies will propagate with different velocities; nevertheless the group-velocity (4.5.59) indicates the propagation velocity of the peak of a wavepacket. Since k(ω) =

ω n(ω) c

(4.5.61)

for a complex index of refraction the group-velocity can be written in the form   d((n)) 1 d((k)) 1 (n) + ω . = = vgr dω c dω

(4.5.62)

Under certain conditions the group-velocity can exceed the vacuum speed of light. E.g., Chiao (1993) considers a gas of inverted two-level atoms where the relative permittivity can be modeled with (ω) = n2 (ω) = 1 −

ωp2 ω02 − ω2 − ig ωω0

(4.5.63)

differing from the Lorentz-model of (4.5.38) by the sign of the second term, thus converting damping into amplification. In the limit of small frequencies, ω  ω0 a wave group propagates with superluminal velocity since  ωp2 1 1 n 1 1− 2 < . vgr c c c ω0

(4.5.64)

This result for small frequencies does not depend on the specific model (4.5.63); from the Kramers-Kronig relations (e.g., Landau and Lifshitz 1960) between Real and Imaginary part of the susceptibility χ one finds that (0) < 1 and hence n(0) < 1 if χ (ω) < 0 (e.g., Chiao 1993) implying that both, phase- and group-velocity, exceed the vacuum speed of light without violating causality. Sommerfeld and Brillouin (Brilluoin 1960) have discussed other propagation velocities beside the phase-, front- and group-velocity: an ‘energy-velocity’ at which energy is transported by the wave and a ‘signal-velocity’ at which the half-maximum amplitude travels. One finds that also these velocities can exceed the vacuum speed of light. It is only the front-velocity of the Sommerfeld precursor that is never larger than c. In recent decades such superluminal wave propagations have been detected in many experiments using single photons (Steinberg and Chiao 1995), at optical

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4 Relativity

frequencies (Spielmann et al. 1994) and using microwaves (Mojahedi et al. 2000a,b; Ranfagni et al. 1991, 1993; Mugnai et al. 1998; Enders and Nimtz 1992, 1993).

4.6 Energy and Momentum We first note that the 4-velocity of some small body has an absolute value that does not change with time. Using metric property 2 (Eq. (4.3.10)) we find gμν uμ uν = gμν

ds 2 dx μ dx ν = = −c2 . dτ dτ dτ 2

(4.6.1)

From this we immediately infer the absolute value of the 4-momentum gμν pμ pν = −p0 p0 + p2 = −m2 c2 .

(4.6.2)

The time component of pμ is related with the body’s energy E by p0 =

E , c

(4.6.3)

i.e., E 2 = (mc2 )2 + p2 c2 .

(4.6.4)

For a small momentum |p|  mc we have E=



m2 c4 + p2 c2 = mc2 +

p2 + ... , 2m

(4.6.5)

where p2 /(2m) is the usual kinetic energy of the body and mc2 is its rest-energy. In words: even for zero momentum a body of mass m possesses an energy of E = mc2 .

(4.6.6)

This relation between mass and energy is of special importance for our theory of gravity since mass-density acts as source of the gravitational field in Newton’s theory. Therefore in any relativistic theory of gravity it must be energy and momentum that produce gravity. If one considers a continuous distribution of noninteracting particles (dust) with 4-velocities uμ one defines an energy-momentum tensor T μν by T μν = ρuμ uν .

(4.6.7)

4.6 Energy and Momentum

153

Here ρ is the rest-mass (energy) density that is measured if one moves together with the ensemble of particles. For an ideal fluid (no shear stresses, anisotropic pressure, viscosity etc.) the energy-momentum tensor reads  p T μν = ρ + 2 uμ uν + pg μν , c

(4.6.8)

where p is the pressure. Generally every continuous distribution of matter or field can be associated with a corresponding symmetric energy-momentum tensor. As further example the energy-momentum tensor of the electromagnetic field takes the form   1 1 (4.6.9) T μν = F μα F ν α − g μν Fαβ F αβ . 4π 4 Evaluating this expression in Minkowskian coordinates we find Fαβ F αβ = 2F0i F 0i + Fij F ij = −2E2 + 2B2 F 0α F 0 α = F 0i F 0i = E2 F 0α F i α = F 0j F ij = ij k Ej Bk F iα F j α = F i0 F j 0 + F ik F j k = −Ei Ej + ikl Bl j km Bm = −Ei Ej + (δij δlm − δim δlj )Bl Bm = −(Ei Ej + Bi Bj ) + B2 δij . Therefore, T 00 =

1 4π

    1 1 1 1 F 0α F 0 α + Fαβ F αβ = E2 − E2 + B2 , 4 4π 2 2

i.e., the energy-density of the electromagnetic field T 00 is given by T 00 =

1 (E2 + B2 ) . 8π

(4.6.10)

For the energy-momentum current one finds T 0i =

1 1 (F 0α F i α ) = (E × B)i , 4π 4π

(4.6.11)

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4 Relativity

known as the Poynting vector of the field. Finally for the Maxwell stress tensor T ij we get T

ij

  1 iα j αβ F F α − Fαβ F 4

1 1 1 = −(Ei Ej + Bi Bj ) + B2 δij + E2 δij − B2 δij 4π 2 2

1 1 −(Ei Ej + Bi Bj ) + (E2 + B2 )δij . = 4π 2

1 = 4π

(4.6.12)

Just as the continuity equation (4.2.3) indicates the conservation of electric charge the equation T μν ,ν = 0

(4.6.13)

indicates the conservation of energy and momentum of some continuous matter or field distribution in some Minkowskian coordinate system. Clearly in an arbitrary coordinate system this relation reads T μν ;ν = 0

(4.6.14)

where we have replaced the partial derivatives by the covariant ones. As an example we consider this law for an ideal fluid with local density ρ, pressure p and temperature T . In Minkowskian coordinates it reads 0=

∂ ∂  μν  p  μ ν μν pη u u . T = + ρ + ∂x ν ∂x ν c2

(4.6.15)

Since ηαβ uα uβ = −c2 (relation (4.3.14)) 0=

∂uα ∂ (ηαβ uα uβ ) = 2ηαβ uβ γ γ ∂x ∂x

and we obtain ∂T αγ ∂x γ  ∂p ∂  = uβ β − γ (ρc2 + p)uγ . ∂x ∂x

0 = ηαβ uβ

(4.6.16)

This result can be written in a very elegant manner (Weinberg 1972). Let us think of the fluid to be composed of infinitely many infinitesimally small particles. As far as mass and energy are concerned these particles will be the baryons of ordinary matter. Let n be the baryon number density that can be measured if one moves with

4.6 Energy and Momentum

155

the matter. The law of baryon number conservation can then be formulated with a baryon current 4-vector N μ = n uμ

(4.6.17)

in the form 0=

∂N μ ∂ = μ (n uμ ) . ∂x μ ∂x

(4.6.18)

Using this law we can rewrite (4.6.16) as

 2 

(ρc2 + p) β ∂p ρc + p ∂p ∂ ∂ β − β −n β 0=u nu = u ∂x β ∂x n ∂x β ∂x n    2 

∂ 1 ρc ∂ + β . (4.6.19) = −nuβ p β ∂x n ∂x n β

We can now employ the first law of thermodynamics in the form  2   ρc 1 T ds = p d +d n n

(4.6.20)

or ∂s ∂ T β =p β ∂x ∂x

   2 1 ρc ∂ + β n ∂x n

where the quantity s is the entropy per baryon. With this the conservation equation (4.6.19) simply reads 0 = uβ

∂s ∂s . = ∂x β ∂τ

(4.6.21)

In other words: entropy per baryon is conserved if one moves together with the fluid. Putting μ = i in (4.6.13) one finds the relativistic Euler equation of hydrodynamics in the form:    1 − v2 /c2 ∂v 1 ∂p  ∇p + 2 v + (v · ∇)v = −  . (4.6.22) ∂t c ∂t ρ + p/c2 Exercise 4.13 Derive the Euler equation (4.6.22) from (4.6.13) by putting μ = i by using the law of entropy conservation.

Chapter 5

Einstein’s Theory of Gravity

5.1 General Relativity Special Relativity can be described as physics in a 4-dimensional space-time manifold M with metric tensor gμν that reduces to ημν = diag(−1, +1, +1, +1) in any global inertial coordinate system. Such selected global coordinates exist because the geometry of Minkowskian space-time is flat, i.e., the curvature and Ricci tensor vanish. Einstein’s theory of gravity is also a structure (M, g), but space-time geometry in the presence of gravitational fields is not longer flat, the curvature tensor describing the tidal actions. For vanishing gravitational fields the structure (M, g) reduces to the Minkowskian space-time; it is fully in accordance with all experiments from Special Relativity. In General Relativity (GR) all aspects of gravitational fields are contained in the space-time metric tensor. A necessary prerequisite this is the Equivalence principle, that also shows the role of Special Relativity in Einstein’s GT. The weak form of the equivalence principle (the universality of free-fall) has already been discussed. The Einstein equivalence principle (EEP) generalizes this to all non-gravitational laws of physics: in any freely falling system all non-gravitational laws of physics take their form from Special Relativity. In some sense certain aspects of gravity disappear in a freely falling reference frame. Such aspects are related with the affine connections of space-time geometry that are not tensors and can be transformed to zero at any point p ∈ M by a suitable coordinate transformation. This, however, by no means implies that some existing gravitational field inside such a freely-falling system is zero; if the curvature tensor has non-vanishing components in one coordinate system then there is no coordinate system where it completely vanishes at any point p ∈ M. This means that the EEP simply says that at each point p of the space-time manifold there are local coordinates such that the metric tensor reduces to the Minkowskian tensor where effects from gravity do not appear. Einstein’s equivalence principle implies that a reasonable theory of gravity should be a metric theory with (M, g) as basic structure and possible additional © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_5

157

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5 Einstein’s Theory of Gravity

fields ψi , taking part in the gravitational interaction. General Relativity is the simplest of all such metric theories, where all additional fields ψi = 0. Sources of the gravitational field, i.e., all forms of energy and momentum as well as gravity fields itself, produce curvature of space-time which again determines the dynamical behavior of the sources.

5.2 Einstein’s Equivalence Principle Einstein’s theory of gravity generalizes the results from Minkowski space-time theory by considering also gravitational fields. A hint of how to incorporate gravity into the space-time structure comes from the phenomenon of gravitational redshift. Let us consider two identical clocks at rest in some gravitational potential U (x). Clock 1 is assumed to be located a distance H above clock 1. Then, because of the gravitational redshift the natural frequencies of the two clocks, f1 and f2 , are related by f2 1 = 1 + 2 [U (x2 ) − U (x1 )] . f1 c

(5.2.1)

It is not difficult to see that the gravitational redshift of electromagnetic waves is a consequence of a certain form of the equivalence principle. This will also make it clear why clocks in a gravitational field are running slower (Fig. 5.1).

Fig. 5.1 Three static clocks in some gravitational field. The larger the gravitational potential the slower the clock runs

5.2 Einstein’s Equivalence Principle

159

Fig. 5.2 Two accelerated clocks as seen from a local freely falling system. The distance between the clocks is H . The first clock emits a first signal, a second one follows after a time interval δt1 . The observer at z2 receives these two signal at times t− and t+

z

z1 δt1

z2 H t−

t+

t

Einstein’s Equivalence Principle Everywhere in the universe and for all times in sufficiently small freely falling laboratories all non-gravitational laws of physics take their form from Special Relativity. In other words: such freely falling systems are locally inertial. Let us now consider two clocks at rest in some external gravitational field (Fig. 5.2). Obviously the two clocks are not freely falling; instead they are at rest in some system that is accelerated upwards, i.e., away from the center of gravitational attraction. With respect to some freely falling local inertial coordinate system x μ = (ct, x i ) the world-lines of the two clocks are depicted in Fig. 5.2. We now consider a light-pulse being emitted from clock 1 in the direction of clock 2. In a first approximation z1 = gt 2 /2 + H and z2 = gt 2 /2 and the velocities are given by v = gt. Since in the accelerated system where the two clocks are at rest the situation is stationary and we might choose for simplicity t = 0 for the emission event. Neglecting (v/c)2 -terms t will agree with the proper times indicated by the two clocks. Then for gH /c2  1 the signal will arrive at clock 2 at t− H /c. The crucial point is that at the point of reception the second clock has a finite velocity v = gt = gH /c in the direction of the first clock. Let the first clock emit a second pulse at t = δt1 immediately after the first one. The arrival time at clock 2 then is t+ =

H v + δt1 − δt1 c c

since during the interval δt1 it has moved a distance v δt1 in the direction of clock 1, i.e., the effective distance is only H − v δt1 instead of H . Hence the time that has elapsed during the reception of the two pulses at clock 2 is  v δt2 = t+ − t− = δt1 1 − c     gH U (x2 ) − U (x1 ) = δt1 1 − 2 δt1 1 − c c2 in accordance with the gravitational redshift formula (5.2.1). Thus from the standpoint of Einstein’s Equivalence Principle the gravitational redshift results from the

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5 Einstein’s Theory of Gravity

first-order Doppler shift of frequencies. Einstein’s form of the equivalence principle has the consequence that gravity can be described by a metric theory, i.e., (see e.g., Will 1993 for more details) – by at least a gμν -field and possibly by “other g-fields”; – these “other g-fields” only couple to the gμν -field but not to matter-fields directly; – at each point in space-time there is a local freely falling system (Einstein’s elevator) where the space-time metric gμν reduces to the flat space-time metric ημν ; – the world-lines of uncharged test particles are geodesics of gμν . The last point follows from the fact that these world-lines are straight lines in a freely falling system, i.e., geodesics with respect to the flat space-time metric. Hence, they must be geodesics with respect to the space-time metric gμν . Metric Property 3 Sufficiently small (uncharged) test bodies move along geodesics of the metric tensor. The gravitational redshift can then be described in a very elegant manner: we incorporate the gravitational potential U into the metric and write in suitable coordinates   2U 2 ds = − 1 − 2 c2 dt 2 + (dx)2 (5.2.2) c or g00 = −1 +

2U ; c2

g0i = 0;

gij = δij .

(5.2.3)

Assuming again metric property 2 (Eq. (4.3.10)) for two clocks at rest (dx = 0) we get for each of the two clocks i: dτi2 = −

  1 2 2U (xi ) dt 2 ds = 1 − c2 i c2

or   U (xi ) dt . dτi 1 − c2 From this we derive f2 dτ1 1 − U (x1 )/c2 = f1 dτ2 1 − U (x2 )/c2 1+ in accordance with (5.2.1).

1 [U (x2 ) − U (x1 )] c2

(5.2.4)

5.3 The Motion of Test Bodies

161

5.3 The Motion of Test Bodies Let us consider the geometry that is determined by the metric (5.2.3) in more detail where we restrict our discussion to terms of order c−2 . The inverse metric tensor in this approximation is given by g 00 = −1 −

2U ; c2

g 0i = 0;

g ij = δij .

(5.3.1)

From this we derive the non-vanishing Christoffel-symbols: i 0i0 = i00 = 00 =−

1 U,i . c2

(5.3.2)

We now come to the geodesic equation ν σ d 2xμ μ dx dx = 0. +  νσ dλ dλ dλ2

Here λ is an affine parameter that might be replaced by the time coordinate t (which is not an affine parameter) in the μ = i equation: d 2xi = dt 2

 

=

dt dλ dt dλ

−1 −2

i = −νσ

d dλ



dt dλ

d 2xi − dλ2



−1

dt dλ

dx i dλ

−3



d 2 t dx i dλ2 dλ

1 0 dx ν dx σ dx i dx ν dx σ + νσ . dt dt c dt dt dt

(5.3.3) (5.3.4)

In more detail this reads (v i ≡ dx i /dt)  j j k d 2xi 2 i i v i v v  +  = −c + 2 j k 00 0j c c c dt 2 j j k vi $ 0 0 v 0 v v + j k . − 00 + 20j c c c c

(5.3.5)

Considering the Christoffel-symbols from (5.3.2) we see that the right hand side i . Keeping only this cof this equation has a term of order c0 resulting from 00 independent term the geodesic equation reads d 2xi i = −c2 00 = U,i . dt 2

(5.3.6)

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5 Einstein’s Theory of Gravity

This, however, is precisely the equation of free-fall of a sufficiently small test body in Newton’s theory of gravity in Galilean coordinates (Cartesian and inertial).

5.4 Einstein’s Theory of Gravity Einstein’s theory of gravity is the ‘simplest’ of all reasonable metric theories of gravity. In Einstein’s theory there are no other g-fields but only one space-time metric that also describes gravity. Metric property 3 indicates an intimate relation between Newton’s theory of gravity and a relativistic one. In both theories test particles move along geodesics of the space-time geometry. As we have seen the Newtonian field equation for the potential U relates the Ricci-tensor of space-time with the field generating source. Now in relativity the source of the gravitational field obviously must be the energymomentum tensor T μν and Einstein’s field equations for the metric tensor take the form F μν (g, ∂g, ∂ 2 g) = κ T μν where F μν is a function of gμν and its first and second partial derivatives with respect to the coordinates x μ . Because of the conservation laws for energy and momentum, Eq. (4.6.14) we have to require F μν ;ν = 0 .

(5.4.1)

Theorem 5.1 (Lovelock 1972) The most general tensor F μν (g, ∂g, ∂ 2 g) that is divergenceless, i.e., obeys Eq. (5.4.1) is of the form F μν = aGμν + bg μν

(5.4.2)

where Gμν are the components of the Einstein-tensor. The usual Einstein’s field equations are obtained with a = 1 and b = 0: Gμν = κ T μν

(5.4.3)

1 R μν − g μν R = κ T μν . 2

(5.4.4)

or

Another important form of the field equations is obtained by contracting equation (5.4.4) with gμν (i.e., by taking its trace): R − 2R = −R = κ T ,

5.4 Einstein’s Theory of Gravity

163

where T ≡ gμν T μν

(5.4.5)

is the trace of the energy-momentum tensor. Inserting this result for the curvature scalar R into Einstein’s field equations leads to the alternative form R μν = κ

  1 T μν − g μν T ≡ κ Tˆ μν . 2

(5.4.6)

Finally we have to determine the coupling constant κ. To this end we consider the ‘Newtonian limit’ of these field equations. In Newton’s theory only the matter density ρ acts as source of the gravitational field. This density to lowest order is contained in the time-time component of the energy-momentum tensor, considering a continuous distribution of energy and momentum. From (4.6.7) we see that T 00 = −T = ρc2 + . . .

(5.4.7)

and for that reason the Newtonian field equation must be contained in the time-time component of (5.4.6): R

00

  1 00 1 00 = κ T − g T = κρc2 . 2 2

The left hand side to order c−2 can be taken from Eq. (3.2.21) keeping in mind that now x 0 = ct and the dimension of the Einstein tensor is (length)−2 : R00 = R 00 = −

U + ... . c2

Hence to lowest order the Einstein field equations lead to 1 U = − κ ρc4 2 and a comparison with the Poisson equation (3.2.19) shows that κ=

8π G . c4

Einstein’s field equations form a complicated set of ten partial differential equations of second order. Because of the Bianchi identities these ten equations are not independent from each other but only six of them. Hence, the equations determine six out of ten degrees of freedom of the metric tensor gμν . Four degrees of freedom for the metric tensor remain, expressing the freedom in the choice of the four space-time coordinates. Of course the field equations cannot tell what

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5 Einstein’s Theory of Gravity

coordinates should be used; instead the coordinates can be fixed by four (more or less) arbitrary conditions for the metric tensor. This is the coordinate or “gauge” freedom of the theory. This gauge freedom is one of the most important differences to the classical Newtonian case. In Newton’s theory time is absolute, so there is a preferred time coordinate which is fixed uniquely up to origin and unit. Out of the many possible spatial coordinates the inertial (Cartesian) ones which in Newton’s theory exist globally are preferred. They are determined uniquely up to origin, unit and orientation in space (determined e.g., by three Euler angles). All these preferred coordinates, however, do not exist in Einstein’s theory of gravity. However, the situation is not too bad for isolated systems with an asymptotically flat space-time. E.g., the solar system might be idealized in this manner: we forget about distant masses and think of the solar system as being isolated. Then far from the solar system the gravitational field will become very small and space-time will approach flat space-time from Special Relativity Theory in this idealized picture. Then in the asymptotic region preferred (inertial and Cartesian) coordinates exist such that there gμν → ημν .

(5.4.8)

If, however, we get closer to the gravitating masses preferred coordinates cease to exist; i.e., many different coordinates have equal rights. If we choose a = 1 and b =  in Lovelock’s Theorem then we end up with field equations of the form with a  term 1 Rμν − gμν R + gμν = κTμν . 2

(5.4.9)

In this case the constant  is called the cosmological constant. It is obvious that the -term can be absorbed in the energy-momentum tensor by replacing Tμν by T˜μν ≡ Tμν − κ −1 gμν .

(5.4.10)

Usually it is assumed that  is related with the energy density of the quantum vacuum pervading the whole universe and might have a value of about 10−52 m−2 (Peebles and Ratra 2003). Metric (5.4.9) plays an important role in modern cosmology.

5.5 The Problem of Observables Since in Einstein’s theory of gravity the coordinates usually have no direct physical meaning the problem of observables is a serious one. It should be clear that observables are independent of any set of coordinates used by some theorist to describe the system of interest. In other words: observables have to be described by scalars, coordinate independent quantities. First one chooses some appropriate coordinate system and draws a coordinate picture of the system of interest. Then one constructs the observables as scalars from such a coordinate picture.

5.5 The Problem of Observables

165

5.5.1 The Ranging Observable Let us consider a typical astronomical measurement in the solar system: lunar laser ranging (LLR). Here laser pulses are emitted from LLR-stations on the Earth to retroreflectors on the lunar surface. A few photons per pulse find their way back into the receiving telescope of the station and one measures the total travel time of a pulse from the station to the Moon and back. This situation is depicted in Fig. 5.3. In the right part of the figure we see the world-line of the clock with the two events E: emission of the pulse and R: reception of the pulse. The observed time interval between E and R is then given by

τ =

R

(5.5.1)

dτ E

with dτ 2 = −

1 2 ds . c2

In practise this indicated time interval τ can then be related with a corresponding interval of some other time scale.

5.5.2 The Spectroscopic Observable We now consider the following problem: one observer emits some monochromatic electromagnetic wave of frequency fE . Another observer receives this signal and measures the frequency fR and we ask about the relation between the two

τR

R R

E

E

τE

Fig. 5.3 Left: A central observable for celestial mechanics, the ranging observable, is a propagation time interval between emission and reception of some electromagnetic pulse. In Lunar Laser Ranging it is a laser pulse that travels from some LLR-station on the Earth to some retroreflector on the lunar surface and back to the ground station. Right: the observable is the proper time interval that has elapsed between the instant of emission and the instant of reception of an electromagnetic pulse

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5 Einstein’s Theory of Gravity

uμR γ∗

R

uμE

E

μ kR

kEμ γE

γR

Fig. 5.4 The spectroscopic observable is the frequency ratio fR /fE . Some observer (emitter) emits some electromagnetic signal of frequency fE . This signal is observed by another observer (receiver) who measures the frequency fR

frequencies. If we concentrate upon one single light-ray propagating from the emitter to the receiver the situation is shown in Fig. 5.4. Here γE is the world-line of μ the emitter, γR that of the receiver, γ ∗ that of the light ray. Let uE be the 4-velocity of μ the emitter at the point of emission, uR that of the receiver at the point of reception. Let k μ be the tangent vector onto γ ∗ then according to (4.3.7) the frequency ratio is given by (gμν k μ uν )R fR = . fE (gμν k μ uν )E

(5.5.2)

Let us analyze this situation in Minkowski space in the absence of gravitational fields. Let us choose a Minkowskian coordinate system such that the receiver is at rest in the event of reception, i.e., μ

uR = (c, 0) . If the emitter has coordinate velocity v at the point of emission then μ

uE = γ (c, v) . Since k μ is a null-vector we can write in Minkowskian coordinates k μ = const. × (1, n)

(5.5.3)

with δij ni nj = 1 . The normalization constant in k μ will not play a role if only frequency ratios are considered. With β≡

v c

(5.5.4)

5.5 The Problem of Observables

167

we then get fR (−k 0 u0 + k i ui )R = = [γ (1 − β · n)]−1 fE (−k 0 u0 + k i ui )E or  1 − β2

fR = fE

1−β ·n

(5.5.5)

.

This is the well-known formula for the Doppler-effect in electromagnetism.

5.5.3 The Astrometric Observable In astrometry the principle observable is the observed angle between two incident light-rays. This situation is depicted in Fig. 5.5. Here γ (λ) is the worldline of the observer, γ1∗ and γ2∗ are two light-rays from two different astronomical sources that are simultaneously observed by the observer in some event O. Let uμ be the 4μ μ velocity of the observer in O, k1 and k2 be the wave vectors of the two incident light-rays. Then Pνμ ≡ δνμ +

1 μ u uν c2

(5.5.6)

is a projection tensor that projects vectors into their components perpendicular to uμ , i.e., Pνμ uν = uμ +

Fig. 5.5 The astrometric observable: the observed angle θ between two incident light-rays γ1∗ and γ2∗ . The observer’s worldline is γ (λ) μ and ki are tangent vectors to the light-rays

1 μ u uν uν = 0 c2

(5.5.7)

γ2∗

γ1∗ uμ k1μ

γ(λ)

k2μ

k¯1

k¯2 θ

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5 Einstein’s Theory of Gravity

since uν uν = −c2 . In some sense uμ points into the time-direction of the observer and the projection operator points into the space ‘experienced’ by the observer. Now μ ki are null-vectors but μ

k = Pνμ k ν

(5.5.8)

is a spacelike vector of non-vanishing length. For uμ = γ c(1, β)

k μ = (1, n)

we find uν k ν = −γ c (1 − β · n)

(5.5.9)

1 μ k = k μ − γ (1 − β · n)uμ . c

(5.5.10)

and therefore

From this it is not difficult to see that μ

μ ν

|k | ≡ (gμν k k )1/2 = γ (1 − β · n) .

(5.5.11)

The observed angle θ between two incident light-rays γ1∗ and γ2∗ is generally given by μ ν

cos θ =

gμν k 1 k 2

μ μ |k 1 | |k 2 |

.

(5.5.12)

In the absence of gravity fields from (5.5.12) we get cos θ =

n1 · n2 − 1 + γ 2 (1 − β · n1 )(1 − β · n2 ) . γ 2 (1 − β · n1 )(1 − β · n2 )

(5.5.13)

This is the aberration formula if gravity fields play no role. A Taylor expansion in terms of c−1 yields  cos θ = n1 · n2 + (n1 · n2 − 1) (n1 + n2 ) · β + (n1 · β)2  + (n2 · β)2 + (n1 · β)(n2 · β) − β 2 + O(c−3 ) .

(5.5.14)

5.6 Tetrads and Tetrad Induced Coordinates

169

5.6 Tetrads and Tetrad Induced Coordinates Consider some massless observer E that moves through empty space with a space capsule and wants to perform some local experiment inside of his spacecraft. Let us describe the motion of E in some coordinate system x μ by some timelike worldline μ LE , given by zE (λ), where λ is some affine parameter. Let us choose this parameter λ μ as the observer’s proper time τ also denoted by T . The tangent vector uμ ≡ dzE /dT then is the observer’s 4-velocity that is normalized according to μ

gμν uμ uν =

gμν dzE dzEν = dT · dT



ds dT

2 = −c2 ,

(5.6.1)

since ds 2 = −c2 dτ 2 along the observer’s world-line. In the following we will denote the unit vector in the direction of uμ by μ

e(0) ≡

1 μ u . c

(5.6.2)

Let μ

a μ ≡ u;ν uν

(5.6.3)

be the observer’s 4-acceleration, a vector that is perpendicular to uμ since gμν a μ uν =

1 (gμν uμ uν );σ uσ = 0 2

(5.6.4)

in virtue of the normalization condition and gμν;σ = 0. μ μ A set of four orthonormal vectors e(α) (α = 0, 1, 2, 3) with e(0) being given by (5.6.2) and μ

ν = ηαβ gμν e(α) e(β)

(5.6.5)

along LE is called a tetrad field along LE . Such tetrad fields are valuable quantities that can be used in different respects, e.g., for the construction of observables. They can also be used to define useful local coordinates Xα = (cT , Xa ) for the observer. First the local time coordinates T will be chosen as proper time τ of the observer whose world-line should be given by Xa = 0, i.e., the observer is located at the spatial origin of his local coordinate system. Next we define: a local system of coordinates Xα is called tetrad-induced if ∂x u μ e(α) = . (5.6.6) ∂Xα LE

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5 Einstein’s Theory of Gravity

From this definition we find that the tetrad vectors in tetrad-induced coordinates (TIC) take a particularly simple form β

E(α) =

∂Xβ μ ∂Xβ ∂x μ e = = δαβ . ∂x μ (α) LE ∂x μ ∂Xα LE

(5.6.7)

Using this condition in TIC we find Gαβ |LE = ηαβ .

(5.6.8)

Hence, TIC are locally Minkowskian. We will now construct certain TIC in the neighbourhood of LE by imposing certain constraints on the Christoffel-symbols. To this end we consider the following quantities ρ

κ σ E(γ (E(γ ) , D(α) E(β) ) ≡ Gρσ E(β);κ E(α) ).

(5.6.9)

Because of the simple form of tetrad vectors in TIC, Eq. (5.6.7), we have ρ

ρ

ρ τ ρ τ E(β) = κτ E(β) E(β);κ = E(β),κ + κτ

that leads to ρ (E(γ ) , D(α) E(β) L = ηργ αβ E

LE

.

(5.6.10)

Lemma 5.1 The Christoffel-symbols in TIC obey the following relations at LE : 0 = 0, 00 a 0 00 = 0a = b = 0a

1 a A , c2

(5.6.11)

1

(a)(b) , c

where Aa are the spatial tetrad components of the 4-acceleration of E, i.e., μ

Aa ≡ gμν e(a) a ν

(5.6.12)

(a)(b) ≡ c · (E(b) , D(0) E(a) ) .

(5.6.13)

and

The quantities (a)(b) are called Ricci-rotation coefficients.

5.6 Tetrads and Tetrad Induced Coordinates

171

Exercise 5.1 Use the orthonormality of tetrad vectors to proof the antisymmetry of rotation coefficients

(a)(b) = − (b)(a) .

(5.6.14)

The proof of Lemma 5.1 follows from (5.6.10), the definition of the 4acceleration and the orthonormality of tetrad vectors. This Lemma implies that α . all Christoffel-symbols of TIC at the observer’s worldline are fixed apart from bc We now have several possibilities to fix these remaining quantities at LE . μ

Exercise 5.2 Suppose the X, Y, Z coordinates lines Y(α) , which are integral curves μ to the tetrad e(α) , are geodesics, parametrized with proper length s. Show that for that case α bc (5.6.15) L = 0. E

Corresponding TIC will be called local geodetic proper coordinates. α . Another one is given by The last Exercise shows one possible choice for bc TIC that are locally harmonic. The harmonicity condition at LE can be written in the form λ Gαβ αβ = 0,

i.e., λ λ aa = 00 .

(5.6.16)

One solution of the harmonicity condition along LE reads 0 bc = 0, a bc =−

1 (δab Ac + δac Ab − δbc Aa ) . c2

(5.6.17)

We will call local TIC with such Christoffel-symbols local harmonic proper coordinates. Because the covariant derivative of the metric tensor vanishes, i.e., δ δ 0 = Gαβ;γ = Gαβ,γ − αγ Gδβ − βγ Gαδ ,

(5.6.18)

the Christoffel-symbols at LE determine the partial derivatives of Gαβ at the worldline of E. Exercise 5.3 Show that condition (5.6.18) for local geodetic proper coordinates proper coordinates leads to Gαβ,0 = 0

Gab,c = 0

G00,a = 2Aa ,

G0a,c =

1 abc b . c

(5.6.19)

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5 Einstein’s Theory of Gravity

Together with Gαβ |LE = ηαβ this leads to a metric tensor of the form   2 G00 = − 1 + 2 A · X + O(|X|2 ) c G0a =

1 abc b Xc + O(|X|2 ) c

(5.6.20)

Gab = δab + O(|X|2 ) with

(a)(b) = abc c .

(5.6.21)

For local harmonic proper coordinates condition (5.6.18) leads to Gαβ,0 = 0

Gab,c =

1 δab Ac c2

G00,a = 2Aa ,

G0a,c =

1 abc b c

(5.6.22)

at the observer’s world-line. Using Gαβ |LE = ηαβ this leads to a metric tensor of the form   2 G00 = − 1 + 2 A · X + O(|X|2 ) c 1 abc b Xc + O(|X|2 ) c   2 = δab 1 − 2 A · X + O(|x|2 ) , c

G0a = Gab

(5.6.23)

where b is again given by relation (5.6.21). Finally, let us try to understand the meaning of our ‘angular velocity’ b . To this end the definition of the Fermi-derivative is useful. Let B μ be some contravariant vector-field along LE with tangent vector field eμ = uμ /c and 4-acceleration a μ . Then the Fermi-derivative DF of B μ is defined by DF B μ ≡ Du B μ +

1 1 (uν B ν )a μ − 3 (aν B ν )uμ 3 c c

(5.6.24)

1 μ ν μ B ;ν u = B μ ;ν e(0) . c

(5.6.25)

with Du B μ ≡

Exercise 5.4 Show that the Fermi-derivative has the following properties: μ

(i) DF e(0) = 0.

5.6 Tetrads and Tetrad Induced Coordinates

173

(ii) Let Aμ and B μ be two contravariant vector-fields along LE with DF Aμ = DF B μ = 0 . Then gμν Aμ B ν |LE = const. (iii) Let Aμ be some contravariant vector-field along LE , perpendicular to uμ , then DF Aμ = (Du Aμ )⊥ , where ⊥ denotes the projection of a vector B μ perpendicular to uμ , i.e.,   1 μ B⊥ ≡ δνμ + 2 uμ uν B ν . c Let us now consider the vector-field μ

C μ ≡ Du e(a) |LE . Obviously we can decompose C μ according to     μ μ σ σ e(0) + Cσ e(b) e(b) C μ = − Cσ e(0) i.e., at the observer’s worldline we get μ

ρ

μ

ρ

μ

σ σ e(0) + gρσ (Du e(a) ) e(b) e(b) . Du e(a) = −gρσ (Du e(a) ) e(0)

(5.6.26)

From the orthonormality condition ρ

σ e(a) = 0 gρσ e(0)

we get ρ

ρ

σ σ + gρσ (Du e(0) )e(a) = 0 gρσ (Du e(a) )e(0)

that we can use to rewrite the first term in the right-hand side of (5.6.26). Adding to this equation a vanishing uμ a μ -term we get along LE : μ

DF e(a) =

1 μ

(a)(b) e(b) . c

(5.6.27)

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5 Einstein’s Theory of Gravity

This relation proofs the following: μ

Theorem 5.2 If the tetrad e(α) along LE is Fermi-Walker transported, i.e., μ

DF e(α) = 0 , the Ricci rotation-coefficients vanish, i.e., (a)(b) = 0. Finally let us study the motion of a test-body in free-fall in the vicinity of the observer. Let ZBα (T ) ≡ (cT , Z a ) denote the world-line of this test-body, given by a geodesic of the form α β d 2Za 1 0 dZ α dZ β dZ a a dZ dZ +  , = − αβ dT dT c αβ dT dT dT dT 2

that we will analyze at LE , i.e., for Z a = 0. By taking into account of the corresponding Christoffel-symbols at Xa = 0 this equation in local harmonic proper coordinates takes the form   d 2Z V2 A (5.6.28) + 2( × V) = − 1 − dT 2 c2 with V ≡ dZ/dT . Hence,  describes nothing but a Coriolis-force due to the rotational motion of spatial axes. The term on the right-hand side of (5.6.28) presents the inertial acceleration due to the 4-acceleration of the observer. Exercise 5.5 Show that in local geodetic proper coordinates the geodesic equation at LE takes the form (see also Misner et al. 1973, Exercise (13.14)) d 2Z 2 + 2( × V) = −A + 2 V(V · A) . dT 2 c

(5.6.29)

Local TIC will be called dynamically non-rotating or locally inertial if  = 0. In that case the local reference system will show no inertial forces due to the rotational motion of spatial basis vectors. Technically speaking this means that G0a = 0 for dynamically non-rotating local coordinates. As we have seen the dynamically nonrotating local proper coordinates result from Fermi-transported tetrad vectors.

5.7 Proper Reference Systems of Accelerated Observers Let us start with inertial Minkowskian coordinates x μ = (ct, x) and consider an observer that is moving along the x-axis with constant 4-acceleration, i.e., a μ aμ = −a 0 a 0 + a 1 a 1 = g 2 .

(5.7.1)

5.7 Proper Reference Systems of Accelerated Observers

175

Together with uμ uμ = −c2 or u0 u0 − u1 u1 = c2

(5.7.2)

u0 a 0 − u1 a 1 = 0

(5.7.3)

and uμ aμ = 0, i.e.,

we get g2 =

a1a1 2 a0a0 2 c = c u0 u0 u1 u1

or g=c

a0 a1 = c . u0 u1

(5.7.4)

Thus, a0 =

g du0 = u1 dτ c

du1 g a = = u0 . dτ c

(5.7.5)

1

A special solution of these two differential equations is given by 0 (τ ) = zobs 1 zobs (τ )

c2 sinh α g

c2 cosh α = g

(5.7.6)

with α=

gτ . c

From this we get u0 (τ ) =

0 dzobs = c cosh α dτ

dz1 u (τ ) = obs = c sinh α dτ 1

(5.7.7)

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5 Einstein’s Theory of Gravity

and a 0 (τ ) =

du0 = g sinh α dτ

(5.7.8)

du1 a (τ ) = = g cosh α . dτ 1

Since cosh2 x − sinh2 x = 1 the trajectory of the observer, Lobs , is given by 2 2 − c2 tobs = xobs

c2 , g2

(5.7.9)

i.e., by a hyperbola in our inertial Minkowskian coordinates. Next we construct a local co-moving tetrad field along Lobs . The observer’s 4-velocity reads uμ =

dzobs = c(cosh α, sinh α, 0, 0) dτ

μ

so that the unit vector e(0) in the direction of uμ is given by μ

e(0) =

1 μ u = (cosh α, sinh α, 0, 0) . c

(5.7.10)

The corresponding spatial tetrad vectors, kinematically non-rotating with respect to the original Minkowskian coordinates, can then be chosen according to μ

e(1) = (sinh α, cosh α, 0, 0) μ

e(2) = (0, 0, 1, 0) μ e(3)

(5.7.11)

= (0, 0, 0, 1) .

It is interesting to note that this tetrad field can easily be obtained from the Minkowskian basic vectors at rest: μ

e¯(α) = δαμ .

(5.7.12)

μ

We first write the tetrads e(α) in terms of the observer’s coordinate velocity dzobs dzobs = v= dt dτ



dt dτ

−1

= c · tanh α .

With β≡

v = tanh α ; c

γ ≡ (1 − β 2 )−1/2 = cosh α

(5.7.13)

5.7 Proper Reference Systems of Accelerated Observers

177

we get μ

e(0) = γ (1, β, 0, 0)

(5.7.14)

μ

e(1) = γ (β, 1, 0, 0) . μ

From this we see that the co-moving tetrads can be obtained from e¯(β) by means of a Lorentz-boost: (β) μ

μ

e(α) = (α) e¯(β)

(5.7.15)

with ⎛

(β)

(α)

γ ⎜ γβ =⎜ ⎝0 0

γβ 1 0 0

0 0 1 0

⎞ 0 0⎟ ⎟. 0⎠

(5.7.16)

1

Nest we consider the coordinate transformation from inertial Minkowskian coordinates x μ = (ct, x) to local co-moving coordinates Xα = (cT , X) with T = τ , the proper-time of the observer, with the ansatz μ

x μ (Xα ) = zμ (T ) + e(a) Xa + ξ μ (T , X) ,

(5.7.17)

where ξ μ is at least of second order in |X|. For the Jacobian of this transformation Aμ ν ≡

∂x μ ∂Xν

(5.7.18)

we get μ

μ

A0 = e(0) +

1 d μ a μ e X + ξ,0 c dT (a)

(5.7.19)

μ

μ Aμ a = e(a) + ξ,a .

Since d 0 g 0 e = e(0) ; dT (1) c

d 1 g 1 e = e(0) dT (1) c

μ

A0 can be written in the form μ

μ

μ

A0 =  e(0) + ξ,0

(5.7.20)

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5 Einstein’s Theory of Gravity

with =1+

gX . c2

Now, our original Minkowskian coordinates are both geodetic and harmonic. For the local coordinates the condition of TIC ensures that the local metric tensor, Gαβ , is Minkowskian at the origin, i.e., Gαβ (X = 0) = ηαβ . Higher order terms in |X|, linear, quadratic and higher are not fixed so far. We can fix them by coordinate conditions that we can impose on the local coordinates or we can specify the transformation functions ξ μ . Let us start with μ

ξG = 0 .

(5.7.21)

Then the metric tensor Gαβ in local coordinates according to the tensor transformation rule takes the form: β

G00 = Aα0 A0 ηαβ = −A00 A00 + Ai0 Ai0 = −2 G0a = Aα0 Aβa ηαβ = −A00 A0a + Ai0 Aia = 0

(5.7.22)

β

Gab = Aαa Ab ηαβ = −A0a A0b + Aia Aib = δab or 

gX ds = − 1 + 2 c 2

2 c2 dT 2 + dX2 .

(5.7.23)

Exercise 5.6 Proof that the spatial coordinate lines X = X(λ); T = τ = const. μ are geodesics, i.e., the coordinates (cT , X) defined by ξG = 0 are geodesic proper coordinates. Next we will assume a special form of the local metric tensor   2gX + O(c−4 ) G00 = − 1 + 2 c G0a = 0 Gab

  2gX + O(c−4 ) . = δab 1 − 2 c

(5.7.24)

In that case G ≡ − det(Gαβ ) = 1 −

4gX + O(c−4 ) c2

(5.7.25)

5.8 The Landau-Lifshitz Formulation of GR

179

and √ GGab = δab + O(c−4 ) ,

(5.7.26)

so that these spatial coordinates are harmonic up to terms of order c−4 . Exercise 5.7 Show that the local metric (5.7.24) can be obtained with ξH0 (T , X) = O(c−3 ) ξHa (T , X)



1 i 1 2 a = 2 e(a) (T ) Aa X − X (A · X) + O(c−4 ) 2 c

(5.7.27)

where μ

Aa = ημν e(a)

ν d 2 zobs = (g, 0, 0) . dT 2

(5.7.28)

5.8 The Landau-Lifshitz Formulation of GR 5.8.1 The Landau-Lifshitz Field Equations Landau and Lifshitz (1941, 1971) have derived a special form of the Einstein field equations, which presents a very useful starting point for solving the field equations with perturbative expansions. The atomic variable of the Landau-Lifshitz (LL) formalism is called hαβ , defined by (5.8.13) and the field equation in harmonic gauge are quasi-linear hyperbolic differential equations of the form hαβ = (16π G/c4 ) τ αβ , where  ≡ ημν ∂μν is the flat space d’Alembertian (the flat space wave operator) and τ αβ is the gravitational source tensor, that itself contains hαβ -terms. Under a condition of ‘no incoming gravitational radiation’, the field equations (in harmonic gauge) can formally be solved in terms of retarded integrals (see (5.8.36) below) over quantities involving the atomic variable itself. To derive explicit results for hαβ , the source term τ αβ can be expanded in terms of small quantities as measures of weak gravitational fields, small velocities and small internal stresses. The MPM-formalism discussed in Chap. 7 presents such a scheme, where suitable expansions of τ αβ lead to fully explicit expressions for hαβ , even at high orders in the small parameters. The LL-formalism is based upon the ‘gothic metric’, defined by gαβ ≡

√ −gg αβ ,

(5.8.1)

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5 Einstein’s Theory of Gravity

where g ≡ det(gαβ ). Note, that gαβ is not a tensor but a tensor density. Let gαβ be the inverse of gαβ and g ≡ det(gαβ ). Then, √ g = det(gαβ ) = det( −gg αβ ) = g 2 det(g αβ ) = g .

(5.8.2)

If we take the inverse matrix of g αβ = (−g)−1/2 gαβ we therefore get √ −g gαβ .

(5.8.3)

H αμβν ≡ gαβ gμν − gαν gβμ .

(5.8.4)

gαβ =

√

−g gαβ =

Let us define (e.g., Poisson and Will 2014)

Now, H αμβν has the same properties as the Riemann tensor, H αμβν = −H μαβν ,

H αμβν = −H μανβ

∂μν H αμβν = 2|g|Gαβ +

H αμβν = +H βναμ . 16π G αβ |g|tLL c4

(5.8.5) (5.8.6)

αβ

where Gαβ = R αβ −(1/2)g αβ R is the Einstein tensor and tLL is the Landau-Lifshitz pseudotensor: 16π G 1 αβ αβ αβ αλ βμ λν ρμ |g|tLL = g,λ gλμ ,μ − g,λ g,μ + g gλμ g,ρ g,ν 2 c4 μρ

μρ

βλ αν νρ αλ βμ − (g αλ gμν gβν ,ρ g,λ + g gμν g,ρ g,λ ) + gλμ g g,ν g,ρ

(5.8.7)

1 ρσ + (2g αλ g βμ − g αβ g λμ )(2gνρ gσ τ − gρσ gντ )gντ ,λ g,μ . 8 Using the Einstein field equations for Gαβ in (5.8.6) we get the exact field equations in the form ∂μν H αμβν =

16π G αβ |g|(T αβ + tLL ) . c4

(5.8.8)

From the symmetry relations (5.8.5) we infer ∂βμν H αμβν = 0

(5.8.9)

5.8 The Landau-Lifshitz Formulation of GR

181

or αβ

,β = 0 ,

(5.8.10)

where the Landau-Lifshitz complex αβ is given by αβ

αβ ≡ [|g|(T αβ + tLL )] .

(5.8.11)

Equation (5.8.10) is equivalent to the local equations of motion: T αβ ;β = 0 . Exercise 5.8 At a certain point p ∈ M choose local coordinates such that the first derivatives of the usual covariant metric tensor vanishes at p, i.e., gαβ,γ (p) = 0. αβ Show that in such coordinates at p: |g|tLL = 0 and ∂μν H αμβν = 2|g|Gαβ in accordance with (5.8.6). Details can be found in Poisson & Will (2014).

5.8.2 Harmonic Gauge The formulation of the harmonic gauge is especially simple in terms of the gothic metric: ∂β gαβ = 0

(harmonic gauge) .

(5.8.12)

Let us define hαβ ≡ gαβ − ηαβ .

(5.8.13)

Then the harmonic condition reads: ∂β hαβ = 0 .

(5.8.14)

Exercise 5.9 Show that the following relations hold: 1 1 (−g) = 1 + h + h2 − hαβ hαβ + O(G3 ) 2 2 √ 1 2 1 αβ 1 −g = 1 + h + h − h hαβ + O(G3 ) 2 8 4 1 1 μ gαβ = ηαβ − hαβ + h ηαβ + hαμ hβ − hhαβ 2 2

(5.8.15) (5.8.16)

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5 Einstein’s Theory of Gravity



 1 2 1 μν h − h hμν ηαβ + O(G3 ) , + 8 4 1 1 αβ g αβ = ηαβ + hαβ − h ηαβ − hhβ 2 2   1 2 1 μν h + h hμν ηαβ + O(G3 ) , + 8 4

(5.8.17)

(5.8.18)

where indices on hαβ are lowered and contracted with the Minkowski metric ημν , thus hαβ ≡ ηαμ ηβν hμν and h ≡ ηαβ hαβ . Solution From the definitions we have h = ηαβ hαβ = −h00 + hkk ;

hαβ hαβ = h00 − 2hi0 hi0 + hij hij .

(5.8.19)

Next we compute −g which is a polynomial of 4th order in G. We get 1 1 −g = −det(gαβ ) = −det(ηαβ + hαβ ) = 1 + h + h2 − hαβ hαβ + O(G3 ) = −g 2 2 and with (1 + x)−1/2 = 1 − x/2 + 3x 2 /8 + . . . , relation (5.8.16) follows. g αβ can then be obtained from g αβ = (−g)−1/2 (ηαβ + hαβ ) and it is easy to check that gαβ is inverse to g αβ , i.e., g αβ gβγ = δαγ + O(G3 ). Exercise 5.10 Assume that h00 is of order c−2 , h0i of order c−3 and hij of order c−4 . Show that (Poisson and Will 2014): g00

  1 00 1 00 3 00 2 5 00 3 1 kk 1+ h (5.8.20) = −1 − h − (h ) − (h ) − h 2 8 16 2 2

1 + h0i h0i + O8 2   1 g0i = h0i 1 + h00 + O7 2

1 1 1 gij = δij 1 − h00 − (h00 )2 − hij + δij hkk + O6 2 8 2 (−g) = 1 − h00 + hkk + O6 .

(5.8.21) (5.8.22) (5.8.23) (5.8.24)

In the harmonic gauge we have ∂μν H αμβν = hαβ + hμν ∂μν hαβ − ∂μ hαν ∂ν hβμ

(5.8.25)

where  ≡ ημν ∂μν

(5.8.26)

5.8 The Landau-Lifshitz Formulation of GR

183

is the flat-space d’Alembertian. Then the field equations take the form hαβ + hμν ∂μν hαβ − ∂μ hαν ∂ν hβμ =

16π G αβ |g|(T αβ + tLL ) c4

(5.8.27)

or hαβ =

16π G αβ τ c4

(5.8.28)

with αβ

αβ

τ αβ = |g|(T αβ + tLL + tH )

(5.8.29)

c4 (∂μ hαν ∂ν hβμ − hμν ∂μν hαβ ) . 16π G

(5.8.30)

and αβ

|g|tH =

Usually the Einstein field equations in harmonic gauge are written in the form hαβ =

16π G |g|T αβ + αβ c4

(5.8.31)

where the ‘gravitational source term’ αβ reads αβ ≡

16π G αβ βμ μν αβ |g|tLL + hαν ,μ h,ν − h h,μν c4

or, written out 1 αβ βμ μν αβ λν ρμ αβ = hαν ,μ h,ν − h h,μν + g gλμ h,ρ h,ν 2 μρ

μρ

βλ αν νρ αλ βμ − (g αλ gμν hβν ,ρ h,λ + g gμν h,ρ h,λ ) + gλμ g h,ν h,ρ

(5.8.32)

1 ρσ + (2g αλ g βμ − g αβ g λμ )(2gνρ gσ τ − gρσ gντ )hντ ,λ h,μ . 8 We see that the gravitational source terms contains products of the metric tensor that are at least quadratic in h and first and second derivatives. We write in obvious notation αβ

αβ

αβ = 2 [h, h] + 3 [h, h, h] + O(h4 )

(5.8.33)

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5 Einstein’s Theory of Gravity

with 1 1 αβ 2 = − hρσ ∂ρσ hαβ + ∂ α hρσ ∂ β hρσ − ∂ α h∂ β h + ∂σ hαρ (∂ σ hβρ + ∂ρ hβσ ) 2 4

1 1 1 − 2∂ (α hρσ ∂ ρ hβ)σ + ηαβ − ∂τ hρσ ∂ τ hρσ + ∂ρ h∂ ρ h + ∂ρ hσ τ ∂ σ hρτ . 4 8 2 (5.8.34)

All indices are lowered and raised with the Minkowski metric ημν ; h ≡ ηαβ hαβ ; the αβ parenthesis around indices indicate symmetrization. Explicit expressions for 3 αβ and 4 can be found in Blanchet and Faye (2001a). Under certain assumptions the field equations (5.8.28) can formally be solved. Usually one imposes some ‘no incoming radiation’ condition of the form ∂ αβ [h (t, x)] = 0 ∂t

for t ≤ −T0 .

(5.8.35)

Under this condition equation (5.8.28) is formally solved by hαβ (t, x) =

16π G −1 αβ R τ c4

(5.8.36)

d 3x f (tR , x ) |x − x |

(5.8.37)

with (−1 R f )(t, x)

1 ≡− 4π

where the retarded time tR is given by tR ≡ t −

|x − x | . c

(5.8.38)

Chapter 6

Exact Solutions—Field Moments

Exact solutions of EFE do not play a central role for the field of Applied General Relativity (if we exclude the field of relativistic astrophysics and cosmology); nevertheless they might serve as a guide to understand certain aspects of practical systems where gravity plays a role and as assistance for the construction of approximative formalisms. A huge number of exact solutions of EFE have been found; the reader is referred to the standard literature (e.g., Stephani et al. 2003; Griffiths and Podolsky 2009).

6.1 Minkowskian Space-Time Clearly the simplest vacuum solution of Einstein’s field equations is the Minkowski space-time, where in Minkowskian coordinates the metric takes the form gμν = ημν = diag(−1, +1, +1, +1) .

(6.1.1)

∇(ν ξμ) = 0 ,

(6.1.2)

The Killing equation

then reads ξμ,ν + ξν,μ = 0 with a solution of the form ξμ = aμ + bμν x ν

© Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_6

(6.1.3)

185

186

6 Exact Solutions—Field Moments

with bμν = −bνμ . This leads to a total of 10 independent Killing vector fields. a μ describes 4 independent infinitesimal space-time translations. bij -terms describe an infinitesimal rotation about the k axis with ij k = 0. E.g., for b12 = −b21 = 1 the Killing vector field has spatial components Rz = (y, −x, 0) that describes an infinitesimal rotation about the z-axis, since the transformation x  = x cos + y sin ,

y  = −x sin + y cos

to first order in can be written as x = x + Rz . We will now show that the remaining three Killing transformations induced by b0i are equivalent to three Lorentz-boosts of the form ct  = γ (ct − βx) ;

x  = γ (x − βct)

or, equivalently ct  = cosh α · ct − sinh α · x ;

x  = − sinh α · ct + cosh α · x

(6.1.4)

where we wrote β = tanh α so that γ = cosh α. Let us argue now in a 2-dimensional space-time with coordinates x μ = (x 0 , x). In the remaining part of this subsection we write x for (x 0 , x). Choosing b01 = −b10 = 1 we get ξ μ = −(x, x 0 ) so that the infinitesimal Killing transformation reads: ˆ x = x + ξ = x − Dx

(6.1.5)

with Dˆ =



01 10

 .

(6.1.6)

For a non-infinitesimal transformation we replace by α and write 

 ∞ m  (−1) m m ˆ ·x= · x. x = exp(−α D) α Dˆ m! 

m=0

(6.1.7)

6.2 Stationary Space-Times

187

Now, each even power of Dˆ equals the unit matrix, whereas each odd power of Dˆ equals Dˆ itself. Therefore, 

x =−

∞  m=0

1 α 2m+1 (2m + 1)!

= − sinh α





   ∞  1 01 2m 1 0 x+ x α 10 01 (2m)!



01 x + cosh α 10





m=0

(6.1.8)

10 x 01

which is equivalent to (6.1.4). Since there can be at most n(n + 1)/2 independent Killing vector-fields the Minkowski space-time is maximally symmetric (e.g., Weinberg 1972).

6.2 Stationary Space-Times A space-time manifold is called stationary if there is a time-like Killing vector-field. Consider a bundle of time-like curves x μ (λ) that is parametrized with a suitably chosen time coordinate λ = x 0 so that its tangent vector-field takes the form ξ μ = (1, 0, 0, 0). Then, [Lξ g]μν =

∂gμν 1 ∂gμν dx σ gμν,σ = = . dλ ∂λ c ∂t

Thus if a space-time is stationary and has a time-like Killing vector-field satisfying the Killing equation (6.1.2) the time coordinate t can be chosen such that the components of the metric tensor are independent of t. The Killing vector field ξ μ then defines a quantity f , f = −ξμ ξ μ = −gμν ξ μ ξ ν > 0 ,

(6.2.1)

the (positive) norm of ξ μ . Moreover, the twist 4-vector ωμ of ξ μ is defined by σ . ωμ = μνλσ ξ ν ∇ λ ξ σ = μνλσ ξ ν g λρ ξ;ρ

(6.2.2)

The norm of the Killing vector field is related with redshift effects between selected clocks. The twist vector measures the extent to which the Killing vector field fails to be orthogonal to a family of 3-surfaces. Lemma 6.1 One finds that ∇[μ ων] = μνλσ ξ λ Rκσ ξ κ .

(6.2.3)

188

6 Exact Solutions—Field Moments

Proof we first re-write ∇[μ ων] as 1 ∇[μ ων] = − μνλσ λσ αβ ∇α ωβ , 4

(6.2.4)

since ∇α μνλσ = 0. Using (6.2.2) we then get 1 ∇[μ ων] = − μνλσ [ λσ αβ βγρτ ∇α ξ γ ∇ ρ ξ τ ] 4 1 = − μνλσ [6∇α (ξ [α ∇ λ ξ σ ] )] 4

1 α λ σ 3 (ξ ∇ ξ + ξ σ ∇ α ξ λ + ξ λ ∇ σ ξ α ) = − μνλσ ∇α 2 3 1 = − μνλσ [∇α ξ α ∇ λ ξ σ + ξ α ∇α ∇ λ ξ σ + ∇α (ξ σ ∇ α ξ λ + ξ λ ∇ σ ξ α )] 2 1 = − μνλσ [ξα R λσαβ ξ β + ∇α ξ σ ∇ α ξ λ + ∇α ξ λ ∇ σ ξ α 2 + ξ σ ∇α ∇ α ξ λ + ξ λ ∇α ∇ σ ξ α )] 1 = − μνλσ (2ξ [σ ∇α ∇ α ξ λ] ) 2 = − μνλσ ξ [σ R λ]α ξ α = μνλσ ξ λ R σα ξ α .

(6.2.5)

The first term of the forth line disappears because of the Killing equation; for the second term of line 4, by using Eq. (2.11.8) we get the first term of line 5 that also disappears because of the antisymmetry of the Riemann tensor; the summation of the second and third terms in line 5 is zero due to the Killing equation; then using Eq. (2.11.8) again we get the final result. Thus in the vacuum region where Rκσ = 0 the twist vector admits a potential ω, ωμ = ∇μ ω ,

(6.2.6)

called the twist-potential. Relations (6.1.2), (6.2.1) and (6.2.2) are the basis of a projection formalism (e.g., Geroch 1971) that relates the differential geometrical structure of the 4-dimensional space-time (M, g) with a corresponding structure in a 3-space M(3) . This 3-space can be chosen as the collection of all trajectories of ξ μ ; that is, an element of M(3) is a curve in M which is everywhere tangent to ξ μ . If ξ μ is hypersurface orthogonal (see below) and ω = 0, then M(3) might be represented by a t = const. hypersurface.

6.2 Stationary Space-Times

189

The metric of a stationary space-time in adapted coordinates can be written in the form (e.g., Israel and Wilson 1972; Kinnersley 1973) ds 2 = −f (c dt + wi dx i )2 + f −1 hij dx i dx j

(6.2.7)

where i, j = 1, 2, 3 and the metric functions f, wi , hij are independent of t. The quantities hij from (6.2.7) define a metric tensor on M(3) . In general, e.g., in non adapted coordinates, there will be a metric tensor hμν = f gμν + ξμ ξν

(6.2.8)

on M(3) with a corresponding covariant derivative denoted by Dμ . In adapted coordinates where ξ μ = (1, 0, 0, 0) the metric of 3-space takes the form hij = −g00 gij + g0i g0j .

(6.2.9)

Let us denote the covariant derivative and Ricci tensor in (M, g) by ∇μ and Rμν and the corresponding quantities in (M(3) , h) by Di and Rij(3) . The Laplacian in (M(3) , h) will be denoted by D 2 ≡ Di D i .

(6.2.10)

The 3-vector w in (6.2.7) is related with the spatial part of the twist 4-vector field by (e.g., Kinnersley 1973) ω = f 2D × w .

(6.2.11)

This relation can be verified by direct calculation in adapted coordinates where ξ μ = (1, 0, 0, 0). The field equations can then be written as (e.g., Tanabe 1976; Bäckdahl and Herberthson 2005; Bäckdahl 2008, Eqs. (11)) f D 2 f = Di f D i f − Di ωD i ω

(6.2.12)

f D ω = 2Di f D ω

(6.2.13)

2

i

(3)

2f 2 Rij = (Di f )(Dj f ) + (Di ω)(Dj ω) .

(6.2.14)

The proof can basically be found in Appendix of Geroch (1978) with a slight modification of hij (called h˜ ij in this paper). Two basic equations are involved in the proof: ∇α ξ β =

1 −1 f αβγ δ ξ γ ωδ + f −1 ξ[β Dα] f 2

∇α ∇β ξγ = Rδαβγ δ ξ δ

(6.2.15) (6.2.16)

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6 Exact Solutions—Field Moments

The first of these equations can be best shown by direct calculation in adapted coordinates where the Killing vector field ξ α = (1, 0, 0, 0) and ξα = −(f, f ωi ), and the second one is left as an Exercise (see below). Applying D 2 to (6.2.1) and using the above two equations we get the field equation (6.2.12) for f . Taking the divergence of (6.2.2), reminding that ωμ = ∇μ ω and using the above two equations again, we get the field equation for the twist potential ω. Of great interest is the static case where ω = 0. With f = e2ψ the vacuum field equations take the form D i Di ψ = 0

(6.2.17)

Rij(3)

(6.2.18)

= 2(Di ψ)(Dj ψ) .

The first field equation (6.2.17) is the (covariant) Laplace equation for the potential (3) ψ. In the Newtonian limit f = −g00 = 1 − 2U/c2 , so that ψ = −U/c2 . Rij is of order 1/c4 . Exercise 6.1 Proof the relation ξσ ;λν = Rμνλσ ξ μ

(6.2.19)

that is used for the derivation of the field equation (6.2.12) (Geroch 1978, (A13)). Proof We use the Killing equation ξσ ;λ = −ξλ;σ , the definition of the curvature tensor ξσ ;λν − ξσ ;νλ = −Rμσ νλ ξ μ and the first Bianchi-identities Rμσ νλ + Rμλσ ν + Rμνλσ = 0 to rewrite the left hand side of (6.2.19): ξσ ;λν = −Rμσ νλ ξ μ + ξσ ;νλ = −Rμσ νλ ξ μ − ξν;σ λ = −Rμσ νλ ξ μ − Rμνσ λ ξ μ + ξλ;νσ = −(Rμσ νλ + Rμνσ λ + Rμλσ ν )ξ μ − ξσ ;λν so that 2ξσ ;λν = −(Rμσ νλ + Rμνσ λ + Rμλσ ν )ξ μ = 2Rμνλσ ξ μ .

6.2 Stationary Space-Times

191

If the 3-metric hij is given the complete 4-dimensional space-time is determined by a single complex valued potential E, E ≡ f + iω ,

(6.2.20)

called the Ernst-potential (Ernst 1968a,b; Israel and Wilson 1972; Kinnersley 1973). With the Ernst potential the field equations for f and ω take the form f D 2 E = Di E D i E

(6.2.21)

with f = (E) =

1 (E + E ∗ ) . 2

(6.2.22)

Equation (6.2.21) is called Ernst-equation. Often, instead of the Ernst potential E a potential ξ with ξ≡

1+E , 1−E

(6.2.23)

E=

ξ −1 , ξ +1

(6.2.24)

so that

is introduced. Then the vacuum field equations take the form (e.g., Hoenselaers and Perjés 1980) θ D 2 ξ = 2ξ ∗ Di ξ D i ξ

(6.2.25)

θ 2 Rij = Di ξ Dj ξ ∗ + Di ξ ∗ Dj ξ = 2(Di ξ Dj ξ ∗ ) (3)

(6.2.26)

with θ ≡ ξ ∗ξ − 1 .

(6.2.27)

The derivation of the above equations is straightforward. Expressing θ and Di ξ in terms of f and ω, one gets θ=

4f , (1 − E)(1 − E ∗ )

Di ξ =

2Di E , (1 − E)2

D2ξ =

2D 2 E(1 − E) + 4Di ED i E . (1 − E)3

192

6 Exact Solutions—Field Moments

Using these relations in (6.2.25) and (6.2.26) we recover the field equations in the form (6.2.12)–(6.2.14). For example, (6.2.25) becomes f [(1 − E)D 2 E + 2Di ED i E] = (1 + E ∗ )Di ED i E , from which we get (6.2.21). The right hand side of (6.2.26) reads Di

8[(Di f )(Dj f ) + (Di ω)(Dj ω)] 1+E 1 + E∗ 1 + E∗ 1+E Dj = + D Dj , i ∗ ∗ 1−E 1−E 1−E 1−E (1 − E)2 (1 − E ∗ )2

from which we get (6.2.14) with the above expression for θ .

6.2.1 Stationary Axially Symmetric Space-Times A stationary space-time is called axially symmetric if it has a space-like Killing vector-field ημ with closed integral curves around some symmetry axis. In canonical Weyl coordinates (t, ρ, z, φ) ((ρ, z) are a special kind of cylindrical coordinates) the metric can be written in Weyl-Lewis-Papapetrou form (e.g., Papapetrou 1953) as:   ds 2 = −f (c dt − W dφ)2 + f −1 e2γ (dρ 2 + dz2 ) + ρ 2 dφ 2 ,

(6.2.28)

where f = f (ρ, z), W = W (ρ, z), γ = γ (ρ, z). The vacuum field equations can be divided into primary and secondary equations. The primary equations take the form 2 + W,z2 ) = 0 f (f,ρρ + ρ −1 f,ρ + f,zz ) − f,ρ2 − f,z2 + ρ −2 f 4 (W,ρ

(6.2.29)

(ρ −1 f 2 W,ρ ),ρ + (ρ −1 f 2 W,z ),z = 0 . (6.2.30) The secondary equations read 1 −2 2 1 2 ρf (f,ρ − f,z2 ) − ρ −1 f 2 (W,ρ − W,z2 ) , 4 4 1 1 γ,z = ρf −2 f,ρ f,z − ρ −1 f 2 W,ρ W,z . 2 2

γ,ρ =

(6.2.31) (6.2.32)

The above equations can be derived from the vacuum field equation Rμν = 0 using the definition of the Ricci tensor. For example the 00-component of Rμν reads R00

  2 + W2) c2 ρ 2 e−2γ f (f,ρρ + ρ −1 f,ρ + f,zz ) − f,ρ2 − f,z2 + ρ −2 f 4 (W,ρ ,z   = , 2 3f 2 W 2 + ρ 2 (6.2.33)

6.2 Stationary Space-Times

193

so that the vacuum relation R00 = 0 reduces to (6.2.29). Similarly, one derives the remaining three vacuum field equations. Exercise 6.2 Prove the remaining field equations (6.2.30), (6.2.31) and (6.2.32) from Rμν = 0. The secondary equations suggest an over-determination of γ . However, taking the partial derivative of (6.2.31) with respect to z and the derivative of (6.2.32) with respect to ρ we get γ,ρz = γ,zρ by using the primary equations. Therefore γ can be solved by quadrature if the functions f and W have been found from the primary equations (e.g., Wald 1984; Islam 1985):

b

γ (b) − γ (a) =

  γ,ρ dρ + γ,z dz .

(6.2.34)

a

As a result, finding the stationary and axisymmetric vacuum solutions of Einstein’s field equations reduces to first solving the primary equations for f and W in ordinary 3D Euclidean space, and then the function γ is obtained from (6.2.34). Clearly, this greatly simplifies the problem of solving the original Einstein field equation for the ten unknown components of gμν . Equation (6.2.30) implies the existence of a potential ω(ρ, z) such that ω,ρ = +ρ −1 f 2 W,z

ω,z = −ρ −1 f 2 W,ρ .

(6.2.35)

This potential ω agrees with the scalar twist potential of the metric (6.2.28). Relation (6.2.35) is often written as ˜ ∇ω = ρ −1 f 2 ∇W

(6.2.36)

with ∇ = (∂ρ , ∂z ) and ∇˜ = (∂z , −∂ρ ). Another way of writing this relation is (Ernst 1968a,b) ∇ω = ρ −1 f 2 nˆ φ × ∇W ,

(6.2.37)

where nˆ φ is a unit vector in azimuthal direction. This is because of nˆ φ × nˆ ρ = −nˆ z and nˆ φ × nˆ z = nˆ ρ . Exercise 6.3 Show that the potential ω from (6.2.35) agrees with the twist potential of the metric (6.2.28). Considering that in cylindrical coordinates =

1 ∂ ρ ∂ρ

  ∂ ∂2 1 ∂2 ρ + 2+ 2 2, ∂ρ ∂z ρ ∂φ

194

6 Exact Solutions—Field Moments

the two primary field equations can then be written in the form 2 2 f f − f,ρ2 − f,z2 + ω,ρ + ω,z =0

f ω − 2f,ρ ω,ρ − 2f,z ω,z = 0

(6.2.38)

and the secondary equations can be written with the Ernst-potential, E ≡ f + iω, as 1 −2 ∗ ∗ ρf (E,ρ E,ρ − E,z E,z ), 4 1 ∗ ∗ γ,z = ρf −2 (E,ρ E,z + E,z E,ρ ). 4

γ,ρ =

(6.2.39)

Often, instead of canonical Weyl coordinates (ρ, z), Weyl spherical or Weyl prolate spheroidal coordinates are employed. Weyl spherical coordinates (r, θ ) are defined as r = (ρ 2 + z2 )1/2 ;

cos θ = z/r .

(6.2.40)

Prolate spheroidal coordinates (PS-coordinates) (μ, ν, ϕ) are defined by x = σ sinh μ sin ν cos ϕ y = σ sinh μ sin ν sin ϕ

(6.2.41)

z = σ cosh μ cos ν , where σ is a constant with the dimension of a length (for the so-called Schwarzschild metric the constant σ will be identifies with m = GM/c2 ; see below). μ is a non-negative real number, ν ∈ [0, π ] and ϕ ∈ [0, 2π ]. Surfaces of constant μ are prolate spheroids, ν = const. surfaces are hyperboloids of revolution with focal points x± = (0, 0, ±σ ) (see Fig. 6.1). Distances from the two focal points are given by (ρ 2 = x 2 + y 2 ): r± ≡



ρ 2 + (z ± σ )2 = σ (cosh μ ± cos ν) .

(6.2.42)

Exercise 6.4 Show that r± = σ (cosh μ ± cos ν). Solution The result is directly obtained with sinh2 μ = cosh2 μ − 1. Often, instead of (μ, ν, ϕ) an alternative set of prolate spheroidal coordinates (ζ, τ, φ) is used, where r+ + r− 2σ r+ − r− τ ≡ 2σ ϕ ≡ arctan(y/x) . ζ ≡

(6.2.43) (6.2.44)

6.2 Stationary Space-Times

195

Fig. 6.1 Lines of constant values of prolate spheroidal coordinates μ and ν for σ = 1. The red curves refer to constant values of μ (μ = 0, 0.2, 0.4, · · · , 1.4) from the center outwards; the blue ones to constant values of ν (ν = 0◦ , 10◦ , 20◦ , 30◦ , · · · , 180◦ ) from top to bottom

There is a unique relation between such PS-coordinates and Cartesian ones:  x = σ (ζ 2 − 1)(1 − τ 2 ) cos ϕ  y = σ (ζ 2 − 1)(1 − τ 2 ) sin ϕ

(6.2.45)

z = σζτ . In the following, following the notation of the standard literature, we will denote the PS-coordinates (ζ, τ ) by (x, y) having in mind that in the following part x and y are not Cartesian coordinates. We now use the relations between the canonical Weyl coordinates (ρ, z) and PS-coordinates (x, y):  ρ = σ (x 2 − 1)(1 − y 2 ) , z = σ xy ,

196

6 Exact Solutions—Field Moments

to derive the relation 

 1 − y2 2 2 x2 − 1 2 2 2 2 2 2 x dx + y dy + x dy + y dx x2 − 1 1 − y2   dy 2 dx 2 + = σ 2 [y 2 (x 2 − 1) + x 2 (1 − y 2 )] x2 − 1 1 − y2   dy 2 dx 2 . (6.2.46) + = σ 2 (x 2 − y 2 ) x2 − 1 1 − y2

dρ 2 + dz2 = σ 2

So in PS-coordinates (t, x, y, φ) the stationary axisymmetric vacuum Weyl-LewisPapapetrou metric takes the form ds 2 = − f (c dt − W dφ)2 (6.2.47) 

 dy 2 dx 2 2 −1 2γ 2 2 2 2 2 e (x − y ) + (x − 1)(1 − y )dφ , + +σ f x2 − 1 1 − y2 where σ is a constant. In the Ernst equation (6.2.21), (E)D 2 E = Di E D i E , the Laplacian is now the usual Laplacian in Euclidian 3-space and D the usual gradient operator. In canonical Weyl coordinates E = E,ρρ + (1/ρ)E,ρ + E,zz ∇E · ∇E = (E,ρ )2 + (E,z )2 .

(6.2.48)

and in PS-coordinates   1 2 2 ∂ E (x − 1)∂ + ∂ (1 − y )∂ x x y y σ 2 (x 2 − y 2 )   1 2 2 2 2 (x . ∇E · ∇E = 2 2 − 1)(E ) + (1 − y )(E ) ,x ,y σ (x − y 2 ) E =

(6.2.49) (6.2.50)

Exercise 6.5 Derive relations (6.2.49) and (6.2.50) that appear in the Ernst equation. Proof E.g., ∂ρ ∂ ∂z ∂ ∂ = + , ∂x ∂x ∂ρ ∂x ∂z

∂ ∂ρ ∂ ∂z ∂ = + , ∂y ∂y ∂ρ ∂y ∂z

∂2 ∂ 2ρ ∂ ∂z 2 ∂ 2 ∂ 2z ∂ ∂ρ 2 ∂ 2 = + , + + ∂x ∂z2 ∂x 2 ∂x 2 ∂ρ ∂x 2 ∂z ∂x ∂ρ 2

6.2 Stationary Space-Times

with

197



 ∂ρ 1 − y2 x2 − 1 = −σy , , 2 ∂y x −1 1 − y2   ∂ 2ρ 1 − y2 x2 − 1 ∂ 2ρ = −σ , = σ , ∂x 2 (x 2 − 1)3 ∂y 2 (1 − y 2 )3 ∂ρ = σx ∂x

∂z = σy , ∂x

∂z = σx , ∂y

∂ 2z ∂ 2z = = 0. ∂x 2 ∂y 2

Corresponding relations involving derivatives with respect to y can be derived. Then, e.g.,   ∂x (x 2 − 1)∂x + ∂y (1 − y 2 )∂y E = 2 2 ∂ 2ρ 2 2 ∂ ρ ∂ E (x − 1) 2 + (1 − y ) 2 + ∂x ∂y ∂ρ 2

2 ∂ 2ρ ∂ρ ∂ρ ∂E 2 2 ∂ ρ (x − 1) 2 + 2x + (1 − y ) 2 − 2y + ∂x ∂y ∂ρ ∂x ∂y 2 2 ∂ 2z 2 2 ∂ z ∂ E (x − 1) 2 + (1 − y ) 2 ∂x ∂y ∂z2   = σ 2 (x 2 − y 2 ) E,ρρ + (1/ρ)E,ρ + E,zz .

(6.2.51)

With the Ernst ξ -potential ξ=

1+E 1−E

the Ernst equation (6.2.25) takes the form (ξ ξ ∗ − 1)ξ = 2ξ ∗ ∇ξ · ∇ξ .

(6.2.52)

We notice that a useful feature of (6.2.49) and (6.2.50) is that both operators appearing there are symmetric under the interchange of the PS-coordinates x and y. Consequently, if ξ(x, y) is a solution of Eq. (6.2.52), then so is ξ(y, x). In this way, when ξ = x is a solution (the Schwarzschild solution, to be discussed later), then ξ = y is a new solution of the vacuum field equations. Therefore, a linear combination of the solutions ξ = x and ξ = y also satisfies (6.2.52). In this way one obtains an important solution to (6.2.52) (Ernst 1968a,b) in the form: ξ = x cos λ − iy sin λ ,

(6.2.53)

where λ is a constant. This is the ξ -potential of the Kerr metric to be discussed later.

198

6 Exact Solutions—Field Moments

Exercise 6.6 Show that ξ = x and ξ = y, where x and y are PS-coordinates are solutions to the vacuum field equations (6.2.52).

6.2.2 The Hartle-Thorne Metric The Hartle-Thorne metric is an ‘exact’ solution of vacuum Einstein field equations that describes the exterior of any slowly and rigidly rotating, stationary and axially symmetric body (Abramowicz et al. 2003). The metric is given with accuracy including second order terms in the dimensionless angular momentum parameter j ≡ S/(Mmc) (where m ≡ GM/c2 ) and terms to first order in the dimensionless quadrupole parameter q = −Q/(Mm2 ) (Hartle 1967; Hartle and Sharp 1967; Hartle and Thorne 1968). In spherical coordinates (ct, r, θ, φ) the Hartle-Thorne metric reads: ds 2 = gtt c2 dt 2 +grr dr 2 +gθθ dθ 2 +gφφ dφ 2 +gφt dφcdt +gtφ cdt dφ

(6.2.54)

with: gtt = −(1 − 2m/r)[1 + j 2 F1 + qF2 ] grr = (1 − 2m/r)−1 [1 + j 2 G1 − qF2 ] gθθ = r 2 [1 + j 2 H1 + qH2 ]

(6.2.55)

gφφ = r sin θ [1 + j H1 + qH2 ] 2

2

2

gtφ = gφt = 2(m2 /r)j sin2 θ and (u = cos θ ) F1 = [8mr 4 (r − 2m)]−1 × [u2 (48m6 − 8m5 r − 24m4 r 2 − 30m3 r 3 − 60m2 r 4 + 135mr 5 − 45r 6 ) +(r − m)(16m5 + 8m4 r − 10m2 r 3 − 30mr 4 + 15r 5 )] + A1 (r) F2 = [8mr(r − 2m)]−1 (5(3u2 − 1)(r − m)(2m2 + 6mr − 3r 2 )) − A1 (r) G1 = [8mr 4 (r − 2m)]−1 ((L − 72m5 r − 3u2 (L − 56m5 r)) − A1 (r) L = 80m6 + 8m4 r 2 + 10m3 r 3 + 20m2 r 4 − 45mr 5 + 15r 6   15r(2 − 2m)(1 − 3u2 ) r A1 = ln r − 2m 16m2 H1 = (8mr 4 )−1 (1 − 3u2 )(16m5 + 8m4 r − 10m2 r 3 + 15mr 4 + 15r 5 ) + A2 (r) H2 = 8mr −1 (5(1 − 3u2 )(2m2 − 3mr − 3r 2 )) − A2 (r)   15(r 2 − 2m2 )(3u2 − 1) r . ln A2 = r − 2m 16m2

(6.2.56)

6.2 Stationary Space-Times

199

For j = q = 0 the Hartle-Thorne metric reduces to the Schwarzschild metric in standard coordinates. Abramowicz et al. (2003) have shown how to get the Kerr metric in Boyer-Lindquist coordinates (see below) to the corresponding accuracy. The Hartle-Thorne metric can be used to describe the exterior gravitational field of rotating neutron stars. Bauböck et al. (2013) gave some empirical fitting for neutron-star parameters that appear in this metric.

6.2.3 Static Axially Symmetric Space-Times A stationary space-time is called static, if the twist-vector ωμ = 0. In that case the space-like Killing vector field is called hypersurface orthogonal. A foliation of space-time into slices of space-like hypersurfaces is given by some scalar function S(x), x ∈ M, that serves as label; a certain space-like hypersurface is given by s = {x|S(x) = s = const.} . A vector-field Vα is said to be hypersurface orthogonal, if it is proportional to the gradient of some scalar function S(x), i.e., if Vα = g(x) · S,α .

(6.2.57)

Then, Vα;β − Vβ;α = g,β S,α − g,α S,β so that V[α;β] is proportional to Vα or Vβ and, consequently, [μνλσ ]Vν;λ Vσ = 0 . Thus a stationary space-time is static if the timelike KVF (Killing vector field) is hypersurface orthogonal. The metric of a static axially symmetric space-time is conveniently written with canonical Weyl coordinates (ct, ρ, z, φ) with f = e2ψ in the form   ds 2 = −e2ψ (c dt)2 + e−2ψ e2γ (dρ 2 + dz2 ) + ρ 2 dφ 2 .

(6.2.58)

200

6 Exact Solutions—Field Moments

We see that ρ, z are a kind of cylindrical coordinates, called Weyl cylindrical coordinates. The potential ψ obey the vacuum field equation ψ,ρρ + ρ −1 ψ,ρ + ψ,zz = 0 ,

(6.2.59)

which is nothing but the Laplace equation in flat space, ψ = 0, in cylindrical coordinates with  = ∂ρρ + ρ −1 ∂ρ + ∂zz . Taking W = 0 and f = −e2ψ in Eqs. (6.2.31), (6.2.32), the potential γ = γ (ρ, z) is determined from the field equations 2 2 − ψ,z ) γ,ρ = ρ(ψ,ρ

(6.2.60)

γ,z = 2ρψ,ρ ψ,z .

(6.2.61)

Again taking W = 0 and f = −e2ψ in the metric (6.2.47), correspondingly in prolate spheroidal coordinates we have: (6.2.62) ds 2 = −e2ψ (cdt)2 + σ 2 e−2ψ 

 dy 2 dx 2 + (x 2 − 1)(1 − y 2 )dφ 2 . + × e2γ (x 2 − y 2 ) 2 x − 1 1 − y2 To get the Schwarzschild metric (see below) with x = r/m − 1 and y = cos θ in usual Schwarzschild coordinates we write σ = m. From Eq. (6.2.49), the field equation for ψ takes the form (e.g., Quevedo and Mashhoon 1985) [(x 2 − 1)ψ,x ],x + [(1 − y 2 )ψ,y ],y = 0 ,

(6.2.63)

which can be solved by separation of variables, ψ(x, y) = F (x)G(y) ,

(6.2.64)

so that (6.2.63) reduces to the Legendre equations for F and G [(x 2 −1)F,x ],x −ν(ν+1)F = 0 ,

[(1−y 2 )G,y ],y +ν(ν+1)G = 0 ,

(6.2.65)

where ν is a constant. To avoid logarithmic singularities at y = ±1, ν must be an integer and since ψ = 0 for x → ∞ the solution for ψ takes the form ∞  ψ= (−1)n+1 qn Qn (x)Pn (y) , n=0

(6.2.66)

6.2 Stationary Space-Times

201

where Pn (y) are Legendre-polynomials of the first kind and Ql (x) are Legendre functions of the second kind for x ≥ 1 P0 (y) = 1 P1 (y) = y P2 (y) =

1 (3y 2 − 1) 2

  x+1 1 Q0 (x) = ln 2 x−1 x Q1 (x) = ln((x + 1)/(x − 1)) − 1 2 Q2 (x) =

3x 2 − 1 3 ln((x + 1)/(x − 1)) − x . 4 2

(6.2.67)

The field equations for the potential γ read: γ,x =

1 − y2 2 2 [x(x 2 − 1)ψ,x − x(1 − y 2 )ψ,y − 2y(x 2 − 1)ψ,x ψ,y ] , x2 − y2

γ,y =

x2 − 1 2 2 [y(x 2 − 1)ψ,x − y(1 − y 2 )ψ,y + 2x(1 − y 2 )ψ,x ψ,y ] . (6.2.68) x2 − y2

Exercise 6.7 Prove the above field equations for γ . Solution Considering γ,x = γ,ρ ρ,x + γ,z z,x , γ,y = γ,ρ ρ,y + γ,z z,y , then use Eqs. (6.2.31), (6.2.32) and E,ρ = (E,x z,y − E,y z,x )/(ρ,x z,y − ρ,y z,x ) E,z = (E,x ρ,y − E,y ρ,x )/(ρ,y z,x − ρ,x z,y ) (6.2.69) with E = f = e2ψ , we can directly get the above field equations for γ in PScoordinates. Lemma 6.2 (Quevedo 1989) Let ψ be asymptotically flat, i.e., lim ψ(x, y) = 0

x→∞

and vanish on the symmetry axis, i.e., γ (x, ±1) = 0, then

γ (x, y) = (x 2 − 1)

y −1

A(x, y) dy , x2 − y2

(6.2.70)

202

6 Exact Solutions—Field Moments

with 2 2 − y(1 − y 2 )ψ,y + 2x(1 − y 2 )ψ,x ψ,y . A(x, y) = y(x 2 − 1)ψ,x

(6.2.71)

In Weyl spherical coordinates the general solution for ψ reads ψ=

∞  al Pl (cos θ ) . l+1 r

(6.2.72)

l=0

Where the (spherical) Weyl-moments al are constant numbers. Then the potential γ is given by γ =

∞  (l + 1)(k + 1) al ak (Pl+1 Pk+1 − Pl Pk ) . l + k + 2 r l+k+2

(6.2.73)

l,k=0

Lemma 6.3 Lemma (Hernández-Pastora and Martin 1993) The spherical and prolate spheroidal Weyl moments, an and qn are related by; an =

T  (−m)n+1 j =0

n! qk (n + k + 1)!!(n − k)!!

(6.2.74)

with k = 2j and T = n/2 for even values of n and k = 2j + 1, T = (n − 1)/2 for odd values of n. If ψ has equatorial symmetry than it has only even Weyl moments, i.e., q2n+1 = a2n+1 = 0.

6.2.3.1

The Schwarzschild Metric

The simplest case for a static metric in prolate spheroidal coordinates is q0 = 1 with (P0 (y) = 1) ψ = −Q0 (x) =

1 x−1 ln . 2 x+1

(6.2.75)

The potential γ takes the form γ =

1 x2 − 1 ln 2 . 2 x − y2

(6.2.76)

6.2 Stationary Space-Times

203

Taking ψ, γ in (6.2.62) and setting σ = m, so that the Schwarzschild metric in prolate spheroidal coordinates reads

x−1 x2 − 1 2 2 2x + 1 2 2 2 2 (cdt) + m dx + dy + (x − 1)(1 − y )dφ ds = − x+1 x−1 1 − y2 (6.2.77) 2

or, using x = r/m − 1 and y = cos θ ,    2m 2m −1 2 2 ds = − 1 − (cdt) + 1 − dr + r 2 (dθ 2 + sin2 θ dφ 2 ) . r r (6.2.78) 

2

Such coordinates (ct, r, θ, φ) are called standard Schwarzschild coordinates. If we write m=

GM c2

(6.2.79)

the parameter M will be identified with the (field) mass of Schwarzschild spacetime. The corresponding spherical Weyl moments read: a2n = −

m2n+1 . 2n + 1

(6.2.80)

The Ernst-potential E in PS coordinates takes the form E =f =

x−1 l−m = = −g00 x+1 l+m

(6.2.81)

with l ≡ mx . The ξ -potential reads ξ=

1+E =x. 1−E

With r± =

 ρ 2 + (z ± m)2

(6.2.82)

204

6 Exact Solutions—Field Moments

the Schwarzschild-metric in canonical Weyl coordinates (ct, ρ, z, φ) takes the form (l = r − m) ds 2 = −

l−m (l + m)2 l+m 2 2 (cdt)2 + ρ dφ . (dρ 2 + dz2 ) + l+m r+ r− l−m

(6.2.83)

The transformation to standard Schwarzschild-coordinates is obtained with  z = (r − m) cos θ (6.2.84) ρ = r 2 − 2mr sin θ ; so that l + m = r, l − m = r − 2m and r± = ±(m cos θ ± (r − m)) . 6.2.3.2

(6.2.85)

The Erez-Rosen Metric

The Erez-Rosen (ER) metric (Erez and Rosen 1959; Doroshkevich et al. 1966; Winicour et al. 1968; Young and Coulter 1969) is an extension of the Schwarzschildmetric by choosing q0 = 1 and q2 = q as non-vanishing PS Weyl moments. Then, ψER = −Q0 (x) − qP2 (y)Q2 (x)

1 x−1 3 1 x−1 q 2 2 + (3y − 1) (3x − 1) ln + x . (6.2.86) = ln 2 x+1 2 4 x+1 2

The corresponding potential γ takes the form (e.g., Bini et al. 2013 and references quoted therein) 1 x2 − 1 + 2q(1 − P2 )Q1 + q 2 (1 − P2 ) · [(1 + P2 )(Q21 − Q22 ) (1 + q)2 ln 2 2 x − y2

1 + (x 2 − 1)(2Q22 − 3xQ1 Q2 + 3Q0 Q2 − Q2 ) (6.2.87) 2

γER =

with Pn ≡ Pn (y), Qn ≡ Qn (x) and Q2 ≡ dQ2 (x)/dx. By relation (6.2.74) the spherical Weyl moments for the Erez-Rosen metric read: a2n = −

  2n m2n+1 1+ q . 2n + 1 2n + 3

(6.2.88)

As we shall see later, the parameter q can be related with a quadrupole moment of the ER space-time. Exercise 6.8 Store the ER metric in PS coordinates (ct, x, y, φ) in a file ER.mpl. Then write a little program using GRTensor to check that it solves the vacuum field equations Rμν = 0. Also check the field equations for the potential γ given the function ψ(x, y) (Fig. 6.2).

6.2 Stationary Space-Times

205

Ndim_ := 4: X1_ := x: X2_ := Y: X3_ := phi: X4_ := t: g11_ := m^2*exp(2*gammap-2*psi)*(x^2-y^2)/(x^2-1): g22_ := m^2*exp(2*gammap-2*psi)*(x^2-y^2)/(1-y^2): g33_ := m^2*exp(-2*psi)*(x^2-1)*(1-y^2): g44_ := -exp(2*psi): psi := - 1/2*ln((x+1)/(x-1)) - q*P2*Q2; gammap := 1/2*(1+q)^2*ln((x^2-1)/(x^2-y^2)) + 2*q*(1-P2)*Q1 + q^2*(1-P2)*((1+P2)*(Q1^2-Q2^2) + 1/2*(x^2-1)*(2*Q2^2 - 3*x*Q1*Q2 + 3*Q0*Q2 - Q2p)); P2 := 1/2*(3*y^2-1); Q0 := 1/2*ln((x+1)/(x-1)); Q1 := x/2*ln((x+1)/(x-1)) - 1; Q2 := 1/4*(3*x^2-1)*ln((x+1)/(x-1)) - 3/2*x; Q2p := 3/2*x*ln((x+1)/(x-1)) - 1/2*(3*x^2-1)/(x^2-1) - 3/2; Info_:=`Erez-Rosen metric in prolate spheroidal coordinates (x = r/m-1, y = cos(theta), phi,t)`: Fig. 6.2 A file ER.mpl for the Erez-Rosen metric

The file ER.mpl could look like this: The corresponding Maple file could read: > >

####################################################### #######################################################

> > >

restart: libname := libname, "D:\\Maple\\grtensor": with(grtensor);

>

grOptionMetricPath := "D:\\Maple\\grtensor\\metrics": qload( ER ): psi; gammap; ########################################################### # check the field equations for gammap ########################################################### px := diff(psi,x): py := diff(psi,y):

> > > > > > > >

206 >

6 Exact Solutions—Field Moments

>

Rx := (1-y^2)/(x^2-y^2)*(x*(x^2-1)*px^2 - x*(1-y^2)*py^2-2*y*(x^2-1)*px*py): difx := simplify(diff(gammap,x) - Rx); Ry := (x^2-1)/(x^2-y^2)*(y*(x^2-1)*px^2 - y*(1-y^2)*py^2+2*x*(1-y^2)*px*py): dify := simplify(diff(gammap,y) - Ry);

>

#######################################################

>

####################################################### # for q = 0 we get the Schwarzschild metric ####################################################### Gtt := grcomponent(g(dn,dn), [t,t]): GStt := simplify(subs(q=0,x = r/m-1,y=cos(theta),Gtt));

> > > >

> > > >

> >

> >

> > > > > >

Gxx := grcomponent(g(dn,dn), [x,x]): GSxx := simplify(subs(q=0,x = r/m-1,y=cos(theta),Gxx)); Gyy := grcomponent(g(dn,dn), [y,y]): GSyy := simplify(subs(q=0,x = r/m-1,y=cos(theta),Gyy)); Gphiphi := grcomponent(g(dn,dn), [phi,phi]): GSphiphi := simplify(subs(q=0,x = r/m-1,y=cos(theta),Gphiphi)); ########################################################### grcalc( R(dn,dn) ): grdisplay( R(dn,dn) ): ###########################################################

6.2.3.3

The Quevedo-Mashhoon M-Q-S Metric

The Quevedo-Mashhoon M-Q-S metric is a stationary axisymmetric solution of the vacuum field equations. It has three parameters: M (mass), Q (quadrupole-moment) and S spin and generalizes the Schwarzschild (Q = S = 0), the Erez-Rosen (S = 0) and the Kerr-metric (Q = 0). The QM M-Q-S spacetime in PS-coordinates (ct, x, y, φ) is of the form (6.2.47) with (Quevedo and Mashhoon 1990, 1991; Quevedo 1990; Bini et al. 2009) ξ=

(a+ + ib+ ) + (a− + ib− )e−2ψ (a+ + +b+ ) − (a− + ib− )e−2ψ

(6.2.89)

6.2 Stationary Space-Times

207

with ψ = qP2 (y)Q2 (x) ,

f =

A −2ψ e , B

C 2ψ e , A m 2 A 1 1+ e2χ , = 2 4 σ (x − 1)

W = −2a − 2σ e2γ and A = a+ a− + b+ b− , 2 2 B = a+ + b+ ,

C = x(1 − y 2 )(λ + η)a+ + y(x 2 − 1)(1 − λη)b+ ,

2 1 x −1 2 χ = (1 + q) ln 2 2 x − y2 + 2q(1 − P2 )Q1 + q 2 (1 − P2 )[(1 + P2 )(Q21 − Q22 ) +

1 2 (x − 1)(2Q22 − 3xQ1 Q2 + 3Q0 Q2 − Q2 )]. 2

Furthermore, a± = x(1 − λη) ± (1 + λη) , b± = y(λ + η) ∓ (λ − η) , with λ = αe2qδ+ , η = αe2qδ− ,

(x ± y)2 3 1 δ± = ln + (1 − y 2 ∓ xy) 2 2 x2 − 1

x−1 3 . + [x(1 − y 2 ) ∓ y(x 2 − 1)] ln 4 x+1

(6.2.90)

208

6 Exact Solutions—Field Moments

Moreover, α and σ are constants defined as αa = σ − m,

σ =

 m2 − a 2

with a = S/Mc (α = 0 for a = 0), q = −Q/(Mm2 ) and m ≡ GM/c2 . For q = 0, i.e., the Kerr geometry, the Ernst ξ potential from (6.2.89) reads: ξK =

x(1 − α 2 ) + 2iαy (1 + α 2 )

(6.2.91)

and using (1 − α 2 )/(1 + α 2 ) = σ/m and 2α/(1 + α 2 ) = −a/m we get ξK =

σ x − iay . m

(6.2.92)

In spherical Weyl coordinates (ct, r, θ, φ), the QM metric takes the form ds 2 = −f (c dt − W dφ)2



+

dr 2 e2γ [(r − m)2 − (m2 − a 2 ) cos2 θ ] + dθ 2 f 

+

 2 sin θ dφ 2 , f



(6.2.93)

where  = r 2 − 2mr + a 2 and the relation between PS and spherical Weyl coordinates are  r = σ x2 + y2 − 1 ,  θ = xy/ x 2 + y 2 − 1 . The QM M-Q-S metric can be used to describe the exterior asymptotically flat gravitational field of a rotating body with an arbitrary quadrupole mass-moment; the Hartle-Thorne solution mentioned before is valid for the exterior field of a slowly rotating and slightly deformed object. Finally, we would like to mention that more general stationary and axisymmetric vacuum solutions have been derived that contain a set of infinitely many independent parameters qn , describing relativistic mass multi-pole moments and a single parameter a related with the spin of the central ‘body’ (higher order spin-moments are then uniquely determined by (qn , a)). Examples are the generalized Quevedo-Mashhoon metric (1991) and the MankoNovikov metric (1992).

6.2 Stationary Space-Times

209

Exercise 6.9 Show that the QM M-Q-S metric for vanishes quadrupole parameter q (the Kerr-metric) leads to f =−

m2 − σ 2 x 2 − a 2 y 2 , (σ x + m)2 + a 2 y 2

W = 2ma e2γ =

(σ x + m)(1 − y 2 ) , σ 2 x 2 + a 2 y 2 − m2

(6.2.94)

σ 2 x 2 + a 2 y 2 − m2 . σ 2 (x 2 − y 2 )

From this show that in the Schwarzschild case q = a = 0 the QM M-Q-S metric yields: f = (1 − x)/(1 + x), W = 0 and exp(2γ ) = (x 2 − 1)/(x 2 − y 2 ).

6.2.4 Spherically Symmetric Space-Time A spherically symmetric space-time is invariant under rotations so that coordinates x μ = (ct, r, θ, φ) can be chosen in such a way that the metric on a constant t, constant r hypersurface takes the form dl 2 = r 2 (dθ 2 + sin2 θ dφ 2 ) ≡ r 2 d 2 .

(6.2.95)

The metric in adapted coordinates can be written in the form (e.g., Weinberg 1972; Misner et al. 1973) ds 2 = −A(t, r)c2 dt 2 + B(t, r)dr 2 + r 2 d 2

(6.2.96)

that is often written in the form ds 2 = −e2 c2 dt 2 + e2! dr 2 + r 2 d 2 ,

(6.2.97)

where  = (t, r) and ! = !(t, r). Corresponding Christoffel-symbols, components of the Riemann and Ricci-tensor can be found in Appendix. We now consider the vacuum field equations, Rμν = 0. The equation Rtr = 0 = (2/r)!,0

(6.2.98)

tells us that ! is independent of t, ! = !(r). Then, e2(!−) Rtt + Rrr = (2/r)( + !),r = 0

(6.2.99)

tells us that  + ! is a function of t only:  + ! = f (t) .

(6.2.100)

210

6 Exact Solutions—Field Moments

Then we can define a new time coordinate T by

T =



dt  ef (t )

(6.2.101)

so that the metric takes the form ds 2 = −e−2!(r) c2 dT 2 + e2!(r) dr 2 + r 2 d 2 .

(6.2.102)

Finally the equation Rθθ = 0 leads to e−2! = 1 −

k , r

(6.2.103)

where k is an integration constant. If we put k = 2m we get the standard form of the Schwarzschild metric     2m 2 2 2m −1 2 ds 2 = − 1 − c dt + 1 − dr + r 2 d 2 . r r Thus we have proven Birkhoff’s Theorem: Theorem 6.1 (Birkhoff’s Theorem) A spherically symmetric vacuum space-time is necessarily locally isometric to the static Schwarzschild geometry. This Theorem was first formulated by Jebson in 1921 and later proven by Birkhoff in 1923.

6.2.4.1

Harmonic Coordinates

For important reasons the harmonic gauge plays an important role in many applications. A function f (x μ ) is called harmonic, if 1 √ g f ≡ √ ( −gg μν f,ν ),μ = 0 . −g

(6.2.104)

Note that in three dimensional Euclidean space, R3 , the operator g reduces to the usual Laplacian . Harmonic coordinates x μ satisfy the harmonicity condition √ g x μ = ( −gg μν ),μ = 0 .

(6.2.105)

Our standard Schwarzschild coordinates can be transformed into harmonic ones with a change of the radial coordinate (Klioner and Soffel (2005)). We will construct coordinates Xα = (ct, X, Y, Z) with X = R(r) sin θ cos φ ,

Y = R(r) sin θ sin φ ,

Z = R(r) cos θ

(6.2.106)

6.2 Stationary Space-Times

211

so that they obey the harmonicity condition (6.2.105). For our static spherically symmetric metric in standard coordinates we have: 1 −1 A ∂tt + B −1 ∂rr + r −2 ∂θθ + r −2 (sin θ )−2 ∂φφ c2

g μν ∂μν = −

(6.2.107)

and λ g μν μν ∂λ = −

1 A 1 B 2 cot θ ∂r − ∂r + ∂ r − 2 ∂θ . 2 AB 2 B2 rB r

(6.2.108)

One immediately finds that the standard time-coordinate t is harmonic. For the spatial coordinates Xi we obtain   λ g μν ∂μν − μν ∂λ X i =



Xi BR

 

A 2 B + − 2A r 2B



R  + R  −

2B R , r2 (6.2.109)

i.e., they are harmonic if 

A 2 B + − 2A r 2B



R  + R  −

2B R =0 r2

(6.2.110)

or equivalently d 2 1/2 −1/2  R ) = 2A1/2 B 1/2 R . (r A B dr In these harmonic coordinates the metric takes the form

r2 B r2 (X · dX)2 . − ds 2 = −Ac2 dt 2 + 2 dX2 + R R4 R2R2

(6.2.111)

(6.2.112)

To get the Schwarzschild metric in harmonic coordinates we perform two subsequent coordinate transformations. The first one simply reads r =ρ+m so that 1−

2m 2m ρ−m 1 − m/ρ =1− = = . r ρ+m ρ+m 1 + m/ρ

(6.2.113)

212

6 Exact Solutions—Field Moments

A transformation to Cartesian coordinates with  x2

x = ρ sin θ cos φ

ρ=

y = ρ sin θ sin φ

θ = arccos z/ρ

z = ρ cos θ

φ = arctan y/x

with dρ =

z dρ/ρ − dz dθ =  ; x2 + y2

x i dx i , ρ

dφ =

x dy − y dx x2 + y2

yields the following form of the metric g00 = −

1 − m/r 1 + m/r

g0i = 0

   m 2  m 2 1 + m/r x i x j gij = δij 1 + + r r 1 − m/r r2

(6.2.114)

that turns out to be harmonic. Exercise 6.10 Show that the metric (6.2.114) is indeed harmonic (see e.g., Weinberg 1972). i ,  0 and  i for the harmonic Exercise 6.11 Compute the Christoffel-symbols 00 jk 0i Schwarzschild metric to first order in m = GM/c2 .

Solution The Christoffel-symbols to this order read: i i 00 = 00 =

mx i ; r3

ji k =

 m i k j x . δ − x δ − x δ j k ij ik r3

(6.2.115)

i Exercise 6.12 Compute the m2 -terms of 00 and ji k of the harmonic Schwarzschild metric. i = −4m2 x i /r 4 and Solution One finds 00 1 11 1 12 1 13 1 23 1 22 1 33

= +2x(y 2 + z2 ) = −y(x 2 − y 2 − z2 ) = −z(x 2 − y 2 − z2 ) = −2xyz = −2xy 2 = −2xz2

2 11 2 12 2 13 2 23 2 22 2 33

= −2yx 2 = +x(x 2 − y 2 + z2 ) = −2xyz = +z(x 2 − y 2 + z2 ) = +2y(x 2 + z2 ) = −2yz2

3 11 3 12 3 13 3 23 3 22 3 33

= −2zx 2 = −2xyz = +x(x 2 + y 2 − z2 ) = +y(x 2 + y 2 − z2 ) = −2zy 2 = +2z(x 2 + y 2 ) ,

where all Γjik ’s have to be multiplied with a factor of (m/r 3 )2 .

6.3 The Kerr Metric

213

The transformation from standard to harmonic coordinates is achieved by solving the differential equation (6.2.111). For the Schwarzschild vacuum solution this equation reads 

 2GM dR d r2 1 − 2 = 2R dr dr c r

(6.2.116)

and has the general solution R = C1 (r − m) + C2 F (r)

(6.2.117)

where C1 and C2 are constants, m ≡ GM/c2 and (Bicák and Katz 2005) F (r) ≡ (r − m) ln(1 − 2m/r) + 2m = −m

∞ k  2 (k − 1) k=2

k(k + 1)

(m/r)k .

(6.2.118)

Often in the literature (e.g., Weinberg 1972) the solution with C1 = 1, C2 = 0 is taken, so that R = r − m and the harmonic metric in empty space takes the form 

1 − m/R ds = − 1 + m/R 2



m 2 dX2 − c dt − 1 + R 2

2





1 + m/R 1 − m/R



m2 (X · dX)2 . R4 (6.2.119)

However, we would like to stress, that for the case of some extended body of spherical symmetry the interior solution can be matched to the external vacuum solution so that the metric is continuous at the surface of the body. In that case both constants, C1 and C2 have to be adjusted, so that both, R(r) and R  (r) are continuous there (see e.g., Klioner and Soffel 2014).

6.3 The Kerr Metric 6.3.1 Boyer-Lindquist Coordinates The Kerr metric is another vacuum solution of the field equations which was first found by Kerr (1963). It describes the exterior gravitational field of a spinning body in terms of an exact vacuum solution to Einstein’s field equations. In BoyerLindquist coordinates (ct, r, θ, φ) it is given by (Misner et al. 1973; d’Inverno 1992) ds 2 = −

  sin2 θ 2 [cdt − a sin2 θ dφ]2 + dr 2 + dθ 2 + [(r + a 2 )dφ − a(cdt)]2   

(6.3.1)

214

6 Exact Solutions—Field Moments

with (m ≡ GM/c2 )  ≡ r 2 − 2mr + a 2

 ≡ r 2 + a 2 cos2 θ

(6.3.2)

and a≡

S Mc

(6.3.3)

is some specific angular momentum. Thus the components of the Kerr metric tensor in Boyer-Lindquist coordinates read: 2mr  amr sin2 θ = −2   =  =   2mra 2 sin2 θ sin2 θ . = r 2 + a2 + 

gtt = −1 + gtφ grr gθθ gφφ

(6.3.4)

The corresponding inverse metric, the non-vanishing Christoffel symbols and components of the Riemann curvature-tensor (up to symmetries) are listed in Sect. 13.6 of the Appendix. Exercise 6.13 Derive the expression for the Kerr metric in Boyer-Lindquist coordinates from the corresponding Ernst ξ -potential in prolate spheroidal coordinates ξ = x cos λ − iy sin λ with cos λ = σ/m and sin λ = a/m. Solution We already had derived expressions for f, W and e2γ in Exercise 6.9 in prolate spherical coordinates. With σx = r − m

y = cos θ

(6.3.5)

and σ 2 = m2 − a 2 we get r 2 + a 2 cos2 θ − 2mr , r 2 + a 2 cos2 θ mar sin θ W =2 2 , r + a 2 cos2 θ − 2mr f =

e2γ =

r 2 + a 2 cos2 θ − 2mr r 2 + a 2 cos2 θ − 2mr + m2 sin2 θ

.

6.4 Cosmologically Relevant Spacetimes

215

 Inserting this in the metric (6.2.28), and reminding that ρ = σ (x 2 − 1)(1 − y 2 ) and z = σ xy, we finally get the Kerr metric in Boyer-Lindquist coordinates.

6.4 Cosmologically Relevant Spacetimes 6.4.1 The Cosmological Principle The Cosmological Principle says that when averaged over very large scales our universe looks homogenous and isotropic. When looking at our cosmic neighbourhood we see, however, that it is heavily structured. The Earth is part of our Solar system (Fig. 6.3) that itself is a member of our Milky Way galaxy. The Milky Way has a maximal extension of about 120,000 ly (light years). The Local Group (Fig. 6.4) is the galaxy group that includes the Milky Way. The Local Group comprises more than 54 galaxies, most of them dwarf galaxies. Its gravitational center is located somewhere between the Milky Way and the Andromeda Galaxy. The Local Group covers a diameter of 10 Million ly (Mly) or 3.1 Mpc. The group itself is a part of the larger Virgo Supercluster (Fig. 6.5). At least 100 galaxy groups and clusters are located within its diameter of about 110 Mly or 33 Mpc. A study by Tully et al. 2014 indicates that the Virgo Supercluster is only a lobe of a greater supercluster, named Laniakea (“immeasurable heaven” in Hawaiian) (Fig. 6.6). The Laniakea Supercluster encompasses 100,000 galaxies

Fig. 6.3 Schematic diagram of our solar system (file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)

216

6 Exact Solutions—Field Moments

Fig. 6.4 The local galactic group (Image credit: Andrew Z. Colvin, file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)

stretched out over 520 Mly or 160 Mpc. Figure 6.7 shows the redshift distribution of more than 20,000 galaxies from the Las Campanas redshift survey (Shectman et al. 1996). Plotted are redshift versus angular coordinates (RA, right ascension; dec., declination: Clustering of galaxies is seen on scales smaller than 30 h−1 Mpc (h is the Hubble constant in units of 100 km s−1 Mpc−1 ), but on larger scales the distribution seems to approach homogeneity (Wu et al. 1999). A whole variety of very special huge cosmic structures have been detected in the recent past whose existence might speak against the validity of the Cosmological Principle. An example for that is the Sloan Great Wall with a maximal extension of about 1370 Mly (Einasto et al. 2010). The usual motivation of the Cosmological Principle are the anisotropies of the Cosmic Microwave Background Radiation (CMBR). Figure 6.8 shows the cosmic density fluctuations on different scales. Shown are data from a galaxy survey, deep radio surveys, the X-ray background (XRB) and cosmic microwave background (CMB) observations. The measurements are compared with two popular cold dark matter (CDM) models with two shape parameters  = 0.2 and 0.5. The figure

6.4 Cosmologically Relevant Spacetimes

217

Fig. 6.5 The Virgo supercluster (Image credit: Andrew Z. Colvin, file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)

shows the mean-square density fluctuations < (δρ/ρ)2 > as function of length scale λ in h−1 Mpc. These observations provide strong evidence for the validity of the Cosmological Principle on scales larger than a few hundred Mpc (Wu et al. 1999).

6.4.2 Robertson-Walker Metric The Cosmological Principle implies that on very large scales our cosmic 3-space is maximally symmetric and in suitably chosen coordinates (t, r, θ, φ) the (averaged) cosmological metric can be written in the form $  dr 2 2 2 2 2 2 2 2 2 2 ds = −c dt + a (t) (6.4.1) + r dθ + r sin θ dφ . 1 − kr 2 This is the Robertson-Walker metric (Robertson 1935, 1936a,b; Walker 1937). The metric components in our canonical coordinates read:

218

6 Exact Solutions—Field Moments

Fig. 6.6 Local superclusters including the Virgo supercluster (Image credit: Andrew Z. Colvin, file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license)

g00 = −1 g0i = 0

(6.4.2)

gij = a (t)g˜ ij (x) 2

with g˜ rr = (1 − kr 2 )−1 g˜ θθ = r 2

(6.4.3)

g˜ φφ = r sin θ . 2

2

6.4 Cosmologically Relevant Spacetimes

219

Fig. 6.7 The redshift distribution of more than 20,000 galaxies, from the Las Campanas redshift survey (Image credit: Shectman et al. 1996)

The Christoffel-symbols in these coordinates read (the dot equals ∂ct ): ij0 = a a˙ g˜ ij a˙ i δ a j 1 = g˜ il (g˜ lj,k + g˜ lk,j − g˜ j k,l ) 2

i 0j =

ji k

(6.4.4)

and the components of the Ricci-tensor take the form : R00 = −3

a¨ a

R0i = 0

(6.4.5)

Rij = (a a¨ + 2a˙ + 2k) g˜ ij . 2

220

6 Exact Solutions—Field Moments

Fig. 6.8 Average cosmic density fluctuations as function of length scale. Shown are data from a galaxy survey, deep radio surveys, the X-ray background (XRB) and cosmic microwave background (CMB) experiments. The measurements are compared with two popular cold dark matter (CDM) models. The figure shows mean-square density fluctuations < (δρ/ρ)2 > (Image credit: Wu et al. 1999)

Let us now analyze the field equations Rμν = κ Tˆμν . We will now assume that the averaged cosmic substratum represents a kind of fluid with Tμν = pgμν + (ρ + p/c2 )uμ uν ,

(6.4.6)

where uμ is the 4-velocity of some cosmic fluid element. Let us consider a cosmic fluid that is co-moving with our standard coordinates, i.e., u0 = c, ui = 0. Then, 1 Tˆ00 = (ρc2 + 3p) 2 ˆ T0i = 0 1 Tˆij = (ρc2 − p)a 2 g˜ ij 2

(6.4.7) (6.4.8)

so we end up with the Einstein field equations in the form κ 3a¨ = − (ρc2 + 3p)a 2

(6.4.9)

6.4 Cosmologically Relevant Spacetimes

221

(the tt-equation) and a a¨ + 2a˙ 2 + 2k =

κ (ρc2 − p)a 2 2

(6.4.10)

(the ij-equations). Eliminating a¨ from the two field equations leads to the first-order differential equation a˙ 2 + k =

κ 2 2 ρc a . 3

(6.4.11)

6.4.3 De Sitter Space The de Sitter cosmos is a special solution of Einstein’s field equations (5.4.9) with cosmological constant . For an ideal cosmic fluid we write ˜ μν + (ρ˜ + p/c ˜ 2 )uμ uν T˜ μν = pg

(6.4.12)

with p˜ = p − /κ and ρc ˜ 2 = ρc2 + /κ. This implies that the dynamical equation (6.4.11) with -term takes the form a˙ 2 + k =

κ 2 2 ρc ˜ a . 3

(6.4.13)

The de Sitter space (de Sitter 1917, Levi-Civita 1917) is empty and flat, i.e., ρ = p = k = 0 so that the last equation reduces to a˙ 2 =

1 2 a 3

(6.4.14)

which is solved by  a(t) = a0 exp(H t) ,

H ≡c

 3

1/2 .

(6.4.15)

The de Sitter metric tensor therefore reads in Cartesian canonical coordinates g00 = −1 g0i = 0 gij = δij exp(2H t) .

(6.4.16)

222

6 Exact Solutions—Field Moments

The following transformation of coordinates 

  3 1 2 ln 1 − ρ  3 + *   ρ cT exp − r=  3 1 − 1 ρ 2 1 t=T + 2c

(6.4.17) (6.4.18)

3

or 1 T =t− 2c



  1 3 ln 1 − a 2 r 2  3

ρ = ar

(6.4.19) (6.4.20)

puts the metric tensor in the form: ds 2 = −Ac2 dT 2 + A−1 ρ 2 + ρ 2 (dθ 2 + sin2 θ dφ 2 )

(6.4.21)

with 1 A = 1 − ρ 2 . 3

(6.4.22)

With another transformation R= ρ=

1+



2ρ 1 − ρ 2 /3

R 1 + R 2 /12

(6.4.23)

the de Sitter metric can be written in isotropic form 

G00

1 − R 2 /12 =− 1 + R 2 /12

2

G0i = 0 gij = δij

1 . (1 + R 2 /12)2

(6.4.24)

We call the last two versions of the metric from (6.4.21) and (6.4.24) the de Sitter metric in local coordinates where the cosmic expansion effectively corresponds to a cosmic tidal force. The isotropic local form of the de Sitter metric admits an

6.4 Cosmologically Relevant Spacetimes

223

expansion of the form G00 = −1 +

∞ 

As R 2s

s=1

G0i = 0

*

Gij = δij 1 +

∞ 

+ Bs R

2s

(6.4.25)

s=1

with  A1 = , 3

2 A2 = − , 18

 As = (−1)

s+1

4s

 12

s (6.4.26)

and 

 Bs = (−1) (s + 1) 12

s

2

.

(6.4.27)

6.4.4 Schwarzschild: De Sitter Solution The Schwarzschild—de Sitter (SchdS) space (e.g., Rindler 2001) is a generalization of both the Schwarzschild and the de Sitter space. It is a spherically symmetric and static solution of the vacuum field equations with cosmological constant : Rμν = −gμν .

(6.4.28)

In standard coordinates (t, r, θ, φ) the SchdS-metric is the form ds 2 = −f (r)c2 dt 2 + f −1 (r)dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ) .

(6.4.29)

The differential equation for f (r) can be deduced from the field equation (6.4.28) with μ = ν = θ . Since gθθ = r 2 and (see Appendix) Rθθ = −1 + rf  + f .

(6.4.30)

The equation for f (r) therefore takes the form 1 = rf  + f + r 2

(6.4.31)

with the solution (rs = 2m) f (r) = 1 −

rs 1 − r 2 . r 3

(6.4.32)

224

6 Exact Solutions—Field Moments

6.5 Field Moments For the description of many aspects of gravitating systems expansions in terms of multipole moments are of great importance. In the chapter on Newtonian Celestial mechanics we have expanded the external Newtonian potential U of a body in terms of multipole moments (either spherical or Cartesian) to derive equations of motion and to describe the tidal forces. In GR multipole expansions are important tools to describe external fields and also the gravitational waves emitted from a gravitating system. In principle two different kinds of multipole moments (independent of their representation as spherical or Cartesian moments) can be introduced: field moments and body moments. Field moments are derived from the form of the metric tensor ‘far away’ from the field generating sources; for stationary fields they are defined at spacelike infinity (e.g., Misner et al. 1973), i.e., for |x| → ∞ and t = const. in some suitably chosen coordinates. Such field moments can be defined even for black holes that present pure vacuum solutions of Einstein’s field equations. In many cases such field moments can be related with corresponding body moments as integrals over the gravitational source tensor. This section concentrates on the problem of field moments. Stationary vacuum solutions of Einstein’s field equations with asymptotic flatness are fully characterized by two families of multipole moments, mass-moments ML and spin-moments SL . Such field-moments have been first introduced by Geroch (1970) for static fields and Hansen (1974) for stationary gravitational fields in a coordinate independent manner. In 1980 Thorne (1980) for the stationary case defined field moments in a very different manner. Later, however, in 1983, it was shown by Gürsel (1983) that, aside from normalization, the Geroch-Hansen moments are identical with the Thorne field moments. Though the definition of Thorne moments rests upon certain coordinate systems, i.e., it is coordinate dependent, it is of great practical value.

6.5.1 Geroch-Hansen Moments Let V ≡ M(3) be the 3-space of some stationary space-time with space metric hij . Then V is called asymptotically flat if there exist coordinates x¯ i and a conformal factor (x) ¯ such that h˜ ij ≡ 2 hij is a smooth metric on V˜ ≡ V ∪  ,

(6.5.1)

6.5 Field Moments

225

where  is a single point in x¯ coordinates (spacelike infinity) and D˜ i | = 0,

| = 0,

D˜ i D˜ j | = 2h˜ ij ,

(6.5.2)

where D˜ i is the covariant derivative with respect to h˜ ij . Let us define the Hansen-potentials, ∗M and ∗S by ∗M ≡ −

c2 2 (f + ω2 − 1) ; 4f

∗S ≡ −

c3 ω. 2f

(6.5.3)

In the Newtonian limit f = −g00 = 1−2U/c2 so that ∗M reduces to the Newtonian potential U . ∗S is a kind of gravito-magnetic potential related with the twist of space-time. Next we consider the potentials ˜ ∗A ≡ −1/2 ∗A 

(A = M, S)

(6.5.4)

and the STF-tensor fields ˜ ∗A PA ≡ 



PLA ≡ STFL PiA1 ...il−1 ||il −

(l − 1)(2l − 3) ˜ (3) A Ri1 i2 Pi3 ...il 2

(6.5.5)

,

(6.5.6)

where PiA1 ...il−1 ||il ≡ D˜ il PiA1 ...il−1

(6.5.7)

and D˜ i is the covariant derivative with respect to h˜ ij ; R˜ ij(3) is the corresponding Ricci-tensor in conformal 3-space. The Geroch-Hansen moments, ML∗ , SL∗ , are then defined by ML∗ ≡ G−1 PLM () SL∗ ≡ G−1 PLS () .

(6.5.8)

The Geroch-Hansen moments generalize the Newtonian mass-moments, ML , that we have introduced earlier in the expansion of the Newtonian gravitational potential U =G

 (2l − 1)!! l≥0

l!

ML

nˆ L r l+1

= r −1 Q + r −3 Qi x i + · · · +

1 −2l+1 QL x L + · · · , r l!

(6.5.9)

226

6 Exact Solutions—Field Moments

where QL ≡ G(2l − 1)!! ML is an STF-tensor. The Newtonian mass-moments QL appear as coefficients in a 1/r-expansion of the potential U , i.e., they are defined at spacelike infinity . Therefore, mathematically one makes a conformal transformation, x → x¯ such that  is mapped to the point x¯ = 0, i.e., to the origin of the (conformally) transformed system, e.g., by x¯ = r −2 x ,

= r −2 .

(6.5.10)

˜ ∗ = −1/2 ∗ = rU , i.e., Then, the potential U is mapped into U˜ =  M M 1 1 U˜ = Q + Qi x¯ i + Qij x¯ i x¯ j + · · · + QL x¯ L + · · · , 2 l!

(6.5.11)

so we can get the Newtonian mass-moments QL by taking the derivatives of U˜ at the point : ˜ ∗M | , Q=

˜ ∗M | , Qi = D˜ i 

˜ ∗M | , . . . , Qij = D˜ i D˜ j 

(6.5.12)

which agrees with the Geroch-Hansen construction for ML∗ = (2l − 1)!! ML . Note, that in the Newtonian limit the Ricci-tensor of 3-space vanishes. If the curvature of 3-space has to be considered the construction of multipole moments, ML∗ and SL∗ , involve the (conformally transformed) Ricci-tensor of 3space. This term is chosen such to ensure the same behaviour of the moments under a shift of origin (i.e., under translations) as in the Newtonian limit (see (3.3.76); Geroch 1970; Hansen 1974; Bäckdahl 2008).

6.5.2 Thorne Moments Whereas Geroch and Hansen defined mass- and spin-moments of some stationary and asymptotically flat space-time in a mathematical, i.e., coordinate independent manner over some 3-space associated with the timelike Killing vector-field, Thorne’s definition of field moments is entirely in the physical 4-dimensional spacetime and requires the use of special coordinates, called ACMC (asymptotically Cartesian and mass centered). Asymptotically Cartesian means that gμν → ημν for |x| → ∞; harmonic (De Donder) coordinates are in this class. Mass centered means that in these coordinates the mass dipole moment vanishes. Gürsel (1983) proved that the Geroch-Hansen moments are equivalent to the Thorne moments, i.e., they agree up to a factor.

6.5 Field Moments

227

Thorne’s first structure theorem (1980) reads: Theorem 6.2 There are harmonic coordinates such that any stationary vacuum solution of Einstein’s field equations with asymptotic flatness can be written in the form ∞

g00 = −1 +

2m 2m2 2G  1 − 2 + 2 r r c r l+1 l=2

∞

g0i

4G  1 =− 3 c r l+1 l=1





(2l − 1)!! ML Nˆ L + RL−1 l!

l(2l − 1)!! ij k Sj L−1 nˆ kL−1 + RL−1 (l + 1)!





  2m m2 gij = δij 1 + + 2 (δij + nij ) r r

∞ (2l − 1)!! 2G  1 + 2 n ˆ δ + R M L L ij L−1 , l! c r l+1

(6.5.13)

l=2

where RL−1 is a symbol denoting a quantity which is independent of r and has an angular dependence of order l − 1, l − 2, . . . , 0. A proof can be found in Thorne (1980). Introducing a symbol = ¨ , meaning that the Thorne rest-terms, RL , will not be written out, we have g00 = ¨ −1 +



∞ 2(2l − 1)!! G 1 2m + 2 M n ˆ L L r l! c r l+1 l=2

g0i = ¨ −

4G c3

∞  l=1

1 r l+1



l(2l − 1)!! ij k Sj L−1 nˆ kL−1 (l + 1)!

¨ δij . gij =



(6.5.14)

Then f = g00 and f2 = ¨ 1−

1 4G  (2l − 1)!! ML nL l+1 . l! c2 r

(6.5.15)

l≥0

With ξκ = g0κ one finds ¨ μ0j k g0k,j = ¨ − μ0j k g0j,k . ωμ = μ0λσ g λτ ∇τ (g σ κ ξκ ) =

(6.5.16)

228

6 Exact Solutions—Field Moments

Therefore (Gürsel 1983), ,n -

4l(2l − 1)!! G mL−1 − (6.5.17) S ij k j lmk lL−1 (l + 1)! c3 r l+1 ,k l≥1

4l(2l − 1)!! , nL G − = ω,i . SL l+1 = 3 (6.5.18) (l + 1)! c r ,i

ωi = ¨ ij k g0j,k = ¨

l≥1

The twist potential is therefore given by: ω= ¨



−4l(2l − 1)!! G 1 . S n L L (l + 1)! c3 r l+1

(6.5.19)

l≥1

From this we end up with

 1 (2l − 1)!! ML nL = ¨ G l! r l+1 l≥0

 1 2l(2l − 1)!! ∗ SL nL . S = ¨ G (l + 1)! r l+1

∗M

(6.5.20)

(6.5.21)

l≥1

Comparing this with the Geroch-Hansen convention:

 1 1 ∗ M nL = ¨ G r l+1 l! L l≥0

 1 1 ∗ ∗ S nL . S = ¨ G r l+1 l! L

∗M

(6.5.22)

(6.5.23)

l≥1

we see that the Thorne-moments are related with the GH-moments by (Gürsel 1983) ML = SL =

1 M∗ (2l − 1)!! L

(6.5.24)

(l + 1) S∗ . 2l(2l − 1)!! L

(6.5.25)

For a more rigorous and complete derivation of the last relations and the construction of conformal mappings the reader is referred to Gürsel (1983). There it is also shown that starting from the vacuum metric in ACMC (harmonic) coordinates the connection coefficients of the (conformally transformed) covariant derivative of 3space and the Ricci tensor terms do NOT contribute to the moments.

6.5 Field Moments

229

In this paper also an algorithm is presented that allows to calculate the Thornemetric explicitly to any desired order. The lowest order terms read explicitly: 2m 2m2 2m3 3GMab nˆ ab − 2 + 3 + + O(1/r 4 ) r r r c2 r 3 2G ij k Sj nk  m  4G ij k Sj l nk nl − g0i = − 1 − + O(1/r 4 ) r c3 r 2 c3 r 3   2m m2 + 2 (δij + ni nj ) gij = δij 1 + r r

g00 = −1 +

+

3GMab nˆ ab 2m3 ni nj + δij + O(1/r 4 ) . r3 c2 r 3

(6.5.26)

Since the R-terms are at least quadratic in G, we have the following Lemma: Lemma 6.4 The metric (6.5.13) from Thorne’s structure theorem, linearized to first order in G reads: ∞

g00

2m 2G  1 (2l − 1)!! + 2 ML nˆ L + O(G2 ) = −1 + r l! c r l+1 l=2

g0i = −

4G c3

∞  l=1

1 l(2l − 1)!! ij k Sj L−1 nˆ kL−1 + O(G2 ) (l + 1)!

r l+1

  ∞ 2m 2G  1 (2l − 1)!! ML nˆ L δij + O(G2 ) . (6.5.27) gij = δij 1 + + 2 r l! c r l+1 l=2

or simply g00 = −1 +

2 ext V c2

4 ext V c3 i   2 ext gij = δij 1 + 2 V c

g0i = −

(6.5.28) (6.5.29) (6.5.30)

with V ext = G

 1 (2l − 1)!! ML nˆ L l! r l+1

(6.5.31)

l≥0

V ext = G

 l(2l − 1)!! l≥1

(l + 1)!

ij k Sj L−1 nˆ kL−1 .

(6.5.32)

230

6 Exact Solutions—Field Moments

Exercise 6.14 From (6.5.28) compute the components of hαβ = to first order in G. Solution We have g

00

√

√ −gg αβ − ηαβ

(6.5.33)

−g = 1 + 2V ext /c2 , and

2 = −1 − 2 V ext , c

g

0i

= g0i ,

  2V ext g = δij 1 − 2 c ij

(6.5.34)

so that h00 = −

4 ext V , c2

h0i = −

4 ext V , c3 i

h00 = 0 ,

(6.5.35)

plus terms of order G2 . Both, the GH- and the Thorne-constructions to derive mass- and spin-multipole moments for some stationary space-time with asymptotic flatness have some flexibility. As far as the GH construction is concerned instead of the Hansenpotentials, ∗A , a whole class of other potentials lead to the same field moments with the same construction procedure (Simon and Beig 1983; Beig and Schmidt 2000). Theorem 6.3 (Simon and Beig 1983) Let A ≡ FA (XB ) (XB = B , B = M, S) be two smooth functions of XB in a neighborhood of  with (a)

∂FA = δAB ∂XB 

(b)

FA (XM , XS ) = −FA (−XM , −XS ) ,

then (M , S ) and (M , S ) have the same multiple moments according to the GH procedure. The proof can be found in Simon and Beig (1983).

6.5.3 The FHP Theorem Often one uses the inverse Ernst ξI -potential to define potentials M and S , from which the field moments are derived: ξI ≡ ξ −1 =

1−E = M + iS , 1+E

(6.5.36)

6.5 Field Moments

231

i.e., the potentials M and S are determined by real and imaginary part if the inverse Ernst potential ξI . Since ∗A =

A , 1 − 2M − 2S

(6.5.37)

A yield the same field moments as the Hansen-potentials ∗A according to the Simon-Beig Theorem. Exercise 6.15 Show that relations (6.5.37) hold. Theorem 6.4 (Fodor et al. 1989) Consider a stationary, axially symmetric spacetime in canonical Weyl-coordinates (t, ρ, z, φ) with ξI being the inverse Ernst ξ potential; (i) transform (ρ, z) according to ρ¯ =

ρ2

ρ ; + z2

z¯ =

ρ2

z + z2

(6.5.38)

(ii) make a conformal transformation ξ˜I =

1 ξ; r¯

r¯ ≡

 ρ¯ 2 + z¯ 2

(6.5.39)

(iii) expand ξ˜I on the z-axis about the point ρ¯ = 0 (which is space-like infinity, ) ξ˜I (ρ¯ = 0) =



ml z¯ l

(6.5.40)

l≥0

then M0

= m0 ;

M1

= m1

M2

= m2 ;

M3

= m3

M4 = m4 − 17 M2,0 m∗0 ;

M 5 = m5 −

(6.5.41) 1 1 M2,0 m∗1 − M3,0 M0∗ 21 3

where Mi,j ≡ mi mj − mi−1 mj +1

(6.5.42)

Ml = Ml + iSl .

(6.5.43)

and

232

6 Exact Solutions—Field Moments

This means that the Ernst ξI -potentials on the z-axis, the symmetry axis of the assumed axial symmetry, yields the ξI -moments mn that fully determine the GerochHansen field moments. Up to l = 3 the ξ -moments agree with the GH ones but for l ≥ 4, correction terms Mij have to be considered. In Fodor et al. (1989) (FHP89) expressions for Ml are given up to l = 10. In Filter (2008) the expression for l = 11 is given. It is clear that as long as we identify the mass-monopole with the field mass of the geometry we can multiply ξ or ξI with any non-zero factor. As an Example we will calculate the field-moments of the Kerr metric with ξ = x(σ/m) − i(a/m)y in PS-coordinates, so that ξI =

m . xσ − iay

(6.5.44)

On the z-axis where y = 1 we write this as (z = xσ ) ξI (ρ¯ = 0) =

m . z − ia

(6.5.45)

The conformally transformed ξI -potential then takes the form ξ˜I (ρ¯ = 0) =

 m =m (ia z¯ )l . 1 − ia z¯

(6.5.46)

l≥0

Therefore all correction terms Mij vanish identically, Mij = 0 and the fieldmoments of the Kerr metric are given by Ml = MlGH + iSlGH = m(ia)l .

(6.5.47)

As we have seen a static axially symmetric space-time is characterized by its Weyl-moments. One can then employ the FHP89 procedure to get the field moments. One finds (Hernández-Pastora 2010) 1 M2 = −a2 + a02 , M3 = −a3 + a1 a02 3 8 19 6 a 5 + a0 a12 M4 = −a4 + a2 a02 − 7 105 0 7 4 19 12 2 M5 = −a5 + a3 a02 − a1 a04 + a2 a0 a1 + a13 (6.5.48) 3 21 7 7 M0 = −a0

M1 = −a1

Such relations have been obtained by Hernández-Pastora and Martin (1993) by means of a Thorne-procedure, i.e., relating the spherical Weyl coordinates (r, θ ) with corresponding harmonic coordinates (ˆr , θˆ ) in which the field moments can be read off the metric components. In this way these authors obtained relations of the

6.5 Field Moments

233

form (6.5.48) up to order 20. A list of such relations for the first 10 mass multipolemoments can be found in Appendix. For some applications if might be useful to invert such relations, i.e., to give the Weyl-moments as function of the mass multipole moments (e.g., Hernandez-Pastora MSA coordinates): a0 = −M0 a1 = −M1 1 a2 = − M03 − M2 3 a3 = −M1 M02 − M3 1 8 6 a4 = − M05 − M02 M2 − M0 M12 − M4 5 7 7

(6.5.49)

Remember that for the pure mass-monopole solution (the Schwarzschild spacetime) we have a2n = −M 2n+1 /(2n + 1). Exercise 6.16 Compute the first ten mass-moments for the Erez-Rosen metric from its spherical Weyl-moments. Solution Due to the equatorial symmetry all odd mass-moments vanish, i.e., M2n+1 = 0. Using the spherical Weyl-moments of the Erez-Rosen metric a2n

  2n m2n+1 1+ q . =− 2n + 1 2n + 3

in the relations to the mass-moments one obtains: M0 = M 2 3 M q 15 4 M4 = − M 5q 105   16 16 7 q+ M6 = −M q 1155 3465   32 64 2 1184 9 q − q+ M8 = −M q 96525 675675 45045   256 71936 2 2368 q + q− . M10 = M 11 q 43648605 3357585 2078505 M2 =

Chapter 7

The Post-Newtonian and MPM Formalisms

7.1 The Post-Newtonian Expansion The idea of the post-Newtonian formalism is to employ the fact that in the solar system velocities of astronomical bodies are small and gravitational fields are weak. The PN-formalism is a slow motion, weak field approximation to Einstein’s theory of gravity. The small parameter of this expansion is =

v c

(7.1.1)

and everywhere in the solar system we have  v 2 U p ∼ ∼ # ∼ ∼ 2 < 10−5 . c c2 ρc2

(7.1.2)

Here, p denotes the pressure, ρ the density of matter and # the specific internal energy density (internal energy density divided by the rest energy density). An upper limit for U/c2 will be given by GM/(c2 R), where M is the mass of some gravitating body and R its radius. For example, (E: Earth, S: Sun) GME 6.9 × 10−10 , c 2 RE

GMS 2.1 × 10−6 . c 2 RS

For the orbital velocity of the Earth about the Sun we have  v 2 E

c



30 300,000

2

10−8 .

Though the small parameter is dimensionless one usually uses c−1 as formal bookkeeping parameter to classify orders of magnitude. For celestial mechanical © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_7

235

236

7 The Post-Newtonian and MPM Formalisms

problems c−2 -terms in g00 are considered to be Newtonian and the first postNewtonian approximation usually neglects terms of order O(6, 5, 4) in 1/c for (g00 , g0i , gij ). The history of the post-Newtonian approximation goes back to Einstein (1915), Droste (1916), de Sitter (1916) and Lorentz and Droste (1917). It was modified and extended however by numerous authors such as Fock (1959), Chandrasekhar (1965), Chandrasekhar and Nutku (1969), Chandrasekhar and Esposito (1970) and many others. Note, that in such classical treatments usually the theory is formulated in a single coordinate system and matter is restricted to the case of an ideal fluid. Meanwhile, however, it is understood, that a physical adequate description of certain physical systems like a gravitational N -body system, requires the use of several different coordinate systems (e.g., Damour et al. 1991 (DSX-I)) and the framework becomes much more elegant if formulated for arbitrary matter (i.e., also for non ideal fluids) by using the basic variables σ = (T 00 + T ss )/c2 as gravitational source density and σ i = T 0i /c as gravitational current density.

7.2 The General Form of the Metric We will assume the existence of certain coordinates x μ = (ct, x i ) such that the following PN-assumptions hold g00 = −1 + O(c−2 ),

g0i = O(c−3 ),

gij = δij + O(c−2 ) .

(7.2.1)

In the following we will frequently encounter such order symbols that we will often abbreviate by On ≡ O(c−n ) .

(7.2.2)

Moreover, we will assume, in accordance with the results from Special Relativity, that T 00 = O(c2 ),

T 0i = O(c+1 ),

T ij = O(c0 )

(7.2.3)

and ∂ = O1 |∂i | . |∂0 | ≡ ∂ct

(7.2.4)

7.2 The General Form of the Metric

237

With assumptions (7.2.1) we can write the metric tensor in the form   2 g00 = − exp − 2 w c g0i = −

4 wi c3

(7.2.5) 

gij = γij exp +

2 w c2



with γij = δij + O2 .

(7.2.6)

Below we will show that the spatial field equations Gij = κ Tij

(7.2.7)

γij = δij + O4 .

(7.2.8)

will be satisfied by

Our canonical form of the metric therefore reads   2 2w 2w 2 g00 = − exp − 2 w = −1 + 2 − 4 + O6 c c c 4 wi c3   2 gij = δij 1 + 2 w + O4 . c

g0i = −

(7.2.9)

A few comments are in place with respect to this form of the metric tensor. Note that the metric tensor is completely specified by two potentials: a scalar potential w(t, x) and a vector potential wi (t, x) just as it was the case for Maxwell’s theory of electromagnetism. A comparison with the ‘Newtonian’ limit of the metric reveals that the scalar potential w generalizes the Newtonian gravitational potential U . As will become clear from the following a split of w into a Newtonian and some postNewtonian part is not meaningful; it only gives rise to confusion. The potential w determines the time-time and the space-space part of the metric tensor. The time-space component, g0i , that is determined by the vector potential wi described gravito-magnetic type effects, i.e., effects that arise from matter-currents (moving or rotating masses). What is called the post-Newtonian metric depends upon the problem of interest and the corresponding dynamical equations of motion. As we

238

7 The Post-Newtonian and MPM Formalisms

shall see later for celestial mechanical problems g00 = −1 +

2w 2w 2 − 4 c2 c

4 wi c3   2 gij = δij 1 + 2 w c

g0i = −

(7.2.10)

is called the post-Newtonian metric where we dropped the order symbols. We will now compute the differential geometrical quantities to post-Newtonian accuracy for celestial mechanical problems: 2

g ≡ + det gμν = −e4w/c + O4 √ 2 −g = e2w/c + O4 .

(7.2.11)

The inverse metric g μν is given by   2 2w 2w 2 g 00 = − exp + 2 w = −1 − 2 − 4 + O6 c c c 4 wi + O5 c3     2 2 = = δij exp − 2 w = δij 1 − 2 w + O4 . c c

g 0i = − g ij

The proof follows from a direct calculation: 1 = δ00 = g 0μ g0μ = g 00 g00 + g 0i g0i = g 00 g00 + O6 0 = δ0i = g iμ g0μ = g i0 g00 + g ij g0j = −g i0 + g0i + O5 δji = g iμ gj μ = g i0 gj 0 + g ik gj k = g ik gj k + O6 . Next we compute the Christoffel-symbols α βγ =

1 αλ g (gλβ,γ + gλγ ,β − gβγ ,λ ) , 2

e.g., 0 00 =

1 00 1 w,0 g g00,0 + g 0i (gi0,0 + gi0,0 − g00,i ) = − 2 + O5 . 2 2 c

(7.2.12)

7.2 The General Form of the Metric

239

α In the same way we find all components of βγ

w,0 + O5 c2 w,i 0i0 = − 2 + O6 c w,0 4 ij0 = δij 2 + 3 w(i,j ) + O5 c c w,i w w,i 4 i 00 = − 2 + 4 4 − 3 wi,0 + O6 c c c 4 w,0 i 0j = − 3 w[i,j ] + 2 δij + O5 c c w w w,i ,j ,k jik = δij 2 + δik 2 − δj k 2 + O4 c c c 0 00 =−

(7.2.13)

where 1 (wi,j + wj,i ) 2 1 ≡ (wi,j − wj,i ) . 2

w(i,j ) ≡

(7.2.14)

w[i,j ]

(7.2.15)

The direct calculation of the components of the Riemann tensor is left as an exercise. Exercise 7.1 Proof by direct calculation that the following relations hold: 1 1  w,ij − 4 −2ww,ij + δij w,tt + 2∂t (wi,j + wj,i ) c2 c  − 3w,i w,j + δij w,k w,k + O6 (7.2.16)

R0i0j = −

R0ij k =

  δij w,tk − δik w,tj + 2∂i (wj,k − wk,j ) + O5

Rij kl

  δil w,j k − δj l w,ik + δkj w,il − δki w,j l + O4

1 c3 1 = 2 c

and R 0 ij k = −R0ij k + O5 = R i 0j k + O5 R i j kl = Rij kl + O4 R 0 i0j =

w,ij 1  + 4 δij w,tt + 2∂t (wi,j + wj,i ) 2 c c  − 3w,i w,j + δij w,k w,k + O6

(7.2.17)

240

7 The Post-Newtonian and MPM Formalisms

1 1  −4ww,ij + δij w,tt + 2∂t (wi,j + wj,i ) w − ,ij c2 c4  − 3w,i w,j + δij w,k w,k + O6 .

R i 0j 0 = −

From this we derive the components of the Ricci-tensor Rμν = R σ μσ ν . E.g., R00 = R σ 0σ 0 = R i 0i0 . In this way we get  1 1  w − 4 −4w w + 3w,tt + 4∂ti wi + O6 2 c c   1 R0i = 3 −2w,ti + 2(wi − ∂ij wj ) + O5 c 1 Rij = − 2 δij w + O4 c

R00 = −

(7.2.18)

and 1 3 4 w − 4 w,tt − 4 ∂ti wi + O6 c2 c c  2  = − 3 wi − ∂ij wj − ∂ti w + O5 c 1 = − 2 δij w . c

R 00 = − R 0i R ij

(7.2.19)

From this we find the curvature scalar R = g μν Rμν = g 00 R00 + 2g 0i R0i + g ij Rij , i.e., R=−

2 w + O4 . c2

(7.2.20)

7.3 Field Equations and the Gauge Problem

241

7.3 Field Equations and the Gauge Problem We will now show that the spatial field equations are satisfied to O4 by γij = δij + O4 . According to (7.2.3) Tij = O0 and we have to prove that 1 8π G Rij − gij R = 4 Tij = O4 . 2 c The left hand side of this equation reads −

1 2 1 δij w + O4 + δij 2 w + O4 = O4 2 c2 c

as was to be shown. This implies that only the field equations G00 = κ T00 ,

G0i = κ T0i

or   8π G 1 00 00 αβ T g − g T αβ 2 c4   8π G 1 0i 0i αβ T − g gαβ T = 2 c4

R 00 =

(7.3.1)

R 0i

(7.3.2)

remain to be solved. Taking into account the order of T αβ one finds 4π G 00 (T + T ss ) + O6 c4 8π G = 4 T 0i + O5 . c

R 00 =

(7.3.3)

R 0i

(7.3.4)

Inserting expressions (7.2.19) we get the field equations in the form: w +

3 4 w,tt + 2 ∂ti wi = −4π Gσ + O4 c2 c

wi − ∂ij wj − ∂ti w = −4π Gσ i + O2

(7.3.5) (7.3.6)

with σ ≡

T 00 + T ss ; c2

σi ≡

T 0i . c

(7.3.7)

Here, σ acts as active gravitational mass-energy density generalizing the density ρ in Newton’s theory of gravity. σ i is the active gravitational mass-current density

242

7 The Post-Newtonian and MPM Formalisms

that does not act as a field generating source in Newton’s theory. The field equations (7.3.7) have a very remarkable property: they are linear in the metric potentials w μ ≡ (w, wi ) .

(7.3.8)

This results from the special form of the metric tensor and the various approximations involved. Note that Einstein’s field equations are non-linear in general and the achieved linearity simplifies the formalism tremendously. The choice of spatially isotropic coordinates with γij = δij + O4 correspond to three gauge conditions for the three spatial coordinates x i . Lemma 7.1 The spatial coordinates x i of the metric (7.2.9) are harmonic up to terms of O4 , i.e., g x i = O4 .

(7.3.9)

Proof: Let f = x i , then f,ν = δiν and √

√ √ √ −g g x i = ( −gg μi ),μ = ( −gg 0i ),0 + ( −gg j i ),j  

 2w 2w 1 − 2 δij = 1+ 2 + O4 = O4 . c c ,j

Because of that Lemma only the gauge freedom of the time coordinate is left. The field equations simplify further if also the time coordinate is chosen to be harmonic, i.e., if g x 0 = O5 .

(7.3.10)

We have √ √ √ √ −g g x 0 = ( −gg μ0 ),μ = ( −gg 00 ),0 + ( −gg i0 ),i =−

4w,0 4wi,i − 3 + O5 . 2 c c

Hence we can write the condition for the time coordinate t to be harmonic in the form 0=−

4 (∂t w + ∂i wi ) + O5 . c3

(7.3.11)

7.3 Field Equations and the Gauge Problem

243

With this harmonic gauge the field equations take the form w = −4π Gσ + O4

(7.3.12)

wi = −4π Gσ + O2 ,

(7.3.13)

i

where  is the flat space d’Alembertian ≡−

1 ∂2 + . c2 ∂t 2

We can even combine the source- and the field-variables σμ ≡ (σ, σi );

wμ ≡ (w, wi )

(7.3.14)

and write in obvious notation  wμ = −4π Gσμ + O(4, 2) .

(7.3.15)

A comparison with Maxwell’s equations for the electromagnetic potential Aμ in the Lorentz-gauge Aα = −

4π μ j c

shows that formally there are not different. The gravitational constant G appears in the coupling constant of the gravitational field equations whereas the speed of light c appears in Maxwell’s equations. Of course the electric charge density can be positive and negative whereas the gravitational mass density σ will be positive for any reasonable kind of matter. In any case it should be obvious that the post-Newtonian field equations in the harmonic gauge, writing the metric in our canonical form are not more complex than Maxwell’s equations of electromagnetism. As already mentioned they are linear in the field variables. If we consider one isolated system (e.g., an idealized solar system) with no gravitational sources outside this system we can require gμν −→ ημν

|x| → ∞

(7.3.16)

i.e., we consider our space-time manifold to be asymptotically flat. For our potentials this implies (w, wi ) −→ 0

|x| → ∞ .

(7.3.17)

244

7 The Post-Newtonian and MPM Formalisms

With such a condition for asymptotic flatness the retarded solution of our field equations reads

μ

wR (t, x) = G

d 3x

σ μ (tR ; x ) , |x − x |

(7.3.18)

where tR again is the retarded time tR ≡ t −

|x − x | . c

Another possible solution is the advanced one with tR being replaced by tA with tA ≡ t +

|x − x | . c

Still another solution is μ

wM (t, x) =

1 μ μ [w (t, x) + wA (t, x)] . 2 R

(7.3.19)

This mixed solution that might also be called time-symmetric solution is in fact usually used in post-Newtonian theories. The reason for this is the following: if we expand σ μ around the coordinate time t we encounter a sequence of time derivatives and the first time derivative is related with irreversible processes such as the emission of gravity waves that do not occur in the first post-Newtonian approximation to Einstein’s theory of gravity. With r ≡ |x − x | we get

σ μ (t ∓ r/c; x ) σ μ (t; x ) = G d 3x =G d x r r

G ∂2 G ∂ ∓ d 3 x  σ μ (t; x ) + 2 2 d 3 x  σ μ (t; x )r . c ∂t 2c ∂t

μ wR/A (t, x)

3 

If we take the time symmetric solution then in the expansion the first time derivative terms cancel automatically, i.e., μ

wM (t, x) = G

d 3x

G ∂2 σ μ (t; x ) + |x − x | 2c2 ∂t 2

d 3 x  σ μ (t; x )|x − x | .

(7.3.20)

Note that the retarded, advanced or mixed solutions of the harmonic field equations are not the only ones. If w, wi solve these equations then also w = w −

1 ∂t λ; c2

1 wi = wi + ∂i λ 4

(7.3.21)

7.3 Field Equations and the Gauge Problem

245

corresponding to a change of the time variable of the form t −→ t  :

t  = t − c−4 λ .

(7.3.22)

This transformation implies: g00 =

∂x μ ∂x ν  ∂x 0 ∂x 0  g = g + O6 μν ∂x 0 ∂x 0 ∂x 0 ∂x 0 00

or      2 2w  2w 1 − 2 + ... − 1 − 2 + . . . = − 1 − 4 ∂t λ c c c   2w 2 = − 1 − 2 − 4 ∂t λ + . . . , c c i.e., w = w +

1 ∂t λ . c2

Similarly we get g0i =

∂x μ ∂x ν  ∂x 0   g = g + g + O5 0i ∂x i 00 ∂x 0 ∂x i μν

 = g0i −

∂x 0 + O5 , ∂x i

i.e., 1 wi = wi − ∂i λ . 4 Let wμ ≡ (w, wi ) satisfy the harmonic gauge condition (7.3.11). A change of the time variable according to (7.3.22) then leads to 1 ∂t  w  + ∂i wi = ∂t w + ∂i wi + λ + O2 , 4 i.e., the new potentials wμ also satisfy the harmonic gauge conditions provided λ = O2 . Then the new potentials also satisfy the harmonic field equations.

(7.3.23)

246

7 The Post-Newtonian and MPM Formalisms

7.4 The External Post-Newtonian Field of a Body The next part deals with an isolated matter distribution in some asymptotically flat space-time. We ask the question: how can the external field of the matter distribution be described to first post-Newtonian order. The harmonic gauge will be used, the metric is parametrized with the two potentials w and w i , and the mixed μ solution wM from (7.3.19)) that solves the harmonic field equations (7.3.15) will be employed. The Blanchet-Damour Theorem below says that there is a special class of (skeletonized) harmonic coordinates where the two metric potentials, and hence the post-Newtonian metric in the external region admit a simple canonical multipole expansion in terms of two families of multipoles: ML (mass-multipole moments) and SL (spin multipole moments). In this Theorem the mass-moments, ML are defined to PN-order, thus they explicitly contain 1/c2 -terms, and the spinmoments, SL , are defined to Newtonian order. They are called: BD-moments. Theorem 7.1 (Blanchet and Damour 1989) Outside of some isolated matter distribution the functions

wμ (t, x) = G

d 3x

σμ (t± ; x ) |x − x |

(7.4.1)

with t± ≡ tmixed =

1 (tret + tadv ) . 2

(7.4.2)

admit an expansion of the form (r = |x|) w(t, x) = G

  1 ∂L r −1 ML (t± ) + 2 ∂t  + O4 l! c

 (−1)l l≥0

   (−1)l d ∂L−1 r −1 MiL l! dt l≥1   1 l ij k ∂j L−1 r −1 SkL−1 − ∂i  + O2 . + l+1 4

(7.4.3)

wi (t, x) = −G

(7.4.4)

Here,  ≡ 4G

μL ≡

 (−1)l 2l + 1 ∂L (r −1 μL (t± )) (l + 1)! 2l + 3

(7.4.5)

d 3 x xˆ iL σ i (t, x)

(7.4.6)

l≥0

7.4 The External Post-Newtonian Field of a Body

247

and

d2 1 d 3 x xˆ L x2 σ 2(2l + 3)c2 dt 2

4(2l + 1) d − (l ≥ 0) d 3 x xˆ iL σ i (l + 1)(2l + 3)c2 dt

SL (t) ≡ d 3 x ij i σ j , (l ≥ 1)

ML (t) ≡

d 3 x xˆ L σ +

(7.4.7) (7.4.8)

where the brackets indicate the STF-part for those indices that are enclosed by the brackets. The Cartesian STF-tensors ML and SL are the post-Newtonian mass- and current multipole moments that have been first introduced in a paper by Blanchet and Damour (1989) and are called BD-moments. The proof of the BD-theorem proceeds in several steps. In a first step one shows that w(t, x) = G

 (−1)l l≥0

wi (t, x) = −G

l!

  ∂L r −1 λL (t± ) + O4

(7.4.9)

   (−1)l , ∂L−1 r −1 KiL−1 (t± ) l! l≥1

 l ij k ∂j L−1 r −1 SkL−1 (t± ) l+1    (−1)l 2l − 1 ∂iL−1 r −1 μL−1 (t± ) + O2 . +G l! 2l + 1 +

(7.4.10)

l≥1

Here,

d 3 x  xˆL σ  +

λL ≡

KL ≡ l

d2 1 2 2(2l + 3)c dt 2

d 3 x  xˆL x2 σ 

 d 3 x  xˆ

(7.4.11) (7.4.12)

where the sharp brackets again denote the STF-part with respect to the enclosed indices. Here we only derive this partial result for w(t, x)

w(t, x) = G

G d2 σ (t, x ) 3  d x + 2 2  |x − x | 2c dt

σ (t, x )|x − x | d 3 x  .

248

7 The Post-Newtonian and MPM Formalisms

Let r ≡ |x|; a Taylor expansion then yields w(t, x) = G

 (−1)l l!

l≥0

1 d2 + 2 2 2c dt

  1 d x σ x ∂L r

3 

3 

 L

 L



d x σ x (∂L r) .

Since (1/r) = 0 we can write ∂L (1/r) = ∂ˆL (1/r) and, therefore, we can replace x L in the first term on the right hand side of this equation by xˆ L . In the second term we use (3.3.41) for the STF-part of x L : xˆ L = x L +

[l/2] 

(−1)n

n=1

l!(2l − 2n − 1)!! (l − 2n)!(2l − 1)!!(2n)!!

× δ(i1 i−2... δi2n−1 i2n xi2n+1 ...il )j1 j1 ...jn jn . Since r = 2/r we see that r = 0 hence we need this formula for xˆ L only up to n = 1. That results from the fact that every n-term leads to a n r term because of the ∂L operator. This implies that in the second term we can use xˆ L = x L − = x L −

l!(2l − 3)!! δ(i i x  + ... (l − 2)!(2l − 1)!! 2 1 2 i3 ...il )jj l(l − 1) δ(i i x  + ... , 2(2l − 1) 1 2 i3 ...il )jj

so that

3 

 L

d x σ x (∂L r) = +

d 3 x  σ  xˆ L (∂L r) l(l − 1) 2(2l − 1)

d 3 x  σ  x L−2 x2 (∂L−2 r) .

Note that in the second term the sum over l starts from l = 2 leading to a contribution to w of the form  

1 d2 (−1)l G  3    2 x σ x ˆ x ∂ d L−2 L−2 (l − 2)!(2l − 1) dt 2 r 2c2 l≥2

 

1 G  (−1)l d 2 3    2 = 2 d x σ xˆL x ∂L l!(2l + 3) dt 2 r 2c l≥0

7.4 The External Post-Newtonian Field of a Body

249

where we have replaced the dummy index l − 2 by l. Altogether we get w(t, x) = G

 (−1)l l!

l≥0

3 

d x

σ  xˆL ∂L

  1 r

 (−1)l d 2 +G d 3 x  σ  xˆL (∂L r) l! dt 2 l≥0

+

 

G  (−1)l d 2 1 3    2 . x σ x ˆ x ∂ d L L 2 2 l!(2l + 3) dt r 2c l≥0

From this we get the result from (7.4.9). Note, that t± produces the second term of the above relation as second term of a Taylor expansion in terms of r/c. In a second step one proves that KL =

d ML + O2 . dt

(7.4.13)

Exercise 7.2 Use the continuity equation σ,t + σ i ,i = O2 and the following relations l δi 2l + 1

(7.4.14)

r∂i nˆ L = (l + 1)ni nˆ L − (2l + 1)nˆ iL

(7.4.15)

ni nˆ L = nˆ iL +

to prove relation (7.4.13). Finally we will rewrite the expression (7.4.9) for w(t, x) w(t, x) = G

 (−1)l l≥0

+G

 (−1)l l≥0

=G

l!

4(2l + 1) d 2 (l + 1)(2l + 3)c dt



d x xˆ σ 3

iL i

  1 ∂L r −1 ML (t± ) + 2 ∂t  + O4 . l! c

 (−1)l l≥0

l!

  ∂L r −1 ML (t± )   ∂L r −1 + O4

250

7 The Post-Newtonian and MPM Formalisms

The final result for wi is obtained by rewriting the last term in (7.4.9) G

   (−1)l 2l − 1 ∂iL−1 r −1 μL−1 l! 2l + 1 l≥1

= −G

   (−1)l 2l + 1 ∂iL r −1 μL (l + 1)! 2l + 3 l≥0

1 = − ∂i  . 4 This completes our proof of the BD-theorem. One might wonder why the last term in expression (7.4.7) for the BD-mass moments ML has to be included. However, the expression for wi indicates that the −term has to be understood as a gauge term that should not appear in the expressions for the BD-moments. Exercise 7.3 Organize of copy of Blanchet and Damour (1989) and complete the remaining parts of the proof of the BD-theorem. It is not difficult to see that  is a harmonic function, i.e.,  = 0. For that reason we can remove the -terms by means of a gauge transformation; in such a skeletonized harmonic gauge the potentials w and wi also satisfy our harmonic field equations (7.3.12). The use of the skeletonized harmonic gauge is useful if the gravitational potentials are needed only outside the gravitating matter as is the case for many celestial mechanical problems. If one deals with gravity fields at the surface or inside some gravitating body usually multipole expansions and hence the skeletonized harmonic gauge become useless. It is interesting to note that the physical implications of the gravito-magnetic potentials wi (t, x) are determined by a corresponding gravito-magnetic field: Bi = −4 ij k ∂j wk .

(7.4.16)

If only the spin-moments are considered, one has Bi = 4G

 (−1)l l≥1

l!

l ij L−1 Sj L−1 l+1

with ij L−1 ≡ ∂ij L−1

  nˆ ij L−1 1 = (−1)l+1 (2l + 1)!! l+1 . r r

(7.4.17)

7.4 The External Post-Newtonian Field of a Body

251

For the spin-dipole with l = 1, Bi = −

6G nˆ ij Sj . r3

(7.4.18)

For many applications we can neglect the time derivatives of ML . That is the case for bodies with approximate axial symmetry rotating approximately around their symmetry axis. We will also neglect the higher spin-moments SL for l > 1. With these approximations in the skeletonized harmonic gauge the metric potentials read: w(t, x) G

 (−1)l l!

l≥0

wi (t, x)

ML ∂L (r −1 ) + O4

(7.4.19)

G G ij k ∂j (r −1 Sk ) = − ij k Sj ∂k (r −1 ) 2 2

=−

G (x × S)i + O2 . 2 r3

This result for w is remarkable indeed because formally it agrees with the Newtonian one. Note, however, that here we are dealing with solutions of the post-Newtonian field equations! In other words: the post-Newtonian BD-mass moments ML have been introduced in such a clever way that the solution w of a relativistic field equation practically looks Newtonian. If one thinks about a parameter formalism whose parameters are determined from observational data then the BD-moments are the quantities that parametrize the gravitational field outside some matter distribution in our post-Newtonian framework; they are directly measurable (e.g., by an analysis of satellite orbits). Especially it makes no sense to split the ML ’s into some ‘Newtonian-part’ plus c−2 -corrections. Let us finally show that for an isolated system the BD-mass ML with l = 0

M=

1 d2 d xσ + 2 2 6c dt

3

4 d d xx σ − 2 3c dt 3

2

d 3 x x i σ i + O4

(7.4.20)

agrees with the ADM-Tolman mass

M

ADM

=

d 3 x ρT + O4

(7.4.21)

where ρT is the Tolman-density  w  T 00 . ρT = 1 + 2 2c c2

(7.4.22)

252

7 The Post-Newtonian and MPM Formalisms

Using the Newtonian continuity equation σ,t + σi,i = O2 one finds d dt





d 3 x x2 σ =

d 3 x x2 σ,t = −

d 3 x x2 σi,i + O2

=2

d 3 x x i σi + O2 .

For that reason 4 d − 2 3c dt

2 d2 d x x σi = − 2 2 3c dt 3

i

d 3 x x2 σ + O2

and the BD-mass can be written as

1 M = d 3 x σ − 2 V(t) + O4 . c

(7.4.23)

Here V is the so-called virial 1 d2 V= 2 dt 2

d 3 x x2 σ .

(7.4.24)

One proceeds with a Newtonian virial theorem in the form

V=

1 d 3 x T ss − σ w + O2 . 2

(7.4.25)

The proof involves the Newtonian Euler equation j

σ i ,t + ∂j (Ti ) = σ w,i + O2 . Using this relation we get

1 d d d 3 x x2 σ,t = d 3 x x i σi + O2 2 dt dt

  = d 3 x x i σ i ,t + O2 = d 3 x x i −∂j T ij + σ w,i + O2

V=

=

  d 3 x T ss + x i σ w,i + O2 .

(7.4.26)

7.5 The Multi-Polar, Post-Minkowskian (MPM) Formalism

253

The proof of the virial theorem is completed with



d x x σ w,i = − 3

i



3

3 

d x d x σσ

x



− xi ) |x − x |3

i (x i



x i (x i − x i ) |x − x |3 

i i i 3 3   x (x − x ) + d xd x σ σ |x − x |3



 1 1 3 3  σσ =− =− d xd x d 3 x σ w + O2 . 2 |x − x | 2 1 =− 2

d 3x d 3x σ σ 

Using this virial theorem we finally get (σ = (T 00 + T ss )/c2 )



1 1 3 ss M = d x σ − 2 d x T − σ w + O4 2 c 00



00 T 1 T = d 3x + w + O = d 3 x ρT + O4 4 c2 2c2 c2

3

= M ADM + O4 .

7.5 The Multi-Polar, Post-Minkowskian (MPM) Formalism The MPM-formalism considers isolated systems, where the source for the metric field is confined to a spatial region inside some radius RB of some suitably chosen system of coordinates. Harmonic coordinates are used throughout where the Einstein field equations take the form of quasi-hyperbolic differential equations. These field equations are solved by means of approximation series in terms of small parameters. A central assumption is that inside the source gravitational fields are weak, the source is weakly stressed and internal velocities are small, thus the gravitational constant G and the inverse vacuum speed of light, 1/c, formally are considered as small parameters. The expression ‘post-Minkowskian’ refers to the expansion in terms of G; the 1/c-expansion leads to the post-Newtonian hierarchy. The (first) post-Newtonian approximation discussed above is contained in the MPMformalism, though different basic variables are usually used. If the isolated system under consideration is composed of individual ‘bodies’ (regions of compact support where the energy-momentum tensor differs from zero), the interest lies in the external fields which are treated with multipole expansions. For AGR the possibility to derive the external field of a single body with arbitrary shape and composition, rotating and vibrating, in principle up to any order in G and

254

7 The Post-Newtonian and MPM Formalisms

1/c in the frame of the MPM-formalism is fascinating indeed, though the framework becomes very complex with increasing orders. The main motivation for creating the whole MPM-formalism, however, was the problem of gravitational waves created by isolated sources, i.e., the problem to relate the time dependent energy-momentum tensor of some gravitational sources with the metric field in the wave zone. The ultimate aim of the MPM-formalism is to extract from the analytical theory some firm predictions for the outcome of gravitational wave experiments such as LIGO (Blanchet 2006, 2014). Such problems are, however, not treated in this book.

7.6 Several Expansions In this section we address the problem of the exterior metric of some isolated matter distribution. We assume that T αβ = 0 outside some body sphere

B ≡ {(t, x)||x| < RB } in suitable harmonic coordinates x α . Furthermore, we will assume asymptotic flatness lim hαβ = 0

|x|→∞

(7.6.1)

for any t = const. Here the basic variable is hαβ ≡

√

−gg αβ − ηαβ ,

where g αβ is the contravariant metric, g = det(gαβ ) and ηαβ = diag(−1, +1, +1, +1).1 The basic variable obeys the field equations hαβ =

16π G √ −gT αβ + αβ , c4

where  ≡ ηαβ ∂α ∂β is the flat space d’Alembertian, T αβ the energy-momentum tensor and αβ the gravitational source term.

1 Note,

that the basic variable in the paper by Damour and Iyer (1991a) is h¯ αβ = −hαβ .

7.6 Several Expansions

255

Under certain assumptions a solution to this problem can, in principle, be obtained by the multi-polar post-Minkowskian (MPM) formalism, that after pioneering work by Thorne (1980) and many others was essentially worked out by Blanchet, Damour and Iyer. The main references for the MPM-formalism are Thorne (1980), Blanchet and Damour (1986, 1989), Damour and Iyer (1991a), Blanchet (1995, 1998, 2006, 2014) and Blanchet and Faye (2001a). At present the standard reference is the Living Reviews article by Blanchet (2014) that also contains a comprehensive up to date list of relevant references. The MPM framework implies different kinds of expansions. Multi-polar indicates an expansion in terms of B ≡ RB /r , which is a multipole-expansion. Here RB is a coordinate radius chosen such that T αβ (t, x) = 0 for |x| > RB . Post-Minkowskian implies an expansion in terms of G ≡ GM/c2 r , which is assumed to be small; the MPM framework is thus a weak field approximation. Though the gravitational constant G has a dimension it is usually used as a bookkeeping parameter and the MPM formalism is based upon an expansion in terms of G. Usually the matter distribution will emit gravity waves that split the region outside of B into various domains • a near zone with r  λc • a far (or wave) zone with r  λc , where λc is the characteristic wavelength of the g-waves, λc = ctc =

2π c . ωc

(7.6.2)

Here tc is the time for noticeable changes to occur within the source; ωc is the characteristic frequency of emitted waves. The MPM-formalism requires a certain domain outside of B that lies in the inner near zone. This is the case for post-Newtonian sources that are slowly moving and weakly stressed. For such sources # ) T 0i T ij 1/2 U 1/2 c ≡ max 00 , 00 , 2  1. T T c Here, 1/c will be used as formal book-keeping parameter so that O(c−1 ) = O( c ) .

(7.6.3)

256

7 The Post-Newtonian and MPM Formalisms

7.7 First Post-Minkowskian Approximation The first post-Minkowskian approximation can be viewed as the starting point of the √ MPM-formalism. There we can neglect αβ , which is of order G2 and put −g = 1 so that the field equations take the form hαβ =

16π αβ T c4

(7.7.1)

with hαβ ,β = 0 as harmonic gauge condition. I Multipole expansion of the exterior retarded metric A special solution of the field equations with retarded potentials reads αβ hR

16π 4G αβ = 4 −1 =− 4 R T c c

d 3x

T αβ (tR , x ) |x − x |

(7.7.2)

with tR ≡ t −

|x − x | . c αβ

Outside of B this retarded solution hR admits a multipole expansion of the form  αβ  #  αβ ) ∞ FL (u) FL (u) 4G  (−1)l αβ ∂L hR (t, x) = − 4 ≡ S ∂L (7.7.3) l! r r c l=0

with u ≡ t − |x|/c. Here we used the compact notation S[L ] ≡ −

4G  (−1)l L . l! c4

(7.7.4)

l

II Irreducible multipole expansion ij Now, FL ≡ FL00 is STF in all indices, but GiL ≡ FL0i and Hij L ≡ FL are αβ not and have to be decomposed into irreducible STF-pieces. Outside of B hR can be written in terms of a multipole expansion involving ten (10) irreducible moments AL , BL , CL , DL , EL , FL , GL , HL , IL , JL . III The canonical metric These ten irreducible moments are, however, not independent of each other due αβ to the local equations of motion (T,β = 0 at this level of approximation). Therefore, the ten moments can be replaced by only six moments ML , SL , WL , XL , YL , ZL .

7.7 First Post-Minkowskian Approximation

257

These six moments have the following meaning: there is a special system of harmonic coordinates where the exterior metric tensor admits a multipole expansion in terms of two families of multipoles only: mass-moments ML and spin-moments SL . These special coordinates are called canonical (or skeletonized) harmonic coordinates; the corresponding metric tensor is called canonical. The remaining four families of moments WL , XL , YL and ZL determine a (harmonic) gauge transformation between the retarded and the canonical metric tensors. We now come to the details. Theorem 7.2 (Blanchet and Damour 1986) Let

φ(t, x) =

d 3x σ (tR , x ) |x − x |

(7.7.5)

be the retarded solution of the wave equation for a scalar function φ(t, x). Consider a body with source σ that is confined to a coordinate sphere of radius RB . Then, outside of B the solution (7.7.5) admits a multipole-expansion of the form  (−1)l

φ(t, x) =

l≥0

l!

* ∂L

FˆL (u) r

+ (7.7.6)

with r ≡ |x|, y = |y| and u ≡ t − |x|/c .

(7.7.7)

Furthermore, FˆL (u) ≡



d 3 y yˆL

+1

−1

dηδl (η)σ (u + ηy/c, y)

(7.7.8)

with δl (η) ≡

(2l + 1)!! (1 − η2 )l . 2l+1 l!

(7.7.9)

The function δl (η) has the properties

+1 −1

dη δl (η) = 1

lim δl (η) = δ(η)

l→∞

(7.7.10) (7.7.11)

258

7 The Post-Newtonian and MPM Formalisms

where δ(η) is the one-dimensional Dirac distribution. We write FˆL (u) in the form   FˆL (u) = I yˆL δl (η)σ (u + ηx/c, y)

(7.7.12)

with

I [f (y, η)] ≡

3

d y

+1 −1

dη f (y, η) .

(7.7.13)

Proof The proof of this Theorem is rather lengthy and the reader is referred to Blanchet and Damour (1986). A detailed proof was published by Zschocke (2014) which will not be repeated here. Exercise 7.4 Proof that

+1 −1

dη δl (η) = 1 .

Solution The integral reads (2l + 1)!! 2l+1 l!

+1 −1

dη (1 − η2 )l =

(2l + 1)!! (l + 1)(1/2) =1 2l+1 l! (l + 1 + 1/2)

since (l + 1) = l! and (l + 1 + 1/2) = (2l + 1)!!/2l+1 · (1/2). Exercise 7.5 Proof that the following relations hold: 1 δ  + ηδl = 0 2l + 3 l+1

(7.7.14)

and δl − δl−1 +

1 δ  = 0 . (2l + 1)(2l + 3) l+1

Here, δl ≡ d(δl (η))/dη. Solution Relation (7.7.14) is obtained directly so that −

1 δ  = δl − η2 (2l + 1)δl−1 2l + 3 l+1

which leads to relation (7.7.15).

(7.7.15)

7.7 First Post-Minkowskian Approximation

259

αβ

The retarded metric hR then reads αβ hR (t, x)

4G =− 4 c

d 3x

T αβ (tR ; x ) . |x − x |

(7.7.16)

According to Theorem (7.2), outside of B , hαβ admits a multipole expansion of the form #  αβ ) FL (u) αβ M(hR (t, x)) = S ∂L (7.7.17) r with αβ

FL (u) = I [δl (η)T αβ (u + ηy/c; y)] .

(7.7.18)

Note that the symbol M for multipole expansion reminds us that this expression is only valid outside of B , i.e., outside a coordinate sphere that completely encompasses our central matter distribution. We write FL ≡ FL00 ,

GiL ≡ FL0i ,

ij

Hij L ≡ FL

(7.7.19)

so that

$  FL (u) , M(h00 x)) = S ∂ (t, L R r

$  GiL (u) 0i , M(hR (t, x)) = S ∂L r

$  Hij L (u) ij , M(hR (t, x)) = S ∂L r

(7.7.20) (7.7.21) (7.7.22)

where FL (u) ≡ I [yˆL δl (η)T 00 (u + ηy/c; y)]

(7.7.23)

GiL (u) ≡ I [yˆL δl (η)T (u + ηy/c; y)]

(7.7.24)

Hij L (u) ≡ I [yˆL δl (η)T ij (u + ηy/c; y)] .

(7.7.25)

0i

We now come to the decompositions of GiL and Hij L into irreducible pieces. In fact, any tensor TP can be decomposed into a finite sum of terms of the form γPL Rˆ L (Damour and Iyer 1991a), where γPL is made of products of ij k and δij and Rˆ L is STF (Blanchet and Damour 1986). Let TˆL be STF, then Ui TˆL can be taken from

260

7 The Post-Newtonian and MPM Formalisms

relation (13.2.18) of Appendix: (+) Ui TˆL = Rˆ iL +

l 2l − 1 (0) (−) + , sis δi l+1 2l + 1

(7.7.26)

where (+) Rˆ L+1 ≡ U , (0) Rˆ L ≡ Ua Tˆbab , (−) Rˆ L−1 ≡ Us TˆsL−1 .

Using this relation we can decompose GiL as GiL = (+) GiL + aia + δi

(7.7.27)

GL+1 ≡ G ,

(7.7.28)

with (+)

(0)

(−)

l Gabab , l+1 2l − 1 GaaL−1 . ≡ 2l + 1

GL ≡

GL−1

(7.7.29) (7.7.30)

In this way M(h0i ) is decomposed into three pieces involving the three STF-tensors and (+) G. Let us look at the (−) G-part more closely. It leads to a term

(−) G, (0) G

∞  (−1)l l=1

l!

∂L





∞ 1 1 (−) (−1)l ˆ (−) δi (u) = GL−1 (u) ∂iL−1 r l! r l=1

We now use the relation (13.2.3) from Appendix, ∂ˆL =

[l/2]  k=0

(−1)k

(2l − 2k − 1)!! δ{i1 i2 . . . δi2k−1 i2k ∂i2k+1 ...il } k , (2l − 1)!!

(7.7.31)

where the curly bracket implies a summation over the smallest set of permutations of enclosed indices that make the expression fully symmetric in i1 , . . . , il . Because of the STF character of (−) GL−1 only the terms with k = 0 and k = 1 contribute so that





ˆ∂iL−1 1 (−) GL−1 = ∂iL−1 1 (−) GL−1 − 1 δ{ii1 ∂i2 ...il−1 }  1 (−) GL−1 . r r 2l − 1 r

7.7 First Post-Minkowskian Approximation

261

By a corresponding argument we see that from the curly bracket only those (l − 1) terms contribute where the index i is in the Kronecker-delta. Therefore, ∞  (−1)l l=1

l!

∂L



 ∞ 1 1 (−) (−1)l (−) δi = ∂iL−1 GL−1 r l! r l=1

$ 1 l−1 ∂L−2  (−) GiL−2 − 2l − 1 r

and changing the dummy index l to l + 1 the last term reads: ∞  (−1)l l=1

l!

l 1 (−) ∂L−1  GiL−1 (l + 1)(2l + 1) r

Exercise 7.6 Let u ≡ t − |x|/c = t − r/c. Proof that:  1 1 f (u) = f¨(u) ,  r r

(7.7.32)

1 d 2 f (u) . f¨(u) ≡ 2 c du2

(7.7.33)



where

Solution We have (f ≡ f (u)): 

  1 1 f + f,j r ,j r       1 1 1 1 f = f +2 ∂jj f,j + f,jj . r r r ,j r ∂j

1 f r



=

Now, in obvious notation (1/r) = 0, f,j = −f˙ r,j and f = f¨ r,j r,j − f˙ r. With r,j r,j = 1, r = 2/r and (1/r),j = −x j /r 3 we obtain relation (7.7.32). Using relation (7.7.32) we get for the (−) G-part  (−1)l l

l!

# ∂iL−1



(−) G L−1

r



l ∂L−1 + (l + 1)(2l + 1)



(−) G ¨ iL−1

r

)

262

7 The Post-Newtonian and MPM Formalisms

and M(h0i R ) has four terms  #    (+) G (0) G  (−1)l 4G iL aia l M(h0i ∂L + ∂L R) = − 4 l! r r c l    ) (−) G (−) G ¨ iL−1 l L−1 + ∂iL−1 + ∂L−1 . r (l + 1)(2l + 1) r (7.7.34) Consider now the large curly bracket only. Changing from l to l − 1 in the first term and combing it with the 4th term we get ∂L−1

CiL−1 r

CiL−1 = −l (+) GiL−1 +

with

l (−) ¨ GiL−1 . (l + 1)(2l + 1)

Changing from l to l + 1 in the 3rd term we get ∂iL

BL r

with

BL = −

1 (−) GL . l+1

Finally in the second term we put il = b so that ∂L → ∂bL−1 and  ∂bL−1

(0)

aia r



= iab ∂aL−1

DbL−1 r

with

DbL−1 = (0) GbL−1 .

So finally we end up with

$  AL (u) , M(h00 (7.7.35) x)) = S ∂ (t, L R r



$  BL (u) CiL−1 (u) DbL−1 (u) + ∂ + M(h0i ∂ x)) = S ∂ (t, iL iab aL−1 L−1 R r r r

(7.7.36) with AL = FL , l ≥ 0 , 1 (−) GL , l ≥ 0 , l+1 l CL = −l (+) GL + (l + 1) (2l + 1) c2

BL = −

DL =

(0)

GL , l ≥ 1 .

(−)

¨L , l ≥ 1, G (7.7.37)

7.7 First Post-Minkowskian Approximation

263

Using relation (13.2.22) from Appendix M(hij ) can be decomposed into irreducible pieces (Damour and Iyer 1991a):





Gj )L−1 EL FL + δ + ∂ M(hij ∂ x)) = S ∂ (t, ij L ij L L−1(i R r r r



$ ab(i Jj )bL−2 I HbL−1 ij L−2 + ab(i ∂j )aL−1 + ∂L−2 + ∂aL−2 . r r r

(7.7.38) Here, the moments EL , . . . , JL are functions of u ≡ t − |x|/c and if L appears in the form L − i then in the sum that appears in S the summation index l goes from i to infinity. The moments EL , . . . , JL are given by (Damour and Iyer 1991a, Eqs. (5.10))

EL =

1 (−2) HL , (l + 2) (l + 1)

FL = KL − GL =

(0)

HL = −

1 (0) 1 (−2) ¨ HL , HL − 3 (l + 2) (l + 1) (2l + 3)

HL −

2l (−2) ¨ HL , (l + 2) (l + 1) (2l + 3)

1 (−1) HL , l+1

IL = l (l − 1) (+2) H L − JL = −l (+1) H L +

l − 1 (0) ¨ l (l − 1) (−2) .... HL + HL, (2l − 1) (l + 2) (l + 1) (2l + 3) (2l + 1)

l−1 (−1) ¨ HL . (l + 1) (2l + 1)

(7.7.39)

Since the dimension of (i) HL is (length)l−i Mc2 we see that all terms on the right hand side of (7.7.38) are dimensionless as it should be. The ten integrals appearing in (7.7.37) and (7.7.39) read (Damour and Iyer 1991a, Eqs. (5.11))   FL (u) = I δl yˆL T 00 ,   (+) GL (u) = I δl−1 yˆ 0 ,

l (0) δl yˆbab T 0a , GL (u) = I l+1

2l + 1 (−) δl+1 yˆaL T 0a , GL (u) = I 2l + 3

264

7 The Post-Newtonian and MPM Formalisms

  H L (u) = I δl−2 yˆ ,

2 (l − 1) (+1) ˆ H L (u) = I δl−1 STF yˆdL−2 il cd Tcil−1 , L l+1

6l (2l − 1) (0) ˆ δl yˆcc , H L (u) = I (l + 1) (2l + 3)

2l (2l + 1) (−1) δl+1 yˆbcab Tˆca , H L (u) = I (l + 2) (2l + 3)

2l + 1 (−2) δl+2 yˆacL Tˆac , H L (u) = I 2l + 5

1 aa , δl yˆL T KL (u) = I 3 (+2)

(7.7.40)

where the argument of T αβ always is (u + ηy/c; y), Tˆij ≡ Tij − δij Taa /3 and the argument of δl is η, δl ≡ δl (η). The subscript L of the STF-operator refers to the indices L − 2, il−1 and il . Thus, outside of B the retarded metric in harmonic gauge admits a multipole expansion in terms of the ten irreducible moments AL , . . . , JL . These moments, however, are not independent from each other, due to the local conservation equations ∂β T αβ = 0

(7.7.41)

neglecting all terms proportional to the gravitational constant G. Considering the arguments of T αβ we have for G = 0: ∂β T αβ (τ ; y) = 0 with τ = u + η|y|/c. Therefore, ∂ α0 ∂ T (τ, y) = − i T αi (τ, y) . ∂cu ∂y Let d αi ∂ ∂ ηni T αi (τ, y) T (τ, y) = i T αi (τ, y) + i dy ∂y ∂cu y

(7.7.42)

7.7 First Post-Minkowskian Approximation

265

be the total derivative with respect to y i , including the derivative of τ . Then, with arguments (τ, y): ∂ αi ∂ α0 d T = − i T αi + ηni T . ∂cu dy ∂cu

(7.7.43)

Lemma 7.2 Due to the local equations of motion (7.7.43) the following relations (Eqs. (5.16), (5.20), (5.22) and (5.26) of Damour and Iyer 1991a) hold (the dot here stands for the derivative with respect to (cu) so produces the dimension of an inverse length; note, that the dimensions of the ten irreducible moments AL , . . . , JL are Mc2 times (length)i , where i = l (AL , DL , FL , GL ), i = l − 1 (CL , JL ), i = l + 1 (BL , HL ), i = l − 1 (CL , JL ), i = l − 2 (IL ), i = l + 2 (EL )): CL = −A˙ L − B¨L ,

l ≥ 1,

GL = −2 B˙L − 2 E¨L − 2 FL , JL = −2 D˙ L − H¨ L ,

(7.7.44) l ≥ 1,

l ≥ 2.

(7.7.45)

(7.7.46)

Moreover, ... .... IL = A¨L + 2 B L + E L + F¨L ,

l ≥ 2.

(7.7.47)

Here we will prove only relation (7.7.44). With the definitions of AL , BL and CL one has to show that T1 + T2 + T3 = 0 with   T1 = −l (+) GL = −lI δl−1 yˆ 0   (−) 1 1 ¨L = − G I¨ δl+1 yˆiL T 0i (2l + 1) (2l + 3)   T3 = F˙L = I˙ δl yˆL T 00 .

T2 = −

Let us first rewrite T2 with two partial integrations using ∂cu = (1/y)∂η

(7.7.48)

and δl (±1) = 0: T2 = −

  1  y −2 yˆiL T 0i . I δl+1 (2l + 3)

(7.7.49)

266

7 The Post-Newtonian and MPM Formalisms

Using relation (13.2.6) of Appendix in the form: yˆiL = −

l y 2 δi + yi yˆL 2l + 1

T2 can be rewritten in the form T2 =

    1 l   yˆ 0 − y −2 yi yˆL T 0i . I δl+1 I δl+1 (2l + 1)(2l + 3) 2l + 3 (7.7.50)

In T3 we use relation (7.7.43) for α = 0 so that

 ∂  d 0i T3 = −I δl yˆL i T + I δl ηyˆL niy T 0i dy ∂cu

(7.7.51)

and finally, using relation (13.2.12) of Appendix: d yˆL = lδi dy i

(7.7.52)

T3 can be written in the form     T3 = lI δl y 0 + I˙ δl ηyˆL niy T 0i .

(7.7.53)

Again using ∂η = (|y|/c)∂u the second term in T2 can be written as +

 1 ˙  I δl+1 yˆL niy T 0i . 2l + 3

and thus  0 T1 + T2 + T3 = lI yˆ T δl − δl−1 +  +I˙ yˆL niy T 0i

1 2l + 3

1 δ  (2l + 1)(2l + 3) l+1 

 =0 δl+1 + ηδl



in virtue of (7.7.14) and (7.7.15). The proof of (7.7.45)–(7.7.47) is complex and the reader is referred to Damour and Iyer (1991a). By using these four relations (7.7.44)–(7.7.47) we recognize that outside of B the metric tensor admits an expansion in terms of six multipoles only, namely AL , BL , DL , EL , FL , and HL .

7.7 First Post-Minkowskian Approximation

267

It is convenient to introduce six new moments that are STF-tensors as functions of u: ML c2 = AL + 2B˙L + E¨L + FL l 1 SL = −DL − H˙ L l+1 2

(7.7.54) (7.7.55)

and 1 WL = −BL − E˙L 2 1 XL = + EL 2 YL = −B˙L − E¨L − FL 1 ZL = + HL 2

(7.7.56) (7.7.57) (7.7.58) (7.7.59)

Exercise 7.7 Show that the moments AL , BL , DL and EL can be written in the form AL = ML c2 + W˙ L + X¨ L + YL

(7.7.60)

BL = −WL − X˙ L

(7.7.61)

l SL − Z˙ L l+1 EL = +2XL .

DL = −

(7.7.62) (7.7.63)

Exercise 7.8 Show that the four relations following from the local equations of motion (7.7.41) can be written in the form CL = −M˙ L c2 − Y˙L

(7.7.64)

GL = +2YL

(7.7.65)

IL = M¨ L c

(7.7.66)

JL = +2

2

l ˙ SL . l+1

(7.7.67)

Theorem 7.3 (Damour and Iyer 1991a) Outside of B M(hαβ ) is related with a αβ metric hcan by a gauge transformation of the form αβ

hcan = M(hαβ ) + ∂ α ϕ β + ∂ β ϕ α − ηαβ ∂μ ϕ μ

(7.7.68)

268

7 The Post-Newtonian and MPM Formalisms

with

$  ML c 2 (7.7.69) h00 = +S ∂ L can r #  (1) 

) c M S c l bL−1 iL−1 iab ∂aL−1 h0i + (7.7.70) can = −S ∂L−1 r l+1 r ⎧ ⎡ ⎡ ⎤ ⎤⎫ (1) ⎨ ab(i Sj )bL−2 ⎬ Mij(2)L−2 2l ij ⎦+ ⎦ ∂aL−2 ⎣ hcan = +S ∂L−2 ⎣ ⎩ ⎭ r l+1 r (7.7.71) and f (n) (u) ≡

d n f (u) . dun

(7.7.72)

The relation between M(hαβ ) and h αβ = hcan is given by a gauge transformation of the harmonic coordinates of the form αβ

x μ −→ x  = x μ + ϕ μ μ

(7.7.73)

with 

$ WL (u) ϕ 0 = −S ∂L r

 XL (u) i ϕ = +S ∂iL r

$ YiL−1 (u) ZbL−1 (u) + iab ∂aL−1 . + ∂L−1 r r

(7.7.74) (7.7.75) (7.7.76)

Proof Let x  μ = x μ + ϕ μ , where x μ are the original harmonic coordinates in which M(hαβ ) is defined. Then, ∂x  μ μ = δνμ + ϕ,ν ∂x ν and the Jacobian J ≡ det(∂x  μ /∂x ν ) to first order in G is given by μ μ | = 1 + ϕ,μ . J = |δνμ + ϕ,ν

(7.7.77)

7.7 First Post-Minkowskian Approximation

269

We generally have 

αβ μ −g  g  (x  )

∂x √ ∂x  α ∂x  β σρ μ =  −g σ g (x ) ∂x ∂x ∂x ρ

so that ηαβ + h

αβ

=



−g  g 

αβ

=

1 α α β (δ + ϕ,σ )(δρβ + ϕ,ρ )(ησρ + hσρ ) . J σ

(7.7.78)

From this relation (7.7.68) follows to first order in G. Since hαβ is harmonic, i.e., ∂β hαβ , we have ∂β h

αβ

= ∂ α ∂β ϕ β + ∂β ∂ β ϕ α − ηαβ ∂βμ ϕ μ = ∂β ∂ β ϕ α

which vanishes because of the structure of ϕ α and 

f (u) f¨(u) = . r r

Thus the gauge transformation preserves the harmonicity of our fundamental metric variable. Now, 00 0 i h00 can = M(hR ) + ∂0 ϕ − ∂i ϕ    ˙      XL AL WL YL = S ∂L − ∂L − ∂L  − ∂L r r r r 

 2 ML c = S ∂L r

in virtue of (7.7.60) and (XL /r) = X¨ L /r. 0i i 0 h0i can = M(hR ) + ∂0 ϕ − ∂i ϕ        BL CiL−1 DbL−1 + ∂L−1 + iab ∂aL−1 = S ∂iL r r r ˙  ˙ ˙    $ XL YiL−1 ZbL−1 WL + ∂L−1 + iab ∂aL−1 + ∂iL + ∂iL r r r r  ,    = S ∂iL r −1 (BL + X˙ L + WL ) + ∂L−1 r −1 (CiL−1 + Y˙iL−1 )  + iab ∂aL−1 r −1 (DbL−1 + Z˙ bL−1 ) .

270

7 The Post-Newtonian and MPM Formalisms

Relation (7.7.70) follows from BL + X˙ L + WL = 0, (7.7.61), Z˙ bL−1 = H˙ bL−1 /2 (7.7.59) and relations (7.7.64) and (7.7.55). The proof for (7.7.71) is left as an exercise. Exercise 7.9 Show that ij

ij

hcan = M(hR ) − ∂i ϕ j − ∂j ϕ i + δij ∂μ ϕ μ agrees with the expression from (7.7.71). ij

Proof M(hR ) is given by (7.7.38) and



$  2Yj )L−1 2XL 2ZbL−1 + ∂L−1(i + ab(i ∂j )aL−1 −∂i ϕ j − ∂j ϕ i = −S ∂ij L r r r ij

and since 2XL = EL and 2ZL = HL the corresponding terms cancel in hcan . The I and J -terms just give the result from (7.7.38) and all remaining terms together vanish due to GL = 2YL , FL = YL − B˙L − E¨L and B = −W˙ L − X¨ L . The Damour-Iyer Theorem says that there is a special system of harmonic coordinates, called ‘canonical’ or ‘skeletonized harmonic gauge’ where the solution of the field equations outside of some coordinates sphere B admits a convergent multipole expansion that is determined by only two families of multipole moments: mass-moments ML and spin-moments SL . The relation of this canonical form of the metric to the retarded solution (7.7.17) is determined by four families of multipole moments: WL , XL , YL and ZL . Explicitly the mass- and spin-moments read: ML (u) = I δl yˆL σ −

4(2l + 1) (1) δl+1 yˆiL σi (7.7.79) + 1)(2l + 3)

2(2l + 1) (2) δl+2 yˆij L σij + 4 c (l + 1)(l + 2)(2l + 5)  $

(2l + 1) (1) SL (u) = I aba σb − 2 δl+1 yˆL−1>ac σbc c (l + 2)(2l + 3) c2 (l

with σ ≡

T 00 + T ss ; c2

σi ≡

T 0i ; c

σij ≡ T ij

and T αβ = T αβ (u + ηy; y) ,

f (n) =

d nf . dun

(7.7.80)

7.7 First Post-Minkowskian Approximation

271

The remaining moments are given by WL (u) = I

2l + 1 δl+1 yˆiL σi (l + 1)(2l + 3)

2l + 1 (1) δl+2 yˆij L σij − 2 c 2(l + 1)(l + 2)(2l + 5)

2l + 1 XL (u) = I δl+2 yˆij L σij 2(l + 1)(l + 2)(2l + 5) 3(2l + 1) (1) YL (u) = I −δl yˆL σii + δl+1 yˆiL σi (l + 1)(2l + 3)

2(2l + 1) δl+2 yˆij L σij(2) − 2 c (l + 1)(l + 2)(2l + 5)  $

l (2l + 1) . ZL (u) = I abbc σac l + 1 (l + 2)(2l + 3) (7.7.81) Note, that in Blanchet (2014) a factor of l/(l + 1) was moved from ZL into ϕ i . Exercise 7.10 Show that the moments ML , SL , WL , XL , YL and ZL are given by the integral relations (7.7.79) and (7.7.81). Proof The elementary moments FL , (i) GL and (i) HL are explicitly given by relations (7.7.40). The proof then follows from the relations of ML . . . ZL to these elementary moments. E.g., (−2) 1 1 (−) 1 GL − H˙ L . WL = −BL − E˙L = 2 l+1 2(l + 1)(l + 2)

Exercise 7.11 Proof that Il ≡

+1 −1

dη δl (η)f (u + η|y|/c, y) =

∞  k=0

(2l + 1)!! 2k k!(2l + 2k + 1)!!



 |y| ∂ 2k f (u, y) . c ∂u

Solution A Taylor-expansion of Il yields Il =

∞ 

+1

k=0 −1

dη δl (η)

ηk k!



|y| ∂ c ∂u

k f (u, y)

(7.7.82)

272

7 The Post-Newtonian and MPM Formalisms

that vanishes for odd values of k so that Il =

∞ 

+1

k=0 −1

dη δl (η)

η2k (2k!)



|y| ∂ c ∂u

2k f (u, y) .

Now,

+1 −1

dη δl (η)η2k =

(2l + 1)!! +1 (2l + 1)!! (k + 1/2)(l + 1) . dη (1 − η2 )l η2k = l+1 2 l! −1 2l+1 l! (l + k + 1 + 1/2)

Using (k + 1/2) = (2k − 1)!!/2k · (1/2) and 1 (2k − 1)!! = k (2k)! 2 k!

(7.7.83)

(easily shown by induction) the result (7.7.82) is obtained. Using expression (7.7.82) and neglecting 1/c4 -terms we get

+1

−1

dη δl (η)σ (u + η|y|/c, y) = σ (u, y) +

1 2(2l + 3)



|y| ∂ c ∂u

2 σ (u, y) ,

so that the Blanchet-Damour mass moments agree with the post-Newtonian limit of the Damour-Iyer mass-moments. Therefore one obtains the following lemma: Lemma 7.3 In the first post-Newtonian limit the Damour-Iyer moments, ML and SL , reduce to the Blanchet-Damour moments (BD moments):

MLBD

d2 1 = d x xˆL σ + d 3 xx2 xˆL σ 2(2l + 3)c2 dt 2

4(2l + 1) d − d 3 x xˆ aL σ a + O4 (l + 1)(2l + 3)c2 dt 3

(7.7.84)

and

SLBD

=

d 3 x aba σb + O2 .

(7.7.85)

7.8 The MPM Algorithm Blanchet has shown that with certain assumptions a multipolar post-Minkowskian formalism can be established that provides an algorithm to compute the external metric of some body (some isolated matter distribution) to any order in G and c−1 in

7.8 The MPM Algorithm

273

Fig. 7.1 Various regions in the harmonic coordinate system defined by coordinate spheres of radii RB , Re and Ri

Ri

Re RB

Body

matching region

principle. However, working out this formalism to higher and higher orders becomes increasingly complicated. In the MPM formalism the near zone with r  λ will be divided into an inner part and an outer (exterior) zone by i and e

i ≡ {(t, x)| |x| < Ri }

e ≡ {(t, x)| |x| > Re }

(7.8.1)

with RB < Re < Ri (Fig. 7.1). In the inner region the metric is valid inside but also outside the body before |x| reaches Ri . Since Ri > Re there is a common domain, called the matching-region:

match ≡ i ∩ e .

(7.8.2)

Outside of B the MPM-formalism provides an approximate solution to Einstein’s field equations in the vacuum in certain harmonic coordinates of the form αβ

αβ

g αβ = glin + gnon−lin + ∂ϕ αβ .

(7.8.3)

The ‘linear part’ of the external metric is completely determined by two families of multipole-moments, ML and SL , and their time derivatives. The ‘non-linear part’ of the external metric consists of products of multipolemoments, ML and SL , that have been determined completely by the external field to lower order in G. The ‘gauge-functions’ ∂ϕ αβ , given by a gauge-vector ϕ α , are fully determined by four families of moments WL , XL , YL and ZL . √ The field equations are taken in the form (5.8.31) (hαβ = −gg αβ − ηαβ ) hαβ =

16π G |g|T αβ + αβ c4

274

7 The Post-Newtonian and MPM Formalisms

and the gravitational source term is written as αβ

αβ

αβ = 2 + 3 + · · · . Here the index on  refers to the powers of hαβ and NOT to the powers of G.

7.8.1 The First PN Approximation Within the MPM-formalism the first post-Newtonian approximation considers the metric corrections hαβ up to order O(6, 5, 6), for (h00 , h0i , hij ), i.e., it considers also terms of order c−4 in hij .

7.8.1.1

The Inner Metric

The internal metric, valid in i is determined by hαβ =

16π G |g|T αβ + αβ . c4

(7.8.4)

αβ

with h,β = 0. To lowest order one has h00 = h00 = −

4V c2

h0i = h0i = O3 hij = hij = O4 so that h = ηαβ hαβ = 4V /c2 + O4 and |g| = 1 + h + O4 = 1 + 4V /c2 + O4 . From this to post-Newtonian accuracy, i.e., up to terms of order O(6, 5, 6), one gets 00 = N 00 = −

14 V,k V,k c4

(7.8.5)

0i = N 0i = 0

(7.8.6)

ij = N ij

(7.8.7)



4 1 = 4 V,i V,j − δij V,k V,k . 2 c

7.8 The MPM Algorithm

275

Therefore, the field equations to post-Newtonian order read h00 =

  4V 16π G 14 1 + T 00 − 4 V,k V,k + O6 4 2 c c c

(7.8.8)

16π G 0i T + O5 c4

16π G ij 4 1 ij T + 4 V,i V,j − δij V,k V,k + O6 h = 2 c4 c

h0i =

(7.8.9) (7.8.10)

with the solution h00 = −

4 4 V + 4 (W − 2V 2 ) , 2 c c

h0i = −

4 Vi , c3

hij = −

4 Wij . c4 (7.8.11)

Here,

V (t, x) ≡ G

d 3x σ (t − |x − x |/c; x ) |x − x |

d 3x σi (t − |x − x |/c; x ) |x − x |  

d 3x 1 1 Wij (t, x) ≡ G σ V δ (t − |x − x |/c; x ) + V − V V ij ,i ,j ij ,k ,k |x − x | 4πG 2 Vi (t, x) ≡ G

(7.8.12) and σ ≡

T 00 + T ss , c2

σi ≡

T 0i , c

σij ≡ T ij ,

W ≡ Wkk .

(7.8.13)

All of these source terms, σ, σi and σij are considered to be of order c0 . Exercise 7.12 Show that (7.8.11) solves the field equations (7.8.8) to order O(6, 5, 6). Solution To the required order the three potentials, V , Vi and Wij obey V = −4π Gσ Vi = −4π Gσi Wij = −4π G σij +

(7.8.14)

1 1 (V,i V,j − δij V,k V,k ) . 4π G 2

(7.8.15) (7.8.16)

276

7 The Post-Newtonian and MPM Formalisms

From this we get 1 W = Wkk = −4π Gσkk + V,k V,k 2

(7.8.17)

V 2 = 2V,k V,k − 8π GV σ + O2 .

(7.8.18)

and

With these relations the post-Newtonian field equations are found directly. From the Newtonian local equations of motion σ,t + σi,i = O2 σi,t + σij,j = σ V,i + O2 one derives V,t + Vi,i = O2

(7.8.19)

Vi,t + Wij,j = O2 .

(7.8.20)

Exercise 7.13 Proof relations (7.8.19) and (7.8.20) from the Newtonian EOM. The harmonic gauge condition is satisfied approximately: ∂β hαβ = O(5, 6) .

7.8.1.2

The External Metric

Outside the gravitational source in the vacuum region the metric corrections are expanded into powers of G in the form αβ

αβ

hαβ = h(1) + h(2) + . . . .

(7.8.21)

αβ

The form of the linear terms h(1) are known from Thorne’s second structure Theorem. Theorem 7.4 (Thorne’s Second Structure Theorem) The general solution of the linearized vacuum field equations (i.e., in e ) in special harmonic coordinates (canonical or skeletonized harmonic gauge) can be written in the form (u =

7.8 The MPM Algorithm

277

t − |x|/c): 4 ext V (u) c2 4 ext h0i can(1) (t, x) = − 3 Vi (u) c 4 ij hcan(1) (t, x) = − 4 Vijext (u) c

h00 can(1) (t, x) = −

(7.8.22)

with V

ext

 (−1)l   ML $ ∂L (u) = G l! r l≥0

 (−1)l

#

*

 ) SbL−1 l iab ∂aL−1 = −G ∂L + l! r l+1 r l≥1 ⎧ ⎛ ⎛ ⎞ ⎞⎫ (1) (2) ⎬  (−1)l ⎨ S M ab(i 2l ij L−2 ⎠ j )bL−2 ⎠ Vijext (u) = G ∂L−2 ⎝ + ∂aL−2 ⎝ . ⎭ l! ⎩ r l+1 r

Viext (u)

(1)

MiL−1

+

l≥2

(7.8.23) Here, ML = ML (u) and SL = SL (u) are generalized multipole-moments, that reduce to the Thorne-moments in the stationary case. Thorne’s second structure Theorem agrees with the Damour-Iyer Theorem if ML and SL are considered as arbitrary field moments. However, in the Damour-Iyer Theorem ML and SL are considered as body moments. In the following relations between the field moments from the second structure Theorem and body moments as integrals over the field generating sources will be derived by means of a matching procedure. It is interesting to note that the linearized external (vacuum) field equations, αβ hcan(1) = 0, are satisfied everywhere except at the spatial origin r = 0 of our harmonic coordinate system. The harmonic gauge condition αβ

∂β hcan(1) = 0 follows from ∂t V ext + ∂i Viext = 0 ,

(7.8.24)

∂t Viext + ∂j Vijext = 0 .

(7.8.25)

278

7 The Post-Newtonian and MPM Formalisms

Next we consider the exterior metric terms of order G2 that obey αβ

αβ

hcan(2) = 2 (hcan(1) , hcan(1) ) ,

(7.8.26)

αβ

where 2 is given by (5.8.34). In the MPM-formalism one employs a solution to this equation everywhere in space, though it is clear that this equation is correct only in the vacuum region and NOT inside the gravitating source! At first one might try ∗ αβ hcan(2)

= −1 R 2 (hcan(1) , hcan(1) ) , αβ

(7.8.27)

but, unfortunately, this leads to divergent integrals. To see this let us consider e.g., 00 can(2) = −

14 ext ext V V c4 ,k ,k

2 4 and for simplicity take V ext = GM/r. Then 00 can(2) = −14m /r with m ≡ GM/c2 and ∗ 00 hcan(2)

14m2 = 4π

d 3y . |x − y|y 4

Using ∞

 rl 1 < = P (n · ny ) l+1 l x |x − y| r>

(7.8.28)

l=0

and with

d nˆ L y Ps (nx · ny ) =

4π L nˆ δls . 2l + 1 x

(7.8.29)

one obtains ∗ 00 hcan(2)

r

∞ 1 dy dy = 14m + r 0 y2 y3 r 2

(7.8.30)

that diverges at r = 0 due to the first integral. Blanchet and Damour (1986) found a very elegant way to deal with this problem. Let us consider the function   B αβ f (B) = −1 (r/r (7.8.31) )  0 R can(2) ,

7.8 The MPM Algorithm

279

(B)

f (B) converges

Analytic continuation x

x x x

x

x

x (B)

p = kmax − 3

Fig. 7.2 The function f (B) from (7.8.31) is analytic in the complex B-plane with the exception of certain poles on the real axes (marked by crosses)

or αβ

f (B) = (r/r0 )B can(2) ,

(7.8.32)

where B is a complex number. Let us assume that only a finite set of multipole αβ moments l ≤ lmax is considered. Then in can(2) there will be a maximal order of divergencies kmax . Then f (B) will be a convergent integral when (B) > kmax − 3 (Fig. 7.2). Near B = 0, f (B) admits a Laurent expansion of the type f (B) =

∞ 

λp B p ,

p0

where p ∈ Z. Relation (7.8.32) then reads ∞ 

αβ

λp B p = (r/r0 )B can(2)

p0

from which we get λp = 0 for p0 ≤ l ≤ −1 in the limit B → 0. Near B = 0 we have ∞  (ln(r/r0 ))p αβ f (B) = can(2) B p p! p=0

280

7 The Post-Newtonian and MPM Formalisms

so that for p ≥ 0 λp =

(ln(r/r0 ))p αβ can(2) . p!

(7.8.33)

Then λ0 is called the FPB=0 -part (Finite Part for B = 0) of f (B), λ0 ≡ FPB=0 [f (B)]

(7.8.34)

and from (7.8.33) it follows that FPB=0 [f (B)] satisfies the correct field equations and is regular at y = 0. So one defines , αβ B αβ [(r/r )  ] . hcan(2) = FPB=0 −1 0 R can(2)

(7.8.35)

In our example we replace ∗ h00 can(2) by expression (7.8.31) and get an integral of the form r

r κ B−1 1 d(r/r0 ) (r/r0 )B−2 = FPB=0 IB = FPB=0 = −r B − 1 0 0 2 2 so that h00 can(2) = −7m /r . Using this construction with (Blanchet 1995)

00 can(2) = −

14 ext ext V V + O6 c4 ,k ,k

0i can(2) = O5 ij

can(2)

(7.8.36)



4 1 ext ext ext ext = 4 V,i V,j − δij V,k V,k + O6 2 c

one obtains h00 can(2) = −

7 ext 2 (V ) + O6 c4

h0i can(2) = O5 ij

hcan(2) = −

(7.8.37)

4 ext Z + O6 c4 ij

with Zijext

≡

−FPB=0 −1 R

(r/r0 )

B

(V,iext V,jext

1 ext ext − δij V,k V,k ) . 2

(7.8.38)

7.8 The MPM Algorithm

281

Note that the trace of Zijext , or Z ext ≡ Ziiext is given by  

1 3 = (V ext )2 + O2 . Z ext = −FPB=0 y B V,kext V,kext − V,kext V,kext 2 4

7.8.1.3

(7.8.39)

Formal Matching in the Overlap Region

Let us denote the harmonic coordinates in the inner domain i by x α and the μ canonical harmonic coordinates used in e by xcan . Assuming the inner and the exterior metric to be isometric in the overlap domain match = i ∩ e we look for a coordinate transformation of the form μ (x) = x μ + ϕ μ (x) . xcan

(7.8.40)

μ

Since x μ and xcan are both assumed to be harmonic one finds ϕ μ + hαβ (x)∂αβ ϕ μ = 0 .

(7.8.41)

From the transformation rule for g αβ and g  (x  ) = |∂x/∂x  |2 · g(x) one derives at the formal matching equation ημν + hμν can (xcan ) =

  1 μ (δ + ∂α ϕ μ )(δβν + ∂β ϕ ν ) ηαβ + hαβ (x) |J | α

(7.8.42)

with 

∂xcan J ≡ det ∂x

 .

We now assume that ϕ μ = O(3, 4), which will be shown later. Then the formal matching equation can be written in the form: μν μν + O(6, 7, 8) hμν can (x) = h (x) + ∂ϕ

(7.8.43)

∂ϕ μν ≡ ∂ μ ϕ ν + ∂ ν ϕ μ − ημν ∂λ ϕ λ .

(7.8.44)

with

From (7.8.41) one finds ∂ν ∂ϕ μν = ϕ μ = −hαβ ∂αβ ϕ μ = O(7, 8) .

(7.8.45)

282

7 The Post-Newtonian and MPM Formalisms

We now first apply the formal matching equation (7.8.43) to the trace of the space-space part of hαβ and obtain 4 ext 4 W = 4 W + ∂λ ϕ λ + 2∂0 ϕ 0 . 4 c c

(7.8.46)

Using this result for the matching of h00 we get V ext = V + c∂t ϕ 0 + O4 Viext = Vi −

7.8.1.4

(7.8.47)

c3 ∂i ϕ 0 + O2 . 4

(7.8.48)

Matching Equations for V and Vi

Next we replace the inner potentials V and Vi by their multipole expansions, valid in the matching region outside the source: M(V ) = G

∞  (−1)l l=0

M(Vi ) = G

l!

∞  (−1)l l=0

l!

∂L ∂L

1 L V (tR ) r 1 L V (tR ) r i

(7.8.49)

(7.8.50)

with V L (t) = I [δl yˆL σ ] ;

ViL (t) = I [δl yˆL σi ] .

In the overlap region the post-Newtonian matching equations for V and Vi then read V ext = M(V ) + c∂t ϕ 0 + O4 Viext = M(Vi ) −

c3 4

∂i ϕ 0 + O2 .

(7.8.51) (7.8.52)

To first post-Newtonian order we can neglect the retardation terms in ViL and write

ViL (t)

=

d 3 y yˆL σi (y) + O2 .

In the expression for V L we use the expansion

+1

−1

dη δl (η)σ (u + η|y|/c; y) = σ (u, y) +

y2 d2 1 σ (u, y) + O4 2(2l + 3) c2 du2

7.8 The MPM Algorithm

283

so that

V L (t) =

d 3 y yˆL σ (t, y)+

d2 1 2(2l + 3)c2 dt 2

d 3 yy2 yˆL σ (t, y)+O4 .

(7.8.53)

The two matching equations (7.8.51) and (7.8.52) then lead to the following: a multipole expanded expression for ϕ 0 in terms of the moments WL , a postNewtonian expressions for the field moments ML and corresponding Newtonian expressions for SL in terms of integrals over the field generating sources. The matching procedure therefore, turns the field moments into body moments. One first finds the ϕ 0 moments in the form ϕ0 = −



∞ WL 4  (−1)l ∂ L l! r c3 l=0

with WL = I −

2l + 1 δl+1 yˆiL σi + O2 , (l + 1)(2l + 3)

(7.8.54)

so that the matching equation (7.8.51) leads to ML = V L +

4 (1) W + O4 c2 L

or

ML =

d 3 y yˆL σ −

d 4(2l + 1) (l + 1)(2l + 3)c2 dt

d 3 y yˆiL σi +

d2 1 2 2(2l + 3)c dt 2

d 3 yy2 yˆL σ ,

which are the BD mass-moments. The matching equation (7.8.52) leads to the Newtonian expression for SL :

SL =

d 3 y aba σb + O2 .

(7.8.55)

The matching of the non-compact-supported potential Wij is described in detail in Blanchet (1995, 2014); it finally leads to the gauge vector ϕ i in form of a multipole expansion with moments XL , YL and ZL .

7.8.2 The MPM Iteration Scheme In principle with the MPM formalism one is able to derive a solution of the field equations outside some field generating source (body) up to any power of G and

284

7 The Post-Newtonian and MPM Formalisms

1/c. The basic equations are the field equations in the form hαβ =

16π G αβ τ c4

with τ αβ = |g|T αβ +

c4 αβ 16π G

and the harmonic gauge condition ∂β hαβ = 0 .

7.8.2.1

The General External Metric

Outside the gravitational source in the vacuum region one writes αβ

hext =

∞ 

αβ

h(n)

(7.8.56)

n=0 αβ

with h(n) being of order Gn . Theorem 7.5 (Blanchet Theorem) The most general solution of the linearized field equations αβ

h(1) = 0 αβ

∂β h(1) = 0 ,

(7.8.57) (7.8.58)

outside a region B and stationary in the past (Eq. (5.8.35)) reads αβ

αβ

β

α λ h(1) = hcan − ∂ α ϕ(1) − ∂ β ϕ(1) + ηαβ ∂λ ϕ(1) .

(7.8.59)

αβ

hcan is of the form h00 can

h0i can

ij

hcan



∞ 4  (−1)l IL (7.8.60) =− 2 l! r c l=0  (1)  #

) ∞ IiL−1 JbL−1 4  (−1)l l iab ∂aL−1 = 3 ∂L−1 + l! r l+1 r c l=1 ⎧ ⎡ ⎡ ⎤ ⎤⎫ (1) (2) ∞ ⎨ ⎬ l  J I ab(i j )bL−2 (−1) 4 2l ij L−2 ⎦ ⎦ . =− 4 ∂L−2 ⎣ + ∂aL−2 ⎣ ⎭ l! ⎩ r l+1 r c l=2

7.8 The MPM Algorithm

285

IL = IL (u) and JL = JL (u) are arbitrary functions of time except for the α is conservation laws I = const., Ii = const., Ji = const.. The gauge vector ϕ(1) parametrized by four multipole moments WL (u), XL (u), YL (u) and ZL (u) according to (7.7.74). This theorem is an extended version of Thorne’s second structure Theorem (7.4) above. To low post-Newtonian orders IL agrees with the Thorne mass-moments ML and JL with the spin-moments SL . However, at higher PN order, ML and IL will differ by nonlinear correction terms. This first happens at order c−5 . The external metric (7.8.56) is then obtained iteratively by solving the equations αβ

(7.8.61)

αβ

(7.8.62)

h(n) = αβ n [h1 , h2 , . . . , hn−1 ] , ∂β h(n) = 0 ,

where the right hand side of the wave equation (7.8.61) is obtained from inserting αβ the previous iterations, up to order n − 1 into n . Thus, αβ

αβ

αβ

αβ

h(2) = 2 [h1 , h1 ] ,

(7.8.63) αβ

αβ

h(3) = 3 [h1 , h1 , h1 ] + 2 [h1 , h2 ] + 2 [h2 , h1 ]

(7.8.64)

etc. As we have already seen such equations cannot be solved by simply taking the retarded integral of the right hand sides because of divergent integrals. One defines (Blanchet 2014; r˜ = (r/r0 )): B αβ u(n) ≡ FPB=0 −1 R [r n ] , αβ

(7.8.65)

αβ

which solves the wave equation (7.8.61). Thus, u(n) is a special solution to the wave equation (7.8.61). However, this solution in general will not satisfy the harmonic gauge condition, i.e.,

i αβ α Bn αi w(n) B r ˜  ≡ ∂β u(n) = FPB=0 −1 R r n

(7.8.66)

will not vanish in general. However, one might construct a retarded solution of the αβ source free wave equation, v(n) whose divergence exactly cancels the divergence α , i.e., term w(n) αβ

α ∂β v(n) = −w(n) .

286

7 The Post-Newtonian and MPM Formalisms

This goes as follows: there exist four STF tensors NL (u), PL (u), QL (u) and RL (u) of retarded time u = t − r/c such that 0 w(n) =

∞ 

  ∂L r −1 NL (u) ,

(7.8.67)

l=0 i w(n) =

∞ 

∞ ,       ∂iL r −1 PL (u) + ∂L−1 r −1 QiL−1 (u) + iab ∂aL−1 r −1 RbL−1 (u) .

l=0

l=1

αβ

The choice for v(n) with the desired property is not unique but a simple choice reads:   00 v(n) = −r −1 N (−1) + ∂a r −1 (−Na(−1) + Qa(−2) − 3Pa ) , 0i v(n)

=r

−1

(−1) (−Qi

(1) + 3Pi ) − iab ∂a

∞      −1 (−1) r Rb − ∂L−1 r −1 NiL−1 , l=2

v(n) = −δij r −1 P + ij

∞ , 

2δij ∂L−1 [r −1 PL−1 ] − 6∂L−2(i [r −1 Pj )L−2 ]

l=2

(1) (2) + ∂L−2 [r −1 (Nij L−2 + 3Pij L−2 − Qij L−2 )] − 2∂aL−2 [r −1 ab(i Rj )bL−2 ] . (7.8.68) Notice the presence of anti-derivatives like e.g.,

N (−1) (u) ≡

u −∞

dvN (v)

etc. Such integrals over the retarded time have a well defined limit v → −∞; because of the ‘no incoming radiation’ condition the integrands vanish for t ≤ −T0 . αβ

α . Exercise 7.14 Proof that ∂β v(n) = −w(n)

Solution For α = 0 one finds   00 0i ∂0 v(n) + ∂i v(n) = −r −1 N + ∂a r −1 (−Na + Qa(−1) − 3Pa(1) )     (−1) (1) (−1) + 3Pi ) − iab ∂ia r −1 Rb +∂i r −1 (−Qi −

∞ 

  ∂iL−1 r −1 NiL−1

l=2

=−

∞  l=0

  0 ∂L r −1 NL = −w(n) .

7.8 The MPM Algorithm

287

For the i-part, one finds after some resorting   ij (2) i0 + ∂j v(n) = r −1 (−Qi + 3Pi ) − iab ∂a r −1 Rb − ∂i [r −1 P ] ∂0 v(n) +

∞ , 

  (1) − ∂L−1 r −1 NiL−1 + 2∂iL−1 [r −1 PL−1 ]

l=2

+∂j L−2 [r −1 (Nij L−2 + 3Pij L−2 − Qij L−2 )] (1)

(2)

−6∂j L−2(i [r −1 Pj )L−2 ] − 2∂aj L−2 [r −1 ab(i Rj )bL−2 ] . The N-terms cancel each other and writing out the symmetrizations in the last line, one has   ij (2) i0 ∂0 v(n) + ∂j v(n) = r −1 (−Qi + 3Pi ) − iab ∂a r −1 Rb − ∂i [r −1 P ] +

∞ , 

2∂iL−1 [r −1 PL−1 ] + ∂L−1 [r −1 (3PiL−1 − QiL−1 )] (2)

l=2

−3∂iL−1 [r −1 PL−1 ] − 3∂L−2 [r −1 PiL−2 ] −∂aL−1 [r −1 abi RbL−1 ] . (2)

The second R-term vanishes and some re-writing yields the desired relation for α = i. So with αβ

αβ

αβ

h(n) = u(n) + v(n)

(7.8.69)

we have a special solution of the wave equation (7.8.61) that solves the harmonic gauge condition. Theorem 7.6 (Blanchet 2014) The most general solution of the harmoniccoordinates Einstein field equations in the vacuum region outside an isolated source, admitting some post-Minkowskian and multipolar expansions, is given by 4∞ αβ the previous construction as hαβ = n=1 h(n) [IL , JL , WL , XL , YL ]. It depends on two sets of STF-tensorial functions of time IL (u) and JL (u) (satisfying the conservation laws) defined in (7.8.60), and on four supplementary functions WL (u), XL (u), YL (u) and ZL (u) parametrizing a gauge vector ϕ λ . Proof Equation (7.8.69) provides a particular solution of the wave equation and the harmonic gauge condition. To it one should add the most general soluαβ tion of the homogeneous system of equations, obtained by setting (n) = 0. But this homogeneous system of equations is nothing but the linearized vac-

288

7 The Post-Newtonian and MPM Formalisms αβ

uum equations with the most general solution h(1) given by (7.8.59). Therefore, this general solution of the homogeneous system of equations is of the form αβ h(1) [δIL , δJL , δWL , δXL , δYL , δZL ] and we can add these ‘corrections’ to the six moments to the old expressions and call them IL , . . . , ZL . Formally the moments IL . . . ZL are given by relations (7.7.79) and (7.7.81) (with IL and JL being given by the expressions for ML and SL ) with the following modifications: (1) The energy-momentum tensor T αβ has to be replaced by τ αβ from relation (5.8.29), i.e., σ ≡

τ 00 + τ ss ; c2

σi ≡

τ 0i ; c

σij = τ ij

and (2) the definition of I [f (y, η)] now includes the regularization procedure, i.e.,

I [f (y, η)] ≡ FPB=0

d 3y

+1 −1

dη f (y, η) .

Chapter 8

First Applications of the PN-Formalism

We will now study some first simple applications of the formalism as it has been developed so far. Note that we have not yet treated the gravitational N-body problem and precisely for that reason the discussions in this chapter are incomplete and will be continued later. Nevertheless, considerable insight into some basic problems can be obtained in this way. The dominant effects from General Relativity are in the time domain, effects that are related with the gravitational redshift. Special Relativity and the gravitational redshift enter the way how time is treated for practical purposes: various timescales are in use to deal with a variety of applications: Terrestrial Coordinate Time TCG, Terrestrial Time TT, International Atomic Time TAI, Coordinated Universal Time UTC and, if the motion of solar system objects plays a role, e.g., for interplanetary spacecraft navigation or a highly precise determination of tidal effects, then barycentric time scales such as Barycentric Coordinate Time TCB or TDB are important. The stability and accuracy of optical clocks has increased drastically in the recent past. With ions like Al+ , Yb+ , Sr+ in electromagnetic traps or neutral atoms like Sr, Yb or Hg one has achieved stabilities and accuracies at the level of 10−18 (e.g., Bloom et al. 2014; Nichelson et al. 2015; Huntemann et al. 2016) with applications in a wide range of sectors from basic science and metrology to geodesy, satellite navigation and environmental monitoring. E.g., with such highly accurate optical clocks the height of the geoid can be determined at the cm level (Bondarescu et al. 2012). Transportable optical clocks (Grotti et al. 2018) and fiber links connecting observing stations with such accurate clocks, located at National laboratories, offer exciting new possibilities. In this chapter we discuss, at the first post-Newtonian level, problems such as equipotential surfaces of the Earth’s gravity field, clock synchronization, time scales, simple astrometric problems and the gravitational time delay (the propagation of light-rays in a highly symmetric gravitational field), the motion of inertial axes (e.g., realized by gyroscope axes) and finally the relativistic motion of satellites in a simplified gravitational field of the Earth. © Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_8

289

290

8 First Applications of the PN-Formalism

The motion of inertial axes in GR differs significantly from that in Newton’s theory where absolute space determines the properties of inertial systems completely. In Newton’s theory some inertial device will keep its orientation with respect to very remote objects in the universe forever. In Einstein’s theory, however, the orientation e.g., of idealized (torque-free) gyroscope axes depends upon the dynamics of gravitational fields at the gyro’s location. If we consider an idealized gyro in orbit about some central gravitating body it will precess with respect to the fixed stars if the central body rotates, an effect which is called ‘dragging of inertial frames’. Though the motion of a satellite in the spherical field of the Earth can be treated exactly in GR approximate treatments are of great importance since both, the measuring accuracy of satellite orbits and the modelling of the various forces (gravitational and non-gravitational) acting on satellites are restricted. Some of these time-dependent forces like, e.g., charged particle drag are extremely difficult to model and there will be a noise limit beyond which accurate modelling becomes impossible (Soffel and Frutos 2016). For that reason the use of exact solutions for modelling satellite orbits, e.g., in the Schwarzschild-field (or Kerr geometry) of the Earth, is irrelevant.

8.1 Equipotential Surfaces and Relativistic Geoid In Sect. 7.2 we have discussed stationary space-times and saw that in adapted coordinates (t, x) the metric tensor can be written in the form, (6.2.7), ds 2 = −f (cdt + wa dx a )2 + f −1 hab dx a dx b ,

(8.1.1)

where f, wa and hab are independent of t and f = −gμν ξ μ ξ ν is the norm of the time-like Killing vector field. In this case we can define a redshift potential φ(x) (Philipp et al. 2017) by f = e−2φ(x)/c . 2

(8.1.2)

The redshift potential φ(x) foliates the 3-dimensional space into surfaces of constant redshift, called isochronometric surfaces. On a specific isochronometric surface all (idealized) clocks run at the same speed. To see this let us choose coordinates such that ξ μ = (1, 0, 0, 0) and f = −g00 . Let us consider two clocks at rest in such a coordinate system. Then the ratio of the two clock frequencies is given by √ f2 −g00 |1 (dτ )1 2 = =√ = e[φ(x2 )−φ(x1 )]/c ≡ 1 + z , f1 (dτ )2 −g00 |2

(8.1.3)

where z is the redshift between the two clocks. Thus on a specific isochronometric surface all (idealized) clocks run at the same rate. This allows a relativistic definition

8.1 Equipotential Surfaces and Relativistic Geoid

291

of the geoid (Bjerhammar 1985, 1986, see also Soffel et al. 1988, Kopeikin 1991, Philipp et al. 2017) Definition The relativistic geoid is the level surface of the redshift potential φ that is closest to mean sea level.

8.1.1 Post-Newtonian Equipotential Surfaces Let us have a look on such isochronometric surfaces in the first post-Newtonian formalism. In the last chapter we introduced a canonical form of the post-Newtonian metric. It applies for an isolated matter distribution and we have assumed the metric to be asymptotically flat. That means the corresponding coordinate system x μ = (ct, x i ) to cover the whole space-time manifold and gμν −→ ημν

for |x| → ∞ .

Such a coordinate system with an asymptotically flat metric tensor is called globally non-rotating and non-accelerated. For many applications, however, rotating coordinates are very useful, e.g., when we want to discuss gravitational problems related with a rotating astronomical body. This is especially the case for a discussion of isochronometric surfaces related with co-rotating clocks. For that purpose we start with a global coordinate system x μ = (ct, x i ) and introduce a coordinate system x μ = (ct, x i ) that rotates with constant angular velocity  with respect to our global coordinates. I.e., we consider a coordinate transformation x μ → x ν of the form t = t,

x i = R ij x j ,

(8.1.4)

where R ij is a time-dependent rotation matrix satisfying dR ij = j kl k R il . dT

(8.1.5)

For applications one might approximately choose the angular velocity as i =

(0, 0, 1)T , so that ⎛

⎞ cos ϕ sin ϕ 0 R ij = R3 (ϕ) = ⎝− sin ϕ cos ϕ 0⎠ 0 0 1

(8.1.6)

with ϕ = t. It can be checked directly that ∂x 0 = 1, ∂x 0

∂x 0 = 0, ∂x i

vi ∂x i =− , 0 c ∂x

∂x i = R ij . ∂x j

(8.1.7)

292

8 First Applications of the PN-Formalism

and, therefore, ∂x 0 = 1, ∂x 0

∂x 0 = 0, ∂x i

∂x i vi = , 0 c ∂x

∂x i = Rj i ∂x j

(8.1.8)

with v i = ij k j x k and v i = R il v l . The metric tensor in x μ is given by (7.2.9) and the transformation rules for gμν yield the components of the metric tensor in rotating coordinates x μ :   2w + v 2 + O6 g 00 = − exp − c2   2w + v 2 2 1 2 2 = −1 + − 4 w+ v + O6 2 c2 c 4 vi − 3 w i + O5 c c     2w 2w + O4 = δij 1 + 2 + O4 , = δij exp c2 c

g 0i = g ij

(8.1.9)

where w≡w+

2wv 2 4w i v i v4 − + 2 2 2 c c 4c

(8.1.10)

and   1 j wj − v w . wi ≡ R 2 ij

(8.1.11)

Exercise 8.1 Derive these expressions from the transformation rules for the metric tensor. Let us write 1 wG ≡ w + v 2 2

(8.1.12)

so that the time-time component of the metric tensor in rotating coordinates takes the form   2w G . (8.1.13) g 00 = − exp − 2 c

8.2 The Problem of Time in the Vicinity of the Earth

293

The surfaces of constant w G are just the post-Newtonian isochronometric surfaces of constant redshift (constant clock rate). Note that the post-Newtonian geopotential w G contains the gravitational potential and the centrifugal potential and that we have neglected the tidal forces resulting from the gravitational influence of other gravitating astronomical bodies.

8.2 The Problem of Time in the Vicinity of the Earth 8.2.1 Synchronization of Nearby Clocks Imagine two nearby clocks, 1 and 2, in the same laboratory establishing local time scales, TS(1) and TS(2). A good way to find the difference, TS(1) − TS(2) is by means of a two-way transfer technique using a cable connection between the two clocks and two time interval counters (TIC; Fig. 8.1). Each clock starts the local TIC by a 1PPS (pulse per second) (see Fig. 8.6 below) and transmits a corresponding signal at the same time to the remote clock, where it stops the TIC. Then the TIC at clock 1 measures the time difference between the 1PPS from clock 1 and the 1PPS from clock 2, delayed by the cable delay d21 . Thus TIC(1) = TS(1) − TS(2) + d21 TIC(2) = TS(2) − TS(1) + d12 so that 2 × [TS(1) − TS(2)] = [TIC(1) − TIC(2)] + [d12 − d21 ] .

(8.2.1)

If the paths are reciprocal, i.e., d12 = d21 , than the clock differences are just half the differences of the TIC-readings (Hanson 1989) (Fig. 8.2).

Fig. 8.1 The time interval and frequency counter SR620 from Stanford Research Systems (Image credit: SRS)

294

8 First Applications of the PN-Formalism

Fig. 8.2 Time comparisons of two local clocks with cable connections

Clock 1

Clock 2

d21

TIC 1

d12

TIC 2

8.2.2 Rates of Clocks in the Earth’s Vicinity We now come to the problem of time in the vicinity of the Earth. To this end we first neglect the tidal forces and consider the Earth to be isolated. For the comparison of clock rates (proper frequencies of atomic clocks) to a first approximation it will be sufficient to consider a simplified version of the metric (8.1.9) g 00 = −1 +

2wG + O4 c2

vi + O3 c = δij + O2 ,

g 0i = g ij

(8.2.2)

with 1 wG = w + v 2 . 2

(8.2.3)

For the following we consider the mass monopole term (the mass of the Earth, M) and the Earth’s oblateness J2 in the expression for w. Let (r, θ, φ) be polar coordinates corresponding to x i , i.e., x = r sin θ cos φ,

y = r sin θ sin φ,

z = r cos θ .

Then we first consider GM w(x) r

+  2 R J2 P2 (cos θ ) 1− r

*

with R = 6.37814 × 108 cm (equatorial radius of the Earth) J2 = 1.083 × 10−3 .

(dimensionless quadrupole moment)

(8.2.4)

8.2 The Problem of Time in the Vicinity of the Earth

295

and P2 (cos θ ) =

1 1 (3 cos2 θ − 1) = − (1 − 3 sin2 !) . 2 2

Here ! is the geographic latitude approximately given by ! = 90◦ − θ . Let f ≡1−

Rp R

(8.2.5)

be the flattening of the Earth where Rp is its polar radius. Then r0 R(1 − f sin2 !)

(8.2.6)

approximately describes the geoid, i.e., the level surface of wG at mean sea level. We write r = r0 + h , where h is the height above the geoid. Taking  in z-direction so that v 2 = 2 (x 2 + y 2 ) = 2 r 2 sin2 θ = 2 r 2 (1 − sin2 !) the geopotential approximately reads GM wG r

+  2 R 1 2 J2 (1 − 3 sin !) + 2 r 2 (1 − sin2 !) . r 2

*

1 1+ 2

(8.2.7) Using this the gravity acceleration at the geoid is given by g(!) = −

∂wG . ∂r r=r0

(8.2.8)

So to first order in f, J2 and the centrifugal term, we get  g(!)

GM r2

*

GM + 2 r

1 1+ 2

+  2 R 2 J2 (1 − 3 sin !) r

  2 R 2 2 2 J2 (1 − 3 sin !) − r(1 − sin !) r

GM 3 GM (1 + 2f sin2 !) + J2 (1 − 3 sin2 !) 2 2 R2 R

r=r0

296

8 First Applications of the PN-Formalism

− 2 R(1 − f sin2 !)(1 − sin2 !)

GM 3 GM + J2 − 2 R 2 2 R2 R

2GM 9 GM 2 + sin2 ! f − J +

R 2 2 R2 R2

[9.78027 + 0.05192 sin2 !] × 102 cm/s2 .

(8.2.9)

(8.2.10)

Using this result g 00 can be written as g 00 −1 +

 2wG 2  0 w −1 + − g(!)h , G c2 c2

(8.2.11)

where 0 wG = wG |r=r0

denotes the geopotential at the geoid. For the comparison of earthbound clocks with frequencies, f1 and f2 and dx = 0 we therefore find √ (dτ )1 −g 00 |1 f2 1 − wG /c2 |1 [1 + g(!)h/c2 ]1 = =√ = = . 2 (dτ )2 f1 1 − wG /c |2 [1 + g(!)h/c2 ]2 −g 00 |2

(8.2.12)

For g 9.81 m/s2 , the clock rates change with height h above the geoid by 1.09 × 10−16 times h in meter. E.g., if we compare a clock at the NBS (Boulder, Colorado; h = 1634 m) with one at the PTB (Brunswick, Germany; h = 73 m) one finds fPTB (1 + 1.8 × 10−13 ) fNBS leading to a difference in clock reading of 5.4 μs per year.

8.2.3 Synchronization of Clocks in the Vicinity of the Earth We consider two clocks located at neighbouring points A and B. B emits a light signal to A that is instantaneously reflected back to B. With respect to coordinates (t, x) the signal propagation is described by 0 = ds 2 = g00 (dx 0 )2 + 2g0i dx 0 dx i + gij dx i dx j

8.2 The Problem of Time in the Vicinity of the Earth

297

and we can solve this equation for dx 0 : 0 dx±



 1 i i j −g0i dx ± (g0i g0j − gij g00 )dx dx . = g00

(8.2.13)

Now, according to the Einstein synchronization procedure x 0 in A is synchronous to 1 0 0 0 [(x + dx− ) + (x 0 + dx+ )] 2 or x0 −

g0i i dx g00

at B. We see that for the case g0i = 0 as is the case in our rotating coordinates a global synchronization of clocks according to Einstein’s prescription is not possible since the procedure becomes path dependent.

8.2.4 Coordinate Time Synchronization For practical purposes one defines a different synchronization procedure based on the coordinate time (coordinate time synchronization). Definition Two clocks with proper times τ1 and τ2 are called (coordinate time) synchronous, if the above relation τ (t) leads to the same coordinate time t. To see the implications of this definition we write the metric in the form   2wG ds 2 = − 1 − 2 (c dt)2 + 2( × x) · dx dt + (dx)2 c

(8.2.14)

or (c2 dτ 2 = −ds 2 )

2 0 c2 dτ 2 = 1 − 2 (wG − g(!)h) (c dt)2 − 2( × x) · dx dt − (dx)2 . c

(8.2.15)

Solving approximately for dt as function of dτ we get

t

⎡

⎤ * +2 1 0 1 V 1 ( × x) · V ⎦ dτ ⎣1 + 2 (wG − g(!)h) + + 2 2 c c c c2

(8.2.16)

298

8 First Applications of the PN-Formalism

with ( × x) · V VE = R cos ! 2 . 2 c c

(8.2.17)

Here V is the velocity of the clock in rotating coordinates and V E is its eastward component. The synchronization of clocks in the vicinity of the Earth can be realized e.g., by clock transport. If we move a portable clock form one earthbound clock to another slowly and e.g., at low height above the geoid, then the elapsed proper time interval during that journey is related to the elapsed coordinate time difference according to

R t τ + 2 c ∗

dτ V E cos ! ,

(8.2.18)

where   1 0 τ = τ 1 + 2 wG . c ∗

(8.2.19)

From this we see that in a slow transport along a meridian the time indicated by the portable clock basically needs no correction, i.e., t = τ ∗ . For a corresponding transport along a parallel circle we get: t τ ∗ +

R LE cos ! , c2

(8.2.20)

if LE denotes the distance covered in eastward direction. If e.g. we move a clock slowly and at a low height along the equator once around the Earth then its readings will differ by an amount of 207 ns from that of a stationary clock (Sagnac effect in the time domain).

8.2.5 The Relation Between Coordinate and Proper Time In the future it will become increasingly important to have a high-precision relation between coordinate time t (which will be TCG in practise) and the proper time τ of some accurate clock (e.g., Petit and Wolf 2005). The desired accuracy will be of order 10−18 . At this level of accuracy in suitably chosen quasi-inertial geocentric coordinates (GCRS-coordinates) it is sufficient to consider the metric in the form g00 = −1 + 2w/c2 , g0i = 0 and gij = δij so that from dτ 2 = −ds 2 /c2 one has

dτ 1 dτ 1 = = 1 − 2 w(xC ) + v2C , dT d(TCG) 2 c

(8.2.21)

8.2 The Problem of Time in the Vicinity of the Earth

299

where xC and vC are the (GCRS) coordinate position and velocity of the clock. We write w = wE + wGD + wT ,

(8.2.22)

where wE is the gravitational potential of the Earth, wGD the geodesic-deviation term that will be discussed later, and wT the tidal potential. The wGD -term gives rise to a correction of less than a few parts in 1019 (Wolf and Petit 1995) and will be neglected.

8.2.5.1

Clock’s on the Earth’s Surface

For earthbound clocks we combine wE with the v2C -term as above and write 1 0 wE (xC ) + v2C = wG (xC ) = wG − 2

h

g dh .

(8.2.23)

0

The tidal potential due to external masses B to sufficient accuracy can be written as  * +   GMB 3r i r j i j EB EB − δij xC xC , wT (xC ) (1 + k2 − h2 ) 3 2 rEB 2rEB B=E

(8.2.24)

i = x i − x i . The quantities k and h are Love-numbers describing the where rEB 2 2 E B solid Earth tides with (1 + k2 − h2 ) = 0.69 (Farrell 1972). Ocean tides can be taken into account by means of the global ocean tide model of Schwiderski (1983) with loading coefficients given by Farrell (1972). Effects from atmospheric loading can be computed with a model by Merriam (1992) and surface load Love numbers kn and hn from Farrell (1972). If we write

v2C = ( × xC )2 = ( ρC cos C )2 ,

(8.2.25)

where φ is latitude then lod-variations (δ ) and polar motion induced latitude variations (δ) can be considered by writing

= 0 + δ

C = 0C + δ

(8.2.26)

leading to corrections of the form 1 ( ρC2 cos2 C · δ − (1/2) 2 ρC2 sin(2C ) · δ) . c2

(8.2.27)

300

8.2.5.2

8 First Applications of the PN-Formalism

Clocks Onboard Terrestrial Satellites

For clocks onboard terrestrial satellites the Earth’s gravitational potential, wE , should be modelled by some Earth’s model with potential coefficients Clm and Slm . Then effects from solid Earth, oceanic and pole tides can be included in the model as small variations of the potential coefficients (Wolf and Petit 1995). Instead of the Taylor expansion of the tidal potential about the geocenter we may write wT (xC ) = w(xC ) − w(xE ) − w ,k (xE )xCk   k rk  rEB 1 1 = GMB − + 3 CE , rCB rEA rEA

(8.2.28)

B=E

k = x k − x k is the vector from the tide raising body B to the position of where rCB C B the clock.

8.2.6 Clock Comparisons: Practical Aspects Next we will discuss practical methods for the synchronisation of clocks operating in widely separated laboratories. The Global Navigation Satellite Systems (GNSS) provide a satisfactory solution to the problem of time transfer (e.g., Gurevich et al. 1994). The two systems—the US Global Positioning System (GPS) and the Russian Global Navigation Satellite System (GLONASS), composed of respectively 30 and 24 non-geostationary satellites—are designed for positioning, but have the particular feature that the satellites are equipped with atomic clocks which broadcast time signals (Figs. 8.3 and 8.4). The signal received from one satellite in a laboratory allows the time difference between the local time scale and the GNSS system time to be determined with uncertainty of a few nanoseconds when averaging over 15 min (SI Brochure 2014). Two-way satellite time and frequency transfer (TWSFTF) is used regularly for comparing more than ten timing centers world-wide (e.g., Bauch et al. 2011).

8.2.6.1

TWSTFT

A very accurate method for time and frequency transfer between two earthbound stations is the TWSTFT method. Two-Way Satellite Time and Frequency Transfer (TWSTFT) is based on the exchange of timing signals through geostationary telecommunication satellites and involves the transmission and reception of radio frequency (RF) signals consisting of binary phase-shift modulations (pseudorandom noise (PRN) codes (Bauch 2013)). Modems designed specifically for that purpose (e.g., the Satellite and Time

8.2 The Problem of Time in the Vicinity of the Earth

301

Fig. 8.3 Constellation of GPS satellites (Image credit: NOAA, Navstar: GPS Satellite Network, E. Howell, Space.com)

and Ranting Equipment (SATRE) of TimeTech GmbH, Stuttgart) generate the modulation at the 70-MHz intermediate frequency (IF), which is then transmitted after up-conversion to the RF-band (e.g., Ku-band, 11–14 GHz or X-band, 8–90 GHz). Each station uses a dedicated PRN code in its transmitted signal. The phase modulation is synchronized with the local clock (Fig. 8.5) and the modem generates a one-pulse-per-second (1PPS) output, synchronous with the emitted signal, called 1PPSTX. This 1PPSTX opens the gate of a Time Interval Counter (TIC). The same process goes on at the remote station. The received RF-signal from the remote station is down-converted and the modem detects the modulation at the IF level and reconstitutes a 1PPS tick from the received signal, called 1PPSRX, that closes the gate of the TIC (Fig. 8.6). We say that the local clock generates a local time scale TS, so for two stations, 1 and 2, we are interested in the difference TS(1) − TS(2). The observed quantities are the readings from the two TICs, TIC(1) and TIC(2). Taking into account delays from the transmitters (dTi ), atmospheric delay in the uplink to the satellite (diS ), delay in the satellite (dSij ), atmospheric delays in the downlink (dSj ) and delays from the receivers (dRi ) we get in a coordinate system

302

8 First Applications of the PN-Formalism

Fig. 8.4 The Russian system of GLONASS satellites (from: http://www.spacecorp.ru/en/ directions/glonass/orbital)

rotating with the Earth: TIC(1) = TS(1) − TS(2) + dT2 + d2S + dS21 + dS1 + dR1 + SCU(2) + SCD(1) TIC(2) = TS(2) − TS(1) + dT1 + d1S + dS12 + dS2 + dR2 + SCU(1) + SCD(2) . (8.2.29) In these equations the SC-terms result from the Sagnac-effect in the rotating system. Combining the two equations in (8.2.29) we get 2 × [TS(1) − TS(2)] = +[TIC(1) − TIC(2)] (TIC readings) +[dT1 − dR1 ]

(equipment delays at station 1)

−[dT2 − dR2 ]

(equipment delays at station 2)

+[d1S − dS1 ]

(up/down difference at station 1)

8.2 The Problem of Time in the Vicinity of the Earth

−[d2S − dS2 ]

303

(up/down difference at station 2)

+[dS12 − dS21 ]

(Satellite delay difference)

−[SCD(1) − SCU(1)] (Sagnac correction for station 1) +[SCD(2) − SCU(2)] (Sagnac correction for station 2) . (8.2.30)

Fig. 8.5 Schematic diagram of the two-way satellite time and frequency transfer (TWSTFT, Image credit: Bauch et al. 2011)

PPS

Master clock N 1.Count Fig. 8.6 A 1PPS (pulse per second) synchronized to the output of a clock

304

8 First Applications of the PN-Formalism

8.2.7 TAI, TT and UTC The realization of Geocentric Coordinate Time, TCG, is achieved with the time scales TAI (International Atomic Time), TT (Terrestrial Time) and UTC (Coordinated Universal Time). Conceptually TT is a timescale differing from T = TCG only by a constant rate. This rate should agree at some level of approximation with that of a clock on the geoid. For earthbound clocks, we had dτ g(ψ) · h U0 . 1− 2 + dT c c2 On the geoid with h = 0, we have   U0 dτ = 1 − 2 dT = kE dT ≡ d(T T ) c with kE = 1 −

U0 . c2

For present achievable accuracies the precise definition and realization of the geoid, however, presents a problem. For that reason, the timescales TT (and TAI) are related with TCG by a defining constant kE : kE = 1 − 6.969290134 × 10−10 .

(8.2.31)

Thus we have the relation TT ≡ kE T = kE TCG .

(8.2.32)

From a practical point of view TAI (or, equivalently UTC) is the central timescale. Responsibility for TAI lies in the hands on the Bureau International des Poids et Mesures (BIPM) since January 1st 1988. TAI is processed in two steps (SI Brochure 2014): – A weighted average based on more than 400 clocks in about seventy laboratories is first calculated. The BIPM applies a special algorithm (ALGOS) (e.g., Audoin and Guinot 2001; Azoubib et al. 1977) to first determine the rate instabilities of all of the clocks involved. The smaller the instability, the greater the assigned statistical weight with which a clock contributes to this average. TAI is a deferred-time timescale, available with a delay of a few weeks. The relative frequency stability of TAI is a few parts in 1016 for mean durations of 1 month. In this process the weighted average is actually performed for the Coordinated Universal Time, UTC, that differs from TAI by a whole number of seconds (the leap seconds; see below). The more than 70 national time-service laboratories,

8.2 The Problem of Time in the Vicinity of the Earth

305 ISSN 11431393

CIRCULAR T 339 2016 APRIL 07, 13h UTC BUREAU INTERNATIONAL DES POIDS ET MESURES ORGANISATION INTERGOUVERNEMENTALE DE LA CONVENTION DU METRE PAVILLON DE BRETEUIL F-92312 SEVRES CEDEX TEL. +33 1 45 07 70 70 FAX. +33 1 45 34 20 21 [email protected]

The contents of the sections of BIPM Circular T are fully described in the document " Explanatory supplement to BIPM Circular T " available at ftp://ftp2.bipm.org/pub/tai/publication/notes/explanatory_supplement_v0.1.pdf 1 - Difference between UTC and its local realizations UTC(k) and corresponding uncertainties. From 2015 July 1, 0h UTC, TAI-UTC = 36 s. Date 2016 0h UTC MJD Laboratory k AOS (Borowiec) APL (Laurel) AUS (Sydney) BEV (Wien) BIM BIRM (Beijing) BY

(Minsk)

CAO CH

(Cagliari) (Bern-Wabern)

CNM (Queretaro) CNMP (Panama)

FEB 26 MAR 2 MAR 7 MAR 12 MAR 17 MAR 22 MAR 27 Uncertainty/ns Notes 57444 57449 57454 57459 57464 57469 57474 uA uB u -0.1 18.4 32.6 -14.6 3502.5 -35.2

-0.4 24.3 56.0 -40.4 3513.2 -36.2

12.4

10.8

[UTC-UTC(k)]/ns -0.7 -1.1 0.0 30.9 11.9 12.8 77.0 106.3 135.3 -63.1 -54.6 -37.3 3534.9 3576.3 3582.2 -38.6 -39.8 -43.2 8.8

10.9

-1.9

1.1 15.3 158.1 -32.1 3603.3 -47.9 -0.1

1.9 14.6 169.3 -38.0 3633.1 -

0.3 0.3 0.3 0.3 1.5 1.5

2.9 10.9 5.9 5.1 8.7 20.0

2.9 10.9 5.9 5.1 8.8 20.1

2.4 1.5 8.7 8.8

-11411.1 -11523.7 -11633.3 -11753.4 -11860.4 -11959.5 -12071.3 8.0 20.0 21.6 2.8 3.5 4.1 3.6 3.2 3.0 2.0 0.3 1.5 1.5 -2.6 6.9

-0.9 4.8

4.1 5.7

8.6 23.9

10.0 9.8

5.0 -4.2

2.1 3.0 11.2 11.6 2.7 3.5 11.2 11.8

1016.0 -22.7

1192.7 0.8

1363.9 10.1

1549.8 21.0

1726.0 9.9

1910.5 11.8

- 0.7 20.0 20.0 6.0 0.3 7.4 7.4

DTAG (Frankfurt/M) EIM (Thessaloniki)

189.6 13.8

191.3 11.9

200.1 1.0

201.3 8.9

204.8 7.8

200.8 14.8

203.4 0.3 7.7 7.7 16.7 5.0 7.9 9.3

ESTC HKO

0.3 131.6

-0.9 135.7

-1.5 146.7

-4.1 151.4

-5.7 153.9

-3.4 161.1

-3.0 0.3 5.5 5.5 167.8 0.3 7.4 7.4

-887.3 220.7

-890.1 217.8

-900.8 211.3

-896.6 218.7

-895.1 220.9

-905.5 220.1

-913.3 0.3 5.4 5.4 219.8 0.3 7.2 7.2

DFNT (Tunis) DMDM (Belgrade)

(Noordwijk) (Hong Kong)

IFAG (Wettzell) IGNA (Buenos Aires) IMBH (Sarajevo)

Fig. 8.7 A part of BIPM Circular T 339 from 2016 April 07, 13 h

which contribute to the formation of TAI, maintain an approximation of UTC, known as UTC(k) for laboratory k. These UTC(k) values are the basis for the mentioned weighted average that presents a free atomic timescale called EAL (Echelle Atomic Libre). UTC itself is disseminated monthly through the publication of the offsets [UTC − UTC(k)] and their uncertainties at five days interval in the BIPM Circular T (see Fig. 8.7). – The frequency accuracy of the EAL is then evaluated by comparing its scale unit with various realizations of the SI second produced by primary frequency standards (PFS) (presently these PFS are located at LNE-SYRTE, Laboratoire National de Métrologie et Essais, Paris, France; INRIM, Instituto Nazionale di Ricerca Metrologia, Torino, Italy; NIST, National Institute of Standard and Technology, Boulder (Colorado), USA; NPL, National Physical Laboratory,

306

8 First Applications of the PN-Formalism

11 primary frequency standards (laboratory)

reference frequency

400 atomic clocks (commercial)

free timescale (EAL)

correction procedure

TAI

Fig. 8.8 Scheme how TAI is realized (Image credit: Soffel and Langhans 2013)

Middlesex, GB and PTB, Physikalisch Technische Bundesanstalt, Brunswick, Germany). This comparison requires the application of corrections to compensate for the relativistic frequency shift between the location of the PFS and a point on the equipotential surface of the Earth implicitly defined by U0 or kE . The scale unit of the EAL is then steered to the SI second through the applications of corrections, of magnitude a few parts in 1016 , every month (see Fig. 8.8). For practical reasons also, TT is derived from atomic time TAI. The IAU has decided that TT = TAI + 32.184 s . The constant 32.184 s is simply a convention that was chosen for historical reasons. Coordinated Universal Time, UTC; is defined to differ from TAI by a certain integral number of (leap) seconds, i.e., TAI = UTC + N s , if N denotes a positive integer. Leap seconds are introduced so that the difference |UTC − UT1| < 0.9 s . This has the advantage that over a very long time span, UTC is related with the Sun via the Earth’s orientation in space and thus, remains useful for ordinary life (Soffel and Langhans 2013). Since 1972, a leap second is introduced, if necessary, with preference at 30 June or 31 December of a year, always at 23h 59m 59s UTC. This way a new offset between UTC and TAI is valid at 0h 00m 00s of 1 July or 1 January respectively. Table 8.1 shows the instants of time when new leap seconds became valid.

8.3 Barycentric Timescales TCB, Teph , TDB The timescales introduced in the last section are all related with the geocenter; they are geocentric timescales to be used in the vicinity of the Earth. For certain applications, however, such as solar system ephemerides or interplanetary spacecraft navigation barycentric timescales have to be used where the barycenter refers to the solar system’s center of mass. The basic barycentric timescale is Barycentric

8.3 Barycentric Timescales TCB, Teph , TDB

307

Table 8.1 Moments of time when new offsets N (TAI-UTC in seconds) between TAI and UTC became valid by the introduction of leap seconds Julian date JD 2441317.5 JD 2441499.5 JD 2441683.5 JD 2442048.5 JD 2442413.5 JD 2442778.5 JD 2443144.5 JD 2443509.5 JD 2443874.5 JD 2444239.5 JD 2444786.5 JD 2445151.5 JD 2445516.5

Date 1 Jan 1972 1 Jul 1972 1 Jan 1973 1 Jan 1974 1 Jan 1975 1 Jan 1976 1 Jan 1977 1 Jan 1978 1 Jan 1979 1 Jan 1980 1 Jul 1981 1 Jul 1982 1 Jul 1983

N +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22

Julian date JD 2446247.5 JD 2447161.5 JD 2447892.5 JD 2448257.5 JD 2448804.5 JD 2449169.5 JD 2449534.5 JD 2450083.5 JD 2450630.5 JD 2451179.5 JD 2453736.5 JD 2454832.5 JD 2456109.5 JD 2457204.5

Date 1 Jul 1985 1 Jan 1988 1 Jan 1990 1 Jan 1991 1 Jul 1992 1 Jul 1993 1 Jul 1994 1 Jan 1996 1 Jul 1997 1 Jan 1999 1 Jan 2006 1 Jan 2009 1 Jul 2012 1 Jul 2015

N +23 +24 +25 +26 +27 +28 +29 +30 +31 +32 +33 +34 +35 +36

Coordinate Time TCB. For the definition of TCB one might imagine some fictitious clock at the geocenter neglecting the gravity field of the Earth. For this case the first post-Newtonian relation between TCB and TCG reads d(TCG) Uext (zE ) 1 v2E − 1− d(TCB) 2 c2 c2 or 1 (TCG) (TCB) − 2 c

 

1 2 Uext (zE ) + vE dt . 2

Here, t = TCB, Uext is the gravitational potential of Sun and planets excluding the Earth and zE , vE the barycentric coordinate position and velocity of the geocenter. For some point x close to the geocenter this relation is given by TCB − TCG = c

−2

t  

1 2  Uext (zE ) + vE dt + vE · (x − zE ) . 2 t0

(8.3.1)

The second term on the right hand side results from the fact that two events that are simultaneous w.r.t. TCB generally are not simultaneous in a geocentric system (i.e., w.r.t. TCG). The moment of time t0 where TCB = TCG is given by t0 = JD 2 443 144.5 .

308

8 First Applications of the PN-Formalism

TCB and TCG differ not only by periodic terms but also by a secular drift given by (TCB − TCG)sec = LC (JD − t0 ) × 86,400 s with LC = 1.480827 × 10−8 . Some of the best solar system ephemerides, the DE ephemerides of the Jet Propulsion Laboratory (JPL), do not employ TCB as basic time variable. The original idea was to use a timescale that differs from TT practically only by periodic terms. Strictly speaking because of arbitrarily long periods contained in the motion of the solar system such a timescale cannot be realized with ultimate precision. Consequently, the timescales Teph used in barycentric ephemerides show a relation with TCB or TCG depending slightly upon the ephemeris itself. In the past the DE ephemerides used the Fairhead–Bretagnon series (Fairhead and Bretagnon 1990) to derive Teph from TT. Meanwhile another barycentric timescale called TDB has been defined by TDB = TCB − LB × (JDTCB − T0 ) × 86,400 s + TDB0 ,

(8.3.2)

with T0 = JD 2443144.5003725, LB = 1.550519768 · 10−8 ,

TDB0 = −6.55 × 10−5 s .

The value for LB was chosen so to minimize the linear drift between TDB and TT for the ephemeris DE405. On the surface of the Earth this difference is less than 2 ms for the forthcoming centuries. In the planetary ephemerides developed at Paris Observatory, as described in Fienga et al. (2009), TDB is used for the description of planetary motion and the differences between TT and TDB are estimated by solving (8.3.1) at the geocenter at each step of the integration. By using the INPOP values for TT − TDB in the data analysis and the fit of the planetary ephemerides, there is a complete consistency between the positions and velocities of the planets and TT − TDB, all provided to users since INPOP08.

8.4 Fairhead–Bretagnon Series Analytical approximations for T ≡ Teph − TT are based upon (semi-)analytical ephemerides of the solar system. The Fairhead–Bretagnon approximation is based upon the planetary theory VSOP82 (Bretagnon 1982) and the lunar theory ELP2000

8.5 Light-Rays in the PN-Field of a Single Body

309

Table 8.2 Some terms in the Fairhead–Bretagnon series for T i 1 2 3 4 5 6 7 8 9 10

Ai [ μs] 1656.674564 22.417471 13.839792 4.770086 4.676740 2.256707 1.694205 1.554905 1.276839 1.193379

ωai [ rad/1000 y] 6283.075943033 5753.384970095 12566.151886066 529.690965095 6069.776754553 213.299095438 −3.523118349 77713.772618729 7860.419392439 5223.693919802

φai [ rad] 6.240054195 4.296977442 6.196904410 0.444401603 4.021195093 5.543113262 5.025132748 5.198467090 5.988822341 3.649823730

Period [ y] 1.0000 1.0921 0.5000 11.8620 1.0352 29.4572 1783.4159 0.0809 0.7993 1.2028

(Chapront-Touzé and Chapront 1983). This series is written in the form T = C0 TT2 +   Ai sin(ωai TT + φai ) + TT Bi sin(ωbi TT + φbi ) + i

+ TT2



i

Ci sin(ωci TT + φci ) + TT3

i



Di sin(ωdi TT + φdi ) .

i

(8.4.1) Here, TT is counted in millennia since J2000.0 and T ≡ Teph − TT is in microseconds. The first 10 coefficients are given in Table 8.2. The full set of 127 coefficients can be found in Fairhead and Bretagnon (1990).

8.5 Light-Rays in the PN-Field of a Single Body As we have seen earlier light-rays are null geodesics, i.e., curves x μ (λ) satisfying the geodetic equation and the condition gμν

dx μ dx ν =0 dλ dλ

that we can also write with coordinate time t in the form gμν

dx μ dx ν = 0. dt dt

(8.5.1)

310

8 First Applications of the PN-Formalism

Writing the geodetic equation also with coordinate time t in place of the affine parameter λ we have from (5.3.3)  ν σ  i d 2xi dx dx 0 1 dx i − νσ . = νσ c dt dt dt dt 2

(8.5.2)

For the light-rays we now make the following ansatz x(t) = x0 + n c(t − t0 ) + xP ≡ xN + xP .

(8.5.3)

Here the index p refers to ‘post-Newtonian’ and indicates the terms in addition to xN that are of lowest order in 1/c. Considering dx 0 = c, dt

dx i = c ni + . . . dt

we find 1 0 dx i dx 0 dx 0  = O1 c 00 dt dt dt 1 1 0 dx i dx 0 dx j dx j dx i 0j = − 2 w,j + O2 c dt dt dt dt dt c 1 0 dx i dx j dx k  = O1 c j k dt dt dt i 00

dx 0 dx 0 = −w,i + O2 dt dt

i 0j

dx 0 dx j = O1 dt dt

jik

 dx j dx k 1  dx j dx k = 2 δij w,k + δik w,j − δj k w,i + O2 dt dt dt dt c * + 2 dx dx k dx i 1 − w,i + O2 . = 2 2w,k dt dt dt c

Inserting these expressions into the geodetic equation (8.5.2) we get * +   d 2xi 1 dx 2 4 dx i dx · ∇w + O1 . = w,i 1 + 2 − 2 dt 2 c dt c dt dt

(8.5.4)

Similarly one finds from (8.5.1) 

   2w 2 2w dx i dx j 0 = −1 + 2 c + δij 1 + 2 + O1 . dt dt c c

(8.5.5)

8.5 Light-Rays in the PN-Field of a Single Body

311

Using dx i dx i = c ni + P + . . . dt dt this leads to the simple equation n·

1 dxP 2w =− 2 . c dt c

(8.5.6)

The geodetic equation can be written as d 2 xP = 2 [∇w − 2n(n · ∇w)] dt 2

(8.5.7)

where we dropped the order symbols in the last two equations. In Euclidean vector space notation (this is really only a notation and nothing more) let us define xP (t) = n · xP (t) xP (t)⊥ = (1 − n ⊗ n)xP (t) .

(8.5.8)

We have   d 2 xP⊥ d 2 xP d 2 xP = − n n · dt 2 dt 2 dt 2 = 2∇w − 4n(n · ∇w) − 2n(n · ∇w) + 4n(n · ∇w) i.e., d 2 xP⊥ = 2 [∇w − n(n · ∇w)] dt 2

(8.5.9)

1 dxP 2w =− 2 . c dt c

(8.5.10)

and

We now consider the gravitational potential w to be generated by some ‘roughly spherical’ matter distribution. I.e., outside the matter we consider only the mass monopole M and write w

GM r

where it is sufficient to write r rN = |x0 + c n(t − t0 )| .

312

8 First Applications of the PN-Formalism

With ∇w = −

GM x r3

we find d 2 xP⊥ 2 GM 2 GM = − 3 [x − n(n · x)] = − 3 d dt 2 r r where d = n × (x × n) = n × (x0 × n)

(8.5.11)

is a constant vector. One might interpret the vector d as pointing from the gravitating mass to the point of closest approach of the unperturbed light ray. Since r 2 = x2 = x20 + 2c n · x0 (t − t0 ) + c2 (t − t0 )2 we get 1 2 GM x˙ P⊥ = − d c c

−3/2  , dt x20 + 2c n · x0 (t − t0 ) + c2 (t − t0 )2

i.e., we encounter an integral of the form

−3/2  = dx ax 2 + bx + c

2(2ax + b) √  ax 2 + bx + x   where  ≡ 4ac − b2 . In our case  = 4c2 x20 − (n · x0 )2 . Since d 2 = [n × (x0 × n)]2 = [x0 − n(n · x0 )]2 = x20 − (n · x0 )2 we have simply  = 4c2 d 2 . Furthermore, 2ax + b = 2c2 (t − t0 ) + 2c n · x0 = 2c xN · n . Altogether we obtain 2 GM d 1 x˙ P⊥ = − 2 c c d2



xN (t) · n x0 · n − rN (t) r0

 (8.5.12)

.

Together with (8.5.10) this leads to   1 2m x˙ = 1 − n− c r   2m n− 1− r

2md d2



xN (t) · n x0 · n − rN (t) r0

2md (cos χ + 1) , d2



(8.5.13)

8.5 Light-Rays in the PN-Field of a Single Body

313

Fig. 8.9 Geometry in the problem of light deflection in the gravitational field of some mass M

Observer

n

χ

d

M

where m = GM/c2 and χ denotes the “unperturbed” angle between the directions towards the gravitating body and the light-source (Fig. 8.9). A second integration yields the desired equation of motion x(t) = x0 + c n(t − t0 ) (8.5.14)  

d r +x·n + 2 [rN − r0 − (x0 · n/r0 )c(t − t0 )] . −2m n ln r0 + x0 · n d Note that the log-term results from the parallel part, i.e., from an integration of −1/2 1  2 = ax + bx + c . r The first two terms are the ‘Newtonian part’ of the equation of motion. The logarithmic term is in the direction of the unperturbed light ray; as we shall see it describes the so-called gravitational time delay (Shapiro time delay, Shapiro 1964). Finally the last part, proportionally to d, describes the gravitational light deflection. For very remote light sources, |x0 | → ∞, and x0 · n → −1 r0 so that   1 2m x˙ = 1 − n− c r   2m n− = 1− r

 2md  x · n + 1 r d2 2md (cos χ + 1) d2

(8.5.15)

and x(t) = x0 + c n(t − t0 ) (8.5.16)  

d r +x·n + 2 [r − r0 + c(t − t0 )] . −2m n ln r0 + x0 · n d

314

8 First Applications of the PN-Formalism

8.5.1 The Celestial Sphere Let k μ = dx μ /dλ be a tangent vector to some null geodesics γ . Then in our asymptotically Cartesian Minkowskian coordinate system lim k μ |γ

t→−∞

defines a ‘point on the celestial sphere’. Here we followed the light ray back in space and time in the direction of null rays. This asymptotically flat region of space-time is called past null infinity. In relativity past null infinity might be viewed as ‘celestial sphere’. Two light rays are said to originate from one and the same star if μ

μ

lim k1 |γ1 = lim k2 |γ2 in our coordinate system, i.e., if n1 = n2 .

8.5.2 The Astrometric Observable Writing the four-velocity of some observer in our coordinate system as uμ = c

dt (1, β) dτ

with β = v/c and a wave vector k μ as tangent vector to some light ray as kμ =

dx μ = const (1, x˙ /c) dλ

the observed angle θ between two incident light rays γ1 and γ2 is given by  cos θ = n1 · n2 + (n1 · n2 − 1) (n1 + n2 ) · β + (n1 · β)2  +(n2 · β)2 + (n1 · β)(n2 · β) − β 2     d1 · n2 x · n1 x0,1 · n1 2m − − d1 d1 r r0,1   $  d2 · n1 x · n2 x0,2 · n2 2m , − + d2 d2 r r0,2

(8.5.17)

8.5 Light-Rays in the PN-Field of a Single Body

315

Fig. 8.10 Unperturbed light rays in the problem of light deflection where the gravitating mass M is one of the two light sources. We see that cos α = sin θ0 = d1 · n2 /d1

S1

Observer

n1 θ0

d1 α

S2

n2

M where x is the position of the observer and r0,1 is r0 = |x0 | of light ray number 1 etc. The β-dependent terms describing the effects of aberration caused by the motion of the observer in our coordinate system have already been derived earlier in (5.5.14); the m-dependent parts describe the effects of gravitational light bending. For the following let us assume that also the observer is at rest in our coordinate system so that the aberration terms in (8.5.17) vanish. Let us furthermore assume that the light source 2 (e.g., our Sun) is identical with the gravitating mass M. Then d2 = 0 and the last term in Eq. (8.5.17) vanishes. Let cos θ0 = n1 · n2

(8.5.18)

describe the ‘unperturbed’ angle between M and the light source 1. Then from Fig. 8.10 we see that d1 · n2 = sin θ0 . d1

(8.5.19)

We can then rewrite (8.5.17) (d = d1 , n = n1 etc.) in the form cos θ = cos θ0 −

2m sin θ0 d



x · n x0 · n − r r0

 .

Let δθ ≡ θ − θ0

(8.5.20)

be the deflection angle, then with cos θ = cos(θ0 + δθ ) = cos θ0 − δθ sin θ0 + . . . we get δθ =

2m d



x · n x0 · n − r r0

 .

For light rays being emitted from remote astronomical sources r0  r,

x0 · n −1 r0

(8.5.21)

316

8 First Applications of the PN-Formalism

and x·n n1 · n2 = cos θ0 . r We, therefore, finally get 4m δθ = d



1 + cos θ0 2

 (8.5.22)

.

If the gravitating body is the Sun, m = 1476 km so that for light rays just grazing the Sun (θ0 0, d R 6, 96 × 105 km) the maximal deflection angle reads δθmax = 1. 75 .

(8.5.23)

8.5.3 The Gravitational Time Delay Actually the light deflection in the gravitational field of a body is accompanied by a retardation in the time domain. This effect was first discussed by Shapiro (1964) and is often called the Shapiro effect. If we consider xP  (t) = −2m ln

rN + xN · n r0 + x0 · n



and the light ray equation x(t) = x0 + c n(t − t0 ) + xP we obtain the coordinate time interval (t − t0 ) by multiplication of the last equation with n: 1 (8.5.24) [(x − x0 ) · n + (t, t0 )] c



r + r0 + |x − x0 | r +x·n (t, t0 ) = 2m ln = 2m ln . r0 + x0 · n r + r0 − |x − x0 | t − t0 =

The last equality results from |x − x0 | = n · (x − x0 ) and x2 − (n · x)2 = x20 − (n · x0 )2 = h2 , where h is the length of the perpendicular from the coordinate origin onto the (unperturbed) light-ray. This formula is fundamental e.g., for radar ranging measurements in the solar system where the gravitational potential of the Sun (S) cannot be neglected. For a signal emitted from the Earth (E), reflected off a planet

8.5 Light-Rays in the PN-Field of a Single Body

317

or spacecraft at xP and received back at Earth the excess roundtrip travel time δt is given by δt = 4mS ln

rP + xP · n . rE + xE · n

Since d 2 = rE2 − (xE · n)2 = (rE + xE · n)(rE − x · n) we can write this as



(rP + xP · n)(rE − xE · n) δt = 4mS ln d2

(8.5.25)

where the motion of the Earth and the planets have been ignored during the round trip of the signal. The gravitational time delay δt will be maximal for superior conjunction of the planet or spacecraft (see Fig. 8.11) when xE · n −rE ;

xP · n rP

and d will be close to the solar radius RS . In this case δt 4m ln(4rE rP /d 2 )   2  A.U. d . 240 μs − 20 μs ln RS rp

(8.5.26)

In the local geocentric reference system with T = TCG at spatial coordinates X (see Chap. 9) we write the gravitational time-delay in the form c(T −T0 ) = |X−X0 |+T with T = E + Tid

(8.5.27)

n Planet

d  RS

Earth

Sun Fig. 8.11 Geometry in the problem of gravitational time delay: a light-rays passes close to the limb of the Sun

318

8 First Applications of the PN-Formalism

where

2GME R + R0 + |X − X0 | E = ln R + R0 − |X − X0 | c2

(8.5.28)

is the gravitational time delay caused by the mass monopole of the Earth. Contributions from higher multipole moments of the Earth will be discussed later. Tid is the gravitational time delay caused by external bodies in the geocentric system. Writing the tidal potential in the form wTid

1 Gij Xi Xj 2

the corresponding Shapiro time-delay is of the form (e.g., Klioner 1992) 1 j Gij X0i X0 (T − T0 ) + Gij X0i N j (T − T0 )2 + (c/3)Gij N i N j (T − T0 )3 + · · · c (8.5.29) where N = (X − X0 )/|X − X0 |.

Tid =

8.6 The PN Motion of a Torque-Free Gyroscope Let1 LG be the world-line of a torque-free gyroscope whose spin axis presents an inertial direction in space. According to the Equivalence Principle there will be local inertial coordinates (LIC), Xα , with origin at LG , such that d S = 0, dτ

(8.6.1)

where S is the spin vector in Xα . In some global, non co-moving coordinate system we might define a 4-spin vector S μ which reduces to S μ → (0, S) in the local rest frame of the gyro, thus S μ is perpendicular to the gyro’s 4-velocity: S μ uν = 0 .

(8.6.2)

In LIC we find  Du S μ ≡ S μ ;ν uν =

1 This

dS 0 dS ; dτ dτ



section is based upon Straumann (2012) and Soffel (1989).

= 0.

(8.6.3)

8.6 The PN Motion of a Torque-Free Gyroscope

319

From (8.6.2) we infer that (Du S μ )uμ = −S μ Du uμ = −S μ aμ , where a μ is the 4-acceleration of the gyro. From this we find that (uμ uμ = −c2 ) Du S μ =

1 ν (S aν )uμ . c2

Because of condition (8.6.2) this implies that S μ is Fermi-Walker transported along its world-line: DF S μ = Du S μ −

1 1 (aν S ν )uμ + 2 (uν S ν )a ν = 0 . c2 c

(8.6.4)

In our global coordinates we thus get Sμ;ν uν =

1 uμ a λ Sλ c2

or   1 λ Sμ,ν − μν Sλ uν = 2 gμν uν a λ Sλ , c that leads us to dSμ 1 λ = μν Sλ uν + 2 gμν uν a λ Sλ . dτ c

(8.6.5)

Condition (8.6.2) leads to dt 1 dx i S0 = − Si dτ c dτ or S0 = −

vi Si . c

(8.6.6)

vi ai . c

(8.6.7)

In the same way from a μ uμ = 0 we get a0 = −

320

8 First Applications of the PN-Formalism

Hence, j k k j 1 dSi j j v 0 v 0 v v = 0i Sj − 0i Sj + ik Sj − ik Sj c dt c c c c    j vk 1 0v j −a Sj + a Sj . + 2 gi0 + gik c c c

Neglecting terms of order c−3 we get  j k dSi 1 j j v 0 v = c 0i − 0i + ik Sj + 2 v i a j Sj . dt c c c

(8.6.8)

Inserting the Christoffel-symbols to PN order 4 w,0 w[j,i] + 2 δij 3 c c w,i 0 i0 = − 2 c w,j w,k w,i j ik = δij 2 + δj k 2 − δik 2 c c c j

0i = −

we end up with 1  dSi = 2 −4w[j,i] Sj + w,t Si + 2(v · S)w,i + (v · ∇w)Si dt c  −v i (S · ∇w) + v i (a · S) .

(8.6.9)

This describes the motion of the spin-axis in our global coordinate system. We will now derive the corresponding dynamical equation in co-moving coordinates that are kinematically non-rotating with respect to the global ones (fixed-star oriented tetrad). As we shall see these co-moving coordinates will not be inertial. To this end we construct a co-moving tetrad along LG , kinematically non-rotating with respect to the global system. We start with a tetrad field for our global rest system:   w e˜(0)μ = 1 − 2 ; −g0j c   w  e˜(i)μ = 0; 1 + 2 δμi . c

(8.6.10)

8.6 The PN Motion of a Torque-Free Gyroscope

321 μ

μ

The co-moving, fixed-star oriented tetrad, e(α) , is obtained from e˜(α) by means of a μ Lorentz-boost such that e(0) = uμ /c. To this end we need the 3-velocity of the gyro μ with respect to e˜(α) (not the coordinate velocity v): v˜ i ≡

uμ e˜(i)μ u˜ i (1 + w/c2 )ui = uμ e˜(0)μ u˜ 0 (1 − w/c2 )u0

or   2w v˜ i 1 + 2 v i . c

(8.6.11)

Let: Sμ: S˜ μ : S μ:

global coordinate components of the spin vector; μ components with respect to e˜(α) ; μ components with respect to e(α) .

Then,  w S˜ i = S μ e˜(i)μ = 1 + 2 S i c and v)S ν S˜ μ = μ ν (−˜ with 00 (−˜v) 1 +

1 2

j

 2 v˜ c

0j (−˜v) = 0 (−˜v) ij (−˜v) δji +

v˜ j c

1 v˜ i v˜ j . 2 c2

This leads us to * ˜μ

S =

v˜j j i 1 v i vj j S ;S + S c 2 c2

+ .

This relation can be used to relate S i with S i :   w w 1 vi S i 1 − 2 S˜ i 1 − 2 S i + (v · S) 2 c2 c c

322

8 First Applications of the PN-Formalism

or  w 1 vi (v · S) . Si 1 + 2 Si + 2 c2 c Inverting this last relation yields  w 1 vi Si 1 − 2 Si − (v · S) . 2 c2 c

(8.6.12)

From this we get the time derivative of Si (remember that S˙i = O(2)) to PN-order: S˙i = S˙i − Si



w˙ v · ∇w + 2 c c2

 −

1 i 1 v˙ (v · S) − 2 v i (˙v · S) . 2 2c 2c

i c2 v˙ i − w Now, to ‘Newtonian order’ uμ = (c, v i ) and a i = ui;μ uμ v˙ i + 00 ,i so that

v˙ i w,i + a i . Using this relation and inserting expression (8.6.9) for S˙i we finally end up with the PN-relation:  1  S˙i = 2 v[i aj ] − 4w[j,i] − 3v[i w,j ] Sj , c

(8.6.13)

that, because of (A × B) × C = B(A · C) − A(B · C), can be written in the form dS = gyro × S dτ

(8.6.14)

3 1 1 v × ∇w − 2 ∇ × g − 2 v × a 2 2c 2c 2c

(8.6.15)

g i ≡ −4wi .

(8.6.16)

with gyro = and

The first term in (8.6.15) is the geodetic precession, the second term containing the gravito-magnetic vector potential g is that of the Lense-Thirring precession and the last term containing the non-gravitational acceleration a (the spatial part of the 4-acceleration, not the coordinate acceleration) is the angular velocity of Thomasprecession that vanishes if the gyro is in free-fall (Fig. 8.12).

8.7 Geodesic Motion in the PN-Schwarzschild Field

323

Fig. 8.12 In the Gravity Probe B experiment ‘ideal’ gyroscopes on board a satellite 642 km above the ground move by 6606 mas/y due to geodetic precession and by 39 mas/y due to Lense-Thirring precession with respect to distant fixed stars (Image credit: Everitt et al. 2011)

8.7 Geodesic Motion in the PN-Schwarzschild Field We now come to the problem of motion of some satellite or planet or Moon in the gravitational field of some spherically symmetric central mass that is described by wi = 0 and w=

μ GM ≡ . r r

According to (5.3.5) the geodesic equation takes the form  j d 2xi vj vk 2 i i v 00 jik =−c + 20j 2 c c c dt

$ j vj vk vi 0 0 v + j0k . + 20j − 00 c c c c Inserting our post-Newtonian Christoffel-symbols the satellite equation takes the form  d 2 xSi 1  j i 2 i j −4ww = w + − 4w v v + w v − 3w v + 4w + 8w v ,i ,i ,j ,i ,t i,t [i,j ] dt 2 c2 + O4 , (8.7.1)

324

8 First Applications of the PN-Formalism

where w[i,j ] ≡ (1/2)(wi,j − wj,i ) and v i is the coordinate velocity of the satellite. For a mass-monopole we have w = μ/r, wi = 0, so that the equation of motion takes the form

xi d 2xi xi μ xi 2 vi 4μ = −μ + − v + 4 (x · v) . (8.7.2) dt 2 r3 c2 r r3 r2 r2 It has several advantages to derive this equation of motion from a Lagrangian L. As we have seen earlier the geodesic equation satisfies an extremal principle of the form

  ds 0 = δ ds = δ dt dt and instead of ds we can equally well use the proper time interval dτ . For that reason one usually defines a Lagrangian L via dτ = 1 − c−2 L . dt

(8.7.3)

From g00 = −1 +

2μ2 2μ − 4 2 + ··· 2 c r c r

g0i = 0

  2μ + ··· gij = δij 1 + 2 c r

and metric property 2 we get

1/2 2μ 2μ2 v2 2μ v2 dτ = 1− 2 + 4 2 − 2 − 2 2 dt c r c r c c rc

(8.7.4)

and the Lagrangian takes the form L=

μ 1 2 1  μ 2 3 μ  v 2 1 v 4 + v − 2 + + r 2 r 2r c 8 c2 2c

(8.7.5)

up to terms of O4 . Exercise 8.2 Show that the post-Newtonian equation of motion (8.7.2) can be obtained as Euler-Lagrange equations ∂L d ∂L − =0 i ∂x dt ∂v i from the Lagrangian (8.7.5).

(8.7.6)

8.7 Geodesic Motion in the PN-Schwarzschild Field

325

The main advantage of the Lagrangian is the possibility to construct explicitly first integrals of motion connected with the conservation of (specific) energy E and angular momentum J. These two conserved quantities are given by ∂L −L ∂v  1 μ 3  v 2 μ μ = v2 − + v2 + 3v2 + 2 2 r 8 c 2c r r

E =v

(8.7.7)

and J = x×

∂L ∂v



3μ 1 v2 + 2 = (x × v) 1 + 2 2c c r

 .

(8.7.8)

Exercise 8.3 Use the equations of motion to prove that E and J are conserved quantities, i.e. integrals of motion. Note that the conservation of angular momentum implies that the orbit is confined to a coordinate plane. Using polar coordinates in the orbital plane with x = r er ,

x˙ = r˙ er + r φ˙ eφ

and |x × x˙ | = r 2 φ˙ ,

v2 = r˙ 2 + r 2 φ˙ 2

the specific energy E and the absolute value of the orbital angular momentum J = |J| can be written as 3 1 2 μ (˙r + r 2 φ˙ 2 ) − + 2 (˙r 2 + r 2 φ˙ 2 )2 2 r 8c  μ μ 2 + 3(˙r + r 2 φ˙ 2 ) + 2 2c r r   1 3μ . J = r 2 φ˙ 1 + 2 (˙r 2 + r 2 φ˙ 2 ) + 2 2c c r E=

(8.7.9)

This leads us to first order differential equations of motion   1 3μ 1 − 2 (˙r 2 + r 2 φ˙ 2 ) − 2 2c c r   E 4μ = J 1− 2 − 2 c c r

r 2 φ˙ = J

(8.7.10)

326

8 First Applications of the PN-Formalism

and r˙ 2 = −r 2 φ˙ 2 + −

2μ 3μ 3 + 2E − 2 (˙r 2 + r 2 φ˙ 2 )2 − 2 (˙r 2 + r 2 φ˙ 2 ) r 4c c r

μ2 c2 r 2

= −r 2 φ˙ 2 +

E2 2μ μE μ μ + 2 E − 3 2 − 12 . − 10 r r c2 r c2 r c

(8.7.11)

Eliminating the φ˙ 2 term from the last equation we get r˙ 2 = A +

C 2B D + 2+ 3 r r r

(8.7.12)

with E2 c2 E B = μ − 6μ 2 c   2E μ2 C = −J 2 1 − 2 − 10 2 c c μ D = 8J2 2 . c A = 2E − 3

Using  2  dr(φ(t)) 2 dr ˙ φ r˙ = = dt dφ     d(1/r) 2 μ E 1−8 2 −2 2 = J2 dφ c r c 

2

the radial equation can be written in the form 

d(1/r) dφ

2

= A +

C 2B  + 2 r r

(8.7.13)

8.7 Geodesic Motion in the PN-Schwarzschild Field

327

with   1E 1+ 2 c2   E μ B = 2 1 + 4 2 J c 2E A = 2 J 

C  = −1 + 6

μ2 c2 J 2

.

Let us now write the radial equation in the form 

d(1/r) dφ

2

 =C

1 1 − r a(1 + e)



1 1 − a(1 − e) r

 (8.7.14)

with C =1−6

μ2 . c2 J 2

(8.7.15)

From this representation we see that r± = a(1 ± e) represent the minimal and maximal values for r, i.e. a and e have the meaning as semi-major axis and numerical eccentricity of the post-Newtonian orbit and might be considered as integration constants alternative to E and J . We have A =

2E J2

  1E 1 1+ =− 2 C 2 2c a (1 − e2 )

and B =

μ J2

 1+4

E c2

 =

C a(1 − e2 )

from which we derive

6 μ μ J 2 = μa(1 − e2 ) 1 − 2 2 + c a 1 − e2 c2 a

7μ μ 1− 2 . E =− 2a 4c a

(8.7.16) (8.7.17)

The solution of (8.7.14) is then simply given by r=

a(1 − e2 ) 1 + e cos f

(8.7.18)

328

8 First Applications of the PN-Formalism

where the ‘true anomaly’ f obeys the relation 

df dφ

2 =C =1−6

μ , a(1 − e2 )c2

(8.7.19)

i.e., f = 1−3

μ a(1 − e2 )c2

(φ − φ0 ) .

(8.7.20)

This implies that the post-Newtonian orbit is that of a precessing ellipse, the secular drift of the argument of the pericenter per revolution being given by φ = 2π

3μ . a(1 − e2 )c2

(8.7.21)

Eliminating E and J from (8.7.10) we obtain r 2 φ˙ =

 

 3 μ a 1 − 4 μa(1 − e2 ) 1 + − + 2 (1 − e2 ) r c2 a

(8.7.22)

or together with (8.7.19) 

μa(1 − e2 )dt

1 μ μ . = r df 1 + 4 2 + 2 c2 a c r 2

For a circular orbit with r = a, e = 0 we have μ  μ  φ˙ 2 ≡ n2 = 3 1 − 3 2 a c a

(8.7.23)

(8.7.24)

defining the mean motion n of the post-Newtonian orbit and the mean anomaly M via the relation M = n(t − t0 ) = nt + M0 .

(8.7.25)

If one introduces as in the usual Newtonian Kepler theory an eccentric anomaly E via sin f =

(1 − e2 )1/2 sin E ; 1 − e cos E

cos f =

cos E − e 1 − e cos E

(8.7.26)

8.8 Celestial Mechanical Perturbation Theory

329

so that √ 1 − e2 df = dE 1 − e cos E

(8.7.27)

r = a(1 − e cos E)

(8.7.28)

and

then the integration of (8.7.23) leads to a corresponding Kepler equation in the form   μ  μ  M = 1+3 2 E− 1− 2 e sin E . c a c a

(8.7.29)

The time dependence of the orbital point is then obtained by means of the Kepler equation via t → M → E → f .

8.8 Celestial Mechanical Perturbation Theory 8.8.1 Post-Newtonian Schwarzschild Effects Although in the last chapter we obtained a solution to the problem of satellite orbits in the PN-Schwarzschild field in closed form it is useful also to apply classical mechanical perturbation theory to the equations of motion (8.7.1)  dv μ  x μx = − 3 + 2 3 4μ − v2 x + 4v(x · v) . dt r r c r

(8.8.1)

Let us write this equation in the form μx dv = − 3 + aP , dt r

(8.8.2)

i.e., let us formally interpret (8.8.1) as the equation of motion of a perturbed Keplerian problem in our chosen coordinate system that we can treat with the means of classical perturbation theory. It should be clear that we deal with coordinate objects only. We can interpret aP as perturbing function and decompose into the usual components S, T and W. For our PN-Schwarzschild problem we use the relations x = rn ;

v = r˙ n + r f˙m

330

8 First Applications of the PN-Formalism

to get the perpendicular part of the perturbing function as T =4

μ r˙ (r 2 f˙) . c2 r 3

Using the Keplerian relations r=

p ; 1 + e cos f

r˙ =

C e sin f ; p

r 2 f˙ = C

(8.8.3)

with p = a(1 − e2 ) C 2 = μp ,

(8.8.4)

this result can be written in the form T =4

μ μ (1 + e cos f )3 e sin f . c2 p 3

(8.8.5)

For the radial and normal part we get S= =

 μ  μ 4 − v2 + 4˙r 2 r

c2 r 2

μ2 (1 + e cos f )2 [3 + 3e2 + 2e cos f − 4e2 cos2 f ] , c2 p 3

(8.8.6)

W = 0. W = 0 indicates that, due to the post-Newtonian conservation of angular momentum, the motion is restricted to a plane. This also implies that, I = 0 ,  = 0 .

(8.8.7)

Integration of the S, T , W-perturbation equations over the time variable t or the true anomaly f with dt =

1  r 2 1 df √ n a 1 − e2

and r = a(1 − e cos E)

8.8 Celestial Mechanical Perturbation Theory

331

gives t μe 2 [(14 + 6e ) cos f + 5e cos 2f ] 2 2 2 c (1 − e ) t0

t μ 5 (3 + 7e2 ) cos f + e cos 2f e = − 2 2 c p t0

 t  5 3 μ − e sin f − sin 2f 3f − ω = 2 e 2 c p t0 a = −

 = (1 −



1 − e2 )ω

(8.8.8)

t μ  2 + 2 (6 1 − e E − 12f + 8e sin f ) c p t0

and

t t0

3μ n dt = 2 c a

    ) t a 2 a f− 5 −3 M −3E + √ 2 r0 r0 1−e t

#

5

(8.8.9) 0

with

M = δ − ω +

t

n dt . t0

From (8.8.8) we immediately obtain the secular drift in the argument of perigee per revolution by < ω >rev = 6π

m p

(8.8.10)

where m ≡ GM/c2 and p = a(1 − e2 ), in accordance with (8.7.21). Again the reader is reminded that the orbit z(t) or the corresponding orbital elements are pure coordinate objects and our results refer to harmonic coordinates. Exercise 8.4 Use Ashby’s formula below (Ashby 1986) for the exact anomalous perihelion precession per revolution in standard Schwarzschild coordinates in the Schwarzschild field to compute higher order terms in m. Ashby’s formula reads: < ω >rev = 4K(z)D −1/2 − 2π

(8.8.11)

with D = 1 − 2m(3 − e)/p, z = 4me/(pD) and K(z) is the complete elliptic integral of the first kind

π/2

K(z) = 0

with |z| < 1.

(1 − z sin2 θ )−1/2 dθ

(8.8.12)

332

8 First Applications of the PN-Formalism

Solution Using the expansion (Abramowitz and Stegun 1970, expression (17.3.11)) K(z) =



1 9 25 3 π 1 + z + z2 + z + ··· 2 4 64 256

(8.8.13)

one obtains: < ω >rev = 6π

m 3π m2 45π m3 + (18 + e2 ) 2 + (6 + e2 ) 3 + · · · , p 2 2 p p

(8.8.14)

(see e.g., Poveda and Marin 2018). The ‘semi-latus rectum’ p is given by p = a(1 − e2 ). Here a and e are defined via the smallest and largest value for the radial coordinate in the orbit: r± = a(1 ± e). Exercise 8.5 Compute the relativistic (post-Newtonian) perihelion advance for all planets all the way to Saturn and compare the results with corresponding values from the literature. E.g., Cugusi and Proverbio (1978) give for < ω >century : 42. 98 (Mercury), 8. 63 (Venus), 3. 84 (Earth), 1. 35 (Mars), 0. 06 (Jupiter) and 0. 01 (Saturn). Exercise 8.6 Approximate the satellite equation of motion in the harmonic Schwarzschild field by d 2 xS = a0 + a2 + a4 , dt 2

(8.8.15)

where the Newtonian acceleration reads a0 = −μn/r 2 with μ = GM and n = x/r and the first post-Newtonian acceleration is given by  μ  a2 = 2 2 n(−v2 + 4μ/r) + 4(n · v)v . c r Using the Christoffel symbols of Exercise 6.12 compute the post-post Newtonian acceleration a4 . Solution One obtains (see e.g., Damour 1987b): a4 =

μ μ  1  μ  2(n · v)2 n − 2(n · v)v − 9 n . 2 2 2 r r c r c

(8.8.16)

8.8.2 The Lense-Thirring Effect Let us now investigate how the spin-dipole field of the Earth will modify the motion of a satellite. To this end we consider an equation of the form d 2 xSi i = aN + aSi , dt 2

8.8 Celestial Mechanical Perturbation Theory

333

where we consider only the mass-monopole term in the ‘Newtonian acceleration’, aN , and the spin contribution, according to (8.7.1), is given by aSi =

4 (wi,j − wj,i )v j c2

that we can write as 1 v×B c2

aS =

(8.8.17)

with Bi = −

6G nˆ ij Sj . r3

Let S = Sez , then ⎛ ⎞ +(x 2 + y 2 − 2z2 )v y + 3yzv z 2GS ⎝ aS = 2 5 −(x 2 + y 2 − 2z2 )v x − 3xzv z ⎠ . c r 3z(xv y − yv z )

(8.8.18)

The decomposition of this perturbing function into radial-, perpendicular- and normal part, S, T and W is then given by 1 (v × B) · n c2 1 T = aS · m = 2 (v × B) · m c 1 W = aS · k = 2 [(x · v)(B · v) − v2 (B · x)] , c C √ where (C = x × v, C = |x × v| = GMp) S = aS · n =

k≡

C ; C

m=

(8.8.19) (8.8.20) (8.8.21)

C×x . |C × x|

For S, T and W one finds (Lense and Thirring 1918) (u = ω + f ) cos I r4 e cos I sin f T = −K pr 3   er sin f cos u sin I W = K 4 2 sin u + p r S=K

(8.8.22)

334

8 First Applications of the PN-Formalism

with K=

2G CS . c2

Exercise 8.7 Derive the expressions (8.8.22) for S, T and W for the spin-dipole field of a body. Solution Using nˆ 3k = n3 nk − (1/3)δ3k we get S=

2G cos I (x × v) · S = K 4 . 2 4 c r r

For T we find T =

1 1 (v × B) · (C × x) = − 2 (n · x)(B · C) . c2 Cr c C

Now, n · x = r˙ =

Ce sin f p

and B·C=−

6GS 2GS nˆ 3i C i = 3 C cos I , 3 r r

proving relation (8.8.22) for T . To derive the expression for W we note that n · v = n˙r , B·v=−

6GS 2GS (n · v)n3 + 3 v3 , 3 r r

v2 =

C2 + (n · v)2 r2

and B·n=−

4GS n3 . r3

Inserting all of these expressions into (8.8.19) we get expression (8.8.22) for W.

8.8 Celestial Mechanical Perturbation Theory

335

For the change of orbital elements one then finds to first order (Lense and Thirring 1918) (ξ = K/(C 2 p) = [2GS/(c2 μ1/2 p3/2 )] a = 0 e = −ξ cos I (1 − e2 ) cos f 1 I = − ξ sin I (cos 2u + 2e cos f cos2 u) 2   1 + e2 sin f + (1 − cos I )

 = −ξ cos I 2f + e  

1 1  = ξ f − sin 2u + e sin f − sin 2u cos f 2 2  e2  = −2 1 − e2 ξ cos I (f + e sin f ) +  √ 1 + 1 − e2  I + 2 1 − e2 sin2  . 2

(8.8.23)

For the secular perturbations one then gets per revolution <  >LT = 2π ξ < ω >LT = −3 cos I <  >LT

(8.8.24)

and <  >LT =<  >LT = (1 − 3 cos I ) <  >LT .

(8.8.25)

Hence, the gravito-magnetic field of the rotating Earth induces an additional perigee precession of satellite orbits and a secular drift of the nodes of the same order of magnitude. This effect was first described by Lense and Thirring (1918); Thirring (1918) and is frequently referred to as the Lense-Thirring effect. For the orbit of the lasergeodynamical satellite LAGEOS this secular drift of the node is of order 2 × 10−5 per revolution, roughly comparable with the effect from the l = 12 zonal harmonic.

Chapter 9

Astronomical Reference Systems

9.1 The Problem of Celestial Mechanics The dynamical behaviour of N interacting bodies of arbitrary shape and composition, rotating and vibrating under their mutual influence of their gravitational attractions is central for the field of celestial mechanics. Principally, this gravitational N -body problem can be divided into three parts: (1) the local problem, (2) the global problem and (3) the way how the local and the global problems are related. The local problem deals with the physics of the various bodies involved, i.e., their individual gravitational fields and how the various local physical interactions between the dynamics of the local sub-systems (atmosphere, ocean, solid sphere, fluid cores etc.) influence their time behaviour. This local problem should be treated in a local coordinate system that is moving with the body under consideration. The global problem deals with the overall translational and rotational motions of the bodies in some ‘global’ coordinate system that encompasses all N bodies of the system. Finally, one needs a framework that matches the various local systems with the global system: an adequate theory of (astronomical) reference systems. Such a theory of reference systems is unproblematic in the Newtonian space-time due to its absolute properties. However, already in Special Relativity (and especially in GR) the construction of such an important theory of local and global reference systems is highly problematic. In this chapter a theory of astronomical reference systems is formulated at the first post-Newtonian approximation to Einstein’s theory of gravity. Based on many previous works (Brumberg and Kopejkin 1989a,b; see also Kopeikin 1988, Klioner and Voinov 1993) Damour-Soffel and Xu (DSX-I) have introduced this formalism in 1991.

© Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_9

337

338

9 Astronomical Reference Systems

9.2 Transformation Between Global and Local Systems For the description of the gravitational N -body problem we will consider a total of N + 1 different coordinate systems: one global coordinate system x μ = (ct, x i ) in which all N bodies are contained and in which the global dynamics of the system a ), A = 1, . . . , N, where the can be described and N local charts XA = (cTA , XA α system XA is assumed to move with body A of the system (Fig. 9.1). In the following we will often speak about two coordinates systems, a global one, glob , with coordinates x μ = (ct, x i ) and a local one, loc , with coordinates Xα = (cT , Xa ). Mostly the global one will be the Barycentric Celestial Reference System (BCRS) and the local one will be the Geocentric Celestial Reference System (GCRS). We now assume the corresponding metric tensors to be of the following canonical form g00 = − exp(−2w/c2 ) g0i = −

4 wi c3

(9.2.1)

gij = δij exp(+2w/c2 ) + O4

Fig. 9.1 One global and N local coordinate systems are used for the description of the gravitational N -body problem (Image credit: Damour et al. 1992b)

9.2 Transformation Between Global and Local Systems

339

in glob and G00 = − exp(−2W/c2 ) G0a = −

4 Wa c3

(9.2.2)

Gab = δab exp(+2W/c2 ) + O4 in loc , i.e., we assume the metric tensors to be of the same form but with metric potentials w μ ≡ (w, wi ) in the global system and different ones W α ≡ (W, W a ) in the local system. Moreover, we assume the usual conditions for the energymomentum tensor T 00 = O2 ,

T 0i = O1 ,

T ij = O0

(9.2.3)

and ∂0 ≡

∂ = O1 · ∂i ∂ct

(9.2.4)

in the global system and corresponding conditions in loc . We again write the transformation Xα → x μ in the general form x μ (Xα ) = zμ (T ) + eaμ (T )Xa + ξ μ (T , Xa ) , where ξ μ is at least quadratic in Xa . Here zμ (T ) described the world-line of some suitably selected point associated with the body under consideration. This worldline will be called the central world-line of the corresponding body; later it will be chosen as the body’s post-Newtonian center of mass. μ

Lemma 9.1 Let Aα ≡ ∂x μ /∂Xα and assume A00 = 1 + O2 ,

Ai0 = O1 ,

A0a = O1 ,

Aia = δia + O2 .

(9.2.5)

Then under these assumptions we have e00 (T ) ≡

1 dz0 = 1 + O2 , c dT

1 dzi i e + O3 , c dT a    1 1 j e00 (T )eai (T ) = 1 + 2 v2 δij + 2 v i v j Ra (T ) + O4 , 2c 2c ea0 (T ) =

(9.2.6) (9.2.7) (9.2.8)

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9 Astronomical Reference Systems

ξ 0 (T , Xa ) = O3 , ξ i (T , Xa ) =





1 i 1 e (T ) Aa X2 − Xa (A · X) + O4 , 2 c2 a

(9.2.9) (9.2.10)

where vi =

dzi dzi = + O2 dt dT

Aa = eai

d 2 zi d 2 zi = eai 2 + O2 2 dT dt

and Rai (T ) is a slowly time-dependent rotation matrix with j

Rai Ra = δij ,

Rai Rbi = δab

(9.2.11)

and dRai = O2 . dT

(9.2.12)

μ

Proof The quantities eα (T ) can be obtained as eαμ (T ) = Aμ α |Xa =0 . The transformation of the metric tensor can then be written as ν μ ν Gαβ = ηαβ + Hαβ = Aμ α Aβ gμν = Aα Aβ (ημν + hμν )

with H00 = O2 = h00 , H0a = O3 = h0i and Hab = O2 = hij . Thus ν ημν Aμ α Aβ = ηαβ + (PN)αβ

or explicitly, −(A00 )2 + Ai0 Ai0 = −1 + O2 −A00 A00 + Ai0 Aia = O3 −A0a A0b + Aia Aib = δab + O2 . With our assumptions we get A00 = 1 + O2 ,

Aia Aib = δab + O2 ,

−A0a + Ai0 Aia = O3 .

(9.2.13)

9.2 Transformation Between Global and Local Systems

341

Furthermore, A0a (T , X) = O1 = ea0 (T ) +

∂ξ 0 ∂Xa

and since the last term is of order X we find ea0 = O1 = ξ 0 . Then, A00 = e00 (T ) +

1 dea0 (T ) a 1 ∂ξ 0 X + = 1 + O2 c dT c ∂T

and since the last two terms in the middle are O2 we finally get the first relation (9.2.6) e00 =

1 dz0 = 1 + O2 . c dT

In a similar way one finds the second and fourth relation, (9.2.7) and (9.2.9) and ξ i = O2 ,

deai = O2 , dT

eai (T )ebi (T ) = δab + O2 .

The remaining relations (9.2.8) and (9.2.10) are shown in the following exercise. Exercise 9.1 Show that the condition of spatial isotropy of the metric tensors can be written in the form √ −gg ij = δij + O4 ;

√ −GGab = δab + O4

(9.2.14)

that leads to the relation ij ηαβ Aiα Aiβ = |Aμ γ |δ + O4 .

(9.2.15)

From this one derives ηαβ Aiα Aiβ = (2 + ηαβ A0α A0β ) δ ij + O4 and j eai ea

eai



vi vj V2 ij 0 2 − 2 = δ 2 − (e0 ) + 2 + O4 , c c

i ∂ξ j A·X j ∂ξ + e = −2e00 2 δ ij + O4 , a ∂Xa ∂Xa c

(9.2.16) (9.2.17)

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9 Astronomical Reference Systems

where v i = dzi /dT + O2 = eai V a + O2 . Show that relation (9.2.8) then follows from (9.2.16) and (9.2.10) follows from (9.2.17). Let is now consider the transformation law for the metric potentials. Theorem 9.1 From the transformation of the metric tensors one finds    2v2 c2  0 0 4 w = 1 + 2 W + 2 v a Wa + ln A0 A0 − A0a A0a + O4 2 c c c3 w i = Rai W a + v i W + (A00 Ai0 − A0a Aia ) + O2 , 4

(9.2.18)

where v i ≡ Rai V a

V a ≡ Ria v i .

or μ

The proof follows directly from g μν (x) = Aα Aνβ Gαβ (X). Note that the post-Newtonian transformation of metric potentials (W, W a ) → (w, wi ) is linear, i.e., of the form α μ w μ (x) = Aμ α (T )W (X) + B (X) .

(9.2.19)

Here, A00 = 1 +

2v2 ; c2

A0a =

4 a v ; c2

Ai0 = v i ;

Aia = Rai

(9.2.20)

and B0 =

 c2  0 0 ln A0 A0 − A0a A0a ; 2

Bi =

c3 0 i (A A − A0a Aia ) . 4 0 0

(9.2.21)

In the following we shall also need the inverse of Eq. (9.2.18) namely Wα = A−1 αμ (wμ − Bμ ) ,

(9.2.22)

which reads explicitly

4 2 2 W = 1 + 2 V (w − B0 ) − 2 v i (wi − Bi ) + O4 c c Wa = −V (w a

− B0 ) + Rai (wi

− Bi ) + O2 .

(9.2.23)

9.3 Split of Local Potentials, Multipole-Moments

343

9.3 Split of Local Potentials, Multipole-Moments The following two Theorems are the heart of the DSX-formalism. Let us consider the metric tensor Gαβ in the local system defined by the metric potentials Wα ≡ (W, Wa ). In the gravitational N -body system these local potentials can be split into two parts W (T , X) = Wself (T , X) + Wext (T , X) a a W a (T , X) = Wself (T , X) + Wext (T , X) .

(9.3.1)

In the following we will often use the notation a

W ≡ Wext ;

a W ≡ Wext .

(9.3.2)

If the local system is associated with body E (e.g., the Earth) then the self parts α , result from the gravitational action of body E of the metric potentials, Wself α , result from the action itself whereas the external parts of the potentials, Wext of all other bodies of the system and inertial terms that appear in the local Esystem. a further. They can be It might be useful to split the external parts Wext and Wext written in the form Wext = Winer + Wtidal ;

a a a Wext = Winer + Wtidal ,

(9.3.3)

where the inertial contributions are linear and the tidal terms are at least quadratic in the local spatial coordinate Xa . As we already know from Sect. 5.6 Winer results from the 4-acceleration of the origin of the local system, i.e., from a deviation of the a -term gives rise to a kind of Coriolis-term in local system from free-fall. The Winer case the spatial axes are not Fermi-Walker transported. Only for such inertial axes a -term will vanish. the Winer In mathematical terms the self-parts of the local metric potentials are defined by

d 3 X

Wself (T , X) = G

a (T , X) Wself

E

d 3 X

=G E

(T , X ) G ∂2 + |X − X | 2c2 ∂T 2

d 3 X (T , X )|X − X | ,

E

 a (T , X ) , |X − X | (9.3.4)

where the integrals extend over the support of body E only. Clearly, if there is only one body in the system, then these expressions reduce to the ones from (7.3.20).

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9 Astronomical Reference Systems

Theorem 9.2 (Extended Blanchet-Damour Theorem) In the local system of body α , admit a convergent E, outside of E, the self-parts of the metric potential, Wself expansion of the form (R = |X|) Wself (T , X) = G

 (−1)l l!

l≥0

a Wself (T , X)

∂L [R −1 MLE (T± )] +

1 ∂T E + O4 c2

  −1 d E ∂L−1 R M = −G l! dT iL l≥1

l 1 −1 E ij k ∂j L−1 (R SkL−1 ) − ∂i E + O2 . + l+1 4  (−1)l

(9.3.5)

Here, E ≡ 4G

l≥0

μEL

 (−1)l 2l + 1 ∂L (R −1 μEL (t± )) (l + 1)! 2l + 3

≡

3

ˆ bL

(9.3.6)

b

d X X  (T , X) E

and

d2 1 = d 3 X Xˆ L X2  2(2l + 3)c2 dT 2 E E

d 4(2l + 1) d 3 X Xˆ aL  a (l ≥ 0) − (l + 1)(2l + 3)c2 dT E

(l ≥ 1) SLE (T ) = d 3 X ij i  j , d X Xˆ L  +

MLE (T )

3

(9.3.7)

E

where the brackets again indicate the STF-part for those indices that are enclosed by the brackets and f (T± ) = [f (T + R/c) + f (T − R/c)]/2. The proof is analogous to the one for the Blanchet-Damour Theorem. MLE and SLE are the BD mass- and current multipole moments of body E that reduce to the corresponding moments in case that there is only one single body. Note, that a post-Newtonian center of mass of body E can be introduced by the vanishing of the BD mass-dipole, i.e., by MaE = 0 .

(9.3.8)

9.3 Split of Local Potentials, Multipole-Moments

345

Theorem 9.3 Let

d 3x

wE (t, x) = G

wEi (t, x)

E

G ∂2 σ (t, x ) + |x − x | 2c2 ∂t 2

3 σ

=G

d x

d 3 x  σ (t, x )|x − x | ,

E

i (t, x )

(9.3.9)

|x − x |

E

be the metric potentials in the global system induced by body E, then μ

α + O(4, 2) , wE = Aμα (T )Wself,E

(9.3.10)

or explicitly 

2v2 wE = 1 + 2 c wEi

=

a Rai Wself,E

 Wself,E +

4 a a v Wself,E + O4 c2

(9.3.11)

+ v Wself,E + O2 i

α are the self-parts of the metric potentials in the local E-system and where Wself,E the velocity v refers to the central point of that system (e.g., the barycentric velocity of the geocenter).

Proof Though relations (9.3.10) look simpler than (9.2.19) the proof is considerably more complicated. It can be proven in several independent ways. Let us indicate some important steps of one proof. Start from our standard solution in the harmonic gauge, where in glob

wEi (t, x)

d 3x

=G E

σ i (t, x ) . |x − x |

We now relate each quantity on the right hand side with corresponding quantities in the local E-system. Since we need wEi only to Newtonian order this is fairly simple. From x i = zi + eai Xa + O2 , eai = Rai + O2 one infers |x − x | = |X − X | + O2 and d 3 x = d 3 X + O2 . From the transformation rule of the energy-momentum tensor one derives σi =

T 0a (X) T 0i (x) = v i T 00 (X) + Rai c c

leading to the second part of (9.3.11). The proof for the transformation rule for wE is obviously more complicated. Let us write the standard form of the metric potential wE in the global system as

d 3x

wE (t, x) = G E

σ (t, x ) 1 E − χ |x − x | 2 ,tt

(9.3.12)

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9 Astronomical Reference Systems

where the superpotential χ E (t, x) is given by

d 3 x  σ (t, x )|x − x | .

χ E (t, x) = −G

(9.3.13)

E

We then have to transform d 3 x, σ and |x − x |−1 with post-Newtonian accuracy, for E a Newtonian transformation is sufficient. From the chain the transformation of χ,tt rule we get ∂t = ∂T − v a ∂a and since χ E (x) = χ E (X) + O2 we infer E E = χ,T χ,tt T −

aEa E vEa E vEa vEb E χ χ (X) − 2 (X) + χ (X) . c ,T a c2 ,a c2 ,ab

For the 3-dimensional volume element we get d 3x =

1 4  1 |Aμ |d 3 X . d x =   dt (dt /dT  ) α

Now, e2 1 1 a dt  = 1 + 2 + 2 AE · X + 2 V  vEa + O4 dT  c c c where we wrote e00 = 1 +

1 e2 + O4 . c2

(9.3.14)

AE is the acceleration of the origin of the local E-system and V is the velocity of some material element in the local system. Together with |Aμ α| = 1 − 2

v2E e2 AE · X + − 2 + O4 c2 c2 c2

one finds *

v2 e2 AE · X 1 a d x = 1 − 3 2 + E2 − 3 − 2 V  vEa 2 c c c c 3 

+ d 3 X .

(9.3.15)

The transformation of the energy-momentum tensor leads to 

 va a e2 AE · X v2E σ = 1 + 2 2 + 2 2 + 2  + 4 E 2 + O4 . c c c c

(9.3.16)

The most complicated part of the proof is the transformation of the inverse distance. To this end let us consider the three events in Fig. 9.2 denoted by eX , eT

9.3 Split of Local Potentials, Multipole-Moments Fig. 9.2 One event eX is described in two charts, i.e., with local coordinates (cT , Xa ) and local coordinates (ct, x i ). The T = const. hyper-surface hits the central world-line LE at the point eT at local time Tsim and the t = const. hyper-surface at the point et at global time tsim

347

eT

T =c onst.

ex nst.

et

t = co LE

and et . In the local E-system eX has coordinates (cT , Xa ) related to (ct, x i ) by the general transformation rule (9.2.19), eT (et ) denotes the intersection of the T = const. (t = const.) hyper-surface through eX with the world-line LE of the origin of the local E-system (e.g., the geocenter), given by Xa = 0. These two events have coordinates eT : (tsim , zEi (tsim )

(T , 0) ,

et : (t, zEi (t))

(Tsim , 0) .

(9.3.17)

Using the general transformation rule one finds that Tsim = T + tsim

1 ea0 a X + O4 c e00

(9.3.18)

1 = t − ea0 Xa + O4 c

and with zsim (t) ≡ z(Tsim ) *

e0i ea0

i x i − zsim (t) = eai −

e00

+ Xa + ξ i (T , Xa ) + O4 .

(9.3.19)

T

Using that equation for a second point one arrives at * i

x −x = i

eai

−

e0i ea0 e00

*

+

V  a vEb b b (X − X ) X − X + c2

+

a

a

ξ i (T , Xa ) − ξ i (T  , X ) a

(9.3.20)

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9 Astronomical Reference Systems

and finally  1 1 1 v2E (vE · N)2 e2 = − + 1 + |x − x | |X − X | 2 c2 c2 2c2 (9.3.21) +

(V · N)(vE · N) AE · X AE · (X − X ) + − c2 c2 2c2



with N≡

X − X . |X − X |

Inserting all of these expressions into wE (t, x) completes the proof for our standard form of the metric tensor. Another independent proof can be found in DSX-I.

9.4 Local Harmonic Proper Coordinates μ

So far the quantities eα have been constrained but not fully specified. The will now be done by requiring these quantities to represent an orthonormal tetrad along the worldline LE of the origin of the local E-system given by Xa = 0 with respect to the external metric ext μ ν eα eβ |LE = ηαβ . gμν

(9.4.1)

ext , es defined by Here, the external metric tensor, gμν ext = −e−2w/c , g00 2

ext g0i =−

4 i w , c3

(9.4.2)

2

gijext = δij e2w/c + O4 with wμ =



μ

wA .

(9.4.3)

A=E

Thus, considering the external part of the metric only, the local coordinates Xα will be chosen as tetrad induced coordinates. One consequence is α (T , 0) = 0 Wext

(9.4.4)

9.4 Local Harmonic Proper Coordinates

349

and the external part of the metric is Minkowskian at the spatial coordinate origin, i.e., Gext αβ |Xa =0 = ηαβ .

(9.4.5)

The gravitational influence of external bodies is effaced and relation (9.4.4) is sometimes called weak effacement condition. Theorem 9.4 From the tetrad condition (9.4.2) one infers that   1 1 2 v + w(zE ) =1+ 2 2 E c   5 1 3 4 1 vE + (w(zE ))2 + w(zE )v2E − 4w i (zE )vEi + O6 + 4 8 2 2 c    $  i 4 i 1 1 2 0 i vE ea (T ) = Ra − 3 w (zE ) + O5 1+ 2 v + 3w(zE ) c c 2 E c    1 1 j j eai (T ) = 1 − 2 w(zE ) δ ij + 2 vEi vE Ra + O4 . c 2c e00 (T )

(9.4.6)

Exercise 9.2 Proof the last theorem by direct calculation using the tetrad condition (9.4.2). From this we see that e.g., the quantity e2 defined by (9.3.14) is given by e2 =

1 2 v + w(zE ) . 2 E

(9.4.7)

We can use this result together with (9.3.21) to get the transformation rule for the inverse distance in explicit form. With X = V = 0 and N = X/|X| we find 1 1 = r R

  1 1 1 1 + 2 w(zE ) + 2 (vE · N)2 + 2 AE · X , c 2c 2c

(9.4.8)

where r ≡ |x − zE (t)| and R ≡ |X|. The inverse of this relation can be written in the form   1 1 1 1 1 2 (9.4.9) = 1 − 2 w(zE ) − 2 (vE · nE ) − 2 aE · rE R r c 2c 2c where rE ≡ x − zE (t), nE = rE /|rE | and aE ≡ d 2 zE /dt 2 .

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9 Astronomical Reference Systems

Let us now look more closely at the harmonic gauge condition. Any harmonic function ϕ satisfies √ √ 1 1 0 = g ϕ = √ ∂μ ( −gg μν ∂ν ϕ) = g μν ∂μν ϕ + √ [∂μ ( −gg μν )]∂ν ϕ −g −g (9.4.10) and this equation has to be valid in any coordinate system. If we require our local and global coordinates, x μ and Xα , to be harmonic the following conditions must hold √ 1 g x λ = √ ∂μ ( −gg μλ ) = 0 −g

(9.4.11)

and G X α = √

1 −G

√ ∂β ( −GGβα ) = 0

(9.4.12)

∂ 2 Xα =0 ∂x μ ∂x ν

(9.4.13)

but also g Xα = g μν (x)

where relation (9.4.11) was used and similarly G x μ = Gαβ (X)

∂ 2xμ = 0. ∂Xα ∂Xβ

(9.4.14)

Exercise 9.3 Proof that √

−GG x μ = x μ + O4 ,

(9.4.15)

where ≡−

1 ∂2 + X . c2 T 2

(9.4.16)

Combining that relation with our fundamental transformation rule (9.2.19) and using X x μ (Xα ) = X ξ μ together with 1 ∂2 μ α 1 d 2 zμ (T ) x (X ) = + O4 c2 ∂T 2 c2 dT 2

9.4 Local Harmonic Proper Coordinates

351

we get for μ = i X x i = X ξ i =

1 d 2 zi + O4 . c2 dT 2

(9.4.17)

It is easy to show that expression (9.2.10) for ξ i indeed satisfies that relation. For ξ 0 (T , X) = O3 we find the following constraint: X ξ 0 =

1 d 2 ea0 1 de00 + 2 + O5 c dT c dT 2

(9.4.18)

and using expressions (9.2.6) and (9.2.7) for e00 and ea0 we get X ξ 0 =

 1 ˙ ˙E ·X , w(zE ) + vE · AE + A 3 c

(9.4.19)

where the dot stands for d/dT . The solution of this equation reads ξ 0 (T , X) =

1 c3



 1˙ 1 1 ˙ w(zE ) + vE · AE + A · X X2 + ξL0 , E 6 6 10

(9.4.20)

where ξL0 = O3 = O(X2 ) is an arbitrary solution of the Laplace equation X ξL0 = 0 that can be written in the form  GL XL , (9.4.21) ξL0 = l≥2

where GL is some arbitrary symmetric and trace-free (STF) tensor. In the following we will use a special choice for GL resulting from the requirement that Xα are local harmonic proper coordinates. From the definition of such coordinates W a,b (zE ) + W b,a (zE ) = 0 .

(9.4.22)

Using the transformation rules for the metric potentials one finds that rela0 . tion (9.4.22) determines the second spatial derivatives of ξ 0 , i.e., ξ,ab Exercise 9.4 Show that the transformation rule for the external potentials lead to the following relation: 0 c3 ξ,ab = 4w (a,b) − w ,t δab + 4v(a w ,b) − 2v(a Ab) + (v · A − v c w ,c )δab ,

(9.4.23)

where v = vE , A = AE and all external potentials and their derivatives have to be taken at zE .

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9 Astronomical Reference Systems

Using result (9.4.23) from Exercise (9.4) we may choose ξH0 in the form 1 c3 ξH0 (T , X) = − 2w a,b Xa Xb − w ,t X2 + v · X(w ,a Xa ) + v · X(w ,a − Aa )Xa 2 1 1 ˙ · X)X2 . + (v · A − v a w ,a )X2 + (A 2 10 (9.4.24) That this form of ξ 0 is compatible with relation (9.4.20) shows the following Exercise. Exercise 9.5 Show that (9.4.20) agrees with (9.4.24) for the case Gab = 2v (zE ) − v − 2w (zE ) .

(9.4.25)

9.5 The Standard x μ → X α Transformation According to (9.2.19) the transformation between global coordinates x μ = (ct, x i ) and local ones Xα = (cT , Xa ) for one and the same event is given by ct = z0 (T ) + ea0 (T )Xa + ξ 0 , x i = zi (T ) + eai (T )Xa + ξ i .

(9.5.1)

Since, e.g., solar-system ephemerides are given in global coordinates for practical applications it is useful to invert these relations in the form Xα = Xα (x μ ). Let us invert the relation for spatial coordinates first by using relation (9.3.19) * i x i − zsim (t) = eai −

e0i ea0 e00

+ Xa + ξ i (T , Xa ) + O4 . T

Now, e0i (T ) = dzi (T )/(c, dT ) and, therefore, e0i /e00 = dzi /(c dt), we get by inserting the expressions for eai , e0i and ξ i and solving for Xa 

$ 1 1 i 1 v (v · r) + w(zE )r i + r i (a · r) − a i r 2 + O4 , Xa = Rai r i + 2 2 c 2 (9.5.2) where r(t) ≡ x − zE (t) and a(t) is the acceleration of zE in global coordinates. The derivation of the T = T (t) relation is more complicated. First we will derive this relation for an event on the central world-line, i.e., for Xa = 0 where we have

9.5 The Standard x μ → Xα Transformation

353

t = z0 (T )/c and therefore dt 1 dz0 (T ) = = e00 (T ) dT c dT or dT = (e00 (T ))−1 . dt Let f be some function defined at the central world-line, LE , then for some event on LE we have f (t) = f (T ) though the values for t and T will differ to post-Newtonian order. From the last relation we therefore get for events at Xa = 0 1 dB(t) dT 1 dA(t) + 4 + O5 =1− 2 (9.5.3) dt LE c dt c dt with d 1 A(t) = v2E + w(zE ) , dt 2 d 1 3 1 B(t) = − v4E − w(zE )v2E + 4vEi w i (zE ) + w 2 (zE ) . dt 8 2 2

(9.5.4)

Next we consider some event outside the central world-line where we have ct = z0 (T ) + ea0 (T )Xa + ξ 0 (T , Xa ) . If f is again some function on the central world-line we have f (T ) f (t − v · r/c2 ) = f (t) − f,t · (v · r/c2 ) + O3 and ∂T ∂t + v i ∂i . We get ea0 (T )Xa



1 1 1 2 2 i i = v · r + 3 (v · r)v + 4w(v · r) − (v · a)r − 4w r + O5 , c 2 c (9.5.5)

where again the indices E have been dropped and the metric potentials have to be taken at LE . Here the right hand side of (9.5.5) refers to t and not to T (we have used v(T ) · r v(t) · r − (a · r)(v · r)/c2 ).

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9 Astronomical Reference Systems

Inserting expression (9.4.24) for ξ 0 we finally get 1 [A(t) + v · r] c2 1 + 4 [B(t) + B i (t)r i + B ij (t)r i r j + C(t, x)] + O5 , c

T =t−

(9.5.6)

where 1 B i (t) = − v2 v i + 4w i − 3v i w , 2 ∂ ∂ 1 ˙ ij B ij (t) = −v i Raj Qa + 2 j w i − v i j w + wδ . ∂x ∂x 2

(9.5.7)

The dot on w indicated the total time derivative, i.e., ˙ ≡ w ,t + v i w ,i w and 1 2 r (˙a · r) , 10   ∂ i . Qa (t) = Rai w − a ∂x i

C(t, x) = −

(9.5.8)

Relation (9.5.6) is the coordinate transformation that is recommended by the International Astronomical Union (Soffel et al. 2003).

9.6 The Description of Tidal Forces 9.6.1 Post-Newtonian Tidal Moments We will now introduce a useful post-Newtonian generalization of the Newtonian tidal expansion of the effective potential describing the gravitational action of external bodies in a local system co-moving with some body E together with the inertial forces appearing in that system. As it will become obvious later in the discussion of the equations of motion of astronomical bodies the following external gravito-electric and gravito-magnetic fields defined by E a (T , x) ≡ ∂a W +

4 ∂T W a , c2

B a (T , X) = −4 abc ∂b W c ,

(9.6.1)

9.6 The Description of Tidal Forces

355

will play a central role. For that reason they will be considered as post-Newtonian analogues of ∇U tidal . In a more compact notation we write E = ∇W +

4 ∂T W , c2

(9.6.2)

B = −4∇ × W . It is easy to see that the gravito-electric and gravito-magnetic external fields, E and B, are invariant under gauge transformations of the external metric potentials of the form W = W −

1 ∂T  ; c2

1 W a = W a + ∂ . 4

Under such gauge transformations the E and B fields transform according to 

E a = ∂a W  +

4 1 4 1 ∂T W a = ∂a W − 2 ∂T a  + 2 ∂T W a + 2 ∂T a  = E a 2 c c c c

and  Ba

=

abc ∂b (−4W c )

  1 = −4 abc ∂b W c + ∂c  = − abc ∂b W c = B a , 4

i.e., they are gauge invariant in the sense defined above. Let us remember that the external metric potentials satisfy the homogeneous field equations ΔX W −

1 ∂2 4 ∂ b (∂T W + ∂b W ) = O4 W+ 2 c2 ∂T 2 c ∂T X W a − ∂a (∂T W + ∂b W b ) = O2 .

Lemma 9.2 In virtue of these field equations and the definitions of the external ‘gauge invariant’ E and B fields, they satisfy the following homogeneous postNewtonian dynamical Maxwell-like equations: ∇ ×E = −

1 ∂T B , c2

∇ × B = 4∂T E + O2 , ∇ ·E = −

3 2 ∂ W + O4 , c2 T

∇ · B = 0,

(9.6.3) (9.6.4) (9.6.5) (9.6.6)

356

9 Astronomical Reference Systems

∇ 2E =

1 2 ∂ E + O4 , c2 T

∇ 2 B = O2 .

(9.6.7) (9.6.8)

The proof follows by direct calculation. E.g., ∇ ×E=

4 ∂ 1 ∇ × W = − 2 ∂T B c2 ∂T c

or ∇ · E = W +

4 3 ∂T a W a = − 2 ∂T2 W . 2 c c

The proof of the remaining dynamical equations for the external E and B fields is left as an exercise. Next we characterize these fields by two corresponding (gravito-electric and gravitomagnetic) sets of post-Newtonian tidal moments GL (T ) ≡ [∂sec = −

7 + 3e2 A1 P 2 rad/revolution . 4π

(9.7.5)

These secular perturbations for ω and M0 amount to 10−5 microarcseconds per century for Pluto and three orders of magnitude smaller for Mercury. Periodic perturbations for eccentricity and semi-major axis are given by 1 − e2 A1 P 2 4π 2 2e 3 a/a = e = A1 P 2 2 1−e 2π 2 e =

(9.7.6)

Amplitudes for these periodic perturbations are about 10−17 for Pluto and 10−23 for Mercury. Therefore, these cosmological perturbations are completely negligible for the dynamics of our solar system. Many publications using local coordinates, considering e.g., the Einstein de Sitter space, come to the same conclusion (e.g., Hackmann et al. 2008a,b, 2010).

Chapter 10

The Gravitational N-Body Problem

We now come to a relativistic formulation of the gravitational N -body problem which can be described exactly only in the frame of numerical relativity. In the general case not even the concept of a ‘body’ can be formulated rigorously because of the non-linearities of GR (the distinction between ‘self- and external-field’ of a body is a real problem). A lot of work has been done for the motion of compact binaries especially with respect to the problem of emitted gravitational waves. A compact binary consists of two compact stars (Neutron stars or black holes) orbiting each other due to their gravitational forces. Usually considered were two structure-less point masses, characterized by their two masses, M1 and M2 (and possibly their spins) and the equations of motion are formulated at some post-Newtonian level of approximation: in the first post-Newtonian (1PN) approximation terms of order 1/c2 are kept. In the n-th PN approximation terms of order c−(2n) are kept in the equations of motion and higher order 1/c−terms are neglected. Usually the equations of motion are formulated in harmonic coordinates. Up to the 2PN-level the equations of motion are conservative and total energy and momentum are conserved. At the 2.5PN order non-conservative effects appear that are related with the emission of gravitational waves (radiation effects). Here is a short list of references where explicit equations of motion have been derived at the n-th PN level: 2PN Level Ohta et al. (1973a,b,c, 1974a, b) 2.5PN Level Blanchet et al. (1998), Damour (1982a,b, 1983, 1987a,b), Damour and Deruelle (1981a,b), Grishuk and Kopeikin (1985), Itoh et al. (2001), Kopeikin (1985) 3PN Level Jaranowski and Schäfer (1998, 1999, 2000), Damour et al. (2000, 2001a,b), Blanchet and Faye (2000a,b, 2001a,b), de Andrade et al. (2001), Itoh (2004), Itoh and Futamase (2003).

© Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_10

367

368

10 The Gravitational N-Body Problem

In the following we discuss the gravitational N -body problem in the DSXformalism, that, at the first PN approximation, employs a total of N + 1 different astronomical reference systems: a local one for each of the N bodies that is comoving with the body under consideration and a single global coordinate system, where the overall motion of the system is described. All N + 1 coordinate systems will be chosen to be harmonic and in each of these systems the PN metric tensor is parametrized with two metric potentials, (w, wi ) in the global system and (WA , WAa ) in the local system of body A.

10.1 Local Evolution Equations The global equations of motion in the gravitational N-body problem result from the local evolution equation ν ν ν σ = Tμ,ν + νσ Tμσ − μν Tσν = 0 . Tμ;ν

(10.1.1)

Lemma 10.1 In any local system the local evolution equations (10.1.1) take the following form: ∂ ∂ 1 ∂ bb 1 ∂ + T − 2 W + O4 . a = 2 a ∂T ∂X ∂T c ∂T c

(10.1.2)

This equation for μ = 0 is the energy-equation. The μ = a equation reads ∂ ∂T



 

 ∂ 4 4 ab 1 + = F a + O4 . 1 + 2 W a + W T ∂Xb c c2

(10.1.3)

This is the post-Newtonian Euler-equation. Here, F a = Ea +

1 Bab  b c2

(10.1.4)

is the gravitational Lorentz-force, where 4 ∂T Wa , c2 = abc Bc = ∂a (−4Wb ) − ∂b (−4Wa ) .

Ea = ∂a W + Bab

(10.1.5) (10.1.6)

We will show (10.1.3). The post-Newtonian energy equation (10.1.2) follows similarly. Since (G ≡ det(Gαβ )) 1 ∂ √ ν νσ =√ −G −G ∂Xσ

10.1 Local Evolution Equations

369

the μ = a equation reads ν Ta;ν

 ∂ √ σ −G Taσ − aν Tσν √ −G ∂Xσ 1 ∂ √ σ =√ ( −GTaν ) − aν Tσν = 0 −G ∂Xν ∂ = Tν + ∂Xν a



1

or √ ∂ √ σ ( −GTaν ) = −GGσ ν aμ T μν . ∂Xν Since σ Gσ ν aμ T μν =

1 μν T ∂a Gμν 2

we obtain: ∂ √ 1√ ( −GTaν ) = −GT μν ∂a Gμν . ν ∂X 2 Since

(10.1.7)

√ G = 1 + 2W/c2 + O4 , the right hand side of (10.1.7) reads:    1 2W  00 T ∂a G00 + 2T 0b ∂a G0b + T bc ∂a Gbc 1+ 2 2 c 

 2W W T 0b b 1 00 ss 2(T + T )∂a 2 − 8 3 W,a + O4 1+ 2 = 2 c c c = W,a −

4 b  ∂a W b + O4 . c2

From this we get the Blanchet-Damour-Schäfer (BDS) form of the post-Newtonian Euler-equation (Blanchet et al. 1990): ∂ a ∂ √ 4 # + ( −GTab ) = W,a − 2  b ∂a W b + O4 , b ∂T ∂X c

(10.1.8)

where # =c a

−1

√

−GTa0

  4 a 4W = − 2 W  + 1 + 2 + O4 . c c

Relation (10.1.3) then follows from the BDS-form with √

  4W 4 −GTab = 1 + 2 T ab + 2 W a  b c c

(10.1.9)

370

10 The Gravitational N-Body Problem

and ∂ a a b + W a ,T = W,T − W a ,b + O2 . (W a ) = W,T ∂T

10.2 The Translational Motion We will now fix the central world-line of a body E, member of gravitational N-body system, by choosing the origin of the local system, Xa = 0, to coincide with the post-Newtonian center of mass by MaE (T ) = 0 .

(10.2.1)

As in the Newtonian case a corresponding d’Alembert criterion will lead to the global translational equations of motion for the N-body problem. Theorem 10.1 From the local evolution equations (10.3.9) and (10.1.3) one obtains: dM E = F0 + O4 dT

(10.2.2)

d 2 MaE = Fa + O4 dT 2

(10.2.3)

dSaE = Da + O2 dT

(10.2.4)

with  $ dML dGL 1 1 (l + 1)ML +l GL + O4 F0 = − 2 l! dT dT c l≥0

1 dHcL l 1 ML GaL + 2 SL HaL + 2 abc MbL Fa = l! dT c (l + 1) c (l + 2) l≥0

4 1 dMbL dGcL dSbL 4(l + 1) HcL − 2 − 2 GcL abc abc + 2 abc SbL dT dT dT c (l + 1) c (l + 2)2 c (l + 2) d 2 GL 2l 3 + 7l 2 + 15l + 6 2l 3 + 5l 2 + 12l + 5 dMaL dGL MaL − 2 2 dT dT c (l + 1)(2l + 3) dT c2 (l + 1)2 ) l 2 + l + 4 d 2 MaL − 2 GL + O 4 , c (l + 1) dT 2

−

Da =

1 abc MbL GcL + O2 . l! l≥0

(10.2.5)

10.2 The Translational Motion

371

The proof of this theorem is left as an Exercise; the Newtonian expression for the external torque has already been derived above; see (3.6.3). In the following we will study the consequences of Theorem 10.1 for a system of N mass-monopoles, i.e., we will assume for l = 0: MLA = SLA = 0 for all bodies A from the system. Since G(T ) = 0, one finds that for the massmonopole model dM A = O4 , dT

(10.2.6)

i.e., the masses of the N bodies are conserved to post-Newtonian order. From (10.2.3) we then get from the d’Alembert equation 0=

d 2 MaE = Fa = ME Ga , dT 2

or Ga = 0 .

(10.2.7)

Implicitly the translational equations of motion is given by the vanishing of the external gravito-electric tidal dipole-moment Ga . Lemma 10.2 Equation (10.2.7) is equivalent to the geodesic equation duλ λ +  μν uμ uν = O4 dτ

(10.2.8)

in the external metric. Here, μ

μ

uμ = uE =

μ

dz dzE = E , dτ dT

where we have assumed, according to the weak effacement condition, that T = τ and −c2 dτ 2 = g μν dx μ dx ν |X=0 . Proof We have   1 Ga = E a |X=0 = ∂a W + 2 ∂T W a |X=0 . c

372

10 The Gravitational N-Body Problem

The local external metric potentials, W and W a , can be expressed in terms of the corresponding global ones, w and wi . From Theorem 9.1 and the transformation rule (9.2.23) we get W α = A−1 αμ (w μ − Bμ )

(10.2.9)

or explicitly   2 2 4 W = 1 + 2 v (w − B) − 2 v i (w i − Bi ) + O4 c c W a = −v (w a

− B) + Rai (wi

(10.2.10)

− Bi ) + O2 .

Direct calculation leads to (Aa ≡ eai d 2 zEi /dT 2 ) 2 4 Ga = (w ,i eai − Aa ) + 2 v2 w ,a + 2 v i (w a,i − w i,a ) c c 2 3 2 1 w ,t v a − 2 v a (v i w ,i ) − 2 wAa + 2 (v · A)v a c2 c c c

Duν 4 + 2 w a,t = −g μν eaμ . dT c X=0 −

(10.2.11)

Thus the gravito-electric tidal dipole-moment has a direct geometrical meaning: it is (minus) the 4-acceleration of zE with respect to the external metric. Theorem 10.2 then follows from the fact that the vanishing of this 4-acceleration implies geodetic motion. Inserting the Christoffel-symbols of the global external metric as seen by body E, given by wE and w Ei , the translational equations of motion for the center of mass of body E in the mass-monopole model read:

d 2 zEi 4 1 2 4 4 = 1 − 2 w E + 2 v ∂i w E + 2 ∂t w Ei − 2 (∂i w Ej − ∂j wEi )v j 2 dt c c c c −

1 (3∂t w E + 4v j ∂j wE )v i + O4 , c2 (10.2.12)

where v i = vEi is the global coordinate velocity of body E. Finally we need the external metric potential related with body E explicitly wμ =

 A=E

wμA .

10.2 The Translational Motion

373

In the local A-system we simply have A Wself =

GMA ; RA

WaB,self = 0 .

(10.2.13)

Transformation into the global system with (9.3.10) yields   GMA 2 ; w A = 1 + 2 v2A RA c

wiA =

GMA i v . RA A

(10.2.14)

Using the transformation rule (9.4.9) for the inverse distance from the center of body A,

1 w(zA ) 1 1 1 2 1− = − (v · n ) − a · r A A A A RA rA c2 2c2 2c2 we get   v2A w(zA ) GMA 1 1 2 − 2 (vA · nA ) − 2 aA · rA w = 1+2 2 − rA c c2 2c 2c A

(10.2.15)

GMA i v . wiA = rA A Inserting these potentials into (10.2.12) we finally end up with the Lorentz-Droste Einstein-Infeld-Hoffmann (LD-EIH) equations of motion (for the history of these equation of motion see e.g., Damour et al. 1991; DSX-I) for body A: i d 2 zA i(LD−EIH) = aA (zA , vA ) + O4 , 2 dt

(10.2.16)

where the LD-EIH acceleration is given by (LD−EIH) aA



 GMB 1 3 2 2 2 =− nAB 1 + 2 vA + 2vB − 4vA · vB − (nAB · vB ) 2 c r2 B=A AB ⎤

 GMC  GMC 1 rAB −4 1+ − nAB · nCB ⎦ 2 rCB c2 rAC c2 rBC C=A

−

C=B

G2 MB MC 7  nBC 2 2 c2 rAB rBC B=A C=B

+



(vA − vB )

B=A

GMB (4nAB · vA − 3nAB · vB ) , 2 c2 rAB (10.2.17)

374

10 The Gravitational N-Body Problem

where rAB ≡ |zA (t) − zB (t)| ,

nAB ≡ [zA (t) − zB (t)]/rAB .

10.2.1 The LD-EIH Lagrangian The LD-EIH acceleration of body A can be derived from a Lagrangian LLD−EIH . It can be obtained by starting from the fact that the motion of each body can be derived from the individual geodesic action

SA = −MA c2

LA

dτ A =

LA (zA , vA ) dt ,

(10.2.18)

with 

 μ ν 1/2 dz dz A A LA = −MA −g A μν (zA ) dt dt

1 1 3 = −MA c2 + MA v2A 1 + 2 v2A + 2 wA 2 4c c

1 4 A i 2 +MA w A − 2 (w A ) − 2 wi vA , 2c c

(10.2.19)

and then by symmetrizing, over the body labels, the explicit expressions in terms of the zB ’s and vB ’s of the individual Lagrangians. In this way one finds up to a constant LLD−EIH =

1 A

2

MA v2A +

  GMA MB  1 + MA v4A 2rAB 8c2 A B=A

A

  3GMA MB v2 A + 2c2 rAB A B=A   GMA MB − [7vA · vB + (nAB · vA )(nAB · vB )] 4c2 rAB A B=A

−

   G2 MA MB MC . 2c2 rAB rAC A B=A C=A

(10.2.20)

10.2 The Translational Motion

375

The LD-EIH acceleration of body A can then be obtained from the Euler-Lagrange equations: d ∂L ∂L = . dt ∂vA ∂zA

(10.2.21)

Exercise 10.1 First derive the LD-EIH Lagrangian (10.2.20) as indicated above; then use the Euler-Lagrange equations (10.2.21) to derive the LD-EIH equation of motion for body A. To Newtonian order only the first two terms in (10.2.20) contribute and with ∂L/∂vA = MA vA , ∂L/∂zA = MA ∇A w A one ends up with the Newtonian equation of motion: aA = ∇A w A .

10.2.2 Laws of Motion As we have already seen above the d’Alembert equilibrium condition in the local E-system leads to 0=

1 1 d 2 MaE M E GE + F (1PN) = 2 l! L aL c2 a dTE l≥0

(1PN)

where Fa , given by (10.2.5), is a function of MLE , SLE ; GEL and HLE and time derivatives thereof. This equation can be turned into the translational equation of motion for the (post-Newtonian) center of mass of body E in the global system. Coming back to the laws of motion, we will assume Ma (T ) = 0 = G(T ) for body E. For the mass, mass- and spin-dipole of body E we then get the equations dM = 0 + O4 , dT − MGa +

(10.2.22)

2 dGc dSb 1 3 d2 + G S + (Mab Gb ) = a + O4 abc b abc c dT dT c2 c2 c2 dT 2 (10.2.23) dSa = Da + O2 (10.2.24) dT

376

10 The Gravitational N-Body Problem

with

dML dGL 1 1 (l + 1)ML +l GL , 0 = − 2 l! dT dT c l≥2

1 d 2 GL 2l 3 + 7l 2 + 15l + 6 a = ML GaL − 2 MaL l! c (l + 1)(2l + 3) dT 2 l≥2

l 2 + l + 4 d 2 MaL 2l 3 + 5l 2 + 12l + 5 dMaL dGL − 2 GL 2 2 dT dT c (l + 1) c (l + 1) dT 2 1 l dHcL 1 + SL HaL + 2 abc MbL 2 l! c (l + 1) dT c (l + 2)



−

(10.2.25)

l≥1

dMbL dGcL 4(l + 1) 1 HcL − 2 abc abc SbL 2 dT dT + 1) c (l + 2)

dSbL 4 GcL , abc − 2 dT c (l + 2) 1 Da = abc MbL GcL . l! +

c2 (l

l≥1

It is important to notice that the right-hand sides of (10.2.25) contains only the mass multipole moments of order l ≥ 2, the spin multipole moments of order l ≥ 1 and the tidal moments of order l ≥ 2. In (10.2.23) we can consistently replace Ga in PN terms by its Newtonian limit Ga = −N a /M + O2

(10.2.26)

1 ML GaL . l!

(10.2.27)

where N a =

l≥2

The equilibrium condition (10.2.23) can therefore be written in the form ˆ a + O4 − MGa = 

(10.2.28)

with ˆ a = a + 

d 1 abc Sb 2 dT c



N c M



2 3 d2 dSb N c + 2 abc + 2 dT M c c dT 2

  N Mab . M (10.2.29)

10.2 The Translational Motion

377

If one considers only the mass monopole and quadrupole and the spin dipole one gets: 1 dMab dGab 3 − 2 Gab , Mab (10.2.30) 2 dT 2c c dT

1 d Mde dSb Mde 1 ˆ a = a +  G Gcde (10.2.31) S abc b cde + 2 abc 2 dT M dT M 2c c   3 d 2 Mab Mcd (10.2.32) + 2 G bcd , M 2c dT 2 0 = −

Da = abc Mbd Gcd

(10.2.33)

with dHcd 1 1 1 a = Mbc Gabc + 2 Sb Hab + 2 abc Mbd 2 dT 2c 3c 1 dMbd Hcd . + 2 abc dT 2c

(10.2.34)

10.2.3 Equations of Motion The laws of motion, (10.2.22)–(10.2.24), do not form a closed evolution system because the time evolution of the higher (l ≥ 2) multipole moments is left unspecified. In principle, the exact way to complete the system would be to solve the 1PN local evolution equations for the matter distribution of each body as seen in its corresponding local frame, and then to compute the integrals defining the multipole moments (DSX II), but this is not the usual way how to proceed in celestial mechanics. One way to proceed would be to assume the time dependence of all multipole moments of a body A to result from pure rotations induced by some angular velocity vector  in the local A system and to add some dynamical equation for . We call this the ‘rigidly rotating multipole model’. It is not clear if such a model is in fact compatible with the 1PN approximation of Einstein’s theory of gravity. However, it certainly presents a useful starting point for dealing with further models for elastic deformable bodies. A drastic formal way to derive a closed system of translational equations of motion is to limit oneself to mass monopoles and spin dipoles only. In this mass-monopole, spin-dipole model 0 = 0 , a =

1 Sb Hab 2c2

Da = 0

(10.2.35)

378

10 The Gravitational N-Body Problem

so that the mass and the spin of any body is conserved: dM = O4 dT dSa = O2 . dT

(10.2.36)

Because of the spin conservation we can write the D’Alembert law in the local E system in the form −MGa =

1 Sb Hab 2c2

or as MAa = Fa

(10.2.37)

with Fa =



A/E MGa

A=E

1 A/E + 2 Sb Hab 2c

 (10.2.38)

.

For the A/E-parts of the tidal moments we get A/E

= Ga

A/E

= Ha

Ga Ha

A/E

A/E

(M A ) + Ga

A/E

(S A )

A/E

(M A ) + Ha

(10.2.39)

(S A )

with    [1 + 2(V AE )2 /c2 ]M A 4G E M A VaAE + 2 DT + O4 , ρA ρA c ± ± * + * + A∗ V AE A∗ Sbc S 2G 2G A/E A c ab + 2 DTE ∂b Ga (S ) = − 2 ∂ab ρA ρA c c ± ± * + * + AE A bcd M Vd acd M A VdAE A/E A Hab (M ) = − 2G ∂ac − 2G∂bc + O2 ρA ρA A/E Ga (M A )

μ =GeE a ∂μ



 A/E

Hab (S A ) = − 2G ∂abc

A

Sc ρA

±

±

±

+ O2 . (10.2.40)

10.2 The Translational Motion

379

A = δ . From this Here, to make the equations more readable we have chosen Ria ia we see that the ‘force’ Fa can be written in the form

Fa =

, Fa (M E × M A ) + Fa (M E × S A ) + Fa (S E × M A ) + Fa (S E × S A ) . A=E

(10.2.41) With this the global translational equation of motion for body E assumes the form: ME

, d 2 zEi LD E A E A E A f = f + (M × S ) + f (S × M ) + f (S × S ) . i i i i dt 2 A=E

(10.2.42) From the above it is clear that i(LD)

fiLD = M E aE

(10.2.43)

.

Exercise 10.2 Proof by direct calculation that (10.2.43) is correct. In that way the EIH-equations of motion are derived with the general formalism of post-Newtonian celestial mechanics of N extended and arbitrary shaped bodies. The further ‘spin-orbit’ and ‘spin-spin’ terms can be calculated from (10.2.38) to (10.2.40) and from the Newtonian-accurate value (F /ρA )± = F (t)/|x − zA (t)| + O2 . Their expressions are fi (M E × S A ) = 2

G E ij k E M sA vEA ∂j k c2



1

 +2

G E jk k E M sA vEA ∂ij c2



1



, rEA     1 G 1 G ij k jk k + 2 2 M A sE vEA , fi (S E × M A ) = 2 2 M A sE vEA ∂jEk ∂ijE rEA rEA c c   1 G j k E E A . fi (S × S ) = − 2 sE sA ∂ij k rEA c (10.2.44) rEA

k = v k − v k and Here, vEA E A E E sEi ≡ Ria Sa , ij

sE ≡ ij k sEk ,

A A sAi ≡ Rib Sb , ij

sA ≡ ij k sAk ,

(10.2.45)

380

10 The Gravitational N-Body Problem

where the explicit expressions for the partial derivatives (∂iE ≡ ∂/∂zEi ) appearing in Eq. (10.2.44) are  ∂ijE

1

 =

rEA 

∂ijE k

1 rEA

3 n 3 rEA EA

 =− =−

=

1 3 rEA

j

[3niEA nEA − δ ij ] ,

15 nEA 4 rEA 1 4 rEA

j

j

[15niEA nEA nkEA − 3niEA δ j k − 3nEA δ ki − 3nkEA δ ij ] . (10.2.46)

The spin-orbit terms coincide with the results of Tulczyjew (1959) based on the formal use of distributional sources, and with the results of Damour (1982b) obtained for binary systems of strongly self-gravitating bodies. The spin-spin term in (10.2.44) coincides with the result of Barker and O’Connell (1975) heuristically derived from a quantum interaction Hamiltonian (DSX-II).

10.3 The PN Two-Body Problem The translational equations of motion for a system of mass-monopoles are given by the LD-EIH accelerations or the corresponding Lagrangian (10.2.20). For the celestial mechanical two-body problem we consider two mass-monopole under the influence of their mutual gravitational attraction. The LD-EIH Lagrangian (10.2.20) then reduces to L = L0 +

1 LPN c2

(10.3.1)

with 1 1 GM1 M2 M1 v21 + M2 v22 + (10.3.2) 2 2 r 1 1 = M1 v41 + M2 v22 8 8

GM1 M2 GM 2 2 ˆ 2 · n) ˆ − 3(v1 + v2 ) − 7v1 · v2 − (v1 · n)(v , + 2r r

L0 = LPN

10.3 The PN Two-Body Problem

381

where M ≡ M 1 + M2 ;

nˆ ≡

x1 − x2 ; r

r ≡ |x1 − x2 | .

One finds, that the total momentum P of the system can be obtained in the usual manner from ∂L/∂v1 + ∂L/∂v2 and is given by: v2 v2 1 1 P = M1 v1 + M2 v2 + M1 v1 12 + M2 v2 22 (10.3.3) 2 2 c c GM1 M2 ˆ nˆ · (v1 + v2 )]) . (6(v1 + v2 ) − 7(v1 + v2 ) − n[ + 2c2 r According to the equations of motion the center of mass X, X=

(M1∗ x1 + M2∗ x2 ) , (M1∗ + M2∗ )

(10.3.4)

with v2 1 1 GM1 M2 MA∗ ≡ MA + MA A2 − , 2 2 r c

(10.3.5)

is not accelerated and the center of mass velocity is proportional to P. Therefore, we can go into a post-Newtonian center of mass system, where P = X = 0 and  

νδM M2 GM 2 x1 = + v − + x M r 2Mc2  

νδM M1 GM 2 v − + x x2 = − M r 2Mc2

(10.3.6)

with x ≡ x1 − x2 ;

v ≡ v1 − v2 ;

δM ≡ M1 − M2 ;

ν ≡ M1 M2 /M 2 .

Using (10.3.6) the motion of each body is immediately obtained once the relative motion is known. For the relative motion of the two bodies one then finds (μ = GM):   dv μnˆ μnˆ μ 3 =− 2 + 2 2 (4 + 2ν) − v2 (1 + 3ν) + ν(nˆ · v)2 dt r 2 r c r μ (10.3.7) + 2 2 v(nˆ · v)(4 − 2ν) . c r

382

10 The Gravitational N-Body Problem

For the restricted two-body problem, M = M1 , nˆ = x/r and ν = 0, this equation reduces to the PN satellite equation (8.7.2) for the motion of M2 : a = −μ

 x μ  x 2 − xv 4μ + + 4v(x · v) . r r3 c2 r 3

The Lagrangian that corresponds to (10.3.7) reads: L=

v4 μ  μ 1 2 μ 1 v + + (1 − 3ν) 2 + 2 (3 + ν)v2 + ν(nˆ · v)2 − . 2 r 8 r c 2c r

(10.3.8)

Thus Lagrangian is especially useful for the evaluation of the integrals of motion. For the specific post-Newtonian energy E and angular momentum J one finds: 1 v4 ∂L μ 3 − L = v2 − + (1 − 3ν) 2 ∂v 2 r 8 c  μ  μ + 2 (3 + ν)v2 + ν(nˆ · v)2 + r 2c r

E=v

(10.3.9)

and   ∂L 1 μ 2 = |x × v| 1 + 2 (1 − 3ν)v + (3 + ν) 2 J = x × . ∂v 2c c r

(10.3.10)

A solution to the dynamical equations (10.3.7) for the relative motion on the PN two-body problem accurate to post-Newtonian order can be obtained in different ways. In the following we will deal with the representations of Brumberg (1972), of Wagoner and Will (1976) and Damour and Deruelle (1985).

10.3.1 The Brumberg Representation In Sect. 13.7 we had studied the solution of the restricted two-body problem in the Brumberg representation. All derivations can be made analogously in the full twobody problem with ν = 0. The specific energy E and orbital angular momentum J can be written as 1 2 3 μ (˙r + r 2 φ˙ 2 ) − + 2 (1 − 3ν)(˙r 2 + r 2 φ˙ 2 )2 2 r 8c  μ μ + (3 + 2ν)˙r 2 + (3 + ν)r 2 φ˙ 2 ) + 2 2c r r   1 μ . J = r 2 φ˙ 1 + 2 (1 − 3ν)(˙r 2 + r 2 φ˙ 2 ) + (3 + ν) 2 2c c r E=

(10.3.11) (10.3.12)

10.3 The PN Two-Body Problem

383

The following central equations generalize those of the restricted two-body problem: The radial equation is of the form 

d(1/r) dφ

2

 =

1 1 − r a(1 + e)



1 1 − a(1 − e) r

  C2 C1 + r

(10.3.13)

with C1 = 1 − (6 − ν) C2 = −ν

μ c2 a(1 − e2 )

,

μ c2

and a solution of the form r=

a(1 − e2 ) . 1 + e cos f

(10.3.14)

The post-Newtonian ‘true anomaly’ f is given by f = F (φ − φ0 ) −

μ ν e sin[F (φ − φ0 )] 2 c2 a(1 − e2 )

(10.3.15)

with F =1−3

μ , c2 a(1 − e2 )

(10.3.16)

so that the secular precession of the pericenter is the same as for the restricted two-body problem. The time dependence in the orbit is then obtained from a postNewtonian Kepler equation in the form:  

 μ μ  3 ν−1 2 M = 1+3 2 E− 1+ e sin E 2 c a c a

(10.3.17)

and the ‘eccentric anomaly’ E is related with f by (8.7.26). Exercise 10.3 Derive expressions for E and J , (10.3.11) and (10.3.12) from the general expressions (10.3.9) and (10.3.10). Also derive relations (10.3.13), (10.3.15) and (10.3.17) (for more details see the section on the Damour-Deruelle representation and Soffel 1989).

384

10 The Gravitational N-Body Problem

Finally we would like to note that in the Brumberg representation the integration constants a and e are related with the E and J by μ 1− 2a √ J = μp 1 + E =−

1 μ (7 − ν) 2 4 c a  

μ 6−ν 1 −2 + ν + . 2 1 − e2 c2 a

(10.3.18) (10.3.19)

Exercise 10.4 Solve Eqs. (10.3.18) and (10.3.19) for a and e as functions of E and J to PN-order. Show that the answer can be written in the form

μ 1 E a=− (10.3.20) 1 − (ν − 7) 2 2E 2 c



$1/2  1 2E μ2 E 2 e = 1 + 2 1 + (5ν − 15) 2 J + (ν − 6) 2 . (10.3.21) 2 μ c c

10.3.2 The Wagoner-Will Representation In the Wagoner-Will representation one starts with a solution of (10.3.7) to Newtonian accuracy. In the orbital plane with x = r(cos φ, sin φ, 0) it can be written as (μ = GM, p = a(1 − e2 )): r=

p 1 + e cos(φ − φ0 ) r2

dφ √ = μp . dt

(10.3.22)

(10.3.23)

The post-Newtonian solution of (10.3.7) can then be obtained by setting r2 dx = v= dt

dφ √ = |x × v| = μp (1 + δPN ) dt

 1/2 μ (− sin φ, e + cos φ, 0) + vPN . p

(10.3.24)

(10.3.25)

One finds (φ  = φ − φ0 ) that r

2 dφ

dt

√ = μp

  μe  1 − 2 (4 − 2ν) cos φ c p

(10.3.26)

10.3 The PN Two-Body Problem

385

and x vPN

y

vPN

  1/2    μ μ 21   −3eφ + (3 − ν) sin φ − 1 + ν e2 sin φ  = p 8 c2 p

1 ν + (1 − 2ν)e sin 2φ  − e2 sin 3φ  2 8   1/2    μ μ 31  = −(3 − ν) cos φ − 3 − ν e2 cos φ  p 8 c2 p

1 ν − (1 − 2ν)e cos 2φ  + e2 cos 3φ  2 8

z vPN = 0.

(10.3.27)

An expression for r(φ) can be obtained by inserting the last two relations into the identity:   d 1 x˙v 1 (10.3.28) =− 2 2 ˙ dφ r r r φ and integrating over φ. One finds:     p 9 μ 1  [−(3 − ν) + 1 + ν e2 + (7 − 2ν)e cos φ  = 1 + e cos φ + 2 r 4 2 c p  ν (10.3.29) + 3eφ  sin φ  − e2 cos 2φ  . 4 From this one infers that the secular drift in the argument of the pericenter per revolution is given by φ = 2π × 3μ/(c2 p). This suggests to introduce a ‘true anomaly’ η (Epstein 1977; Haugan 1985) with   3μ η ≡ 1− 2 φ − φ0 (10.3.30) c p as new angular variable. Then (10.3.26) and (10.3.29) take the form:   dφ μe √ = μp 1 − 2 (4 − 2ν) cos η r2 dt c p and

   9 −(3 − ν) + 1 + ν e2 4

ν 1 + (7 − 2ν)e cos η − e2 cos 2η . 2 4

p = 1 + e cos η + r



(10.3.31)

μ c2 p

(10.3.32)

386

10 The Gravitational N-Body Problem

If we furthermore introduce the eccentric anomaly E  that is related with η in the usual manner, i.e. sin η =

(1 − e2 )1/2 sin E  ; 1 − e cos E 

cos η =

cos E  − 3 , 1 − e cos E 

(10.3.33)

the last expression can be written as:  23 μ 1 −3 − e2 + e4 r = a(1 − e cos E ) − 2 2 2 4 2 c (1 − e )    

19 3 2 17 2 3 4  2 + e − (3 + 5e )ν + 1 + e + e ν + e cos E 4 4 2 2     $ 3 5 2 13 1 + e ν + e2 cos 2E  − + e2 + . (10.3.34) 4 2 4 4 

The time dependence in the post-Newtonian relative two-body orbit can be expressed by means of a generalized Kepler equation. Using (10.3.31) and the relations   μ (1 − e2 )1/2 ˙  ˙ φ = 1+3 2 η˙ ; η˙ = (10.3.35) E 1 − e cos E  c p one finds that  

  μ √ cos E  − e μe 1−3 2 r 2 (E  )η˙ , μp = 1 + 2 (4 − 2ν) 1 + e cos E  c p c p (10.3.36) where r(E  ) is given by (10.3.34). Integration of this expression over the time variable t finally leads to the desired Kepler equation in the form: M  = E  − ge sin E  − h sin 2E  ,

(10.3.37)

with 

M =



2π TE 

 (t − t0 )

(10.3.38)

with  TE  = 2π

a3 μ

1/2 1+3

μ μ 2 4 2 4 + [12 + 15e + 6e − (4 + 13e + 7e )ν] c2 p 2c2 a(1 − e2 )2

(10.3.39)

10.3 The PN Two-Body Problem

g = 1+ h=

387

μ 2c2 a(1 − e2 )2

[18 − e2 − 6e4 − (4 + 11e2 − 7e4 )ν] (10.3.40)

μe2 [−13 + 2e2 + (3 + 5e2 )ν] . 4c2 a(1 − e2 )2

(10.3.41)

We finally would like to mention that in this Wagoner-Will representation the integration constants e and p are related with E and J by  

μ μ 2 4 (1 − e2 ) − {19 − 5ν + 2[11 − 9ν]e + 3(1 − 3ν)e } 2p 4c2 p  

μ √ 2 [(7 − ν) + (1 − 3ν)e ] . (10.3.42) J = μp 1 + 2c2 p E=−

10.3.3 The Damour-Deruelle Representation In this section we will discuss the Damour-Deruelle representation of the postNewtonian two mass-monopole orbit. For more details the reader is referred to Damour and Deruelle (1985). From (10.3.12) for the specific angular momentum J one obtains a radial equation in the form C 2B D + 2+ 3 r r r   3 E A = 2E 1 + (3ν − 1) 2 2 c   E B = μ 1 + (7ν − 6) 2 c   E μ2 2 C = −J 1 + 2(3ν − 1) 2 + (5ν − 10) 2 c c r˙ 2 = A +

D = (8 − 3ν)

(10.3.43)

μJ 2 . c2

With the transformation r = r¯ +

D , 2C0

(10.3.44)

388

10 The Gravitational N-Body Problem

where C0 = −J 2 (the non-relativistic limit of C), the radial equation takes the ‘non-relativistic’ form: C 2B 2 + 2 r˙ = A + r r

(10.3.45)

with C=C−

BD . C0

Hence, r can be written as a linear function of cos E, E being some generalized ‘eccentric anomaly’ and the same is true for r. One therefore writes r = ar (1 − er cos E)

(10.3.46)

M ≡ n(t − t0 ) = E − et sin E

(10.3.47)

and

with (−A)3/2 B  

BD 1/2 A et = 1 − 2 C − C0 B n=

D B ar = − + A 2C0   AD et . er = 1 + 2BC0

(10.3.48)

Note that according to (10.3.46) the integration constants ar and er of the DamourDeruelle representation coincide with the corresponding constants a and e of the Brumberg representation. The relations between ar , er with the conserved quantities E and J are therefore given by the right hand sides of (10.3.20) and (10.3.21). For the remaining constants et and n one obtains:



$1/2  1 E 2E μ2 et = 1 + 2 1 + (−7ν + 17) 2 J 2 + (−2ν + 2) 2 2 μ c c (10.3.49)   1 E (−2E)3/2 1 − (ν − 15) 2 . n= μ 4 c

10.3 The PN Two-Body Problem

389

The PN mean motion n can then also be written as  n=

μ ar3

1/2   1 μ 1 + (ν − 9) 2 . 2 c ar

(10.3.50)

Note that  2π = 2π Tf = n

  1 μ 1 + (9 − ν) 2 2 c ar

ar3 μ

(10.3.51)

denotes the anomalous period, i.e., the time of return to the pericenter. We note that to PN-order   er 3 E μ 4 − ν . (10.3.52) = 1 + (3ν − 8) 2 = 1 + et 2 c ar c2 Similarly the dynamical equation for the angular motion can be reduced to an auxiliary Keplerian problem. From (10.3.12) we obtain r φ˙ = J 2

  E μ 1 + (3ν − 1) 2 + (2ν − 4) 2 c c r

(10.3.53)

that we write in the form φ˙ =

H I + 3 2 r r

(10.3.54)

with H =J

  E 1 + (3ν − 1) 2 c

I = (2ν − 4)

μJ . c2

With I 2H

(10.3.55)

r˜ 2 φ˙ = H .

(10.3.56)

r˜ = a(1 ˜ − e˜ cos E)

(10.3.57)

r = r˜ + this can simply be written as

Expressing r˜ as

390

10 The Gravitational N-Body Problem

with  e˜ = 1 +

I ; a˜ = ar − 2H

I 2H ar

 er

(10.3.58)

then together with the Kepler-equation (10.3.47) we obtain dφ =

H 1 − et cos E dE . na˜ 2 (1 − e˜ cos E)2

(10.3.59)

Choosing a new eccentricity eφ with eφ ≡ 2e˜ − et

(10.3.60)

we get to post-Newtonian accuracy 1 − et cos E 1 = 1 − eφ cos E (1 − e˜ cos E)2 and, therefore up to terms of order c−4 dφ =

dE H . na˜ 2 1 − eφ cos E

(10.3.61)

The angular motion is therefore given by φ − φ0 = K f

(10.3.62)

with tan

f = 2



1 + eφ 1 − eφ

1/2 tan

E 2

(10.3.63)

and K=

H na˜ 2 (1 − eφ2 )1/2

.

(10.3.64)

In terms of E and J the constants eφ and K can be expressed as:



$1/2  1 E 2E μ2 eφ = 1 + 2 1 + (ν − 15) 2 J 2 − 6 2 2 μ c c

(10.3.65)

10.3 The PN Two-Body Problem

391

and K =1+3

μ2 J c2

(10.3.66)

√ keeping only PN-terms. Inserting the Newtonian value for J = μp the constant K can be written in the form K = 1 + 3μ/(c2 p). From r = ar (1 − er cos E) it is obvious that the periastron passages occur for E = f = 0, 2π, 4π, . . . , and therefore the advance of the pericenter per revolution, according to (10.3.62) is our old result 3μ . c2 p

φ = 2π(K − 1) = 2π

The form of the relative orbit can be found from r = ar (1 − er cos E) or  r=

er eφ



  er ar (1 − eφ cos E) + ar 1 − eφ

(10.3.67)

and the relation:  eφ = er 1 +

μ 2ar c2

 (10.3.68)

or, solving for ar :   er μ = 2 ar 1 − eφ 2c

(10.3.69)

the polar equation for the relative orbit takes the form: 2  μ μ  1 − eφ r = ar − 2 + 2. 2c 1 + eφ cos f 2c

(10.3.70)

This equation means that the relative orbit is the conchoid of a precessing ellipse, which means that it is obtained from an ellipse by a radial displacement and a precession in the orbital plane. Historically, Eq. (10.3.70) had already been derived by Infeld and Plebanski in 1960. With μ 1 f  = f − er 2 sin f 2 c p

(10.3.71)

this result can also be written as r=

ar (1 − er2 ) . 1 + er cos f 

(10.3.72)

392

10 The Gravitational N-Body Problem

10.4 The Rotational Motion 10.4.1 Landau-Lifshitz and Fock Spin Several authors like Landau and Lifshitz (1971) and Fock (1959) have introduced a concept of a relativistic spin vector for an isolated matter distribution in an asymptotically flat space-time. Such a concept is based upon the Landau-Lifshitz complex (5.8.11) satisfying

μν ,ν = 0 .

(10.4.1)

Let  be some closed spacelike hypersurface with oriented surface (i.e., 3volume) element extending into the asymptotic domain, then the quantities

P ≡ μ

μν

dν , 

μν JLL

2 ≡ c

x [μ ν]λ dλ

(10.4.2)



will be conserved. If one chooses a coordinate system (ct, x) in which  is a constant-time hypersurface then these conserved quantities can be written in the form

2 μν μ 3 μ0 P = d x , JLL = (10.4.3) d 3 x x [μ ν]0 . c P 0 can be related with the total energy of the system, P i with its total momentum, 0i with the motion of the center of mass of the system and J ij with its spin vector JLL LL k by SLL ij k

k JLL = ij k SLL ,

(10.4.4)

i.e., i SLL

1 = ij k c

d 3 x x j 0k .

(10.4.5)

However, it is known (Damour and Iyer 1991a,b) that even in the first postNewtonian approximation this integral is not well defined where

0k

  4U 1 [3U,t U,k + 4U,j (Vj,k − Vk,j ] + O3 . = 1 + 2 T 0k + 4π Gc c

(10.4.6)

10.4 The Rotational Motion

393

Here, U is the Newtonian potential generated by the mass density σ :

σ (x ) , U (t, x) = G d 3 x  |x − x |

(10.4.7)

and

Vj (t, x) = G

d 3x

σ j (x ) . |x − x |

(10.4.8)

Exercise 10.5 Show that to post-Newtonian order in harmonic gauge 00 (8π G) tLL = −7U,j U,j 0i (8π G) tLL =

2 [3U,t U,i + 4U,j (Vj,i − Vi,j ] c

(10.4.9)

ij

(8π G) tLL = 2U,i U,j − δij U,k U,k . 0k , 0k has no compact support as we Note, that because of the contribution from tLL can assume is the case for the matter contribution. This has the consequence that the integral on the right-hand side of (10.4.5) is not absolutely convergent and the i is mathematically ambiguous (Damour and Iyer 1991a). In fact expression for SLL the part coming from the Landau-Lifshitz tensor falls off as r −4 with increasing values of r, whereas the integral contains another factor of r 3 . Often in the literature, when one deals with the first post-Newtonian theory of i an expression for a conserved total spin is the spin (e.g., Will 1993) instead of SLL given which goes back to Fock (1959)    $

4U 1 0k i 3 j k SF = ij k d x x σ 1 + 2 + tF (10.4.10) c c

with tF0k

σ =− c



 7 k 1 k V + Q + O3 , 2 2

(10.4.11)

where

Qk (t, x) = G

d 3 x  σ l (x )

nkxx  nlxx  |x − x |

(10.4.12)

with nkxx  ≡ (x k − x  k )/|x − x |. It is clear, that in contrast to the Landau-Lifshitz expression, the integrand in (10.4.10) has compact support and the integral is welli . A simple and direct way is to defined. There are several ways to relate SFi with SLL employ the relations (see e.g., Will 1993): U = −4π Gσ ,

χ = −2U ,

Vj,j = −U,t ,

394

10 The Gravitational N-Body Problem

use integrations by parts and drop all (possibly ill-defined) surface integrals (see e.g., McCrea and O’Brien 1978). The basic steps can easily be understood by writing 0k (4π Gc)tLL = 3U,t U,k + 4U,j (Vj,k − Vk,j ) → 4U V k − U,t U,k

1 → −16π Gσ V k + U,k χ,jj t . 2

10.4.2 The PN-Spin in the N Body Problem Damour et al. (1993) have succeeded to introduce a reasonable spin vector of a body E that is member of a gravitational N body system. They start with the definition

Sa (T ) ≡ abc

  

 4  7 c 1 c c V + Q d XX  1 + 2W − 2 2 + 2 + c c E 3

b

(10.4.13)

in the local E system. Here,

d 3 X

 c (T , X ) |X − X |

(10.4.14)

d 3 X  d (T , X )

c Nd NXX  XX  |X − X |

(10.4.15)

V+c

≡ E

and

Qc+ ≡

E

with c c   NXX  ≡ (X − X )/|X − X | . c

Using the post-Newtonian Euler equation (10.1.3) one finds dSa = Da dT

(10.4.16)

with a 1PN torque Da given by

Da = abc

3

d XX E

b

 F − ∂T c

 c2



7 c 1 c V + Q 2 + 2 +

$

.

(10.4.17)

The force density, F c = E c + Bcd  d /c2 appearing here on the right-hand side c can be decomposed into self and external parts, i.e., F c = F +c + F . Using the

10.4 The Rotational Motion

395

explicit expression for W + one finds that the self-part of the 1PN torque vanishes to post-Newtonian order (DSX-III) so that

Da = abc

d 3 X Xb F

c

(10.4.18)

E

with b

F =

1 1 L G(2) − 7l − 4 X bL−1 G(2) L GbL + X2 X X bL L−1 l! 2(2l + 3)c2 (2l + 1)c2 l≥0

l cL−1 H (1) + 1 bcd  c X L HdL bcd X dL−1 2 (l + 1)c c2  4l eL−1 c (1)  bcd def X − 2  Gf L−1 , c (l + 1) +

(1)

(10.4.19)

with HL ≡ (d/dT )HL etc. Lemma 10.3 Using the expression (10.4.19) for the external post-Newtonian force density the 1PN torque can be written in the form:  1 d ∗ 1 l+1 Da = abc MbL GcL + 2 abc SbL HcL + S l! dT a c l+2

(10.4.20)

l≥0

with Sa∗ =

 l 1 1 MaL HL 2 l! l + 1 c l≥0

1 HaL NL 2l + 3 4(2l + 3) abc PbL GcL + (l + 2)(2l + 5) −

(10.4.21)

(l + 6) (1) abc (NbL G(1) cL − NbL GcL ) (2(l + 2)(2l + 5) $ 2 d + abc (NbL GcL ) . (l + 2)(2l + 5) dT

+

Here,

L  d 3 X X2 X

NL ≡ E

(10.4.22)

396

10 The Gravitational N-Body Problem

and

PL ≡

aL  a . d 3X X

(10.4.23)

E

The appearance of (d/dT )Sa∗ makes the definition of the 1PN spin of body E non unique since parts thereof or even the complete expression for Sa∗ might be included in the definition of the post-Newtonian spin. This point is extensively discussed in DSX-III. Let us continue our discussion with a torque of the form Da =

 1 1 l+1 abc MbL GcL + 2 abc SbL HcL . l! c l+2

(10.4.24)

l≥0

The first term on the right-hand side is the quasi-Newtonian torque. Note, however, that MbL formally are post-Newtonian Blanchet-Damour moments and our gravitoelectric tidal moments GcL have been defined with post-Newtonian accuracy. Next consider the spin term with l = 0. The gravito-magnetic dipole tidal moment, Ha , in the local E system, can be decomposed into two parts: Ha = Ha + Ha

(10.4.25)

where   Ha = Ria bi − 4 ij k vj ek X=0 + O2

(10.4.26)



dRib Ric + O2 . Ha = abc Vb Ac + c2 dT

(10.4.27)

and

Here, vi ≡ dzEi /dt = Ria Va and ai = d 2 zEi /dt 2 = Ria Aa to sufficient accuracy, and ei = ∂w +

4 ∂twi , c2

(10.4.28)

bi = −4 ij k ∂j w k . Writing dRia Rj a = ab = − ba ≡ abc c dT

(10.4.29)

we see that the gravito-magnetic tidal dipole, Ha , describes an inertial force induced by the rotation of the local spatial axes with respect to local inertial ones where

10.5 Rigidly Rotating Multipoles

397

Ha = 0. If the local E system is chosen to be kinematically non-rotating with respect E = δ , one finds that to the global system, i.e., Ria ia

a ≡ ainer =

1 3 2 1 ∂ ∂ Ha = − 2 aij vEi j w(zE ) + 2 aij j wi (zE ) − 2 aib vEi Gb ∂x ∂x 2c2 2c c 2c

or iner = −

3 2 1 vE × ∇w(zE ) + 2 ∇ × w(zE ) − 2 vE × G . 2 2c c 2c

(10.4.30)

Exercise 10.6 Prove (10.4.30) by direct calculation.

10.5 Rigidly Rotating Multipoles 10.5.1 Angular Velocity The concept of an angular velocity  has proven to be very useful in the Newtonian description of a rotating body. Usually, in a Newtonian inertial coordinate system tied to the body  is introduced by a velocity field of the form v =  × x.

(10.5.1)

In relativity, one might call a body rotating rigidly if in the 4-velocity field uμ =

dt (c, v i ) dτ

(10.5.2)

the coordinate velocity of some material element is given by (10.5.1). However, without specifying the space-time metric this is meaningless. One classical definition of a rigidly rotating body in GRT goes back to Born (1909, 1910). Born-rigidity means that the distance, as determined from the spacetime metric between every pair of neighbouring material elements in the body is independent of time. For the internal flow this implies that the expansion rate μ ρ θ = u,μ and the shear-rate tensor σμν −Pμν ·θ/3 (θμν = Pμσ Pν u(σ ;ρ) ; Pμν = gμν + uμ uν /c2 ) vanish. Clearly in relativity such rigidly rotating bodies can be constructed with v of the form (10.5.1) and  = const., leading to stationary manifolds. Such solutions of Einstein’s field equations have been discussed intensively in the literature (e.g., Hartle 1967; Hartle and Sharp 1967; Hartle and Thorne 1968; Friedman 1995; Friedman et al. 1986, 1989, 1992), where it is shown that all rotating configurations which minimize the total mass-energy (e.g., all stable configurations) must rotate uniformly. It is well known that such a rigidly rotating body in GRT can never change its angular velocity because otherwise signals would propagate with a speed exceeding the velocity of light in vacuum. However, such effects violating

398

10 The Gravitational N-Body Problem

the basic principle from Special Relativity are proportional to (vrot /c)2 ∼ ( L/c)2 , where L is the linear size of the body and one expects that rigidly rotating bodies satisfying the field equations and gauge conditions can be introduced if only first order terms in ( L/c) are kept (slowly rotating bodies). This is in fact the case even for strongly gravitating bodies such as neutron stars as has been demonstrated formally by Thorne and Gürsel (1983).

10.5.2 Rigidly Rotating Multipoles In this section we introduce the model of rigidly rotating multipoles where all relevant quantities are related by one and the same rotation matrix with constant numbers. Let us start from the local E system with coordinates (cT , Xa ), e.g., the GCRS in case of the Earth. The model of rigidly rotating multipoles is then defined by means of a rotation matrix P ab (T ) with corresponding angular velocity  according to

a (T ) =

1 d dc abc P db (T ) P (T ) . 2 dT

(10.5.3)

The model is then defined by the following set of relations: S a = C ab b C ab = P ac P bd C

cd

Ma1 ···al = P a1 b1 · · · P al bl M b1 ···bl SL = C

bL

b



(10.5.4)

(l ≥ 2)

(l ≥ 2)

C ba1 ···al = P bd P a1 c1 · · · P al cl C dc1 ···cl

(10.5.5)

(l ≥ 2)

where all barred quantities are constants, i.e., C

cd

= const. ,

M b1 ·bl = const. ,

C dc1 ···cl = const. .

If this model, is compatible with Einstein’s theory of gravity if (v/c)2 -terms are ignored is not known. However, it presents a well-defined post-Newtonian generalization of the Newtonian model of a rigidly rotating body and might serve as starting point for a relativistic model of a real (an-elastic deformable body with complex structure) body like the Earth. We will assume the local coordinate system, e.g., the GCRS, to be mass centered, i.e., its origin agrees with the post-Newtonian

10.5 Rigidly Rotating Multipoles

399

center of mass. In that case Ma = 0. In Klioner et al. (2001) it is shown that under reasonable assumptions the contribution of higher spin-moments, SL with l ≥ 2 can be neglected in the post-Newtonian expression for the torque. In that case the evolution equation for the spin of body E in its local system takes the form: 1 d (C ab b ) = abc MbL GcL + abc C bd d ciner . dT l! l≥1

(10.5.6)

Chapter 11

Light-Rays

11.1 Historical Remarks The theoretical basis for modelling astrometric measurements is the analysis of the light-ray equation, or the equation of null geodesics. Only for a few gravitational systems such as black-hole space-times, exact solutions for null geodesics are known (e.g., Chandrasekhar 1983; Hackmann 2014; Hackmann et al. 2008a,b, 2010). For more complex situations, e.g., for the propagation of light-rays in the gravitational field of solar system bodies, one might resort to approximation schemes such as the post-Newtonian (PN) or the post-Minkowskian approximation. At the first PN (1PN) level one approximates the exact null-geodesic trajectory xγ (t) in some suitably chosen coordinate system (ct, x) by neglecting c−3 terms, in the 2PN approximation one neglects c−5 terms. In the first post-Minkowskian approximation (1PM) one neglects G2 terms, in the 2PM approximation G3 terms etc.1 1PN Light Propagation in the Field of Monopoles at Rest The largest effect in light deflection in the solar system is due to the masses (monopoles) of the massive solar system bodies. The standard post-Newtonian solution of light trajectory in the gravitational field of a mass monopole at rest has been discussed in Chap. 8. An upper limit for the light deflection angle (the angle between vectors m and k in Fig. 11.1) in the field of a body at rest to 1PN order is given by M ϕ1PN ≤

4GMA , c2 dA

(11.1.1)

where dA is the (constant) distance of closest approach of the unperturbed light-ray to the mass MA .

1 The

following section is based upon private notes by Zschocke (2017).

© Springer Nature Switzerland AG 2019 M. H. Soffel, W.-B. Han, Applied General Relativity, Astronomy and Astrophysics Library, https://doi.org/10.1007/978-3-030-19673-8_11

401

402

11 Light-Rays

1PN Light Propagation in the Field of Quadrupoles at Rest The effects of light deflection in a quadrupole gravitational field at rest have been investigated many times by several authors. However, for the first time the full analytical solution for the light trajectory in a quadrupole field in post-Newtonian approximation has been obtained by Klioner (1991b), where an explicit time dependence of the coordinates of a photon and the solution of the boundary value problem for the geodesic equations has been obtained. These results were confirmed by a different approach by Le Poncin-Lafitte et al. (2008), while simplified expressions for 1 μas astrometric accuracy and rigorous estimates about the magnitude of quadrupole effects on light deflection have been derived by Zschocke and Klioner (2011). 1PN Light Propagation in the Field of Higher Multipole Moments at Rest Light deflection effects from higher mass multipole moments at rest have been studied by Le Poncin-Lafitte et al. (2008). For a single body MA one finds Jn ϕ1PN

4GMA JnA (RA )n ≤ , c2 (dA )n+1

(11.1.2)

where JnA and RA are the dimensionless parameter of zonal harmonics and a (coordinate) radius of the gravitating body A. 1.5PN Light Propagation in the Field Spin-Dipoles at Rest The first explicit 1.5PN (considering also c−3 terms in the photon trajectory) solution of the light trajectory in the gravitational field of massive bodies at rest possessing a spin-dipole has been obtained by Klioner (1991a,b). This solution provides all the details of light propagation, especially the explicit time dependence of the coordinates of the photon and the solution of the corresponding boundary value problem. An upper limit for the 1.5PN spin-dipole light deflection is given by S ϕ1.5PN ≤

4GSA , c3 dA2

(11.1.3)

where SA indicates the spin of the gravitating body. 1.5PN Light Propagation in the Field of Arbitrary Time-Independent Multipoles at Rest A systematic approach to the integration of light geodesic equations in the stationary 1.5PN gravitational field of an isolated body with time-independent multipole moments, ML and SL , is contained in Kopeikin (1997). 1PN and 1.5PN Light Propagation in the Field of Moving Monopoles Since the massive bodies of the Solar system are moving the center of mass coordinates of some body A are functions of time. For to-days astrometric accuracies at the microarcsecond level one has to account for the problem of how to treat the motion of the massive bodies during the time of propagation of light from the point of

11.1 Historical Remarks

403

emission to the point of reception. An analytical integration of 1PN light trajectory in the field of a uniformly moving body has first been derived by Klioner (1989). One has 4GMA , c2 dA (s)

(11.1.4)

|x1 − xA (s)| , c

(11.1.5)

M ϕ1PN ≤

where s indicates retarded time, i.e., s = t1 −

the observer is located at x1 at coordinate time t1 and the spatial position of the body at retarded time is xA (s). So to a first approximation one gets the old result for a mass monopole at rest, with dA being replaced by dA (s). If velocity terms are taken into account (Kopeikin et al. 1999; Kopeikin and Makarov 2007; Zschocke 2015) one obtains M ϕ1.5PN ≤

4GMA vA (s) . c2 dA (s) c

(11.1.6)

1PM Light Propagation in the Field of Moving Monopoles A rigorous solution of the problem of light propagation in the field of arbitrarily moving monopoles and in the first post-Minkowskian approximation has been found by Kopeikin and Schäfer (1999). By applying advanced integration methods introduced in Kopeikin (1997) and further developed by Kopeikin et al. (1999), the authors succeeded in integrating the geodesic equations for photons using retarded potentials, so that the positions of gravitating bodies are computed at the retarded instant of time s according to the light cone equation. Using this rigorous approach Kopeikin and Schäfer (1999) have shown that if the positions and velocities of the bodies are taken at retarded time then the effects of acceleration and the effects due to the time dependence of velocity of the bodies are much smaller than 1 μas in the solar system. 1PN Light Propagation in the Field of Moving Quadrupoles The light deflection at moving massive bodies with mass and quadrupoles has been investigated by Kopeikin and Makarov (2007), where the quadrupole term is taken in the Newtonian limit. Using the elaborated integration methods mentioned above, they succeeded to integrate analytically the geodesic equations by neglecting all terms smaller than 1 μas. 1PM Light Propagation in the Field of Moving Spin-Dipoles Kopeikin and Mashhoon (2002) have derived analytical solutions in postMinkowskian approximation for the case of light propagation in the field of arbitrarily moving bodies possessing a mass monopole and a spin-dipole.

404

11 Light-Rays

1PM Light Propagation in the Field of Time-Dependent Multipoles The case of the propagation of light rays in the field of localized sources which are completely characterized by time-dependent mass and spin multipole moments has been investigated by Kopeikin and Korobkov (2005) and Kopeikin et al. (2006). In particular, they have found an analytical solution for the light propagation in such gravitating systems. 1.5PN Light Propagation in the Field of Moving Multipoles Zschocke (2016a) has recently solved the problem of light propagation in the 1.5PN approximation in the gravitational field of N arbitrarily moving bodies endowed with a full set of intrinsic mass-multipoles and spin-multipoles, employing the Kopeikin integration technique. Maximal orders of magnitude for moving zonal harmonics are   vA (s) 4GMA JnA (RA )n Jn ϕ1.5PN ≤ 1+ . (11.1.7) c c2 (dA (s))n+1 2PN Light Propagation in the Field of a Mass-Monopole at Rest Post-post-Newtonian (2PN) effects on light deflection by some static mass have been investigated exhaustively in the literature (E.g., Epstein and Shapiro 1980; Fischbach and Freeman 1980; Richter and Matzner 1982a,b, 1983; Cowling 1984; Brumberg 1987; Bodenner and Will 2003; Le Poncin-Lafitte et al. 2004; Teyssandier and Le Poncin-Lafitte 2008; Ashby and Bertotti 2010). Accuracies have been determined by comparisons with numerical integrations of the null geodesic equation by Klioner and Zschocke (2010). 2PN Light Propagation in the Field of Arbitrarily Moving Point-Like Body Recently, Zschocke (2016b) has solved the problem of light propagation in the field of a single arbitrarily moving point-like body in the 2PN approximation (see also Zschocke 2018b, 2019). For the 2PN problem with several (point-like) gravitating bodies only very limiting results have been published whose applicability is restricted (e.g., Bruegmann 2005). Relevant physical parameters of solar system bodies are listed in Table 11.1. Orders of magnitude for gravitational light deflection effects in the solar system are presented in Table 11.2. In the following we will exhaustively discuss the 1PN problem of light-rays in the field of a single body at rest that has arbitrary mass and spin moments and then the Kopeikin-Schäfer formalism for 1PM accuracies and a system of N moving gravitating point-like masses.

11.1 Historical Remarks

405

Table 11.1 Numerical values for mass MA , radius RA , actual coefficients of zonal harmonics JnA , distance between observer and body rA1 , orbital velocity vA of Sun, Jupiter and Saturn (JPL 2019) Parameter /c2 [m]

Sun

Jupiter

Saturn

1476

1.4

0.4

PA [m]

696 × 106

71.5 × 106

60.3 × 106

J2A

2 × 10−7

14.696 × 10−3

16.291 × 10−3

J4A

−

−0.587 × 10−3

−0.936 × 10−3

J6A

−

0.034 × 10−3

0.086 × 10−3

J8A

−

−2.5 × 10−6

−10.0 × 10−6

A J10

−

0.21 × 10−6

2.0 × 10−6

SA [kg m2 / s]

1.64 × 1041

4.15 × 1038

7.13 × 1037

vA /c

4 × 10−8

4.4 × 10−5

3.2 × 10−5

GMA

The value for J2A for the Sun is taken from Fienga et al. (2008), while JnA with n = 2, 4, 6 for Jupiter and Saturn are taken from de Pater and Lissauer (2015), while JnA with n = 8, 10 for Jupiter and Saturn are taken from Hubbard and Militzer (2016) and Anderson and Schubert (2007), respectively. The spin angular momenta SA are determined from the moment of inertia IA 2 = 0.059, 0.254, 0.210 for Sun, Jupiter, Saturn, respectively (from NASA with the ratio IA /MA RA planetary fact sheets)

Table 11.2 Numerical magnitudes for light deflection angles in μas in the gravitational field of solar system bodies (Sun, Jupiter or Saturn) according to the upper limits given above

Sun

Jupiter

Saturn

M ϕ1PN

1.75 × 106

16.3 × 103

5.8 × 103

J2 ϕ1PN

1

240

95

J4 ϕ1PN

−

9.6

5.46

J6 ϕ1PN

−

0.56

0.50

J8 ϕ1PN

−

0.04

0.06

J10 ϕ1PN

−

0.003

0.01

M ϕ1.5PN

0.1

0.8

0.2

J2 ϕ1.5PN

−

0.011

0.003

S ϕ1.5PN

0.7

0.2

0.04

SO ϕ1.5PN

−

0.015

0.006

The physical parameters for Sun, Jupiter and Saturn are summarized in Table 11.1. The given light deflection angles are for grazing light-rays, i.e., for dA = RA . For the light deflection in the SO , results of Meichfield of spin-octupole, ϕ1.5PN sner (2015) where used. Blank entries indicate numbers smaller than 1 nas

406

11 Light-Rays

11.2 Light-Rays for 1PN Stationary Multipoles The geodesic equation (8.5.2) can be written in the form

d 2xi 1 0 j k i 2 i i j i j k 0 0 j = −c 00 − 2c0j x˙ − j k x˙ x˙ + c00 + 20j x˙ + j k x˙ x˙ x˙ c dt 2 (11.2.1) with x = xγ and x˙ i ≡

dxγi dt

.

Inserting the post-Newtonian Christoffel symbols with the potential w and wi we get      d 2xi 4  i 4 j x˙ 2 4 dx j = w − w,i x˙ j + 4 w,k x˙ i x˙ j x˙ k . · ∇w x˙ i + 2 w,j 1 + − ,i 2 2 2 dt dt c c c c

(11.2.2) Kopeikin (1997) succeeded to solve the post-Newtonian light-ray equation (11.2.2) for a central body endowed with arbitrary stationary mass- and spin-multipole moments for very remote light sources (note, the sign errors associated with the gravito-magnetic potential wi ). Assuming the gravitating body to be located at the center of our coordinate system the two metric potentials, w and wi are given in the skeletonized harmonic gauge, according to (7.4.3) and (7.4.4): w(x) = G

 (−1)l l≥0

l!

M L ∂L

  1 r

(11.2.3)

and wi (x) = −G

 (−1)l l≥1

l!

l iab S bL−1 ∂aL−1 l+1

  1 . r

(11.2.4)

Since we assume the Newtonian trajectory of be of the form xN (t) = x0 + cn(t − t0 ) we can replace dx i /dt in each post-Newtonian term by cni , so that the propagation equation reads:  d 2xi 4 i 4 j j k i w nj + w,k ni nj nk . = 2w − 4w n n + − w ,i ,k ,j ,i c c dt 2

(11.2.5)

11.2 Light-Rays for 1PN Stationary Multipoles

407

Inserting the expressions for w and w i we end up with  (−1)l d 2xi i j ML ∂ = (2n n − P )G ij l! dt 2 l≥0

+



xj r3



4G  (−1)l l ( iab nj − abc nc Pij )S bL−1 ∂ c (l + 1)!



l≥1

xj r3

 , (11.2.6)

where Pij are the components of the operator P⊥ that projects perpendicular to n: Pij = δij − ni nj = Pji = P ij .

(11.2.7)

This projection operator has only two algebraically independent components and satisfies the relation Pki Pjk = (δik − ni nk )(δkj − nk nj ) = Pji .

(11.2.8)

Let the 3-vector d be defined by projection of x perpendicular to n d = P⊥ x = x − n(n · x) = x0 − n(n · x0 ) = n × (x × n) = n × (x0 × n)

(11.2.9)

that satisfies d · n = 0.

(11.2.10)

This can also be expressed by the relation j

Pji d j = Pji Pk x k = Pki x k = d i .

(11.2.11)

The vector d points from the origin of x to the point of closest approach of the unperturbed light-ray to that origin. The fact that d has only two independent components has a strange looking consequence: ∂d i = ∂k d i = ∂k (Pji x j ) = Pki . ∂d k

(11.2.12)

Often, however, it might be convenient to consider the spatial components of d as formally independent so that ∂d i = δij ∂d j

(11.2.13)

408

11 Light-Rays

with a subsequent projection perpendicular to n, i.e., the rule (11.2.13) can be used if everywhere we replace the operator ∂i⊥ ≡

∂ ∂d i

j ∂ˆi ≡ Pi ∂i⊥ .

by

(11.2.14)

We now use the Kopeikin-parametrization of the unperturbed light-ray equation (Kopeikin 1997) in the form x=d+n·s,

(11.2.15)

s = n · x = c(t − t0 ) + n · x0

(11.2.16)

where

is a time coordinate if we replace x by xN (t). Then the parameter s0 = n · x0 corresponds to the time t ∗ = t0 − n · x0 /c ,

(11.2.17)

which is the time of closest approach of the unperturbed light-ray to the origin of x . Thus, s = c(t − t ∗ ) ;

s0 = c(t0 − t ∗ )

(11.2.18)

τ = t − t∗ .

(11.2.19)

or s = cτ

with

Following Kopeikin (1997) we can now split the partial derivative with respect to x i in the form 

∂i = ∂i⊥ + ∂i

(11.2.20)

with ∂i⊥ ≡

∂ , ∂di



∂i ≡ ni

∂ . ∂s

(11.2.21)

I.e., one splits the partial derivative, ∂/∂x i , into two pieces, one in the direction of the unperturbed light-ray and a second one perpendicular to it. From (r = |x|; ξ = |d|) r 2 = ξ 2 + s2

11.2 Light-Rays for 1PN Stationary Multipoles

409

we get the useful relations r −s =

d2 ; r +s

r0 − s0 =

d2 . r0 + s

(11.2.22)

Exercise 11.1 a) Let F (x) = 1/r 3 = (x 2 + y 2 + z2 )−3/2 . First calculate ∂i F (x); then substitute d + ns for x and compute (∂i⊥ + ni ∂s )F (d + ns) . Compare the two results with each other. b) Do the same for an arbitrary function F (x) = F (|x|) = F (r). Solution a) F,i = −3x i /r 5 . Since r 2 = d j d j + s 2 we have F (d + ns) = (d j d j + s 2 )−3/2 . Then, 3d i r5 3s ∂s F (d + ns) = − 5 r

∂i⊥ F (d + ns) = −

so that (∂i⊥ + ni ∂s )F (d + ns) gives the same as ∂i F (x). b) F,i = F,r (x i /r). F (d + ns) = F [(d i d i + s 2 )1/2 ] so that ∂i⊥ F = F,r (d i /r) and ∂s F = F,r (s/r) and (∂i⊥ + ni ∂s )F (d + ns) = F,i . Lemma 11.1 ∂ =

l  p=0

l! p ⊥ n