| Description |
1 online resource (494 pages) |
| Contents |
Cover -- Half Title -- Title Page -- Copyright Page -- Table of Contents -- Preface -- Notation and conventions -- Chapter 1 Introduction -- 1.1 Non-Euclidean Geometries -- 1.1.1 The metric tensor in different coordinate frames -- 1.1.2 The Gaussian curvature -- 1.1.3 Pseudo-Euclidean geometries and spacetime -- 1.2 Newtonian Theory and Its Shortcomings -- 1.3 The Role of the Equivalence Principle -- 1.4 Geodesic Equations As a Consequence of the Equivalence Principle -- 1.5 Locally Inertial Frames -- Chapter 2 Elements of differential geometry -- 2.1 Topological Spaces, Mapping, Manifolds |
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2.1.1 Topological spaces -- 2.1.2 Mapping -- 2.1.3 Manifolds and differentiable manifolds -- 2.1.4 Dieomorphisms -- 2.2 Vectors -- 2.2.1 The traditional definition of a vector -- 2.2.2 A geometrical definition -- 2.3 One-Forms -- 2.3.1 One-forms as geometrical objects -- 2.3.2 Vector fields and one-form fields -- 2.4 Tensors -- 2.4.1 Geometrical definition of a tensor -- 2.4.2 Symmetries of a tensor -- 2.5 The Metric Tensor and Its Properties -- Chapter 3 Affine connection and parallel transport -- 3.1 The Covariant Derivative of Vectors -- 3.2 The Covariant Derivative of Scalars and One-Forms |
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3.3 Symmetries of Christoffel's Symbols -- 3.4 Transformation Rules for Christoffel's Symbols -- 3.5 The Covariant Derivative of Tensors -- 3.6 Christoffel's Symbols in Terms of the Metric Tensor -- 3.7 Parallel Transport -- 3.7.1 Parallel transport of a vector along a closed path on a two-sphere -- 3.8 Geodesic Equation -- 3.9 Fermi Coordinates -- 3.10 Non-Coordinate Bases -- Chapter 4 The curvature tensor -- 4.1 Parallel Transport Along a Loop -- 4.2 Symmetries of the Riemann Tensor -- 4.3 The Riemann Tensor Gives the Commutator of Covariant Derivatives -- 4.4 The Bianchi Identities |
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4.5 The Equation of Geodesic Deviation -- Chapter 5 The stress-energy tensor -- 5.1 The Stress-Energy Tensor in Flat Spacetime -- 5.2 Is Taß a Tensor? -- 5.3 Does Taß Satisfy a Conservation Law? -- 5.4 Is Taß -- ß= 0 a Conservation Law? -- Chapter 6 The Einstein equations -- 6.1 Geodesic Equations in the Weak-Field, Stationary Limit -- 6.2 Einstein's Field Equations -- 6.3 Gauge Invariance of Einstein's Equations -- 6.4 The Harmonic Gauge -- Chapter 7 Einstein's equations and variational principles -- 7.1 Eulerlagrange's Equations in Special Relativity |
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7.2 Eulerlagrange's Equations in Curved Spacetime -- 7.3 Einstein's Equations in Vacuum -- 7.4 Einstein's Equations with Sources -- 7.4.1 The stress-energy tensor in some relevant cases -- 7.5 Einstein's Equations in the Palatini Formalism -- Chapter 8 Symmetries -- 8.1 Killing Vector Fields -- 8.2 Killing Vector Fields and the Choice of Coordinate Systems -- 8.3 Killing Vector Fields and Conservation Laws -- 8.3.1 Conserved quantities in geodesic motion -- 8.3.2 Conserved quantities from the stress-energy tensor -- 8.4 Hypersurface-Orthogonalvector Fields -- 8.4.1 Frobenius' theorem |
| Notes |
Print version record |
| Form |
Electronic book
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| Author |
Gualtieri, Leonardo
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Pani, Paolo
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| ISBN |
9780429957802 |
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0429957807 |
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