| Formatted Contents Note: |
Chapter 1 A Quick Review of Special Relativity 1 -- Frame of Reference 5 -- Clock Synchronization 5 -- Inertial Frames 6 -- Galilean Transformations 7 -- Events 7 -- Interval 8 -- Postulates of Special Relativity 9 -- Three Basic Physical Implications 13 -- Light Cones and Spacetime Diagrams 17 -- Four Vectors 19 -- Relativistic Mass and Energy 20 -- Chapter 2 Vectors, One Forms, and the Metric 23 -- Vectors 23 -- New Notation 25 -- Four Vectors 27 -- Einstein Summation Convention 28 -- Tangent Vectors, One Forms, and the Coordinate Basis 29 -- Coordinate Transformations 31 -- Metric 32 -- Signature of a Metric 36 -- Flat Space Metric 37 -- Metric as a Tensor 37 -- Index Raising and Lowering 38 -- Index Gymnastics 41 -- Dot Product 42 -- Passing Arguments to the Metric 43 -- Null Vectors 45 -- Metric Determinant 45 -- Chapter 3 More on Tensors 47 -- Manifolds 47 -- Parameterized Curves 49 -- Tangent Vectors and One Forms, Again 50 -- Tensors as Functions 53 -- Tensor Operations 54 -- Levi-Cevita Tensor 59 -- Chapter 4 Tensor Calculus 60 -- Testing Tensor Character 60 -- Importance of Tensor Equations 61 -- Covariant Derivative 62 -- Torsion Tensor 72 -- Metric and Christoffel Symbols 72 -- Exterior Derivative 79 -- Lie Derivative 81 -- Absolute Derivative and Geodesics 82 -- Riemann Tensor 85 -- Ricci Tensor and Ricci Scalar 88 -- Weyl Tensor and Conformal Metrics 90 -- Chapter 5 Cartan's Structure Equations 93 -- Holonomic (Coordinate) Bases 94 -- Nonholonomic Bases 95 -- Commutation Coefficients 96 -- Commutation Coefficients and Basis One Forms 98 -- Transforming between Bases 100 -- A Note on Notation 103 -- Cartan's First Structure Equation and the Ricci Rotation Coefficients 104 -- Computing Curvature 112 -- Chapter 6 Einstein Field Equations 122 -- Equivalence of Mass in Newtonian Theory 123 -- Test Particles 126 -- Einstein Lift Experiments 126 -- Weak Equivalence Principle 130 -- Strong Equivalence Principle 130 -- Principle of General Covariance 131 -- Geodesic Deviation 131 -- Einstein Equations 136 -- Einstein Equations with Cosmological Constant 138 -- An Example Solving Einstein's Equations in 2 + 1 Dimensions 139 -- Energy Conditions 152 -- Chapter 7 Energy-Momentum Tensor 155 -- Energy Density 156 -- Momentum Density and Energy Flux 156 -- Stress 156 -- Conservation Equations 157 -- Dust 158 -- Perfect Fluids 160 -- Relativistic Effects on Number Density 163 -- More Complicated Fluids 164 -- Chapter 8 Killing Vectors 167 -- Derivatives of Killing Vectors 177 -- Constructing a Conserved Current with Killing Vectors 178 -- Chapter 9 Null Tetrads and the Petrov Classification 180 -- Null Vectors 182 -- A Null Tetrad 184 -- Extending the Formalism 190 -- Physical Interpretation and the Petrov Classification 193 -- Chapter 10 Schwarzschild Solution 203 -- Vacuum Equations 204 -- A Static, Spherically Symmetric Spacetime 204 -- Curvature One Forms 206 -- Solving for the Curvature Tensor 209 -- Vacuum Equations 211 -- Meaning of the Integration Constant 214 -- Schwarzschild Metric 215 -- Time Coordinate 215 -- Schwarzschild Radius 215 -- Geodesics in the Schwarzschild Spacetime 216 -- Particle Orbits in the Schwarzschild Spacetime 218 -- Deflection of Light Rays 224 -- Time Delay 229 -- Chapter 11 Black Holes 233 -- Redshift in a Gravitational Field 234 -- Coordinate Singularities 235 -- Eddington-Finkelstein Coordinates 236 -- Path of a Radially Infalling Particle 238 -- Eddington-Finkelstein Coordinates 239 -- Kruskal Coordinates 242 -- Kerr Black Hole 244 -- Frame Dragging 249 -- Singularity 252 -- A Summary of the Orbital Equations for the Kerr Metric 252 -- Chapter 12 Cosmology 256 -- Cosmological Principle 257 -- A Metric Incorporating Spatial Homogeneity and Isotropy 257 -- Spaces of Positive, Negative, and Zero Curvature 262 -- Useful Definitions 264 -- Robertson-Walker Metric and the Friedmann Equations 267 -- Different Models of the Universe 271 -- Chapter 13 Gravitational Waves 279 -- Linearized Metric 280 -- Traveling Wave Solutions 284 -- Canonical Form and Plane Waves 287 -- Behavior of Particles as a Gravitational Wave Passes 291 -- Weyl Scalars 294 -- Review: Petrov Types and the Optical Scalars 295 -- pp Gravity Waves 297 -- Plane Waves 301 -- Aichelburg-Sexl Solution 303 -- Colliding Gravity Waves 304 -- Effects of Collision 311 -- More General Collisions 312 -- Nonzero Cosmological Constant 318. |
