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Preface | |

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A Quick Review of Special Relativity | |

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Frame of Reference | |

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Clock Synchronization | |

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Inertial Frames | |

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Galilean Transformations | |

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Events | |

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The Interval | |

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Postulates of Special Relativity | |

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Three Basic Physical Implications | |

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Light Cones and Spacetime Diagrams | |

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Four Vectors | |

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Relativistic Mass and Energy | |

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Quiz | |

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Vectors, One Forms, and the Metric | |

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Vectors | |

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New Notation | |

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Four Vectors | |

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The Einstein Summation Convention | |

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Tangent Vectors, One Forms, and the Coordinate Basis | |

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Coordinate Transformations | |

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The Metric | |

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The Signature of a Metric | |

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The Flat Space Metric | |

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The Metric as a Tensor | |

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Index Raising and Lowering | |

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Index Gymnastics | |

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The Dot Product | |

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Passing Arguments to the Metric | |

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Null Vectors | |

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The Metric Determinant | |

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Quiz | |

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More on Tensors | |

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Manifolds | |

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Parameterized Curves | |

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Tangent Vectors and One Forms, Again | |

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Tensors as Functions | |

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Tensor Operations | |

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The Levi-Cevita Tensor | |

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Quiz | |

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Tensor Calculus | |

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Testing Tensor Character | |

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The Importance of Tensor Equations | |

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The Covariant Derivative | |

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The Torsion Tensor | |

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The Metric and Christoffel Symbols | |

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The Exterior Derivative | |

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The Lie Derivative | |

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The Absolute Derivative and Geodesics | |

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The Riemann Tensor | |

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The Ricci Tensor and Ricci Scalar | |

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The Weyl Tensor and Conformal Metrics | |

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Quiz | |

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Cartan's Structure Equations | |

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Introduction | |

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Holonomic (Coordinate) Bases | |

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Nonholonomic Bases | |

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Commutation Coefficients | |

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Commutation Coefficients and Basis One Forms | |

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Transforming between Bases | |

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A Note on Notation | |

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Cartan's First Structure Equation and the Ricci Rotation Coefficients | |

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Computing Curvature | |

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Quiz | |

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The Einstein Field Equations | |

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Equivalence of Mass in Newtonian Theory | |

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Test Particles | |

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The Einstein Lift Experiments | |

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The Weak Equivalence Principle | |

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The Strong Equivalence Principle | |

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The Principle of General Covariance | |

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Geodesic Deviation | |

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The Einstein Equations | |

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The Einstein Equations with Cosmological Constant | |

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An Example Solving Einstein's Equations in 2 + 1 Dimensions | |

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Energy Conditions | |

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Quiz | |

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The Energy-Momentum Tensor | |

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Energy Density | |

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Momentum Density and Energy Flux | |

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Stress | |

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Conservation Equations | |

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Dust | |

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Perfect Fluids | |

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Relativistic Effects on Number Density | |

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More Complicated Fluids | |

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Quiz | |

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Killing Vectors | |

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Introduction | |

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Derivatives of Killing Vectors | |

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Constructing a Conserved Current with Killing Vectors | |

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Quiz | |

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Null Tetrads and the Petrov Classification | |

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Null Vectors | |

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A Null Tetrad | |

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Extending the Formalism | |

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Physical Interpretation and the Petrov Classification | |

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Quiz | |

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The Schwarzschild Solution | |

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The Vacuum Equations | |

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A Static, Spherically Symmetric Spacetime | |

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The Curvature One Forms | |

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Solving for the Curvature Tensor | |

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The Vacuum Equations | |

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The Meaning of the Integration Constant | |

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The Schwarzschild Metric | |

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The Time Coordinate | |

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The Schwarzschild Radius | |

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Geodesics in the Schwarzschild Spacetime | |

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Particle Orbits in the Schwarzschild Spacetime | |

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The Deflection of Light Rays | |

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Time Delay | |

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Quiz | |

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Black Holes | |

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Redshift in a Gravitational Field | |

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Coordinate Singularities | |

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Eddington-Finkelstein Coordinates | |

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The Path of a Radially Infalling Particle | |

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Eddington-Finkelstein Coordinates | |

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Kruskal Coordinates | |

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The Kerr Black Hole | |

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Frame Dragging | |

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The Singularity | |

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A Summary of the Orbital Equations for the Kerr Metric | |

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Further Reading | |

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Quiz | |

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Cosmology | |

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The Cosmological Principle | |

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A Metric Incorporating Spatial Homogeneity and Isotropy | |

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Spaces of Positive, Negative, and Zero Curvature | |

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Useful Definitions | |

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The Robertson-Walker Metric and the Friedmann Equations | |

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Different Models of the Universe | |

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Quiz | |

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Gravitational Waves | |

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The Linearized Metric | |

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Traveling Wave Solutions | |

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The Canonical Form and Plane Waves | |

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The Behavior of Particles as a Gravitational Wave Passes | |

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The Weyl Scalars | |

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Review: Petrov Types and the Optical Scalars | |

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pp Gravity Waves | |

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Plane Waves | |

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The Aichelburg-Sexl Solution | |

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Colliding Gravity Waves | |

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The Effects of Collision | |

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More General Collisions | |

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Nonzero Cosmological Constant | |

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Further Reading | |

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Quiz | |

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Final Exam | |

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Quiz and Exam Solutions | |

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References and Bibliography | |

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Index | |