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equations:einstein_equation

$G_{\mu \nu} = 8 \pi G T_{\mu \nu}$

# Einstein Equation

## Intuitive

Einstein's equation describes how spacetime gets curved through mass and energy.

## Concrete

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

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## 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 $R =R_{\nu}^\nu$ $$G_{\mu \nu} = R_{\mu\nu}-\frac{1}{2}Rg_{\mu \nu}$$ where the Ricci Tensor $R_{\mu\nu}$ is defined in terms of the Christoffel symbols $\Gamma^\mu_{\nu \rho}$

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