General Relativity – Physics Cafe

Outline of Topics
1. Overview
2. Special Relativity
3. The Equivalence Principle
4. Coordinate systems and coordinate transformations
5. Tensors and Invariant Equations
6. Tensor Math
7. Parallel Transport
8. The Tensor Derivative
9. The Schwarzschild solution
The mathematical objects of interest
- $g_{\mu\nu}$ – the Metric Tensor
- $\Gamma^{\mu}{}_{\alpha\beta}$ – The Christoffel (Connection) Symbol
- $R^{\alpha}{}_{\rho\gamma\beta}$ – The Riemann (Curvature) Tensor
- $R_{\mu\nu}$ – The Ricci Tensor
- $R$ – The Ricci Scalar
- $T$ – The Energy-Momentum Tensor
- $G_{\mu\nu}$ – The Einstein Tensor
The Equations
- Einstein’s Equation: $R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi GT_{\mu\nu}$
- The Geodesic Equation: $\frac{d^2x^{\mu}}{d\lambda^2} + \Gamma^{\mu}_{\rho\sigma} \frac{dx^{\rho}}{d\lambda}\frac{dx^{\sigma}}{d\lambda} = 0$
Coordinate Systems
- Boyer-Lindquist
- Eddington-Finkelstein
- Fermi normal
- Kerr-Schild
- Kruskal
- Painleve-Gullstrand
- Reimann normal
Spacetimes
- de Sitter
- Friedmann-Robertson-Walker
- Kerr
- Kerr-Newman
- Majumdar-Papapetrou
- Minkowski
- Schwartschild
- Vaidya
Horizons
- Apparent
- Cauchy
- Event
- Killing
- Trapping