Lectures on General Relativity

FOREWORD ......Page 3
A. TRAUTMAN -- FOUNDATIONS AND CURRENT PROBLEMS OF GENERAL RELATIVITY ......Page 7
1.2 Gravitational and Inertial Masses ......Page 13
1.3 First Order Corrections to Particle Motion ......Page 16
1.4 Gravitational Radiation ......Page 19
1.5 Quantum Effects Connected with Gravitation ......Page 25
2.2 Algebraic Structures ......Page 31
2.3 Equivalence Relations and Morphisms ......Page 35
2.4 Vector Spaces ......Page 36
2.5 The Dual Space ......Page 39
2.7 Tensor Product of Vector Spaces ......Page 42
2.8 Multiforms and Multivectors ......Page 46
2.9 Orientation of Vector Spaces and Subspaces ......Page 49
2.10 Densities ......Page 52
2.11 Complexification of a Real vector Space ......Page 53
2.12 Euclidean Vector Spaces ......Page 54
3.1 Minkowski Vector Spaces ......Page 60
3.2 Lorentz Trans formations ......Page 62
3.3 Null Tetrad and the Special Linear Group SL(2,C) ......Page 63
3.4 Lorentz Trans formations and Bilinear Transformations of the Complex Plane ......Page 67
3.5 Spinors ......Page 69
4.1 Topological Spaces ......Page 73
4.2 Products of Topological Spaces ......Page 74
4.4 Differentiable Manifolds ......Page 75
4.5 Coordinate Transformations ......Page 77
4.7 Products of Differentiable Manifolds ......Page 79
4.8 Tangent Vectors ......Page 80
4.9 The Dragging Along of a Vector by a Differentiable Mapping ......Page 84
4.10 Vector Fields ......Page 85
4.11 Differential Forms ......Page 86
4.12 Tangent Tensors and Tensor Fields ......Page 89
4.13 Geometric Objects ......Page 90
4.14 One-Parameter Groups of Diffeomorphisms ......Page 93
4.16 Affine Connections on a Differentiable Manifold ......Page 95
4.17 The Torsion Tensor ......Page 100
4.18 The Curvature Tensor ......Page 101
4.19 Metric Tensor Fields. Riemannian Geometry ......Page 102
4.20 Integration in a Differentiable Manifold ......Page 104
5.1 Space-Time as a Differentiable Manifold ......Page 107
5.2 The Affine Connection in Physics ......Page 109
5.3 Newtonian Mechanics in the Absence of Gravitation ......Page 110
5.4 Newtonian Mechanics in the Presence of Gravitation ......Page 111
5.5 Axiomatic Formulation of the Geometry of Newtonian Gravitation Theory ......Page 112
5.6 The Interpretation of gab Privileged Classes of Coordinate Systems ......Page 121
5.7 Relativistic Theories of Space, Time, and Gravitation ......Page 124
6.1 The Principle of General Covariance ......Page 128
6.2 The Principle of Equivalence ......Page 129
6.3 Correspondence with Newtonian Theory ......Page 133
6.4 Stationary and Static Gravitational Fields ......Page 136
6.5 Propagation of Light in a Gravitational Field ......Page 140
6.6 Local Reference Frames and Fermi Transport ......Page 143
6.7 The Physical Distinction between Static and Stationary Space-Times ......Page 149
6.8 Evaluation of the Curvature Tensor at the Earth's Surface ......Page 151
6.9 The Gravitational Field Equations ......Page 154
6.10 The Perihelion Precession ......Page 157
7.1 Lie Groups ......Page 164
7.2 Lie Groups of Trans formations ......Page 168
7.3 Infinitesimal Coordinate Transformations ......Page 170
7.4 The Variational Principle ......Page 171
7.5 Conservation Laws in Riemannian Space-Time ......Page 182
7.6 Conservation Laws in General Relativity ......Page 188
7.7 Relation between Field Equations and Equations of Motion ......Page 195
7.8 The Motion of Particles in a Given Field ......Page 198
7.9 The Canonical Formalism for Relativistic Particles ......Page 202
8.2 Mathematical Formulation of the Criteria ......Page 207
8.3 Integration of the Isometry Condition in M 4 ......Page 213
8.4 The Isometry Condition in V_4 ......Page 219
8.5 The Born Condition in V_4 ......Page 221
8.6 The Herglotz-Noether Theorem ......Page 228
8.7 Dynamics ......Page 230
8.8 Integrability Conditions ......Page 232
9.1 Introduction ......Page 234
9.2 Newtonian Cosmology ......Page 236
9.3 Relativistic Cosmology ......Page 244
9.4 Propagation of Light in Relativistic Cosmology ......Page 246
9.5 Distance in Cosmology ......Page 248
9.6 The Event Horizon of the de Sitter Universe ......Page 250
9.7 Epilogue ......Page 254
F.A.E. PIRANI -- INTRODUCTION TO GRAVITATIONAL RADIATION THEORY ......Page 255
1.1 Introduction ......Page 259
1.2 Notation and Conventions ......Page 260
1.3 The Principle of Equivalence ......Page 262
1.4 Non-Local Experiments; the Curvature Tensor ......Page 268
1.5 Gravitational Wave-Fronts ......Page 277
2.1 Introduction ......Page 285
2.2 Linearized Theory; Gauge Transformations ......Page 286
2.3 Digression on Trace-Free Symmetric Tensors ......Page 291
2.4 The Electromagnetic Multipole ......Page 298
2.5 The Gravitational Multipole ......Page 308
3.2 Spinors and Tensors ......Page 313
3.3 Some Spinor Algebra ......Page 318
3.4 Spinor Equivalents of Certain Vectors and Tensors ......Page 320
3.5 Geometrical Interpretation of 1-Spinors ......Page 325
3.6 Canonical Decomposition of A Symmetric Spinor - the Petrov Classification ......Page 327
3.7 Spinor Analysis ......Page 332
3.8 Second Derivatives. Maxwell's Equations. Ricci and Bianchi Identities ......Page 334
4.2 Congruences of Null Geodesics ......Page 339
4.3 Propagation of Shadows ......Page 342
4.4 Null Frames and Null Rotations. The Optical Scalars ......Page 345
4.5 Spinor Equations for Congruences ......Page 350
4.6 Spin Frames and Spin Coefficients ......Page 354
5.1 Plane-Fronted Waves in Maxwell's Theory ......Page 361
5.2 Plane-Fronted Waves in Einstein's Theory ......Page 362
5.3 Propagation of Algebraically Special Fields ......Page 368
5.4 Asymptotic Solutions ......Page 373
5.5 Some Conclusions and Remarks ......Page 375
BIBLIOGRAPHY ......Page 377
H. BONDI -- SOME SPECIAL SOLUTIONS OF THE EINSTEIN EQUATIONS ......Page 383
1.2 Newton's Laws of Dynamics ......Page 387
1.3 The Concept of the Inertial Observer ......Page 389
1.4 The Copernican Controversy ......Page 390
1.5 The Principles of Relativity of Newton and Einstein ......Page 391
2.2 Measuring Devices and Measurements ......Page 394
2.3 The k-Factor in k-Calculus ......Page 395
2.4 The Clock Paradox ......Page 399
2.5 The Relativity of Simultaneity ......Page 402
2.6 Relativistic Dynamics ......Page 405
2.7 The Lorentz Transformation ......Page 406
2.8 The Role of Acceleration in Special Relativity ......Page 408
3.1 Review of Newtonian Gravitation Theory ......Page 415
3.2 Derivation and Interpretation of the Field Equations of General Relativity ......Page 417
3.3 Singularities in General Relativity ......Page 419
3.4 Negative Mass in General Relativity ......Page 421
3.5 Gravitational Red Shift ......Page 426
4. GRAVITATIONAL WAVES ......Page 428
5.2 Equilibrium Configuration of Spheres ......Page 443
5.3 Collapsing Spheres ......Page 454
BIBLIOGRAPHY ......Page 467