GENERAL RELATIVITY AND ITS APPLICATIONS : black holes, compact stars and gravitational waves.

494 Pages·2020·19.748 MB·English

by  FERRARI, VALERIA. GUALTIERI LEONARDO. PANI PAOLO

General Relativity
and its Applications

General Relativity
and its Applications
Black Holes, Compact Stars and
Gravitational Waves
Valeria Ferrari
Leonardo Gualtieri
Paolo Pani
CRC PressFirst edition published 2021
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Library of Congress Cataloging-in-Publication Data
Names: Ferrari, Valeria, 1952- author. | Gualtieri, Leonardo, 1971- author.
| Pani, Paolo, 1984- author.
Title: General relativity and its Applications : Black Holes, Compact Stars
and Gravitational Waves
Valeria Ferrari, Leonardo Gualtieri, Paolo Pani.
Description: Boca Raton : CRC Press, 2020. | Includes bibliographical
references and index.
Identifiers: LCCN 2020028364 | ISBN 9780367625320 (paperback) | ISBN
9781138589773 (hardback) | ISBN 9780429491405 (ebook)
Subjects: LCSH: General relativity (Physics) | Gravitation. | Gravitational
waves.
Classification: LCC QC173.6 .F45 2020 | DDC 530.11--dc23
LC record available at https://lccn.loc.gov/2020028364
ISBN: 978-1-138-58977-3 (hbk)
ISBN: 978-0-367-62532-0 (pbk)
ISBN: 978-0-429-49140-5 (ebk)
Typeset in Computer Modern font
by KnowledgeWorks Global Ltd.
Contents
Preface xiii
Notation and conventions xvii
hapter
C 1(cid:4) Introduction 1
1.1 NON-EUCLIDEANGEOMETRIES 1
1.1.1 The metric tensor in different coordinate frames 4
1.1.2 The Gaussian curvature 5
1.1.3 Pseudo-Euclidean geometries and spacetime 5
1.2 NEWTONIANTHEORYANDITSSHORTCOMINGS 6
1.3 THEROLEOFTHEEQUIVALENCEPRINCIPLE 9
1.4 GEODESICEQUATIONSASACONSEQUENCEOFTHEEQUIVALENCE
PRINCIPLE 11
1.5 LOCALLYINERTIALFRAMES 13
hapter
C 2(cid:4) Elements of differential geometry 17
2.1 TOPOLOGICALSPACES,MAPPING,MANIFOLDS 17
2.1.1 Topological spaces 17
2.1.2 Mapping 18
2.1.3 Manifolds and differentiable manifolds 22
2.1.4 Diffeomorphisms 26
2.2 VECTORS 27
2.2.1 The traditional definition of a vector 27
2.2.2 A geometrical definition 29
2.3 ONE-FORMS 39
2.3.1 One-forms as geometrical objects 39
2.3.2 Vector fields and one-form fields 43
2.4 TENSORS 46
2.4.1 Geometrical definition of a tensor 46
2.4.2 Symmetries of a tensor 51
2.5 THEMETRICTENSORANDITSPROPERTIES 54
v
vi (cid:4) Contents
hapter
C 3(cid:4) Affine connection and parallel transport 63
3.1 THECOVARIANTDERIVATIVEOFVECTORS 63
3.2 THECOVARIANTDERIVATIVEOFSCALARSANDONE-FORMS 67
3.3 SYMMETRIESOFCHRISTOFFEL’SSYMBOLS 68
3.4 TRANSFORMATIONRULESFORCHRISTOFFEL’SSYMBOLS 69
3.5 THECOVARIANTDERIVATIVEOFTENSORS 70
3.6 CHRISTOFFEL’SSYMBOLSINTERMSOFTHEMETRICTENSOR 72
3.7 PARALLELTRANSPORT 76
3.7.1 Parallel transport of a vector along a closed path on a two-
sphere 78
3.8 GEODESICEQUATION 81
3.9 FERMICOORDINATES 82
3.10 NON-COORDINATEBASES 84
hapter
C 4(cid:4) The curvature tensor 87
4.1 PARALLELTRANSPORTALONGALOOP 87
4.2 SYMMETRIESOFTHERIEMANNTENSOR 91
4.3 THE RIEMANN TENSOR GIVES THE COMMUTATOR OF COVARIANT
DERIVATIVES 91
4.4 THEBIANCHIIDENTITIES 92
4.5 THEEQUATIONOFGEODESICDEVIATION 92
hapter
C 5(cid:4) The stress-energy tensor 97
5.1 THESTRESS-ENERGYTENSORINFLATSPACETIME 97
5.2 ISTαβ ATENSOR? 100
5.3 DOESTαβ SATISFYACONSERVATIONLAW? 104
5.4 ISTαβ = 0ACONSERVATIONLAW? 107
;β
hapter
C 6(cid:4) The Einstein equations 109
6.1 GEODESICEQUATIONSINTHEWEAK-FIELD,STATIONARYLIMIT 110
6.2 EINSTEIN’SFIELDEQUATIONS 112
6.3 GAUGEINVARIANCEOFEINSTEIN’SEQUATIONS 117
6.4 THEHARMONICGAUGE 118
hapter
C 7(cid:4) Einstein’s equations and variational principles 121
7.1 EULER-LAGRANGE’SEQUATIONSINSPECIALRELATIVITY 121
7.2 EULER-LAGRANGE’SEQUATIONSINCURVEDSPACETIME 122
7.3 EINSTEIN’SEQUATIONSINVACUUM 124
7.4 EINSTEIN’SEQUATIONSWITHSOURCES 128
7.4.1 The stress-energy tensor in some relevant cases 129
7.5 EINSTEIN’SEQUATIONSINTHEPALATINIFORMALISM 131
Contents (cid:4) vii
hapter
C 8(cid:4) Symmetries 135
8.1 KILLINGVECTORFIELDS 135
8.2 KILLING VECTOR FIELDS AND THE CHOICE OF COORDINATE SYS-
TEMS 139
8.3 KILLINGVECTORFIELDSANDCONSERVATIONLAWS 142
8.3.1 Conserved quantities in geodesic motion 142
8.3.2 Conserved quantities from the stress-energy tensor 143
8.4 HYPERSURFACE-ORTHOGONALVECTORFIELDS 144
8.4.1 Frobenius’ theorem 145
8.4.2 Hypersurface-orthogonal vector fields and the choice of coor-
dinate systems 146
8.5 DIFFEOMORPHISMINVARIANCEOFGENERALRELATIVITY 148
hapter
C 9(cid:4) The Schwarzschild solution 151
9.1 STATICANDSPHERICALLYSYMMETRICSPACETIMES 151
9.2 THESCHWARZSCHILDSOLUTION 154
9.3 SINGULARITIESOFTHESCHWARZSCHILDSOLUTION 159
9.4 SPACELIKE,TIMELIKE,ANDNULLHYPERSURFACES 160
9.4.1 Constant radius hypersurfaces in Schwarzschild’s spacetime 162
9.5 SINGULARITIESINGENERALRELATIVITY 163
9.5.1 Geodesic completeness 163
9.5.2 How to remove a coordinate singularity 164
9.5.3 Extension of the Rindler spacetime 166
9.5.4 Extension of the Schwarzschild spacetime 170
9.5.5 Eddington-Finkelstein coordinates 176
9.6 THEBIRKHOFFTHEOREM 178
hapter
C 10(cid:4) Geodesic motion in Schwarzschild’s spacetime 181
10.1 AVARIATIONALPRINCIPLEFORGEODESICMOTION 181
10.2 EQUATIONSOFMOTION 182
10.3 THECONSTANTSOFGEODESICMOTION 185
10.4 ORBITSOFMASSLESSPARTICLES 188
10.5 ORBITSOFMASSIVEPARTICLES 190
10.6 RADIALCAPTUREOFAMASSIVEPARTICLE 193
hapter
C 11(cid:4) Kinematical tests of General Relativity 197
11.1 GRAVITATIONALSHIFTOFSPECTRALLINES 197
11.1.1 Redshift of spectral lines in the weak-field limit 201
11.1.2 Redshift of spectral lines in a strong gravitational field 201
11.2 THEDEFLECTIONOFLIGHT 203
11.3 PERIASTRONPRECESSION 207
viii (cid:4) Contents
11.4 THESHAPIROTIMEDELAY 212
11.5 THESHADOWOFABLACKHOLE 214
11.5.1 The accretion disk of a black hole 218
hapter
C 12(cid:4) Gravitational waves 221
12.1 PERTURBATIVEAPPROACH 221
12.2 GRAVITATIONALWAVESASPERTURBATIONSOFFLATSPACETIME 223
12.3 HOWTOCHOOSETHEHARMONICGAUGE 228
12.4 PLANEGRAVITATIONALWAVES 229
12.5 THETTGAUGE 230
12.6 HOW DOES A GRAVITATIONAL WAVE AFFECT THE MOTION OF A
SINGLEPARTICLE 232
12.7 GEODESICDEVIATIONINDUCEDBYAGRAVITATIONALWAVE 232
12.8 GRAVITATIONALWAVESANDMICHELSONINTERFEROMETERS 239
hapter
C 13(cid:4) Gravitational waves in the quadrupole approximation 243
13.1 THEWEAK-FIELD,SLOW-MOTIONAPPROXIMATION 243
13.2 THEQUADRUPOLEFORMULA 245
13.2.1 The tensor-virial theorem 245
13.2.2 The quadrupole moment tensor 247
13.2.3 Absence of monopolar and dipolar gravitational waves 247
13.3 HOWTOTRANSFORMTOTHETTGAUGE 248
13.4 GRAVITATIONALWAVESEMITTEDBYAHARMONICOSCILLATOR 250
13.5 GRAVITATIONALWAVEEMITTEDBYABINARYSYSTEMINCIRCULAR
ORBIT 252
13.6 ENERGYCARRIEDBYAGRAVITATIONALWAVE 257
13.6.1 The stress-energy pseudo-tensor of the gravitational field 257
13.6.2 Energy flux of a gravitational wave 261
hapter
C 14(cid:4) Gravitational wave sources 267
14.1 EVOLUTIONOFACOMPACTBINARYSYSTEM 267
14.2 GRAVITATIONALWAVESFROMINSPIRALLINGCOMPACTOBJECTS 271
14.2.1 September 14th, 2015: the detection of gravitational waves 273
14.2.2 The chirp mass and the luminosity distance 274
14.2.3 A lower bound for the total mass of the system 277
14.2.4 The final stages of the inspiral 278
14.2.5 Merger and ringdown: identifying the nature of coalescing
compact objects 279
14.2.6 More signals from coalescences 280
14.3 GRAVITATIONALWAVESFROMROTATINGCOMPACTSTARS 284
14.3.1 Stars rigidly rotating around a principal axis 284
14.3.2 Wobbling stars 289
Contents (cid:4) ix
14.4 COSMOLOGICALPARAMETERS 292
14.4.1 The cosmological redshift 293
14.4.2 The Hubble constant 294
14.4.3 Luminosity distance 295
14.4.4 Standard sirens: coalescing binaries as standard candles 297
hapter
C 15(cid:4) Gravitational waves from oscillating black holes 299
15.1 ATOYMODEL:SCALARPERTURBATIONS 299
15.2 PERTURBATIONSOFTHESCHWARZSCHILDSPACETIME 304
15.2.1 Linearized Einstein’s equations in vacuum 304
15.2.2 Harmonic decomposition 305
15.2.3 The Regge-Wheeler gauge 308
15.2.4 Perturbations with l = 0 and l = 1 310
15.2.5 Linearized Einstein’s equations in vacuum 310
15.2.6 Separation of the angular dependence 311
15.3 MASTEREQUATIONSFORAXIALANDPOLARPERTURBATIONS 312
15.3.1 The Regge-Wheeler equation for the axial perturbations 313
15.3.2 The Zerilli equation for the polar perturbations 313
15.4 THEQUASI-NORMALMODESOFASCHWARZSCHILDBLACKHOLE 315
hapter
C 16(cid:4) Compact stars 321
16.1 STELLAREVOLUTIONINANUTSHELL 321
16.2 WHITEDWARFS 324
16.2.1 The discovery of white dwarfs 324
16.2.2 Degenerate gas in quantum mechanics 325
16.2.3 A criterion for degeneracy 327
16.2.4 The equation of state of a fully degenerate gas of fermions 330
16.2.5 The structure of a white dwarf 334
16.2.6 The Chandrasekhar limit 338
16.3 NEUTRONSTARS 343
16.3.1 The discovery of neutron stars 343
16.3.2 The internal structure of a neutron star 343
16.3.3 Thermodynamics of perfect fluids in General Relativity 346
16.3.4 The stress-energy tensor of a perfect fluid 352
16.3.5 The equations of stellar structure in General Relativity 353
16.3.6 The Schwarzschild solution for a homogeneous star 358
16.3.7 Relativistic polytropes 360
16.3.8 Buchdahl’s theorem 365
16.3.9 Stability of a compact star 367