General Relativity

  1. The Einstein Field Equations
    1. Formulation of the Equations
      1. Geometric Side: The Einstein Tensor
        1. Construction from Ricci Tensor and Scalar
          1. Divergence-Free Property
            1. Symmetry
            2. Physical Side: The Stress-Energy-Momentum Tensor
              1. Definition and Physical Meaning
                1. Conservation Properties
                  1. Symmetry and Positivity
                  2. The Field Equations
                    1. Einstein's Original Form
                      1. Coupling Constant
                        1. Units and Conventions
                      2. Matter Sources
                        1. Perfect Fluids
                          1. Energy Density
                            1. Pressure
                              1. Four-Velocity
                              2. Dust
                                1. Pressureless Matter
                                  1. Galaxy Models
                                  2. Electromagnetic Fields
                                    1. Maxwell Stress-Energy Tensor
                                      1. Electromagnetic Field Tensor
                                      2. Scalar Fields
                                        1. Klein-Gordon Field
                                          1. Inflaton Field
                                        2. The Cosmological Constant
                                          1. Historical Introduction
                                            1. Einstein's Static Universe
                                              1. The "Greatest Blunder"
                                              2. Modern Interpretation
                                                1. Dark Energy
                                                  1. Vacuum Energy
                                                    1. Observational Evidence
                                                  2. Properties of the Field Equations
                                                    1. Non-linearity
                                                      1. Self-Interaction of Gravity
                                                        1. Superposition Principle Failure
                                                        2. Local Conservation of Energy and Momentum
                                                          1. Bianchi Identities
                                                            1. Covariant Divergence
                                                            2. Gauge Freedom
                                                              1. Coordinate Invariance
                                                                1. Harmonic Gauge
                                                              2. The Weak-Field Limit
                                                                1. Linearized Gravity
                                                                  1. Perturbation Theory
                                                                    1. Gravitational Waves in Linear Theory
                                                                    2. Recovery of Newtonian Gravity
                                                                      1. Correspondence Principle
                                                                        1. Post-Newtonian Approximation
                                                                        2. Gauge Choices in Linear Theory
                                                                          1. Harmonic Gauge
                                                                            1. Transverse-Traceless Gauge

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                                                                        3. Mathematical Framework: Tensor Calculus and Differential Geometry

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                                                                        5. Exact Solutions to the Einstein Field Equations