Appendices
Appendix 1
Formulas of General Relativity.
Summary of important metric, curvature, and field equations.
\( ds^2 = g_{\mu\nu} dx^\mu dx^\nu \)
\( G_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} \)
Appendix 2
Derivation of the Derivative of the Christoffel Symbols.
Schematic derivation of \( \nabla_\lambda \Gamma^\rho_{\mu\nu} \) and related identities.
Appendix 3
Mathematical Derivation of Schwarzschild.
Step-by-step: from metric to Riemann and Ricci tensors for the Schwarzschild solution.
Appendix 4
The Schwarzschild Formula Extended for Electric Charges.
Connection between surface integral and volume source: divergence theorem in curved space.
Appendix 5
Schwarzschild Solution Inside a Mass.
\( \nabla^2 \Phi = 4\pi G \rho \) as the Newtonian limit of the field equations.
Appendix 6
Derivation of Gauss’s Theorem.
Relation between flux through a surface and volume.
\(\oint_{\partial V} \vec{F}\cdot d\vec{A}=\iiint_V (\vec{\nabla}\cdot\vec{F})\, dV\)
Appendix 7
Derivation of the Laplace and Poisson Equations.
\( \nabla^2 \Phi = 4\pi G \rho \) as the Newtonian limit of the field equations.
Appendix 9
Special Relativity.
Lorentz transformations, time dilation, length contraction, and \( E = mc^2 \).
Appendix 10
Specific Angular Momentum.
Conservation of \(L/m\) in a central potential and in Schwarzschild geometry.
Appendix 12
Derivation of the Euler–Lagrange Equation.
Variational principle: \( \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{x}^\mu} \right) = \frac{\partial L}{\partial x^\mu} \).
Use of the Appendices
- Appendix 1: Quick formula reference.
- Appendix 8: Intuitive understanding of tidal forces and curvature.
- Appendix 9: Refresher on special relativity.
All appendices are in principle self-contained.