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Lectures on General Relativity
Bengt MĂĽnsson
Lectures on General Relativity Bengt MĂĽnsson 14 december 2018
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Other books by the same author (in Swedish): Grundläggande algebra ISBN 9789176999554 Einsteins speciella och allmänna relativitetsteori ISBN 9789176995907 Experimentell matematik med Derive 5.0 ISBN 9197296864
The text is typeset in LATEX 2ε AMS-TEX Typeface Computer Modern 10p
c Bengt Månsson 2018 ! Publisher: BoD – Books on Demand, Stockholm, Sweden Printing: BoD – Books on Demand, Norderstedt, Germany ISBN: 9789177856917
Contents TWO OVERVIEWS OF GENERAL RELATIVITY 5
I
1 First overview
6
1.1
General relativity as a dynamical theory of space-time and gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.2
Einstein spaces . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.3
The large scale structure of space-time . . . . . . . . . . . .
8
1.4
The problem of motion . . . . . . . . . . . . . . . . . . . . .
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The crisis in general relativity . . . . . . . . . . . . . . . . .
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2 Second overview
14
2.1
Differential Geometry . . . . . . . . . . . . . . . . . . . . .
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2.2
2.1.1 Manifolds . . . . . . . . . . 2.1.2 Tensors . . . . . . . . . . . 2.1.3 Metric structure . . . . . . 2.1.4 Connection . . . . . . . . . 2.1.5 Parallel transport . . . . . . 2.1.6 Curvature . . . . . . . . . . 2.1.7 ”Surgery” and embeddings General relativity . . . . . . . . . .
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14 16 19 21 23 24 27 31
2.3
2.2.1 Space-time manifolds . . . 2.2.2 Local causality . . . . . . 2.2.3 Energy-momentum tensor 2.2.4 The field equations . . . . The Cauchy problem . . . . . . .
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31 31 31 32 33
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CONTENTS 2.4
Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . .
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2.5
2.4.1 Killing vectors . . . 2.4.2 Coordinate condition 2.4.3 Maximal symmetry . Spin coefficient formalism .
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33 34 35 36
2.6
2.5.1 Tetrad and spin coefficients . . . . . . . . . . . . 2.5.2 Identities . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Free field equations . . . . . . . . . . . . . . . . . 2.5.4 Robinson-Trautman solutions . . . . . . . . . . . Incompleteness, extensions, and elementary singularities
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36 38 38 41 43
2.7
2.6.1 Incompleteness, extension . . . . . . . . . . . . . . . 2.6.2 Elementary singularities . . . . . . . . . . . . . . . . Singular space-times . . . . . . . . . . . . . . . . . . . . . .
43 45 46
2.7.1 2.7.2 2.7.3
46 48 49
The concept of singularity . . . . . . . . . . . . . . . Background material for a singularity theorem . . . The singularity theorem . . . . . . . . . . . . . . . .
SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS 53 II
3 Timelike geodesics
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4 Perihelion precession
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5 Proper time in orbit
63
5.1
Some numerical examples . . . . . . . . . . . . . . . . . . .
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5.2
The general inequality Δτ |orb < Δτ |obs . . . . . . . . . . . .
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5.3
Circular orbit and weak field . . . . . . . . . . . . . . . . .
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6 Null geodesics
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7 Light deflection
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8 Geodesics inside r = 3m
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CONTENTS 9 Delayed radar signals 9.1
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii 82 85
10 Geodesic precession
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11 Frame dragging
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11.1 Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . .
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11.2 Schwarzschild’s inner solution . . . . . . . . . . . . . . . . .
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Appendix II.A - Elliptic functions and integrals
96
BLACK HOLES AND COSMOLOGICAL SOLUTIONS 99
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12 Introduction
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13 Symmetric spaces
102
13.1 Killing’s equation . . . . . . . . . . . . . . . . . . . . . . . . 103 13.2 Maximal symmetry . . . . . . . . . . . . . . . . . . . . . . . 109 13.3 Homogeneity and isotropy . . . . . . . . . . . . . . . . . . . 111 13.4 Examples of maximally symmetric spaces . . . . . . . . . . 114 13.5 Uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . 117 13.6 Maximally symmetric subspaces . . . . . . . . . . . . . . . . 120 14 Extension of space-times
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14.1 Extensions in general . . . . . . . . . . . . . . . . . . . . . . 123 14.2 Spherically symmetric space-time . . . . . . . . . . . . . . . 124 14.3 Maximal extension . . . . . . . . . . . . . . . . . . . . . . . 126 15 Black holes and gravitational collapse
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15.1 In- and outgoing solutions . . . . . . . . . . . . . . . . . . . 131
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CONTENTS 15.1.1 Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . 135 15.2 Gravitational collapse and creation of black holes . . . . . . 138 15.3 Gravitational collapse in detail . . . . . . . . . . . . . . . . 141 15.4 Singularities and horizons . . . . . . . . . . . . . . . . . . . 151 15.5 Realistic gravitational collapse . . . . . . . . . . . . . . . . 156 15.5.1 I . . 15.5.2 II . . 15.5.3 III . 15.6 Inner size of
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16 Cosmological solutions
156 157 160 160 167
16.1 The cosmological principle and the Robertson-Walker metric 168 16.2 Open and closed universe . . . . . . . . . . . . . . . . . . . 170 16.3 Geodesics and the meaning of the coordinates . . . . . . . . 171 16.4 Using the ďŹ eld equations . . . . . . . . . . . . . . . . . . . . 172 16.4.1 16.4.2 16.4.3 16.5 Mixed
Non-relativistic matter . . Relativistic matter . . . . Vacuum energy . . . . . . matter and critical density
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174 179 181 182
16.6 Comparison with observations . . . . . . . . . . . . . . . . . 183 16.7 The initial singularity. Horizons . . . . . . . . . . . . . . . . 189 16.8 Cosmic Microwave Radiation Background . . . . . . . . . . 195 16.9 Conclusion
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Appendix III.A - Equivalence of Killingâ&#x20AC;&#x2122;s equation and the existence of isometries 198 Appendix III.B - Geodesic deviation
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Appendix III.C - The exit cone
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Appendix III.D - Other cosmological models
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CONTENTS
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ADVANCED TOPICS
17 Exact solutions
213 214
17.1 Einstein spaces . . . . . . . . . . . . . . . . . . . . . . . . . 214 17.2 Weyl’s solutions . . . . . . . . . . . . . . . . . . . . . . . . . 214 17.3 The C-metric . . . . . . . . . . . . . . . . . . . . . . . . . . 219 17.4 Schwarzschild’s solution in isotropic coordinates . . . . . . . 219 17.5 The Einstein-Rosen bridge . . . . . . . . . . . . . . . . . . . 220 17.6 Kerr’s solution . . . . . . . . . . . . . . . . . . . . . . . . . 222 17.7 Robinson-Trautman space-times
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17.7.1 Final remark . . . . . . . . . . . . . . . . . . . . . . 227 17.8 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . 227 17.9 The Resissner-Nordström solution . . . . . . . . . . . . . . 227 17.10Vaidya’s solution . . . . . . . . . . . . . . . . . . . . . . . . 229 17.11Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 17.11.1 Star in equlibrium . . . . . . . . . . . . . . . . . . . 230 17.11.2 Star with constant density . . . . . . . . . . . . . . . 231 17.12Causality violating solutions . . . . . . . . . . . . . . . . . . 235 17.12.1 Gödel’s solution . . . . . . . . . . . . . . . . . . . . 235 17.12.2 The solutions of van Stockum and others . . . . . . 237 18 Cosmic time functions in certain Robinson-Trautman spacetimes 239 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 18.2 RT space-times . . . . . . . . . . . . . . . . . . . . . . . . . 240 18.3 A class of cosmic time functions . . . . . . . . . . . . . . . . 246 18.4 Properties of RT curvature singularities . . . . . . . . . . . 246 18.5 Conclusion
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19 Energy of the gravitational field
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CONTENTS 19.1 An integral theorem . . . . . . . . . . . . . . . . . . . . . . 252 19.2 Energy-momentum complex for the gravitational field . . . 253 19.3 Energy density etc . . . . . . . . . . . . . . . . . . . . . . . 258 19.4 The superpotential . . . . . . . . . . . . . . . . . . . . . . . 260 19.4.1 Schwarzschild’s space-time . . . . . . . . . . . . . . . 260 19.5 Difficulties with the interpretation . . . . . . . . . . . . . . 262 19.5.1 Spatial transformation . . . . . . . . . . . . . . . . . 262 19.5.2 Localization of the energy of a gravitational field . . 263 19.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
20 Equations of motion
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20.1 The principle of equivalence . . . . . . . . . . . . . . . . . . 265 20.2 Weyl’s solution . . . . . . . . . . . . . . . . . . . . . . . . . 266 20.2.1 Principal determination of the metric . . . . . . . . 266 20.2.2 An example with two singularities . . . . . . . . . . 267 20.3 The Einstein-Infeld-Hoffmann method . . . . . . . . . . . . 270 20.3.1 Description of the method . . . . 20.3.2 Application to massive particles . 20.3.3 Integration of the EIH equations 20.4 The Newman-Posadas method . . . . .
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270 273 275 284
20.4.1 Worldlines in Minkowski space-time . . . . . . . . . 284 20.4.2 Relation between F2S-metric and acceleration . . . . 286 20.4.3 Adaption to elementary singularities . . . . . . . . . 286 21 Equations of motion for a class of space-time singularities289 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 21.2 Specification of the Type of Singularity
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21.3 The Equations of Motion . . . . . . . . . . . . . . . . . . . 294 21.4 Covariance of the Infeld-Plebański Equations . . . . . . . . 298 21.5 Particular cases . . . . . . . . . . . . . . . . . . . . . . . . . 304 21.6 Conclusion
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CONTENTS
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22 On the possibility of covariant equations of motion for space-time singularities 311 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 22.2 Coordinate systems and regularization . . . . . . . . . . . . 312 22.3 Application to RT-like singularities . . . . . . . . . . . . . . 314 22.4 Covariance of the IP equations . . . . . . . . . . . . . . . . 316 22.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . 320 22.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 23 Gravitational waves
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23.1 Linearized field equations . . . . . . . . . . . . . . . . . . . 323 23.2 Emitted energy . . . . . . . . . . . . . . . . . . . . . . . . . 325 23.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 335 23.3.1 The binary pulsar PSR 1913+16 . . . . . . . . . . . 336 23.3.2 LIGO . . . . . . . . . . . . . . . . . . . . . . . . . . 336 23.4 Other results on gravitational waves . . . . . . . . . . . . . 338 23.4.1 23.4.2 23.4.3 23.4.4
Linear mass quadropole oscillator . . General form for the emitted power Plane waves . . . . . . . . . . . . . . Exact wave solutions . . . . . . . . .
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24 What is a space-time singularity?
338 340 342 345 347
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 24.2 Problems of definition . . . . . . . . . . . . . . . . . . . . . 348 24.3 Examples of coordinate effects . . . . . . . . . . . . . . . . . 349 24.4 Elimination of ”coordinate singularities” . . . . . . . . . . . 350 24.4.1 (A) Finiteness of measured times and lengths 24.4.2 (B) Finiteness of tidal forces at r = 2m . . . 24.4.3 (C) Well-behaved coordinate systems . . . . . 24.5 Definition of a singular space-time . . . . . . . . . .
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350 352 354 358
24.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . 360
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CONTENTS 24.6 Singularity theorems . . . . . . . . . . . . . . . . . . . . . . 362 24.6.1 24.6.2 24.6.3 24.6.4 24.7 Other
Preliminaries . . . . . . . Statement of the theorem Proof of the theorem . . . Remarks . . . . . . . . . . results on singularities . .
24.8 Conclusion
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362 365 365 368 369
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Appendix IV.A – Weyl solutions
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Appendix IV.B – Tensor densities
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Appendix IV.C - Lagrange density
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Appendix IV.D - Maximal extension of the Schwarzschild space-time 380 Appendix IV.E - Free fall towards r = 2m as viewed from the outside 382 Appendix IV.F - The generalized affine parameter
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Appendix IV.G - The convergence of a set of curves
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Appendix IV.H - Properties of the matrix A = (tαβ )
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Appendix IV.I - Two-dimensional analogy
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Appendix IV.J - Lambert’s W -function
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Appendix IV.K - Hydrostatic equlibrium
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Appendix IV.L - Proof for ! ∂ 0! ∂ ∗0 (ln P0 ) = 1
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Appendix IV.M - Proof for χab = ta;b and χab = χba .
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CONTENTS
V
EINSTEIN...
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25 Time to tie the threads together...
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26 Special Relativity Theory, Einstein 1905, 1907, 1912
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26.1 Background and postulates . . . . . . . . . . . . . . . . . . 401 26.2 Simultaneity definition . . . . . . . . . . . . . . . . . . . . . 402 26.3 The concepts of cause and effect . . . . . . . . . . . . . . . 404 26.4 New axiomatization? . . . . . . . . . . . . . . . . . . . . . . 404 26.5 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 405 26.6 Particle Dynamics . . . . . . . . . . . . . . . . . . . . . . . 406 27 General Relativity Theory, Einstein 1911-1916
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27.1 Background and postulates . . . . . . . . . . . . . . . . . . 407 27.2 Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . 408 27.3 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 408 27.4 Simultaneity in General Relativity . . . . . . . . . . . . . . 410 27.5 Unified field theory . . . . . . . . . . . . . . . . . . . . . . . 411 28 Einstein’s view of the method of theoretical physics
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28.1 Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 28.2 The importance of mathematics . . . . . . . . . . . . . . . . 413 28.2.1 Some examples . . . . . . . . . . . . . . . . . . . . . 414 28.2.2 Calculus of Variation . . . . . . . . . . . . . . . . . . 416 Appendix V.A - Transverse and longitudinal mass
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Appendix V.B - Einstein’s theory for the non-symmetric field423
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CONTENTS
VI
A SHORT BASIC COURSE
431
29 Differential geometry
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29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 29.2 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 29.3 Tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . 433 29.4 Tensor analysis . . . . . . . . . . . . . . . . . . . . . . . . . 436 29.4.1 29.4.2 29.4.3 29.4.4 29.4.5 29.4.6 29.4.7
Affinity . . . . . . . . . . . Covariant derivative . . . . Curvature tensor . . . . . . Geodesic coordinates . . . . Metric connection, Riemann Metric affinity . . . . . . . Geodesics . . . . . . . . . .
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30 General Theory of Relativity
437 437 438 440 440 443 448 453
30.1 Background and postulates . . . . . . . . . . . . . . . . . . 453 30.2 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 457 30.3 Newton’s theory as a first approximation . . . . . . . . . . . 459 30.4 Spherically symmetric gravitational field . . . . . . . . . . . 462 30.5 Schwarzschild’s solution . . . . . . . . . . . . . . . . . . . . 463 30.6 Theory of measurement . . . . . . . . . . . . . . . . . . . . 465 30.6.1 30.6.2 30.6.3 30.6.4
Expressions for dL2 and dT 2 . . . . . . . . Principal determination of the metric tensor Physical interpretation of the coordinates . Timelike, spacelike and null coordinates . .
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466 469 470 471
31 Non-euclidean spatial geometry
476
32 Accelerated reference systems
482
32.1 Superluminal speed . . . . . . . . . . . . . . . . . . . . . . . 482 32.2 The twin paradox . . . . . . . . . . . . . . . . . . . . . . . . 486 32.2.1 Solution of the paradox within special relativity . . . 486
CONTENTS
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32.2.2 Several breaking points . . . . . . . . . . . . . . . . 487 32.2.3 Solution of the paradox within general relativity . . 489
VII
PROBLEMS
33 Problems
491 492
LITERATURE, REFERENCES, AND INDEX 495
VIII
Literature
496
References
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CONTENTS