| Preface | p. xi |
| Geometry and Physics: An Overview | p. 1 |
| Geometry | p. 1 |
| Special relativity | p. 4 |
| General relativity | p. 7 |
| Local, global and infinitesimal | p. 11 |
| The Background Manifold Structure | p. 14 |
| Topological spaces | p. 14 |
| Maps | p. 15 |
| Coordinate neighbourhoods | p. 17 |
| Differentiable manifolds | p. 19 |
| Maps of manifolds | p. 20 |
| The tangent space | p. 21 |
| Bases in the tangent space | p. 24 |
| Transformation properties of vector components | p. 25 |
| The tangent map | p. 27 |
| The cotangent space | p. 28 |
| Bases in the cotangent space | p. 29 |
| The dual tangent map | p. 31 |
| Tensors | p. 32 |
| Symmetry operations on tensors | p. 37 |
| The metric tensor | p. 39 |
| Raising and lowering of tensor indices | p. 47 |
| Alternating tensors | p. 48 |
| Exterior algebra | p. 51 |
| Measure of lengths and the world-function | p. 55 |
| Differentiation | p. 57 |
| Tensor fields and congruences | p. 57 |
| The Lie derivative | p. 62 |
| The connector | p. 68 |
| Parallel propagation and geodesics | p. 72 |
| Transformation properties of the connector | p. 77 |
| The covariant derivative | p. 78 |
| Torsion and normal coordinates | p. 82 |
| Compatibility of the metric with the connection | p. 84 |
| Parallelism | p. 85 |
| Applications of the covariant derivative | p. 86 |
| The exterior derivative | p. 88 |
| Frobenius theorems | p. 90 |
| Isometries on M | p. 99 |
| The Curvature | p. 102 |
| The Riemann tensor | p. 102 |
| Symmetry properties of the Riemann tensor and the Gaussian curvature | p. 106 |
| Significance of a curvature tensor vanishing everywhere | p. 109 |
| The Ricci tensor, the curvature scalar, the Weyl tensor | p. 111 |
| The Bianchi identities | p. 114 |
| The equation of geodesic deviation | p. 117 |
| The covariant derivative of the world-function | p. 120 |
| Maximally symmetric spaces | p. 125 |
| Space-time and Tetrad Formalism | p. 129 |
| The space-time manifold and the physical observer | p. 129 |
| Construction of a tetrad | p. 133 |
| Relations among tetrads and the Lorentz group | p. 135 |
| The propagation laws for tetrads | p. 138 |
| The Ricci rotation coefficients | p. 139 |
| Differential operators related to a tetrad frame | p. 141 |
| Spinors and the Classification of the Weyl Tensor | p. 146 |
| Outline | p. 146 |
| The group SL (2, C) | p. 146 |
| Lie algebras | p. 151 |
| Bivector algebra | p. 155 |
| Spinors | p. 159 |
| The spinor connection | p. 166 |
| The spinor curvature | p. 168 |
| The torsion case | p. 172 |
| Conformal spinors | p. 174 |
| The Weyl spinor and the Petrov classification | p. 177 |
| Coupling Between Fields and Geometry | p. 185 |
| Newtonian fluids | p. 185 |
| Generalization to special relativity | p. 187 |
| Coupling between fields and geometry: the field action | p. 188 |
| The gravitational action and the Einstein equations | p. 191 |
| The energy-momentum tensor of a perfect fluid | p. 195 |
| The energy-momentum tensor of a single particle | p. 198 |
| The energy-momentum tensor of the electromagnetic field | p. 200 |
| The energy-momentum pseudotensor | p. 201 |
| Dynamics on Curved Manifolds | p. 206 |
| Conservation laws | p. 206 |
| The equations of motion of an extended body | p. 211 |
| The centre-of-mass description | p. 216 |
| Motion of a point particle | p. 221 |
| Constants of motion | p. 225 |
| Maxwell's equations for a free electromagnetic field | p. 227 |
| Maxwell's equations in the presence of charges and currents | p. 228 |
| The radiation field | p. 231 |
| The light cone | p. 236 |
| Stationary space-times | p. 238 |
| The geometry of stationary null surfaces | p. 244 |
| Geometry of Congruences | p. 250 |
| Tetrad decomposition of the Riemann tensor | p. 250 |
| The expansion equation | p. 255 |
| The vorticity equation | p. 258 |
| The Einstein equations in tetrad form | p. 259 |
| The geometry of null rays | p. 261 |
| Singularities | p. 267 |
| Physical Measurements in Space-time | p. 274 |
| The concept of measurement | p. 274 |
| The measurement of time intervals and space distances | p. 275 |
| Measurements of angles | p. 281 |
| Curvature effects in the measurement of angles | p. 282 |
| Measurement of frequency | p. 286 |
| Measurement of relative velocities | p. 287 |
| The velocity composition law | p. 295 |
| Energy and momentum of a particle | p. 297 |
| Measurement of electric and magnetic fields | p. 298 |
| The properties of a fluid | p. 300 |
| The equations of motion of a fluid | p. 304 |
| The small curvature limit | p. 306 |
| Gravitational radiation | p. 314 |
| Spherically Symmetric Solutions | p. 320 |
| The spherically symmetric line element | p. 320 |
| The external Schwarzschild solution | p. 322 |
| The internal Schwarzschild solution | p. 326 |
| The global structure of spherically symmetric space-times | p. 329 |
| The extended external Schwarzschild solution | p. 332 |
| Penrose diagrams | p. 338 |
| Time-like geodesics in the external Schwarzschild solution | p. 342 |
| The precession of the apsidal points | p. 347 |
| The plunging-in observer | p. 349 |
| Null geodesics in the external Schwarzschild solution | p. 351 |
| The bending of light rays | p. 354 |
| The Reissner-Nordstrom solution | p. 358 |
| The extended Reissner-Nordstrom solution | p. 361 |
| Particle behaviour near the Reissner-Nordstrom singularity | p. 369 |
| Homogeneous and isotropic cosmology | p. 372 |
| The Friedmann solutions | p. 378 |
| Cosmological effects | p. 384 |
| Axially Symmetric Solutions | p. 389 |
| The axially symmetric line element: the canonical form | p. 389 |
| The Kerr solution | p. 392 |
| Physical interpretation of the Kerr metric | p. 397 |
| The space-time structure | p. 400 |
| Time-like geodesics | p. 406 |
| Rotationally induced effects | p. 408 |
| The angular geodesic equation | p. 413 |
| The equatorial circular geodesics | p. 414 |
| Null geodesics | p. 418 |
| The Kerr-Newman solution | p. 421 |
| The Weyl and the T-S solutions | p. 424 |
| Hawking radiation: an overview | p. 426 |
| Notation | p. 431 |
| References | p. 434 |
| Index | p. 441 |
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