General Relativity

  1. Mathematical Framework: Tensor Calculus and Differential Geometry
    1. Manifolds and Coordinate Systems
      1. Definition of a Differentiable Manifold
        1. Topological Spaces
          1. Smooth Structure
            1. Examples of Manifolds
            2. Coordinate Systems and Charts
              1. Local Coordinates
                1. Atlas of Charts
                  1. Transition Functions
                    1. Coordinate Transformations
                    2. Tangent Spaces
                      1. Tangent Vectors
                        1. Geometric Definition
                          1. Algebraic Definition
                            1. Directional Derivatives
                            2. Basis Vectors
                              1. Coordinate Basis
                                1. Change of Basis
                                2. Cotangent Spaces
                                  1. Dual Vectors
                                    1. One-Forms
                                      1. Coordinate Basis for Cotangent Space
                                  2. Tensor Analysis
                                    1. Introduction to Tensors
                                      1. Vectors as Contravariant Tensors
                                        1. One-forms as Covariant Tensors
                                          1. General Tensors
                                            1. Rank and Type of Tensors
                                              1. Mixed Tensors
                                            2. Tensor Algebra
                                              1. Tensor Addition
                                                1. Scalar Multiplication
                                                  1. Outer Product
                                                    1. Tensor Product
                                                      1. Antisymmetrization and Symmetrization
                                                      2. Contraction
                                                        1. Index Contraction
                                                          1. Trace Operations
                                                          2. Raising and Lowering Indices
                                                            1. Role of the Metric Tensor
                                                              1. Musical Isomorphisms
                                                          3. The Metric Tensor
                                                            1. Definition and Properties
                                                              1. Bilinear Form
                                                                1. Symmetry Properties
                                                                  1. Non-degeneracy
                                                                  2. Calculating Distances and Angles
                                                                    1. Line Element
                                                                      1. Proper Time and Proper Distance
                                                                        1. Arc Length
                                                                        2. The Signature of the Metric
                                                                          1. Lorentzian Signature
                                                                            1. Riemannian vs. Pseudo-Riemannian
                                                                              1. Sign Conventions
                                                                              2. Examples of Metrics
                                                                                1. Minkowski Metric
                                                                                  1. Euclidean Metric
                                                                                    1. Spherical Coordinates
                                                                                  2. Covariant Differentiation
                                                                                    1. The Need for Covariant Derivatives
                                                                                      1. Failure of Ordinary Derivatives
                                                                                        1. Coordinate Independence
                                                                                        2. The Covariant Derivative
                                                                                          1. Definition and Properties
                                                                                            1. Leibniz Rule
                                                                                              1. Linearity
                                                                                              2. Christoffel Symbols
                                                                                                1. Connection Coefficients
                                                                                                  1. Transformation Properties
                                                                                                    1. Metric Compatibility
                                                                                                    2. Parallel Transport
                                                                                                      1. Transporting Vectors Along Curves
                                                                                                        1. Path Dependence
                                                                                                          1. Holonomy
                                                                                                          2. Geodesic Equation
                                                                                                            1. Derivation from Parallel Transport
                                                                                                              1. Variational Principle
                                                                                                                1. Affine Parameter
                                                                                                              2. Curvature Tensors
                                                                                                                1. The Riemann Curvature Tensor
                                                                                                                  1. Definition via Commutator
                                                                                                                    1. Geometric Interpretation
                                                                                                                      1. Symmetries and Properties
                                                                                                                        1. Antisymmetry Properties
                                                                                                                          1. Cyclic Identity
                                                                                                                            1. Bianchi Identities
                                                                                                                          2. The Ricci Tensor
                                                                                                                            1. Contraction of the Riemann Tensor
                                                                                                                              1. Physical Interpretation
                                                                                                                                1. Symmetry Properties
                                                                                                                                2. The Ricci Scalar
                                                                                                                                  1. Further Contraction
                                                                                                                                    1. Scalar Curvature
                                                                                                                                    2. The Weyl Tensor
                                                                                                                                      1. Conformal Curvature
                                                                                                                                        1. Traceless Part of Riemann Tensor
                                                                                                                                          1. Gravitational Waves

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                                                                                                                                      2. The Principles of General Relativity

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                                                                                                                                      4. The Einstein Field Equations