Einstein's theory : a rigorous introduction to general relativity for the mathematically untrained

Cover......Page 1
Contents......Page 4
List of figures......Page 10
Preface by Arne Næss......Page 15
Preface by Øyvind Grøn......Page 20
1.1 Introduction......Page 24
1.2 Vectors as arrows......Page 25
1.3 Vector fields......Page 28
1.4 Calculus of vectors. Two dimensions......Page 33
1.5 Three and more dimensions......Page 45
1.6 The vector product......Page 51
1.7 Space and metric......Page 55
2.1 Differentiation......Page 63
2.2 Calculation of slopes of tangent lines......Page 72
2.3 Geometry of second derivatives......Page 75
2.4 The product rule......Page 76
2.5 The chain rule......Page 79
2.6 The derivative of a power function......Page 82
2.7 Differentiation of fractions......Page 84
2.8 Functions of several variables......Page 86
2.9 The MacLaurin and the Taylor series expansions......Page 94
3.1 Parametric description of curves......Page 103
3.2 Parametrization of a straight line......Page 108
3.3 Tangent vector fields......Page 111
3.4 Differential equations and Newton's 2. law......Page 116
3.5 Integration......Page 118
3.6 Exponential and logarithmic functions......Page 125
3.7 Integrating equations of motion......Page 131
4 Curvilinear coordinate systems......Page 135
4.1 Trigonometric functions......Page 137
4.2 Plane polar coordinates......Page 155
5.1 Basis vectors and dimension of space......Page 162
5.2 Space and spacetime......Page 166
5.3 Transformation of vector components......Page 169
5.4 The Galilean coordinate transformation......Page 176
5.5 Transformation of basis vectors......Page 181
5.6 Vector components......Page 183
5.7 Tensors......Page 186
5.8 The metric tensor......Page 191
5.9 Tensor components......Page 200
5.10 The Lorentz transformation......Page 205
5.11 The relativistic time dilation......Page 219
5.12 The line element......Page 222
5.13 Minkowski diagrams and light cones......Page 232
5.14 The spacetime interval......Page 236
5.15 The general formula for the line element......Page 246
5.16 Epistemological comment......Page 253
5.17 Kant or Einstein......Page 256
6 The Christoffel symbols......Page 265
6.1 Geometrical calculation......Page 267
6.2 Algebraic calculation......Page 280
6.3 Spherical coordinates......Page 285
6.4 Symmetry of the Christoffel symbols......Page 293
7 Covariant differentiation......Page 295
7.1 Variation of vector components......Page 296
7.2 The covariant derivative......Page 301
7.3 Transformation of covariant derivatives......Page 309
7.4 Covariant tensor components......Page 313
7.5 Connection expressed by the metric......Page 315
8 Geodesics......Page 320
8.1 Generalizing `flat space concepts'......Page 321
8.2 Parallel transport: unexpected difficulties......Page 323
8.3 Definition of parallel transport......Page 327
8.4 The general geodesic equation......Page 329
8.5 Local Cartesian coordinates......Page 334
9.1 The curvature of plane curves......Page 337
9.2 The curvature of surfaces......Page 342
9.3 Curl......Page 356
9.4 The Riemann curvature tensor......Page 361
10.1 Introduction......Page 372
10.2 Divergence......Page 375
10.3 The equation of continuity......Page 380
10.4 The stress tensor......Page 384
10.5 The net surface force acting on a fluid element......Page 388
10.6 The material derivative......Page 393
10.7 The equation of motion......Page 396
10.8 Four-velocity......Page 398
10.9 Newtonian energy-momentum of a perfect fluid......Page 403
10.10 The basic conservation laws......Page 407
10.11 Relativistic energy-momentum of a perfect fluid......Page 413
11.1 A new conception of gravitation......Page 415
11.2 The Ricci curvature tensor......Page 418
11.3 The Bianchi identity and Einstein's tensor......Page 426
12.1 Newtonian kinematics......Page 442
12.2 Forces......Page 444
12.3 Newton's theory of gravitation......Page 451
12.4 Special relativity and gravity......Page 453
12.5 The general theory of relativity......Page 463
12.6 The Newtonian limit of general relativity......Page 479
12.7 Repulsive gravitation......Page 489
12.8 `Geodesic postulate' and the field equations......Page 491
12.9 Constants of motion......Page 495
12.10 Conceptual structure of general relativity......Page 498
12.11 General relativity versus Newton's theory......Page 499
12.12 Epistemological comment......Page 502
13 Some applications......Page 505
13.1 Rotating reference frame......Page 507
13.2 The gravitational time dilation......Page 510
13.3 The Schwarzschild solution......Page 517
13.4 The Pound--Rebka experiment......Page 526
13.5 The Hafele--Keating experiment......Page 530
13.6 Mercury's perihelion precession......Page 535
13.7 Gravitational deflection of light......Page 543
13.8 Black holes......Page 548
14 Relativistic universe models......Page 561
14.1 Observations......Page 562
14.2 Homogeneous and isotropic models......Page 566
14.3 Einstein's cosmological field equations......Page 572
14.4 The Friedmann models......Page 577
14.5 The matter and radiation dominated periods......Page 587
14.6 Problems with the standard model......Page 591
14.7 Inflationary cosmology......Page 594
A The Laplacian in a spherical coordinate system......Page 605
B Ricci tensor of a static, spherically symmetric metric......Page 614
C Ricci tensor of the Robertson--Walker metric......Page 624
Bibliography......Page 631
Index......Page 633