groen o., hervik s. einstein''s general theory of relativity

Einstein’s
General Theory of Relativity
Øyvind Grøn and Sigbjørn Hervik
ii
Version 9th December 2004.
c
Grøn & Hervik.
Contents
Preface xv
Notation xvii
I INTRODUCTION:
N
EWTONIAN PHYSICS AND SPECIAL RELATIVITY 1
1 Relativity Principles and Gravitation 3
1.1 Newtonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Galilei–Newton’s principle of Relativity . . . . . . . . . . . . . . 4
1.3 The principle of Relativity . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Newton’s law of Gravitation . . . . . . . . . . . . . . . . . . . . . 6
1.5 Local form of Newton’s Gravitational law . . . . . . . . . . . . . 8
1.6 Tidal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 The principle of equivalence . . . . . . . . . . . . . . . . . . . . . 14
1.8 The covariance principle . . . . . . . . . . . . . . . . . . . . . . . 15
1.9 Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 The Special Theory of Relativity 21
2.1 Coordinate systems and Minkowski-diagrams . . . . . . . . . . 21
2.2 Synchronization of clocks . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Relativistic time-dilatation . . . . . . . . . . . . . . . . . . . . . . 25
2.5 The relativity of simultaneity . . . . . . . . . . . . . . . . . . . . 26
2.6 The Lorentz-contraction . . . . . . . . . . . . . . . . . . . . . . . 28

2.7 The Lorentz transformation . . . . . . . . . . . . . . . . . . . . . 30
2.8 Lorentz-invariant interval . . . . . . . . . . . . . . . . . . . . . . 32
2.9 The twin-paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.10 Hyperbolic motion . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.11 Energy and mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.12 Relativistic increase of mass . . . . . . . . . . . . . . . . . . . . . 38
2.13 Tachyons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.14 Magnetism as a relativistic second-order effect . . . . . . . . . . 40
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
II T
HE MATHEMATICS OF THE
GENERAL THEORY OF RELATIVITY 49
3 Vectors, Tensors, and Forms 51
3.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
iv Contents
3.2 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 One-forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.5 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Basis Vector Fields and the Metric Tensor 63
4.1 Manifolds and their coordinate-systems . . . . . . . . . . . . . . 63
4.2 Tangent vector fields and the coordinate basis vector fields . . . 65
4.3 Structure coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.4 General basis transformations . . . . . . . . . . . . . . . . . . . . 71
4.5 The metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.6 Orthonormal basis . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 Spatial geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.8 The tetrad field of a comoving coordinate system . . . . . . . . . 80
4.9 The volume form . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.10 Dual forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5 Non-inertial Reference Frames 89
5.1 Spatial geometry in rotating reference frames . . . . . . . . . . . 89
5.2 Ehrenfest’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3 The Sagnac effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.4 Gravitational time dilatation . . . . . . . . . . . . . . . . . . . . . 94
5.5 Uniformly accelerated reference frame . . . . . . . . . . . . . . . 95
5.6 Covariant Lagrangian dynamics . . . . . . . . . . . . . . . . . . 98
5.7 A general equation for the Doppler effect . . . . . . . . . . . . . 103
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Differentiation, Connections and Integration 109
6.1 Exterior Differentiation of forms . . . . . . . . . . . . . . . . . . 109
6.2 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.3 Integration of forms . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4 Covariant differentiation of vectors . . . . . . . . . . . . . . . . . 120
6.5 Covariant differentiation of forms and tensors . . . . . . . . . . 127
6.6 Exterior differentiation of vectors . . . . . . . . . . . . . . . . . . 129
6.7 Covariant exterior derivative . . . . . . . . . . . . . . . . . . . . 133
6.8 Geodesic normal coordinates . . . . . . . . . . . . . . . . . . . . 136
6.9 One-parameter groups of diffeomorphisms . . . . . . . . . . . . 137
6.10 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.11 Killing vectors and Symmetries . . . . . . . . . . . . . . . . . . . 143
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7 Curvature 149
7.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.3 The Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . 153
7.4 Extrinsic and Intrinsic Curvature . . . . . . . . . . . . . . . . . . 159
7.5 The equation of geodesic deviation . . . . . . . . . . . . . . . . . 162

7.6 Spaces of constant curvature . . . . . . . . . . . . . . . . . . . . . 163
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Contents v
III EINSTEIN’S FIELD EQUATIONS 175
8 Einstein’s Field Equations 177
8.1 Deduction of Einstein’s vacuum field equations from Hilbert’s
variational principle . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2 The field equations in the presence of matter and energy . . . . 180
8.3 Energy-momentum conservation . . . . . . . . . . . . . . . . . . 181
8.4 Energy-momentum tensors . . . . . . . . . . . . . . . . . . . . . 182
8.5 Some particular fluids . . . . . . . . . . . . . . . . . . . . . . . . 184
8.6 The paths of free point particles . . . . . . . . . . . . . . . . . . . 188
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
9 The Linear Field Approximation 191
9.1 The linearised field equations . . . . . . . . . . . . . . . . . . . . 191
9.2 The Newtonian limit of general relativity . . . . . . . . . . . . . 194
9.3 Solutions to the linearised field equations . . . . . . . . . . . . . 195
9.4 Gravitoelectromagnetism . . . . . . . . . . . . . . . . . . . . . . 197
9.5 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9.6 Gravitational radiation from sources . . . . . . . . . . . . . . . . 202
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
10 The Schwarzschild Solution and Black Holes 211
10.1 The Schwarzschild solution for empty space . . . . . . . . . . . 211
10.2 Radial free fall in Schwarzschild spacetime . . . . . . . . . . . . 216
10.3 The light-cone in a Schwarzschild spacetime . . . . . . . . . . . 217
10.4 Particle trajectories in Schwarzschild spacetime . . . . . . . . . . 221
10.5 Analytical extension of the Schwarzschild spacetime . . . . . . . 226
10.6 Charged and rotating black holes . . . . . . . . . . . . . . . . . . 229
10.7 Black Hole thermodynamics . . . . . . . . . . . . . . . . . . . . . 241
10.8 The Tolman-Oppenheimer-Volkoff equation . . . . . . . . . . . . 248

10.9 The interior Schwarzschild solution . . . . . . . . . . . . . . . . 249
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
IV C
OSMOLOGY 259
11 Homogeneous and Isotropic Universe Models 261
11.1 The cosmological principles . . . . . . . . . . . . . . . . . . . . . 261
11.2 Friedmann-Robertson-Walker models . . . . . . . . . . . . . . . 262
11.3 Dynamics of Homogeneous and Isotropic cosmologies . . . . . 265
11.4 Cosmological redshift and the Hubble law . . . . . . . . . . . . 267
11.5 Radiation dominated universe models . . . . . . . . . . . . . . . 272
11.6 Matter dominated universe models . . . . . . . . . . . . . . . . . 275
11.7 The gravitational lens effect . . . . . . . . . . . . . . . . . . . . . 277
11.8 Redshift-luminosity relation . . . . . . . . . . . . . . . . . . . . . 283
11.9 Cosmological horizons . . . . . . . . . . . . . . . . . . . . . . . . 287
11.10Big Bang in an infinite Universe . . . . . . . . . . . . . . . . . . . 288
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
vi Contents
12 Universe Models with Vacuum Energy 297
12.1 Einstein’s static universe . . . . . . . . . . . . . . . . . . . . . . . 297
12.2 de Sitter’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . 298
12.3 The de Sitter hyperboloid . . . . . . . . . . . . . . . . . . . . . . 301
12.4 The horizon problem and the flatness problem . . . . . . . . . . 302
12.5 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
12.6 The Friedmann-Lemaître model . . . . . . . . . . . . . . . . . . . 311
12.7 Universe models with quintessence energy . . . . . . . . . . . . 317
12.8 Dark energy and the statefinder diagnostic . . . . . . . . . . . . 320
12.9 Cosmic density perturbations . . . . . . . . . . . . . . . . . . . . 327
12.10Temperature fluctuations in the CMB . . . . . . . . . . . . . . . . 331
12.11Mach’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
12.12The History of our Universe . . . . . . . . . . . . . . . . . . . . . 341

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
13 An Anisotropic Universe 359
13.1 The Bianchi type I universe model . . . . . . . . . . . . . . . . . 359
13.2 The Kasner solutions . . . . . . . . . . . . . . . . . . . . . . . . . 362
13.3 The energy-momentum conservation law in an anisotropic uni-
verse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
13.4 Models with a perfect fluid . . . . . . . . . . . . . . . . . . . . . 365
13.5 Inflation through bulk viscosity . . . . . . . . . . . . . . . . . . . 368
13.6 A universe with a dissipative fluid . . . . . . . . . . . . . . . . . 369
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
V A
DVANCED TOPICS 375
14 Covariant decomposition, Singularities, and Canonical Cosmology 377
14.1 Covariant decomposition . . . . . . . . . . . . . . . . . . . . . . 377
14.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 380
14.3 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
14.4 Lagrangian formulation of General Relativity . . . . . . . . . . . 387
14.5 Hamiltonian formulation . . . . . . . . . . . . . . . . . . . . . . . 390
14.6 Canonical formulation with matter and energy . . . . . . . . . . 392
14.7 The space of three-metrics: Superspace . . . . . . . . . . . . . . . 394
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
15 Homogeneous Spaces 401
15.1 Lie groups and Lie algebras . . . . . . . . . . . . . . . . . . . . . 401
15.2 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . 404
15.3 The Bianchi models . . . . . . . . . . . . . . . . . . . . . . . . . . 407
15.4 The orthonormal frame approach to the Bianchi models . . . . . 411
15.5 The 8 model geometries . . . . . . . . . . . . . . . . . . . . . . . 416
15.6 Constructing compact quotients . . . . . . . . . . . . . . . . . . . 418
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
16 Israel’s Formalism: The metric junction method 427

16.1 The relativistic theory of surface layers . . . . . . . . . . . . . . . 427
16.2 Einstein’s field equations . . . . . . . . . . . . . . . . . . . . . . . 429
16.3 Surface layers and boundary surfaces . . . . . . . . . . . . . . . 431
16.4 Spherical shell of dust in vacuum . . . . . . . . . . . . . . . . . . 433
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
Contents vii
17 Brane-worlds 441
17.1 Field equations on the brane . . . . . . . . . . . . . . . . . . . . . 441
17.2 Five-dimensional brane cosmology . . . . . . . . . . . . . . . . . 444
17.3 Problem with perfect fluid brane world in an empty bulk . . . . 447
17.4 Solutions in the bulk . . . . . . . . . . . . . . . . . . . . . . . . . 447
17.5 Towards a realistic brane cosmology . . . . . . . . . . . . . . . . 449
17.6 Inflation in the brane . . . . . . . . . . . . . . . . . . . . . . . . . 452
17.7 Dynamics of two branes . . . . . . . . . . . . . . . . . . . . . . . 455
17.8 The hierarchy problem and the weakness of gravity . . . . . . . 457
17.9 The Randall-Sundrum models . . . . . . . . . . . . . . . . . . . . 459
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
18 Kaluza-Klein Theory 465
18.1 A fifth extra dimension . . . . . . . . . . . . . . . . . . . . . . . . 465
18.2 The Kaluza-Klein action . . . . . . . . . . . . . . . . . . . . . . . 467
18.3 Implications of a fifth extra dimension . . . . . . . . . . . . . . . 471
18.4 Conformal transformations . . . . . . . . . . . . . . . . . . . . . 474
18.5 Conformal transformation of the Kaluza-Klein action . . . . . . 478
18.6 Kaluza-Klein cosmology . . . . . . . . . . . . . . . . . . . . . . . 480
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
VI A
PPENDICES 487
A Constants of Nature 489
B Penrose diagrams 491
B.1 Conformal transformations and causal structure . . . . . . . . . 491

B.2 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . . . . . . 493
B.3 de Sitter spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 493
C Anti-de Sitter spacetime 497
C.1 The anti-de Sitter hyperboloid . . . . . . . . . . . . . . . . . . . . 497
C.2 Foliations of AdS
n
. . . . . . . . . . . . . . . . . . . . . . . . . . . 498
C.3 Geodesics in AdS
n
. . . . . . . . . . . . . . . . . . . . . . . . . . 499
C.4 The BTZ black hole . . . . . . . . . . . . . . . . . . . . . . . . . . 500
C.5 AdS
3
as the group SL(2, R) . . . . . . . . . . . . . . . . . . . . . 501
D Suggested further reading 503
Bibliography 507
Index 515

List of Problems
Chapter 1 17
1.1 The strength of gravity compared to the Coulomb force . . . . 17
1.2 Falling objects in the gravitational field of the Earth . . . . . . . 17
1.3 Newtonian potentials for spherically symmetric bodies . . . . 17
1.4 The Earth-Moon system . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 The Roche-limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 A Newtonian Black Hole . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Non-relativistic Kepler orbits . . . . . . . . . . . . . . . . . . . . 19
Chapter 2 42
2.1 Two successive boosts in different directions . . . . . . . . . . . 42
2.2 Length-contraction and time-dilatation . . . . . . . . . . . . . . 43

2.3 Faster than the speed of light? . . . . . . . . . . . . . . . . . . . 44
2.4 Reflection angles off moving mirrors . . . . . . . . . . . . . . . 44
2.5 Minkowski-diagram . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6 Robb’s Lorentz invariant spacetime interval formula . . . . . . 45
2.7 The Doppler effect . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.8 Abberation and Doppler effect . . . . . . . . . . . . . . . . . . . 45
2.9 A traffic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.10 The twin-paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.11 Work and rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.12 Muon experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.13 Cerenkov radiation . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 3 60
3.1 The tensor product . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2 Contractions of tensors . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 The Lorentz-Abraham-Dirac equation . . . . . . . . . . . . . . . 62
Chapter 4 85
4.1 Coordinate-transformations in a two-dimensional Euclidean
plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.2 Covariant and contravariant components . . . . . . . . . . . . . 86
4.3 The Levi-Civitá symbol . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 Dual forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 5 107
5.1 Geodetic curves in space . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Free particle in a hyperbolic reference frame . . . . . . . . . . . 107
5.3 Spatial geodesics in a rotating RF . . . . . . . . . . . . . . . . . 108
x List of Problems
Chapter 6 147
6.1 Loop integral of a closed form . . . . . . . . . . . . . . . . . . . 147
6.2 The covariant derivative . . . . . . . . . . . . . . . . . . . . . . 147

6.3 The Poincaré half-plane . . . . . . . . . . . . . . . . . . . . . . . 148
6.4 The Christoffel symbols in a rotating reference frame with plane
polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Chapter 7 170
7.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.2 Inverse metric on S
n
. . . . . . . . . . . . . . . . . . . . . . . . . 170
7.3 The curvature of a curve . . . . . . . . . . . . . . . . . . . . . . 170
7.4 The Gauss-Codazzi equations . . . . . . . . . . . . . . . . . . . 171
7.5 The Poincaré half-space . . . . . . . . . . . . . . . . . . . . . . . 171
7.6 The pseudo-sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.7 A non-Cartesian coordinate system in two dimensions . . . . . 172
7.8 The curvature tensor of a sphere . . . . . . . . . . . . . . . . . . 172
7.9 The curvature scalar of a surface of simultaneity . . . . . . . . . 172
7.10 The tidal force pendulum and the curvature of space . . . . . . 172
7.11 The Weyl tensor vanishes for spaces of constant curvature . . . 173
7.12 Frobenius’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 173
Chapter 8 189
8.1 Lorentz transformation of a perfect fluid . . . . . . . . . . . . . 189
8.2 Geodesic equation and constants of motion . . . . . . . . . . . 189
Chapter 9 206
9.1 The Linearised Einstein Field Equations . . . . . . . . . . . . . 206
9.2 Gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . 208
9.3 The spacetime inside and outside a rotating spherical shell . . 209
Chapter 10 251
10.1 The Schwarzschild metric in Isotropic coordinates . . . . . . . 251
10.2 Embedding of the interior Schwarzschild metric . . . . . . . . . 252
10.3 The Schwarzschild-de Sitter metric . . . . . . . . . . . . . . . . 252
10.4 The life time of a black hole . . . . . . . . . . . . . . . . . . . . . 252

10.5 A spaceship falling into a black hole . . . . . . . . . . . . . . . . 252
10.6 The GPS Navigation System . . . . . . . . . . . . . . . . . . . . 253
10.7 Physical interpretation of the Kerr metric . . . . . . . . . . . . . 253
10.8 A gravitomagnetic clock effect . . . . . . . . . . . . . . . . . . . 253
10.9 The photon sphere radius of a Reissner-Nordström black hole . 254
10.10 Curvature of 3-space and 2-surfaces of the internal and the
external Schwarzschild spacetimes . . . . . . . . . . . . . . . . . 254
10.11 Proper radial distance in the external Schwarzschild space . . . 255
10.12 Gravitational redshift in the Schwarzschild spacetime . . . . . 255
10.13 The Reissner-Nordström repulsion . . . . . . . . . . . . . . . . 256
10.14 Light-like geodesics in the Reissner-Nordström spacetime . . . 256
10.15 Birkhoff’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 256
10.16 Gravitational mass . . . . . . . . . . . . . . . . . . . . . . . . . . 257
List of Problems xi
Chapter 11 290
11.1 Physical significance of the Robertson-Walker coordinate system290
11.2 The volume of a closed Robertson-Walker universe . . . . . . . 290
11.3 The past light-cone in expanding universe models . . . . . . . 290
11.4 Lookback time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
11.5 The FRW-models with a w-law perfect fluid . . . . . . . . . . . 292
11.6 Age-density relations . . . . . . . . . . . . . . . . . . . . . . . . 292
11.7 Redshift-luminosity relation for matter dominated universe . . 293
11.8 Newtonian approximation with vacuum energy . . . . . . . . . 293
11.9 Universe with multi-component fluid . . . . . . . . . . . . . . . 293
11.10 Gravitational collapse . . . . . . . . . . . . . . . . . . . . . . . . 293
11.11 Cosmic redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
11.12 Universe models with constant deceleration parameter . . . . . 295
11.13 Relative densities as functions of the expansion factor . . . . . 295
11.14 FRW universe with radiation and matter . . . . . . . . . . . . . 295
Chapter 12 352

12.1 Matter-vacuum transition in the Friedmann-Lemaître model . 352
12.2 Event horizons in de Sitter universe models . . . . . . . . . . . 352
12.3 Light travel time . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.4 Superluminal expansion . . . . . . . . . . . . . . . . . . . . . . 353
12.5 Flat universe model with radiation and vacuum energy . . . . 353
12.6 Creation of radiation and ultra-relativistic gas at the end of the
inflationary era . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.7 Universe models with Lorentz invariant vacuum energy (LIVE).353
12.8 Cosmic strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.9 Phantom Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
12.10 Velocity of light in the Milne universe . . . . . . . . . . . . . . . 356
12.11 Universe model with dark energy and cold dark matter . . . . 357
12.12 Luminosity-redshift relations . . . . . . . . . . . . . . . . . . . . 357
12.13 Cosmic time dilation . . . . . . . . . . . . . . . . . . . . . . . . . 357
12.14 Chaplygin gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
12.15 The perihelion precession of Mercury and the cosmological
constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Chapter 13 371
13.1 The wonderful properties of the Kasner exponents . . . . . . . 371
13.2 Dynamical systems approach to a universe with bulk viscous
pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
13.3 Murphy’s bulk viscous model . . . . . . . . . . . . . . . . . . . 372
Chapter 14 397
14.1 FRW universes with and without singularities . . . . . . . . . . 397
14.2 A magnetic Bianchi type I model . . . . . . . . . . . . . . . . . . 398
14.3 FRW universe with a scalar field . . . . . . . . . . . . . . . . . . 399
14.4 The Kantowski-Sachs universe model . . . . . . . . . . . . . . . 399
Chapter 15 421
15.1 A Bianchi type II universe model . . . . . . . . . . . . . . . . . 421
15.2 A homogeneous plane wave . . . . . . . . . . . . . . . . . . . . 422

15.3 Vacuum dominated Bianchi type V universe model . . . . . . . 423
15.4 The exceptional case, VI
∗
−1/9
. . . . . . . . . . . . . . . . . . . . 423
xii List of Problems
15.5 Symmetries of hyperbolic space . . . . . . . . . . . . . . . . . . 424
15.6 The matrix group SU(2) is the sphere S
3
. . . . . . . . . . . . . 424
Chapter 16 438
16.1 Energy equation for a shell of dust . . . . . . . . . . . . . . . . . 438
16.2 Charged shell of dust . . . . . . . . . . . . . . . . . . . . . . . . 438
16.3 A spherical domain wall . . . . . . . . . . . . . . . . . . . . . . 438
16.4 Dynamics of spherical domain walls . . . . . . . . . . . . . . . 438
Chapter 17 462
17.1 Domain wall brane universe models . . . . . . . . . . . . . . . 462
17.2 A brane without Z
2
-symmetry . . . . . . . . . . . . . . . . . . . 463
17.3 Warp factors and expansion factors for bulk and brane domain
walls with factorizable metric functions . . . . . . . . . . . . . . 463
17.4 Solutions with variable scale factor in the fifth dimension . . . 464
Chapter 18 483
18.1 A five-dimensional vacuum universe . . . . . . . . . . . . . . . 483
18.2 A five-dimensional cosmological constant . . . . . . . . . . . . 484
18.3 Homotheties and Self-similarity . . . . . . . . . . . . . . . . . . 484
18.4 Conformal flatness for three-manifolds . . . . . . . . . . . . . . 484
List of Examples
1.1 Tidal forces on two particles . . . . . . . . . . . . . . . . . . . . 10

1.2 Flood and ebb on the Earth . . . . . . . . . . . . . . . . . . . . . 11
1.3 A tidal force pendulum . . . . . . . . . . . . . . . . . . . . . . . 12
3.1 Tensor product between two vectors . . . . . . . . . . . . . . . 55
3.2 Tensor-components . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Exterior product and vector product . . . . . . . . . . . . . . . . 60
4.1 Transformation between plane polar-coordinates and Carte-
sian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 The coordinate basis vector field of plane polar coordinates . . 67
4.3 The velocity vector of a particle moving along a circular path . 68
4.4 Transformation of coordinate basis vectors and vector compo-
nents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5 Some transformation matrices . . . . . . . . . . . . . . . . . . . 69
4.6 The line-element of flat 3-space in spherical coordinates . . . . 75
4.7 Basis vector field in a system of plane polar coordinates . . . . 76
4.8 Velocity field in plane polar coordinates . . . . . . . . . . . . . 76
4.9 Structure coefficients of an orthonormal basis field associated
with plane polar coordinates . . . . . . . . . . . . . . . . . . . . . 77
4.10 Spherical coordinates in Euclidean 3-space . . . . . . . . . . . . 82
5.1 Vertical free motion in a uniformly accelerated reference frame 100
5.2 The path of a photon in uniformly accelerated reference frame 102
6.1 Exterior differentiation in 3-space. . . . . . . . . . . . . . . . . . 110
6.2 Not all closed forms are exact . . . . . . . . . . . . . . . . . . . . 116
6.3 The surface area of the sphere . . . . . . . . . . . . . . . . . . . 117
6.4 The Electromagnetic Field outside a static point charge . . . . . 118
6.5 Gauss’ integral theorem . . . . . . . . . . . . . . . . . . . . . . . 119
6.6 The Christoffel symbols for plane polar coordinates . . . . . . . 125
6.7 The acceleration of a particle as expressed in plane polar coor-
dinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.8 The acceleration of a particle relative to a rotating reference
frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.9 The rotation coefficients of an orthonormal basis field attached
to plane polar coordinates . . . . . . . . . . . . . . . . . . . . . . 132
6.10 Curl in spherical coordinates . . . . . . . . . . . . . . . . . . . . 135
6.11 The divergence of a vector field . . . . . . . . . . . . . . . . . . 142
6.12 2-dimensional Symmetry surfaces . . . . . . . . . . . . . . . . . 145
7.1 The curvature of a circle . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 The curvature of a straight circular cone . . . . . . . . . . . . . 161
9.1 Gravitational radiation emitted by a binary star . . . . . . . . . 203
10.1 Time delay of radar echo . . . . . . . . . . . . . . . . . . . . . . 218
10.2 The Hafele-Keating experiment . . . . . . . . . . . . . . . . . . 220
10.3 The Lense-Thirring effect . . . . . . . . . . . . . . . . . . . . . . 237
11.1 The temperature in the radiation dominated epoch . . . . . . . 274
xiv List of Examples
11.2 The redshift of the cosmic microwave background . . . . . . . 274
11.3 Age-redshift relation in the Einstein-de Sitter universe . . . . . 276
11.4 Redshift-luminosity relations for some universe models . . . . 285
11.5 Particle horizon for some universe models . . . . . . . . . . . . 288
12.1 The particle horizon of the de Sitter universe . . . . . . . . . . . 299
12.2 Polynomial inflation . . . . . . . . . . . . . . . . . . . . . . . . . 308
12.3 Transition from deceleration to acceleration for our universe . 316
12.4 Universe model with Chaplygin gas . . . . . . . . . . . . . . . . 325
12.5 Third order luminosity redshift relation . . . . . . . . . . . . . . 326
12.6 The velocity of sound in the cosmic plasma . . . . . . . . . . . 336
14.1 A coordinate singularity . . . . . . . . . . . . . . . . . . . . . . 383
14.2 An inextendible non-curvature singularity . . . . . . . . . . . . 383
14.3 Canonical formulation of the Bianchi type I universe model . . 392
15.1 The Lie Algebra so(3) . . . . . . . . . . . . . . . . . . . . . . . . 403
15.2 The Poincaré half-plane . . . . . . . . . . . . . . . . . . . . . . . 406
15.3 A Kantowski-Sachs universe model . . . . . . . . . . . . . . . . 410
15.4 The Bianchi type V universe model . . . . . . . . . . . . . . . . 415

15.5 The Lie algebra of Sol . . . . . . . . . . . . . . . . . . . . . . . . 417
15.6 Lens spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
15.7 The Seifert-Weber Dodecahedral space . . . . . . . . . . . . . . 420
16.1 A source for the Kerr field . . . . . . . . . . . . . . . . . . . . . 436
18.1 Hyperbolic space is conformally flat . . . . . . . . . . . . . . . . 475
18.2 Homotheties for the Euclidean plane . . . . . . . . . . . . . . . 477
“Paradoxically, physicists claim that gravity is
the
weakest
of the fundamental forces.”
Prof. Hallstein Høgåsen– after having fallen
from a ladder and breaking both his arms
Preface
Many of us have experienced the same, fallen and broken something. Yet,
supposedly, gravity is the weakest of the fundamental forces. It is claimed to
be 10
−15
times weaker than electromagnetism. But still, every one of us have
more or less a personal relationship with gravity. Gravity is something which
we have to consider every day. Whenever we loose something on the floor
and whenever we pour something in a cup, gravity is an active participant.
Hadn’t it been for gravity, we could not have done anything of the above.
Thus gravity is part of our everyday-life.
This is basically what this book is about; gravity. We will try to convey the
concepts of gravity to the reader as Albert Einstein saw it. Einstein saw upon
gravity as nobody else before him had seen it. He saw upon gravity as curved
spaces, four-dimensional manifolds and geodesics. All of these concepts will
be presented in this book.
The book offers a rigorous introduction to Einstein’s general theory of rel-
ativity. We start out from the first principles of relativity and present to Ein-

stein’s theory in a self-contained way.
After introducing Einstein’s field equations, we go onto the most impor-
tant chapter in this book which contains the three classical tests of the theory
and introduces the notion of black holes. Recently, cosmology has also proven
to be a very important testing arena f or the general theory of relativity. We
have thus devoted a large part to this subject. We introduce the simplest mod-
els decribing an evolving universe. In spite of their simpleness they can say
quite a lot about the universe we live in. We include the cosmological con-
stant and explain in detail the “standard model” in cosmology. After the main
issues have been presented we introduce an anisotropic universe model and
explain some of it features. Unless one just accepts the cosmological princi-
ples as a fact, one is unavoidably led to the study of such anisotropic universe
models. As an introductory course in general relativity, it is suitable to stop
after finishing the chapters with cosmology.
For the more experienced reader, or for people eager to learn more, we
have included a part called “Advanced Topics”. These topics have been cho-
sen by the authors because they present topics that are important and that
have not been highlighted elsewhere in textbooks. Some of them are on the
very edge of research, others are older ideas and topics. In particular, the last
two chapters deal with Einstein gravity in five dimensions which has been a
hot topic of research the recent years.
All of the ideas and matters presented in this book have one thing in com-
mon: they are all based on Einstein’s classical idea of gravity. We have not
considered any quantum mechanics in our presentation, with one exception:
black hole thermodynamics. Black hole thermodynamics is a quantum fea-
ture of black holes, but we chose to include it because the study of black holes
would have been incomplete without it.
There are several people who we wish to thank. First of all, we would
xvi List of Examples
like to thank Finn Ravndal who gave a thorough introduction to the theory of

relativity in a series of lectures during the late seventies. This laid the founda-
tion for further activity in this field at the University of Oslo. We also want to
thank Ingunn K. Wehus and Peter Rippis for providing us with a copy of their
theses [Weh01, Rip01], and to Svend E. Hjelmeland for computerizing some
of the notes in the initial stages of this book. Furthermore, the kind efforts
of Kevin Reid, Jasbir Nagi, James Lucietti, Håvard Alnes, Torquil MacDonald
Sørensen who read through the manuscript and pointed out to us numerous
errors, typos and grammatical blunders, are gratefully acknowledged.
ØYVIND GRØN
Oslo, Norway
SIGBJØRN HERVIK
Cambridge, United Kingdom
Notation
We have tried to be as homogeneous as possible when it comes to notation
in this book. There are some exceptions, but as a general rule we use the
following notation.
Because of the large number of equations, the most important equations
are boxed, like this:
E = mc
2
.
All tensors, including vectors and forms, are written in bold typeface. A gen-
eral tensor usually has a upper case letter, late in the alphabet. T is a typical
tensor. Vectors, are usually written in two possible ways. If it is more natural
to associate the vector as a tangent vector of some curve, then we usually use
lower case bold letters like u or v. If the vectors are more naturally associated
with a vector field, then we use upper case bold letters, like A or X. How-
ever, naturally enough, this rule is the most violated concerning the notation
in this book. Forms have Greek bold letters, i.e. ω is typical form. All the
components of tensors, vectors and forms, have ordinary math italic fonts.

Matrices are written in sans serif, i.e. like M. The determinants are written
in the usual math style: det(M) = M. A typical example is the metric tensor,
g. In the following notation we have:
g : The metric tensor itself.
g
µν
: The components of the metric tensor.
g : The matrix made up of g
µν
.
g : The determinant of the metric tensor, g = det(g).
The metric tensor comes in many guises, each one is useful for different pur-
poses.
Also, for the signature of the metric tensor, the (− + ++)-convention is
used. Thus the time direction has a − while the spatial directions all have +.
The abstract index notation
One of the most heavily used notation, both in this book and in the physics
literature in general, is the abstract index notation. So it is best that we get
this sorted out as early as possible. As a general rule, repeated indices means
summation! For example,
α
µ
β
µ
≡

µ
α
µ
β

µ
where the sum is over the range of the index µ. Furthermore, the type of index,
can make a difference. Greek indices usually run over the spacetime manifold,
starting with 0 as the time component. Latin indices are usually associated to
xviii Notation
a hypersurface or the spatial geometry. They start with 1 and run up to the
dimension of the manifold. Hence, if we are in the usual four-dimensional
space-time, then µ = 0, , 3, while i = 1, , 3. But no rule without exceptions,
also this rule is violated occasionally. Also, indices inside square brackets,
means the antisymmetrical combination, while round brackets means sym-
metric part. For example,
T
[µν]
≡
1
2
(T
µν
− T
νµ
)
T
(µν)
≡
1
2
(T
µν
+ T
νµ

) .
Whenever we write the indices between two vertical lines, we mean that the
indices shall be well ordered. For a set, µ
1
µ
2
µ
p
, to be well ordered means
that µ
1
≤ µ
2
≤ ≤ µ
p
. Thus an expression like,
T
µν
S
|µν|
means that we shall only sum over indices where µ ≤ ν. We usually use this
notation when S
|µν|
is antisymmetric, which avoids the over-counting of the
linearly dependent components.
The following notation is also convenient to get straight right away. Here,
A
µ ν
is an arbitrary tensor (it may have indices upstairs as well).
e

α
(A
µ ν
) = A
µ ν,α
Partial derivative
∇
α
A
µ ν
= A
µ ν;α
Covariant derivative
£
X
Lie derivative with respect to X
d Exterior derivative operator
d
†
Codifferential operator
⋆ Hodge’s star operator
 Covariant Laplacian
⊗ Tensor product
∧ Wedge product, or exterior product
Part I
INTRODUCTION:
NEWTONIAN PHYSICS AND
SPECIAL RELATIVITY

1

Relativity Principles and Gravitation
To obtain a mathematical description of physical phenomena, it is advanta-
geous to introduce a reference frame in order to map the position of events in
space and time. The choice of reference frame has historically depended upon
the view of human beings of their position in the Universe.
1.1 Newtonian mechanics
When describing physical phenomena on Earth, it is natural to use a coordi-
nate system with origin at the center of the Earth. This coordinate system is,
however, not ideal for the description of the motion of the planets around the
Sun. A coordinate system with origin at the center of the Sun is more natural.
Since the Sun moves around the center of the galaxy, there is nothing special
about a coordinate system with origin at the Sun’s center. This argument can
be continued ad infinitum.
The fundamental reference frame of Newton is called ‘absolute space’. The
geometrical properties of this space are characterized by ordinary Euclidean
geometry. This space can be covered by a Cartesian coordinate system. A
non-rotating reference frame at rest, or moving uniformly in absolute space
is called a Galilean reference frame. With chosen origin and orientation, the
system is fixed. Newton also introduced a universal time which proceeds at
the same rate at all positions in space.
Relative to a Galilean reference frame, all mechanical systems behave ac-
cording to Newton’s three laws.
Newton’s 1st law: Free particles move with constant velocity
u =
dr
dt
= constant
where r is a position vector.
4 Relativity Principles and Gravitation
Newton’s 2nd law: The acceleration a = du/dt of a particle is proportional

to the force F acting on it
F = m
i
du
dt
(1.1)
where m
i
is the inertial mass of the particle.
Newton’s 3rd law: If particle 1 acts on particle 2 with a force F
12
, then 2 acts
on 1 with a force
F
21
= −F
12
.
The first law can be considered as a special case of the second with F =
0. Alternatively, the first law can be thought of as restricting the reference
frame to be non-accelerating. This is presupposed for the validity of Newton’s
second law. Such reference frames are called inertial frames.
1.2 Galilei–Newton’s principle of Relativity
Let Σ be a Galilean reference frame, and Σ
′
another Galilean frame moving
relative to Σ with a constant velocity v (see Fig. 1.1).
Figure 1.1: Relative translational motion
We may think of a reference frame as a set of reference particles with given
motion. A comoving coordinate syste m in a reference frame is a system in which

the reference particles of the frame have constant spatial coordinates.
Let (x, y, z) be the coordinates of a comoving system in Σ, and (x
′
, y
′
, z
′
)
those of a comoving system in Σ
′
. The reference frame Σ moves relative to Σ
′
with a constant velocity v along the x-axis. A point with coordinates (x, y, z)
in Σ has coordinates
x
′
= x − vt, y
′
= y, z
′
= z (1.2)
in Σ
′
, or
r
′
= r − v t. (1.3)
An event at an arbitrary point happens at the same time in Σ and Σ
′
,

t
′
= t. (1.4)
The space coordinate transformations (1.2) or (1.3) with the trivial time trans-
formation (1.4) are called the Galilei-transformations.
If the velocity of a particle is u in Σ, then it moves with a velocity
u
′
=
dr
′
dt
= u − v (1.5)
1.3 The principle of Relativity 5
in Σ
′
.
In Newtonian mechanics one assumes that the inertial mass of a body is
independent of the velocity of the body. Thus the mass is the same in Σ as in
Σ
′
. Then the force F
′
, as measured in Σ
′
, is
F
′
= m
i

du
′
dt
′
= m
i
du
dt
= F. (1.6)
The force is the same in Σ
′
as in Σ. This result may be expressed by saying that
Newton’s 2nd law is invariant under a Galilei transformation; it is written in
the same way in every Galilean reference frame.
All reference frames moving with constant velocity are Galilean, so New-
ton’s laws are valid in these frames. Every mechanical system will therefore
behave in the same way in all Galilean frames. This is the Galilei–Newton prin-
ciple of relativity.
It is difficult to find Galilean frames in our world. If, for example, we
place a reference frame on the Earth, we must take into account the rotation
of the Earth. This reference frame is rotating, and is therefore not Galilean.
In such non-Galilean reference frames free particles have accelerated motion.
In Newtonian dynamics the acceleration of free particles in rotating reference
frames is said to be due to the centrifugal force and the Coriolis force. Such
forces, that vanish by transformation to a Galilean reference frame, are called
‘fictitious forces’.
A simple example of a non-inertial reference frame is one that has a con-
stant acceleration a. Let Σ
′
be such a frame. If the position vector of a particle

is r in Σ, then its position vector in Σ
′
is
r
′
= r −
1
2
at
2
(1.7)
where it is assumed that Σ
′
was instantaneously at rest relative to Σ at the
point of time t = 0. Newton’s 2nd law is valid in Σ, so that a particle which is
acted upon by a force F in Σ can be described by the equation
F = m
i
d
2
r
dt
2
= m
i

d
2
r
′

dt
2
+ a

. (1.8)
If this is written as
F
′
= F − m
i
a = m
i
d
2
r
′
dt
2
(1.9)
we may formally use Newton’s 2nd law in the non-Galilean frame Σ
′
. This
is obtained by a sort of trick, namely by letting the fictitious force act on the
particle in addition to the ordinary forces that appear in a Galilean frame.
1.3 The principle of Relativity
At the beginning of this century Einstein realised that Newton’s absolute space
is a concept without physical content. This concept should therefore be re-
moved from the description of the physical world. This conclusion is in accor-
dance with the negative result of the Michelson–Morley experiment [MM87].
In this experiment one did not succeed in measuring the velocity of the Earth

through the so-called ‘ether’, which was thought of as a ‘materialization’ of
Newton’s absolute space.
6 Relativity Principles and Gravitation
However, Einstein retained, in his special theory of relativity, the Newto-
nian idea of the privileged observers at rest in Galilean frames that move with
constant velocities relative to each other. Einstein did, however, extend the
range of validity of the equivalence of all Galilean frames. While Galilei and
Newton had demanded that the laws of mechanics are the same in all Galilean
frames, Einstein postulated that all the physical laws governing the behavior of the
material world can be formulated in the same way in all Galilean frames. This is Ein-
stein’s special principle of relativity. (Note that in the special theory of relativity
it is usual to call the Galilean frames ‘inertial frames’. However in the gen-
eral theory of relativity the concept ‘inertial frame’ has a somewhat different
meaning; it is a freely falling frame. So we will use the term Galilean frames
about the frames moving relative to each other with constant velocity.)
Applying the Galilean coordinate transformation to Maxwell’s electromag-
netic theory, one finds that Maxwell’s equations are not invariant under this
transformation. The wave-equation has the standard form, with isotropic ve-
locity of electromagnetic waves, only in one ‘preferred’ Galilean frame. In
other frames the velocity relative to the ‘preferred’ frame appears. Thus Max-
well’s electromagnetic theory does not fulfil Galilei–Newton’s principle of rel-
ativity. The motivation of the Michelson–Morley experiment was to measure
the velocity of the Earth relative to the ‘preferred’ frame.
Einstein demanded that the special principle of relativity should be valid
also for Maxwell’s electromagnetic theory. This was obtained by replacing the
Galilean kinematics by that of the special theory of relativity (see Ch. 2), since
Maxwell’s equations and Lorentz’s force law is invariant under the Lorentz
transformations. In particular this implies that the velocity of electromagnetic
waves, i.e. of light, is the same in all Galilean frames, c = 299 792.5 km/s ≈
3.00 ×10

8
m/s.
1.4 Newton’s law of Gravitation
Until now we have neglected gravitational forces. Newton found that the
force between two point masses M and m at a distance r is given by
F = −G
Mm
r
3
r.
(1.10)
This is Newton’s law of gravitation. Here G is Newton’s gravitational constant,
G = 6.67 × 10
−11
m
3
/kg s
2
. The gravitational force on a point mass m at a
position r due to many point masses M
1
, M
2
, . . . , M
n
at positions r
′
1
, r
′

2
, . . . , r
′
n
is given by the superposition
F = −mG
n

i=1
M
i
|r −r
′
i
|
3
(r −r
′
i
). (1.11)
A continuous distribution of mass with density ρ(r
′
) so that dM = ρ(r
′
)d
3
r
′
thus gives rise to a gravitational force at P (see Fig. 1.2)
F = −mG


ρ(r)
r −r
′
|r −r
′
|
3
d
3
r
′
. (1.12)
Here r
′
is associated with positions in the mass distribution, and r with the
position P where the gravitational field is measured.
1.4 Newton’s law of Gravitation 7
Figure 1.2: Gravitational field from a continuous mass distribution.
The gravitational potential φ(r) at the field point P is defined by
F = −m∇φ(r). (1.13)
Note that the ∇ operator acts on the coordinates of the field point, not of the
source point.
Calculating φ(r) from Eq. (1.12) it will be useful to introduce Einstein’s sum-
mation convention. For arbitrary a and b one has
a
j
b
j
≡

n

j=1
a
j
b
j
(1.14)
where n is the range of the indices j.
We shall also need the Kronecker symbol defined by
δ
i
j
=

1 when i = j
0 when i = j.
(1.15)
The gradient of |r − r
′
|
−1
may now be calculated as follows
∇
1
|r −r
′
|
= e
i

∂
∂x
i

(x
j
− x
j
′
)(x
j
− x
j
′
)

−1/2
= −e
i
(x
j
− x
j
′
)
∂x
j
∂x
i


(x
j
− x
j
′
)(x
j
− x
j
′
)

3/2
= −e
i
(x
j
− x
j
′
)δ
i
j
|r −r
′
|
3
= −
(x
i

− x
i
′
)e
i
|r −r
′
|
3
= −
(r −r
′
)
|r −r
′
|
3
. (1.16)
Comparing with Eqs. (1.12) and (1.13) we see that
φ(r) = −G

ρ(r
′
)
1
|r −r
′
|
d
3

r
′
. (1.17)
When characterizing the mass distribution of a point mass mathematically,
it is advantageous to use Dirac’s δ-function. This function is defined by the
following requirements
δ(r − r
′
) = 0, r
′
= r (1.18)
and

V
f(r)δ(r −r
′
)d
3
r
′
=

f(r), r
′
= r is inside V
0, r
′
= r is outside V.
(1.19)