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| Preface | |
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| Geometry and Physics: An Overview | |
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| Geometry | |
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| Special relativity | |
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| General relativity | |
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| Local, global and infinitesimal | |
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| The Background Manifold Structure | |
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| Topological spaces | |
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| Maps | |
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| Coordinate neighbourhoods | |
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| Differentiable manifolds | |
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| Maps of manifolds | |
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| The tangent space | |
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| Bases in the tangent space | |
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| Transformation properties of vector components | |
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| The tangent map | |
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| The cotangent space | |
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| Bases in the cotangent space | |
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| The dual tangent map | |
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| Tensors | |
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| Symmetry operations on tensors | |
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| The metric tensor | |
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| Raising and lowering of tensor indices | |
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| Alternating tensors | |
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| Exterior algebra | |
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| Measure of lengths and the world-function | |
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| Differentiation | |
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| Tensor fields and congruences | |
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| The Lie derivative | |
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| The connector | |
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| Parallel propagation and geodesics | |
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| Transformation properties of the connector | |
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| The covariant derivative | |
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| Torsion and normal coordinates | |
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| Compatibility of the metric with the connection | |
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| Parallelism | |
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| Applications of the covariant derivative | |
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| The exterior derivative | |
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| Frobenius theorems | |
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| Isometries on M | |
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| The Curvature | |
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| The Riemann tensor | |
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| Symmetry properties of the Riemann tensor and the Gaussian curvature | |
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| Significance of a curvature tensor vanishing everywhere | |
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| The Ricci tensor, the curvature scalar, the Weyl tensor | |
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| The Bianchi identities | |
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| The equation of geodesic deviation | |
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| The covariant derivative of the world-function | |
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| Maximally symmetric spaces | |
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| Space-time and Tetrad Formalism | |
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| The space-time manifold and the physical observer | |
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| Construction of a tetrad | |
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| Relations among tetrads and the Lorentz group | |
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| The propagation laws for tetrads | |
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| The Ricci rotation coefficients | |
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| Differential operators related to a tetrad frame | |
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| Spinors and the Classification of the Weyl Tensor | |
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| Outline | |
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| The group SL (2, C) | |
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| Lie algebras | |
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| Bivector algebra | |
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| Spinors | |
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| The spinor connection | |
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| The spinor curvature | |
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| The torsion case | |
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| Conformal spinors | |
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| The Weyl spinor and the Petrov classification | |
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| Coupling Between Fields and Geometry | |
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| Newtonian fluids | |
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| Generalization to special relativity | |
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| Coupling between fields and geometry: the field action | |
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| The gravitational action and the Einstein equations | |
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| The energy-momentum tensor of a perfect fluid | |
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| The energy-momentum tensor of a single particle | |
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| The energy-momentum tensor of the electromagnetic field | |
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| The energy-momentum pseudotensor | |
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| Dynamics on Curved Manifolds | |
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| Conservation laws | |
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| The equations of motion of an extended body | |
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| The centre-of-mass description | |
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| Motion of a point particle | |
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| Constants of motion | |
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| Maxwell's equations for a free electromagnetic field | |
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| Maxwell's equations in the presence of charges and currents | |
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| The radiation field | |
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| The light cone | |
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| Stationary space-times | |
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| The geometry of stationary null surfaces | |
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| Geometry of Congruences | |
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| Tetrad decomposition of the Riemann tensor | |
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| The expansion equation | |
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| The vorticity equation | |
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| The Einstein equations in tetrad form | |
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| The geometry of null rays | |
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| Singularities | |
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| Physical Measurements in Space-time | |
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| The concept of measurement | |
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| The measurement of time intervals and space distances | |
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| Measurements of angles | |
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| Curvature effects in the measurement of angles | |
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| Measurement of frequency | |
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| Measurement of relative velocities | |
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| The velocity composition law | |
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| Energy and momentum of a particle | |
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| Measurement of electric and magnetic fields | |
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| The properties of a fluid | |
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| The equations of motion of a fluid | |
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| The small curvature limit | |
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| Gravitational radiation | |
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| Spherically Symmetric Solutions | |
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| The spherically symmetric line element | |
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| The external Schwarzschild solution | |
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| The internal Schwarzschild solution | |
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| The global structure of spherically symmetric space-times | |
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| The extended external Schwarzschild solution | |
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| Penrose diagrams | |
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| Time-like geodesics in the external Schwarzschild solution | |
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| The precession of the apsidal points | |
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| The plunging-in observer | |
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| Null geodesics in the external Schwarzschild solution | |
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| The bending of light rays | |
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| The Reissner-Nordstrom solution | |
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| The extended Reissner-Nordstrom solution | |
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| Particle behaviour near the Reissner-Nordstrom singularity | |
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| Homogeneous and isotropic cosmology | |
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| The Friedmann solutions | |
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| Cosmological effects | |
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| Axially Symmetric Solutions | |
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| The axially symmetric line element: the canonical form | |
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| The Kerr solution | |
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| Physical interpretation of the Kerr metric | |
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| The space-time structure | |
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| Time-like geodesics | |
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| Rotationally induced effects | |
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| The angular geodesic equation | |
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| The equatorial circular geodesics | |
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| Null geodesics | |
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| The Kerr-Newman solution | |
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| The Weyl and the T-S solutions | |
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| Hawking radiation: an overview | |
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| Notation | |
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| References | |
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| Index | |