General Relativity: An Introduction for Physicists

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

• ISBN13: 9780521829519
• ISBN10: 0521829518
• eBook ISBN(s): 9780511137952
• Format: Hardcover
• Copyright: 2006-03-27
• Publisher: Cambridge University Press
• More Book Details

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.

Table of Contents

Preface	xv
1 The spacetime of special relativity	1	(25)
1.1 Inertial frames and the principle of relativity	1	(2)
1.2 Newtonian geometry of space and time	3	(1)
1.3 The spacetime geometry of special relativity	3	(2)
1.4 Lorentz transformations as four-dimensional 'rotations'	5	(1)
1.5 The interval and the lightcone	6	(2)
1.6 Spacetime diagrams	8	(2)
1.7 Length contraction and time dilation	10	(1)
1.8 Invariant hyperbolae	11	(1)
1.9 The Minkowski spacetime line element	12	(2)
1.10 Particle worldlines and proper time	14	(2)
1.11 The Doppler effect	16	(2)
1.12 Addition of velocities in special relativity	18	(1)
1.13 Acceleration in special relativity	19	(2)
1.14 Event horizons in special relativity	21	(1)
Appendix 1A: Einstein's route to special relativity	22	(2)
Exercises	24	(2)
2 Manifolds and coordinates	26	(27)
2.1 The concept of a manifold	26	(1)
2.2 Coordinates	27	(1)
2.3 Curves and surfaces	27	(1)
2.4 Coordinate transformations	28	(2)
2.5 Summation convention	30	(1)
2.6 Geometry of manifolds	31	(1)
2.7 Riemannian geometry	32	(1)
2.8 Intrinsic and extrinsic geometry	33	(3)
2.9 Examples of non-Euclidean geometry	36	(2)
2.10 Lengths, areas and volumes	38	(4)
2.11 Local Cartesian coordinates	42	(2)
2.12 Tangent spaces to manifolds	44	(1)
2.13 Pseudo-Riemannian manifolds	45	(2)
2.14 Integration over general submanifolds	47	(2)
2.15 Topology of manifolds	49	(1)
Exercises	50	(3)
3 Vector calculus on manifolds	53	(39)
3.1 Scalar fields on manifolds	53	(1)
3.2 Vector fields on manifolds	54	(1)
3.3 Tangent vector to a curve	55	(1)
3.4 Basis vectors	56	(3)
3.5 Raising and lowering vector indices	59	(1)
3.6 Basis vectors and coordinate transformations	60	(1)
3.7 Coordinate-independent properties of vectors	61	(1)
3.8 Derivatives of basis vectors and the affine connection	62	(2)
3.9 Transformation properties of the affine connection	64	(1)
3.10 Relationship of the connection and the metric	65	(2)
3.11 Local geodesic and Cartesian coordinates	67	(1)
3.12 Covariant derivative of a vector	68	(2)
3.13 Vector operators in component form	70	(1)
3.14 Intrinsic derivative of a vector along a curve	71	(2)
3.15 Parallel transport	73	(2)
3.16 Null curves, non-null curves and affine parameters	75	(1)
3.17 Geodesics	76	(1)
3.18 Stationary property of non-null geodesics	77	(1)
3.19 Lagrangian procedure for geodesics	78	(3)
3.20 Alternative form of the geodesic equations	81	(1)
Appendix 3A: Vectors as directional derivatives	81	(1)
Appendix 3B: Polar coordinates in a plane	82	(5)
Appendix 3C: Calculus of variations	87	(1)
Exercises	88	(4)
4 Tensor calculus on manifolds	92	(19)
4.1 Tensor fields on manifolds	92	(1)
4.2 Components of tensors	93	(1)
4.3 Symmetries of tensors	94	(2)
4.4 The metric tensor	96	(1)
4.5 Raising and lowering tensor indices	97	(1)
4.6 Mapping tensors into tensors	97	(1)
4.7 Elementary operations with tensors	98	(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