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: \( \frac{d}{d\tau} \left( \frac{\partial L}{\partial \dot{x}^\mu} \right) = \frac{\partial L}{\partial x^\mu} \)