Lecture notes on general relativity

1.1 Introduction......Page 1
1.2 Newtonian gravity......Page 2
1.3 Gravity and SR......Page 4
2.1 Inertial coordinates......Page 8
2.2 4-vectors......Page 10
2.3 Tensors in Minkowski space......Page 11
3.1 Operations on tensors......Page 13
3.2 The energy-momentum tensor......Page 15
3.3 The continuity equation......Page 17
4.2 The metric......Page 18
5.1 Existence of local inertial coordinates......Page 23
5.2 Particle motion......Page 25
6.1 Transformation of the Christoffel symbols......Page 28
6.2 Manifolds......Page 29
6.3 Vectors and tensors......Page 30
6.4 Summary of the mathematical formulation......Page 32
7.1 Differentiation of vectors......Page 33
7.2 Parallel transport......Page 34
7.3 The wave equation......Page 35
8.1 Covariant derivatives of tensors......Page 37
8.2 Connections......Page 38
8.3 Curvature......Page 39
8.4 Symmetries of the Riemann tensor......Page 40
9.1 Bracket notation......Page 41
9.3 The operator D......Page 42
9.4 Geodesic deviation......Page 43
10.1 Tidal forces......Page 45
10.2 The weak field limit......Page 46
10.3 The non-vacuum case......Page 48
11.1 Spherical symmetry......Page 50
11.2 The curvature tensor......Page 51
12.2 Potential energy......Page 54
12.3 Photons......Page 55
12.4 Gravitational redshift......Page 56
12.5 Killing vectors......Page 57
13.1 Massive particles......Page 59
13.2 Comparison with the Newtonian theory......Page 60
13.3 Newtonian orbits and the relativistic correction......Page 61
13.4 The perihelion advance......Page 63
14.2 The phase portrait......Page 64
14.3 Photon orbits......Page 69
14.4 The bending of light......Page 70
15.2 Eddington--Finkelstein coordinates......Page 73
16.2 Kruskal coordinates......Page 82