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

650 Pages·2002·8.85 MB·English

by  Øyvind Grøn

Einstein’s Theory
a Rigorous Introduction for the
Mathematically Untrained
Øyvind Grøn and Arne Næss
2
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Ad Infinitum
∞
Einstein’s Theory
a Rigorous Introduction to General Relativity
for the Mathematically Untrained
by
Øyvind Grøn
Arne Næss
∞
Ad Infinitum
Oslo
Einstein’s Theory: a Rigorous Introduction to General
Relativity for the Mathematically Untrained
by
Øyvind Grøn and Arne Næss
Copyright (cid:13)c 2002 by Ad Infinitum AS, Oslo, Norway
www.adinfinitum.no
ISBN 82-92261-07-9
Contents
Preface by Arne Næss xv
Preface by Øyvind Grøn xx
1 Vectors 24
1.1 Introduction . . . . . . . . . . . . . . . 24
1.2 Vectors as arrows . . . . . . . . . . . . . 25
1.3 Vector fields . . . . . . . . . . . . . . . . 28
1.4 Calculus of vectors. Two dimensions . 33
1.5 Three and more dimensions . . . . . . 45
1.6 The vector product . . . . . . . . . . . . 51
1.7 Space and metric . . . . . . . . . . . . . 55
2 Differential calculus 63
2.1 Differentiation . . . . . . . . . . . . . . 63
2.2 Calculation of slopes of tangent lines . 72
2.3 Geometry of second derivatives . . . . 75
2.4 The product rule . . . . . . . . . . . . . 76
iv
Contents v
2.5 The chain rule . . . . . . . . . . . . . . 79
2.6 The derivative of a power function . . 82
2.7 Differentiation of fractions . . . . . . . 84
2.8 Functions of several variables . . . . . 86
2.9 The MacLaurin and the Taylor series ex-
pansions . . . . . . . . . . . . . . . . . . 94
3 Tangent vectors 103
3.1 Parametric description of curves . . . . 103
3.2 Parametrization of a straight line . . . 108
3.3 Tangent vector fields . . . . . . . . . . . 111
3.4 Differential equations and Newton’s 2.
law . . . . . . . . . . . . . . . . . . . . . 116
3.5 Integration . . . . . . . . . . . . . . . . 118
3.6 Exponential and logarithmic functions 125
3.7 Integrating equations of motion . . . . 131
4 Curvilinear coordinate systems 135
4.1 Trigonometric functions . . . . . . . . . 137
4.2 Plane polar coordinates . . . . . . . . . 155
5 The metric tensor 162
5.1 Basis vectors and dimension of space . 162
5.2 Space and spacetime . . . . . . . . . . . 166
5.3 Transformation of vector components . 169
5.4 The Galilean coordinate transformation 176
5.5 Transformation of basis vectors . . . . 181
5.6 Vector components . . . . . . . . . . . . 183
Contents vi
5.7 Tensors . . . . . . . . . . . . . . . . . . 186
5.8 The metric tensor . . . . . . . . . . . . . 191
5.9 Tensor components . . . . . . . . . . . 200
5.10 The Lorentz transformation . . . . . . . 205
5.11 The relativistic time dilation . . . . . . 219
5.12 The line element . . . . . . . . . . . . . 222
5.13 Minkowski diagrams and light cones . 232
5.14 The spacetime interval . . . . . . . . . 236
5.15 The general formula for the line element 246
5.16 Epistemological comment . . . . . . . . 253
5.17 Kant or Einstein . . . . . . . . . . . . . 256
6 The Christoffel symbols 265
6.1 Geometrical calculation . . . . . . . . . 267
6.2 Algebraic calculation . . . . . . . . . . 280
6.3 Spherical coordinates . . . . . . . . . . 285
6.4 Symmetry of the Christoffel symbols . 293
7 Covariant differentiation 295
7.1 Variation of vector components . . . . 296
7.2 The covariant derivative . . . . . . . . 301
7.3 Transformation of covariant derivatives 309
7.4 Covariant tensor components . . . . . 313
7.5 Connection expressed by the metric . . 315
8 Geodesics 320
8.1 Generalizing ‘flat space concepts’ . . . 321
Contents vii
8.2 Parallel transport: unexpected difficul-
ties . . . . . . . . . . . . . . . . . . . . . 323
8.3 Definition of parallel transport . . . . . 327
8.4 The general geodesic equation . . . . . 329
8.5 Local Cartesian coordinates . . . . . . . 334
9 Curvature 337
9.1 The curvature of plane curves . . . . . 337
9.2 The curvature of surfaces . . . . . . . . 342
9.3 Curl . . . . . . . . . . . . . . . . . . . . 356
9.4 The Riemann curvature tensor . . . . . 361
10 Conservation laws of classical mechanics 372
10.1 Introduction . . . . . . . . . . . . . . . 372
10.2 Divergence . . . . . . . . . . . . . . . . 375
10.3 The equation of continuity . . . . . . . 380
10.4 The stress tensor . . . . . . . . . . . . . 384
10.5 The net surface force acting on a fluid
element . . . . . . . . . . . . . . . . . . 388
10.6 The material derivative . . . . . . . . . 393
10.7 The equation of motion . . . . . . . . . 396
10.8 Four-velocity . . . . . . . . . . . . . . . 398
10.9 Newtonian energy-momentum of a per-
fect fluid . . . . . . . . . . . . . . . . . . 403
10.10 The basic conservation laws . . . . . . 407
10.11 Relativistic energy-momentum of a per-
fect fluid . . . . . . . . . . . . . . . . . . 413
Contents viii
11 Einstein’s field equations 415
11.1 A new conception of gravitation . . . . 415
11.2 The Ricci curvature tensor . . . . . . . 418
11.3 The Bianchi identity and Einstein’s ten-
sor . . . . . . . . . . . . . . . . . . . . . 426
12 Einstein’s theory of spacetime and gravitation 442
12.1 Newtonian kinematics . . . . . . . . . . 442
12.2 Forces . . . . . . . . . . . . . . . . . . . 444
12.3 Newton’s theory of gravitation . . . . . 451
12.4 Special relativity and gravity . . . . . . 453
12.5 The general theory of relativity . . . . . 463
12.6 The Newtonian limit of general relativity 479
12.7 Repulsive gravitation . . . . . . . . . . 489
12.8 ‘Geodesic postulate’ and the field equa-
tions . . . . . . . . . . . . . . . . . . . . 491
12.9 Constants of motion . . . . . . . . . . . 495
12.10 Conceptual structure of general relativity 498
12.11 General relativity versus Newton’s the-
ory . . . . . . . . . . . . . . . . . . . . . 499
12.12 Epistemological comment . . . . . . . . 502
13 Some applications 505
13.1 Rotating reference frame . . . . . . . . 507
13.2 The gravitational time dilation . . . . . 510
13.3 The Schwarzschild solution . . . . . . . 517
13.4 The Pound–Rebka experiment . . . . . 526
13.5 The Hafele–Keating experiment . . . . 530
Contents ix
13.6 Mercury’s perihelion precession . . . . 535
13.7 Gravitational deflection of light . . . . 543
13.8 Black holes . . . . . . . . . . . . . . . . 548
14 Relativistic universe models 561
14.1 Observations . . . . . . . . . . . . . . . 562
14.2 Homogeneous and isotropic models . . 566
14.3 Einstein’s cosmological field equations 572
14.4 The Friedmann models . . . . . . . . . 577
14.5 Thematterandradiationdominatedpe-
riods . . . . . . . . . . . . . . . . . . . . 587
14.6 Problems with the standard model . . 591
14.7 Inflationary cosmology . . . . . . . . . 594
A The Laplacian in a spherical coordinate system 605
B Ricci tensor of a static, spherically symmetric
metric 614
C Ricci tensor of the Robertson–Walker metric 624
Bibliography 631
Index 633
List of Figures
1.1 Vectors with different directions . . . . . . . 26
1.2 Vectors with different magnitudes . . . . . . 27
1.3 Parallel transportation of a vector . . . . . . 27
1.4 Wind velocity field over Europe on February
3, 1988 . . . . . . . . . . . . . . . . . . . . . . 30
1.5 Grid . . . . . . . . . . . . . . . . . . . . . . . 31
1.6 Basis vectors and vector components . . . . 34
1.7 Vector components . . . . . . . . . . . . . . 35
1.8 Vector addition . . . . . . . . . . . . . . . . . 38
1.9 Vector subtraction . . . . . . . . . . . . . . . 38
1.10 Decomposition of a vector . . . . . . . . . . 39
1.11 Vector addition . . . . . . . . . . . . . . . . . 41
1.12 Vector projection . . . . . . . . . . . . . . . . 42
1.13 Commutation of the dot product . . . . . . 43
1.14 Associative rule . . . . . . . . . . . . . . . . 44
1.15 Cartesian basis vectors in three dimensions 46
1.16 Vector components in three dimensions . . . 47
~ ~
1.17 The vector product of A and B . . . . . . . . 51
x