Intuitive

• How to Invent General Relativity by J. Schwichtenberg

Concrete

• The Meaning of Einstein’s Equation by John C. Baez and Emory F. Bunn explains the Einstein equations perfectly.

The static limit of the Einstein equation is known as Newton's law.

Abstract

Why is it interesting?

The Einstein equation is the fundamental equation of general relativity. It describes how spacetime is curved through the presence of matter and energy.

Definitions

On the right-hand side, Newton's gravitational constant $G$, the speed of light $c$ and the stress-energy tensor $T_{\mu \nu}$.

On the left-hand side, the Einstein tensor $G_{\mu \nu}$ is defined as a sum of the Ricci Tensor $R_{\mu\nu}$ and the trace of the Ricci tensor, called Ricci scalar $R =R_{\nu}^\nu$ $$\begin{equation} G_{\mu \nu} = R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu} \end{equation}$$ where the Ricci Tensor $R_{\mu\nu}$ is defined in terms of the Christoffel symbols $\Gamma^\mu_{\nu \rho}$

$$\begin{equation} R_{\alpha\beta} = \partial_{\rho}{\Gamma^\rho_{\beta\alpha}} - \partial_{\beta}\Gamma^\rho_{\rho\alpha} + \Gamma^\rho_{\rho\lambda} \Gamma^\lambda_{\beta\alpha} - \Gamma^\rho_{\beta\lambda}\Gamma^\lambda_{\rho\alpha} \end{equation}$$ and the Christoffel Symbols are defined in terms of the metric $$\begin{equation} \Gamma_{\alpha \beta \rho} =\frac12 \left(\frac{\partial g_{\alpha \beta}}{\partial x^\rho} + \frac{\partial g_{\alpha \rho}}{\partial x^\beta} - \frac{\partial g_{\beta \rho}}{\partial x^\alpha} \right) = \frac12\, \left(\partial_{\rho}g_{\alpha \beta} + \partial_{\beta}g_{\alpha \rho} - \partial_{\alpha}g_{\beta \rho}\right). \end{equation}$$