Table of Contents
vii
1 Basics of geometry and relativity 1
1.1 Two dimensional geometry 1
1.2 Inertial and gravitational masses 13
1.3 Relativity 16
2 Relativistic dynamics 35
2.1 Relativistic point particle 35
2.2 Current and charge densities 46
2.3 Maxwell's equations in the presence of sources 51
2.4 Motion of a charged particle in EM field 54
2.5 Energy-momentum tensor 56
2.6 Angular momentum 61
3 Principle of general covariance 65
3.1 Principle of equivalence 65
3.2 Principle of general covariance 70
3.3 Tensor densities 76
4 Affine connection and covariant derivative 83
4.1 Parallel transport of a vector 83
4.2 Christoffel symbol 96
4.3 Covariant derivative of contravariant tensors 105
4.4 Metric compatibility 111
4.5 Covariant derivative of covariant and mixed tensors 115
4.6 Electromagnetic analogy 118
4.7 Gradient, divergence and curl 120
5 Geodesic equation 133
5.1 Covariant differentiation along a curve 133
5.2 Curvature from derivatives 135
5.3 Parallel transport along a closed curve 139
5.4 Geodesic equation 149
5.5 Derivation of geodesic equation from a Lagrangian 161
6 Applications of the geodesic equation 167
6.1 Geodesic as representing gravitational effect 167
6.2 Rotating coordinate system and the Coriolis force 172
6.3 Gravitational red shift 180
6.4 Twin paradox and general covariance 190
6.5 Other equations in the presence of gravitation 193
7 Curvature tensor and Einstein's equation 199
7.1 Curvilinear coordinates versus gravitational field 199
7.2 Definition of an inertial coordinate frame 199
7.3 Geodesic deviation 200
7.4 Properties of the curvature tensor 203
7.5 Einstein's equation 214
7.6 Cosmological constant 222
7.7 Initial value problem 223
7.8 Einstein's equation from an action 229
8 Schwarzschild solution 237
8.1 Line element 237
8.2 Connection 240
8.3 Solution of the Einstein equation 244
8.4 Properties of the Schwarzschild solution 257
8.5 Isotropic coordinates 260
9 Tests of general relativity 265
9.1 Radar echo experiment 265
9.2 Motion of a particle in a Schwarzschild background 267
9.2.1 Vertical free fall 273
9.2.2 Circular orbit 277
9.3 Motion of light rays in a Schwarzschild background 280
9.4 Perihelion advance of Mercury 287
10 Black holes 295
10.1 Singularities of the metric 296
10.2 Singularities of the Schwarzschild metric 301
10.3 Black holes 306
11 Cosmological models and the big bang theory 311
11.1 Homogeneity and isotropy 311
11.2 Different models of the universe 314
11.2.1 Close universe 314
11.2.2 Flat universe 315
11.2.3 Open universe 315
11.3 Hubble's law 316
11.4 Evolution equation 320
11.4.1 k = 1 325
11.4.2 k = 0 327
11.4.3 k = -1 328
11.5 Big bang theory and blackbody radiation 329
Index 335
