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The Metric The Signature of a Metric The Flat Space Metric The Metric as a Tensor Index Raising and Lowering Index Gymnastics The Dot Product Passing Arguments to the Metric Null Vectors The Metric Determinant Quiz CHAPTER 3 More on Tensors Manifolds Parameterized Curves Tangent Vectors and One Forms, Again Tensors as Functions Tensor Operations The Levi-Cevita Tensor Quiz Tensor Calculus Testing Tensor Character The Importance of Tensor Equations The Covariant Derivative The Torsion Tensor The Metric and Christoffel Symbols The Exterior Derivative The Lie Derivative The Absolute Derivative and Geodesics The Riemann Tensor The Ricci Tensor and Ricci Scalar The Weyl Tensor and Conformal Metrics Quiz 32 36 37 37 38 41 42 43 45 45 45 47 47 49 50 53 54 59 59 60 60 61 62 72 72 79 81 82 85 88 90 91
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CHAPTER 5 Cartan s Structure Equations Introduction Holonomic (Coordinate) Bases Nonholonomic Bases Commutation Coef cients Commutation Coef cients and Basis One Forms Transforming between Bases A Note on Notation Cartan s First Structure Equation and the Ricci Rotation Coef cients Computing Curvature Quiz The Einstein Field Equations Equivalence of Mass in Newtonian Theory Test Particles The Einstein Lift Experiments The Weak Equivalence Principle The Strong Equivalence Principle The Principle of General Covariance Geodesic Deviation The Einstein Equations The Einstein Equations with Cosmological Constant An Example Solving Einstein s Equations in 2 + 1 Dimensions Energy Conditions Quiz The Energy-Momentum Tensor Energy Density Momentum Density and Energy Flux Stress Conservation Equations 93 93 94 95 96 98 100 103 104 112 120 122 123 126 126 130 130 131 131 136 138 139 152 152 155 156 156 156 157
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Dust Perfect Fluids Relativistic Effects on Number Density More Complicated Fluids Quiz CHAPTER 8 Killing Vectors Introduction Derivatives of Killing Vectors Constructing a Conserved Current with Killing Vectors Quiz Null Tetrads and the Petrov Classi cation Null Vectors A Null Tetrad Extending the Formalism Physical Interpretation and the Petrov Classi cation Quiz The Schwarzschild Solution The Vacuum Equations A Static, Spherically Symmetric Spacetime The Curvature One Forms Solving for the Curvature Tensor The Vacuum Equations The Meaning of the Integration Constant The Schwarzschild Metric The Time Coordinate The Schwarzschild Radius Geodesics in the Schwarzschild Spacetime Particle Orbits in the Schwarzschild Spacetime 158 160 163 164 165 167 167 177 178 178
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The De ection of Light Rays Time Delay Quiz CHAPTER 11 Black Holes Redshift in a Gravitational Field Coordinate Singularities Eddington-Finkelstein Coordinates The Path of a Radially Infalling Particle Eddington-Finkelstein Coordinates Kruskal Coordinates The Kerr Black Hole Frame Dragging The Singularity A Summary of the Orbital Equations for the Kerr Metric Further Reading Quiz Cosmology The Cosmological Principle A Metric Incorporating Spatial Homogeneity and Isotropy Spaces of Positive, Negative, and Zero Curvature Useful De nitions The Robertson-Walker Metric and the Friedmann Equations Different Models of the Universe Quiz Gravitational Waves The Linearized Metric Traveling Wave Solutions The Canonical Form and Plane Waves 224 229 230 233 234 235 236 238 239 242 244 249 252 252 253 254 256 257 257 262 264 267 271 276 279 280 284 287
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The Behavior of Particles as a Gravitational Wave Passes The Weyl Scalars Review: Petrov Types and the Optical Scalars pp Gravity Waves Plane Waves The Aichelburg-Sexl Solution Colliding Gravity Waves The Effects of Collision More General Collisions Nonzero Cosmological Constant Further Reading Quiz Final Exam Quiz and Exam Solutions References and Bibliography Index
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