General Relativity: An Introduction for Physicists

by M. P. Hobson , G. P. Efstathiou , A. N. Lasenby

ISBN13: 9780521829519

ISBN10: 0521829518

Format: Hardcover

Pub. Date: 2006-03-27

Publisher(s): Cambridge University Press

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Summary

General Relativity: An Introduction for Physicists provides a clear mathematical introduction to Einstein's theory of general relativity. It presents a wide range of applications of the theory, concentrating on its physical consequences. After reviewing the basic concepts, the authors present a clear and intuitive discussion of the mathematical background, including the necessary tools of tensor calculus and differential geometry. These tools are then used to develop the topic of special relativity and to discuss electromagnetism in Minkowski spacetime. Gravitation as spacetime curvature is then introduced and the field equations of general relativity derived. After applying the theory to a wide range of physical situations, the book concludes with a brief discussion of classical field theory and the derivation of general relativity from a variational principle. Written for advanced undergraduate and graduate students, this approachable textbook contains over 300 exercises to illuminate and extend the discussion in the text.

Preface

1 The spacetime of special relativity

(25)

1.1 Inertial frames and the principle of relativity

(2)

1.2 Newtonian geometry of space and time

(1)

1.3 The spacetime geometry of special relativity

(2)

1.4 Lorentz transformations as four-dimensional 'rotations'

(1)

1.5 The interval and the lightcone

(2)

1.6 Spacetime diagrams

(2)

1.7 Length contraction and time dilation

(1)

1.8 Invariant hyperbolae

(1)

1.9 The Minkowski spacetime line element

(2)

1.10 Particle worldlines and proper time

(2)

1.11 The Doppler effect

(2)

1.12 Addition of velocities in special relativity

(1)

1.13 Acceleration in special relativity

(2)

1.14 Event horizons in special relativity

(1)

Appendix 1A: Einstein's route to special relativity

(2)

Exercises

(2)

2 Manifolds and coordinates

(27)

2.1 The concept of a manifold

(1)

2.2 Coordinates

(1)

2.3 Curves and surfaces

(1)

2.4 Coordinate transformations

(2)

2.5 Summation convention

(1)

2.6 Geometry of manifolds

(1)

2.7 Riemannian geometry

(1)

2.8 Intrinsic and extrinsic geometry

(3)

2.9 Examples of non-Euclidean geometry

(2)

2.10 Lengths, areas and volumes

(4)

2.11 Local Cartesian coordinates

(2)

2.12 Tangent spaces to manifolds

(1)

2.13 Pseudo-Riemannian manifolds

(2)

2.14 Integration over general submanifolds

(2)

2.15 Topology of manifolds

(1)

Exercises

(3)

3 Vector calculus on manifolds

(39)

3.1 Scalar fields on manifolds

(1)

3.2 Vector fields on manifolds

(1)

3.3 Tangent vector to a curve

(1)

3.4 Basis vectors

(3)

3.5 Raising and lowering vector indices

(1)

3.6 Basis vectors and coordinate transformations

(1)

3.7 Coordinate-independent properties of vectors

(1)

3.8 Derivatives of basis vectors and the affine connection

(2)

3.9 Transformation properties of the affine connection

(1)

3.10 Relationship of the connection and the metric

(2)

3.11 Local geodesic and Cartesian coordinates

(1)

3.12 Covariant derivative of a vector

(2)

3.13 Vector operators in component form

(1)

3.14 Intrinsic derivative of a vector along a curve

(2)

3.15 Parallel transport

(2)

3.16 Null curves, non-null curves and affine parameters

(1)

3.17 Geodesics

(1)

3.18 Stationary property of non-null geodesics

(1)

3.19 Lagrangian procedure for geodesics

(3)

3.20 Alternative form of the geodesic equations

(1)

Appendix 3A: Vectors as directional derivatives

(1)

Appendix 3B: Polar coordinates in a plane

(5)

Appendix 3C: Calculus of variations

(1)

Exercises

(4)

4 Tensor calculus on manifolds

(19)

4.1 Tensor fields on manifolds

(1)

4.2 Components of tensors

(1)

4.3 Symmetries of tensors

(2)

4.4 The metric tensor

(1)

4.5 Raising and lowering tensor indices

(1)

4.6 Mapping tensors into tensors

(1)

4.7 Elementary operations with tensors

(2)

4.8 Tensors as geometrical objects

100

(1)

4.9 Tensors and coordinate transformations

101

(1)

4.10 Tensor equations

102

(1)

4.11 The quotient theorem

103

(1)

4.12 Covariant derivative of a tensor

104

(3)

4.13 Intrinsic derivative of a tensor along a curve

107

(1)

Exercises

108

(3)

5 Special relativity revisited

111

(24)

5.1 Minkowski spacetime in Cartesian coordinates

111

(1)

5.2 Lorentz transformations

112

(1)

5.3 Cartesian basis vectors

113

(2)

5.4 Four-vectors and the lightcone

115

(1)

5.5 Four-vectors and Lorentz transformations

116

(1)

5.6 Four-velocity

116

(2)

5.7 Four-momentum of a massive particle

118

(1)

5.8 Four-momentum of a photon

119

(1)

5.9 The Doppler effect and relativistic aberration

120

(2)

5.10 Relativistic mechanics

122

(1)

5.11 Free particles

123

(1)

5.12 Relativistic collisions and Compton scattering

123

(2)

5.13 Accelerating observers

125

(3)

5.14 Minkowski spacetime in arbitrary coordinates

128

(3)

Exercises

131

(4)

6 Electromagnetism

135

(12)

6.1 The electromagnetic force on a moving charge

135

(1)

6.2 The 4-current density

136

(2)

6.3 The electromagnetic field equations

138

(1)

6.4 Electromagnetism in the Lorenz gauge

139

(2)

6.5 Electric and magnetic fields in inertial frames

141

(1)

6.6 Electromagnetism in arbitrary coordinates

142

(2)

6.7 Equation of motion for a charged particle

144

(1)

Exercises

145

(2)

7 The equivalence principle and spacetime curvature

147

(29)

7.1 Newtonian gravity

147

(1)

7.2 The equivalence principle

148

(1)

7.3 Gravity as spacetime curvature

149

(2)

7.4 Local inertial coordinates

151

(1)

7.5 Observers in a curved spacetime

152

(1)

7.6 Weak gravitational fields and the Newtonian limit

153

(2)

7.7 Electromagnetism in a curved spacetime

155

(2)

7.8 Intrinsic curvature of a manifold

157

(1)

7.9 The curvature tensor

158

(1)

7.10 Properties of the curvature tensor

159

(2)

7.11 The Ricci tensor and curvature scalar

161

(2)

7.12 Curvature and parallel transport

163

(2)

7.13 Curvature and geodesic deviation

165

(2)

7.14 Tidal forces in a curved spacetime

167

(3)

Appendix 7A: The surface of a sphere

170

(2)

Exercises

172

(4)

8 The gravitational field equations

176

(20)

8.1 The energy-momentum tensor

176

(2)

8.2 The energy-momentum tensor of a perfect fluid

178

(1)

8.3 Conservation of energy and momentum for a perfect fluid

179

(2)

8.4 The Einstein equations

181

(2)

8.5 The Einstein equations in empty space

183

(1)

8.6 The weak-field limit of the Einstein equations

184

(1)

8.7 The cosmological-constant term

185

(3)

8.8 Geodesic motion from the Einstein equations

188

(2)

8.9 Concluding remarks

190

(1)

Appendix 8A: Alternative relativistic theories of gravity

191

(2)

Appendix 8B: Sign conventions

193

(1)

Exercises

193

(3)

9 The Schwarzschild geometry

196

(34)

9.1 The general static isotropic metric

196

(2)

9.2 Solution of the empty-space field equations

198

(4)

9.3 Birkhoff' s theorem

202

(1)

9.4 Gravitational redshift for a fixed emitter and receiver

202

(3)

9.5 Geodesics in the Schwarzschild geometry

205

(2)

9.6 Trajectories of massive particles

207

(2)

9.7 Radial motion of massive particles

209

(3)

9.8 Circular motion of massive particles

212

(1)

9.9 Stability of massive particle orbits

213

(4)

9.10 Trajectories of photons

217

(1)

9.11 Radial motion of photons

218

(1)

9.12 Circular motion of photons

219

(1)

9.13 Stability of photon orbits

220

(1)

Appendix 9A: General approach to gravitational redshifts

221

(3)

Exercises

224

(6)

10 Experimental tests of general relativity

230

(18)

10.1 Precession of planetary orbits

230

(3)

10.2 The bending of light

233

(3)

10.3 Radar echoes

236

(4)

10.4 Accretion discs around compact objects

240

(4)

10.5 The geodesic precession of gyroscopes

244

(2)

Exercises

246

(2)

11 Schwarzschild black holes

248

(40)

11.1 The characterisation of coordinates

248

(1)

11.2 Singularities in the Schwarzschild metric

249

(2)

11.3 Radial photon worldlines in Schwarzschild coordinates

251

(1)

11.4 Radial particle worldlines in Schwarzschild coordinates

252

(2)

11.5 Eddington–Finkelstein coordinates

254

(5)

11.6 Gravitational collapse and black-hole formation

259

(1)

11.7 Spherically symmetric collapse of dust

260

(4)

11.8 Tidal forces near a black hole

264

(2)

11.9 Kruskal coordinates

266

(5)

11.10 Wormholes and the Einstein–Rosen bridge

271

(3)

11.11 The Hawking effect

274

(3)

Appendix 11A: Compact binary systems

277

(2)

Appendix 11B: Supermassive black holes

279

(3)

Appendix 11C: Conformal flatness of two-dimensional Riemannian manifolds

282

(1)

Exercises

283

(5)

12 Further spherically symmetric geometries

288

(22)

12.1 The form of the metric for a stellar interior

288

(4)

12.2 The relativistic equations of stellar structure

292

(2)

12.3 The Schwarzschild constant-density interior solution

294

(2)

12.4 Buchdahl's theorem

296

(1)

12.5 The metric outside a spherically symmetric charged mass

296

(4)

12.6 The Reissner–Nordstrom geometry: charged black holes

300

(2)

12.7 Radial photon trajectories in the RN geometry

302

(2)

12.8 Radial massive particle trajectories in the RN geometry

304

(1)

Exercises

305

(5)

13 The Kerr geometry

310

(45)

13.1 The general stationary axisymmetric metric

310

(2)

13.2 The dragging of inertial frames

312

(2)

13.3 Stationary limit surfaces

314

(1)

13.4 Event horizons

315

(2)

13.5 The Kerr metric

317

(2)

13.6 Limits of the Kerr metric

319

(2)

13.7 The Kerr-Schild form of the metric

321

(1)

13.8 The structure of a Kerr black hole

322

(5)

13.9 The Penrose process

327

(3)

13.10 Geodesics in the equatorial plane

330

(2)

13.11 Equatorial trajectories of massive particles

332

(1)

13.12 Equatorial motion of massive particles with zero angular momentum

333

(2)

13.13 Equatorial circular motion of massive particles

335

(2)

13.14 Stability of equatorial massive particle circular orbits

337

(1)

13.15 Equatorial trajectories of photons

338

(1)

13.16 Equatorial principal photon geodesics

339

(2)

13.17 Equatorial circular motion of photons

341

(1)

13.18 Stability of equatorial photon orbits

342

(2)

13.19 Eddington-Finkelstein coordinates

344

(3)

13.20 The slow-rotation limit and gyroscope precession

347

(3)

Exercises

350

(5)

14 The Friedmann-Robertson-Walker geometry

355

(31)

14.1 The cosmological principle

355

(1)

14.2 Slicing and threading spacetime

356

(1)

14.3 Synchronous coordinates

357

(1)

14.4 Homogeneity and isotropy of the universe

358

(1)

14.5 The maximally symmetric 3-space

359

(3)

14.6 The Friedmann-Robertson-Walker metric

362

(1)

14.7 Geometric properties of the FRW metric

362

(3)

14.8 Geodesics in the FRW metric

365

(2)

14.9 The cosmological redshift

367

(1)

14.10 The Hubble and deceleration parameters

368

(3)

14.11 Distances in the FRW geometry

371

(3)

14.12 Volumes and number densities in the FRW geometry

374

(2)

14.13 The cosmological field equations

376

(3)

14.14 Equation of motion for the cosmological fluid

379

(2)

14.15 Multiple-component cosmological fluid

381

(1)

Exercises

381

(5)

15 Cosmological models

386

(42)

15.1 Components of the cosmological fluid

386

(4)

15.2 Cosmological parameters

390

(2)

15.3 The cosmological field equations

392

(1)

15.4 General dynamical behaviour of the universe

393

(4)

15.5 Evolution of the scale factor

397

(3)

15.6 Analytical cosmological models

400

(8)

15.7 Look-back time and the age of the universe

408

(3)

15.8 The distance-redshift relation

411

(2)

15.9 The volume-redshift relation

413

(2)

15.10 Evolution of the density parameters

415

(2)

15.11 Evolution of the spatial curvature

417

(1)

15.12 The particle horizon, event horizon and Hubble distance

418

(3)

Exercises

421

(7)

16 Inflationary cosmology

428

(39)

16.1 Definition of inflation

428

(2)

16.2 Scalar fields and phase transitions in the very early universe

430

(1)

16.3 A scalar field as a cosmological fluid

431

(2)

16.4 An inflationary epoch

433

(1)

16.5 The slow-roll approximation

434

(1)

16.6 Ending inflation

435

(1)

16.7 The amount of inflation

435

(2)

16.8 Starting inflation

437

(1)

16.9 'New' inflation

438

(2)

16.10 Chaotic inflation

440

(1)

16.11 Stochastic inflation

441

(1)

16.12 Perturbations from inflation

442

(1)

16.13 Classical evolution of scalar-field perturbations

442

(4)

16.14 Gauge invariance and curvature perturbations

446

(3)

16.15 Classical evolution of curvature perturbations

449

(3)

16.16 Initial conditions and normalisation of curvature perturbations

452

(4)

16.17 Power spectrum of curvature perturbations

456

(2)

16.18 Power spectrum of matter-density perturbations

458

(1)

16.19 Comparison of theory and observation

459

(3)

Exercises

462

(5)

17 Linearised general relativity

467

(31)

17.1 The weak-field metric

467

(3)

17.2 The linearised gravitational field equations

470

(2)

17.3 Linearised gravity in the Lorenz gauge

472

(1)

17.4 General properties of the linearised field equations

473

(1)

17.5 Solution of the linearised field equations in vacuo

474

(1)

17.6 General solution of the linearised field equations

475

(5)

17.7 Multipole expansion of the general solution

480

(1)

17.8 The compact-source approximation

481

(2)

17.9 Stationary sources

483

(2)

17.10 Static sources and the Newtonian limit

485

(1)

17.11 The energy–momentum of the gravitational field

486

(4)

Appendix 17A: The Einstein–Maxwell formulation of linearised gravity

490

(3)

Exercises

493

(5)

18 Gravitational waves

498

(26)

18.1 Plane gravitational waves and polarisation states

498

(3)

18.2 Analogy between gravitational and electromagnetic waves

501

(1)

18.3 Transforming to the transverse-traceless gauge

502

(2)

18.4 The effect of a gravitational wave on free particles

504

(3)

18.5 The generation of gravitational waves

507

(4)

18.6 Energy flow in gravitational waves

511

(2)

18.7 Energy loss due to gravitational-wave emission

513

(3)

18.8 Spin-up of binary systems: the binary pulsar PSR B1913+16

516

(1)

18.9 The detection of gravitational waves

517

(3)

Exercises

520

(4)

19 A variational approach to general relativity

524

(31)

19.1 Hamilton's principle in Newtonian mechanics

524

(3)

19.2 Classical field theory and the action

527

(2)

19.3 Euler–Lagrange equations

529

(2)

19.4 Alternative form of the Euler–Lagrange equations

531

(2)

19.5 Equivalent actions

533

(1)

19.6 Field theory of a real scalar field

534

(2)

19.7 Electromagnetism from a variational principle

536

(3)

19.8 The Einstein–Hilbert action and general relativity in vacuo

539

(3)

19.9 An equivalent action for general relativity in vacuo

542

(1)

19.10 The Palatini approach for general relativity in vacuo

543

(2)

19.11 General relativity in the presence of matter

545

(1)

19.12 The dynamical energy–momentum tensor

546

(3)

Exercises

549

(6)

Bibliography

555

(1)

Index

556

General Relativity: An Introduction for Physicists

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