| |
| |
| Preface | |
| |
| |
| |
| A Quick Review of Special Relativity | |
| |
| |
| Frame of Reference | |
| |
| |
| Clock Synchronization | |
| |
| |
| Inertial Frames | |
| |
| |
| Galilean Transformations | |
| |
| |
| Events | |
| |
| |
| The Interval | |
| |
| |
| Postulates of Special Relativity | |
| |
| |
| Three Basic Physical Implications | |
| |
| |
| Light Cones and Spacetime Diagrams | |
| |
| |
| Four Vectors | |
| |
| |
| Relativistic Mass and Energy | |
| |
| |
| Quiz | |
| |
| |
| |
| Vectors, One Forms, and the Metric | |
| |
| |
| Vectors | |
| |
| |
| New Notation | |
| |
| |
| Four Vectors | |
| |
| |
| The Einstein Summation Convention | |
| |
| |
| Tangent Vectors, One Forms, and the Coordinate Basis | |
| |
| |
| Coordinate Transformations | |
| |
| |
| 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 | |
| |
| |
| |
| 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 | |
| |
| |
| |
| Cartan's Structure Equations | |
| |
| |
| Introduction | |
| |
| |
| Holonomic (Coordinate) Bases | |
| |
| |
| Nonholonomic Bases | |
| |
| |
| Commutation Coefficients | |
| |
| |
| Commutation Coefficients and Basis One Forms | |
| |
| |
| Transforming between Bases | |
| |
| |
| A Note on Notation | |
| |
| |
| Cartan's First Structure Equation and the Ricci Rotation Coefficients | |
| |
| |
| 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 | |
| |
| |
| Dust | |
| |
| |
| Perfect Fluids | |
| |
| |
| Relativistic Effects on Number Density | |
| |
| |
| More Complicated Fluids | |
| |
| |
| Quiz | |
| |
| |
| |
| Killing Vectors | |
| |
| |
| Introduction | |
| |
| |
| Derivatives of Killing Vectors | |
| |
| |
| Constructing a Conserved Current with Killing Vectors | |
| |
| |
| Quiz | |
| |
| |
| |
| Null Tetrads and the Petrov Classification | |
| |
| |
| Null Vectors | |
| |
| |
| A Null Tetrad | |
| |
| |
| Extending the Formalism | |
| |
| |
| Physical Interpretation and the Petrov Classification | |
| |
| |
| 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 | |
| |
| |
| The Deflection of Light Rays | |
| |
| |
| Time Delay | |
| |
| |
| Quiz | |
| |
| |
| |
| 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 Definitions | |
| |
| |
| 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 | |
| |
| |
| 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 | |