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equations:einstein_equation

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

*see also General Relativity *

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

- How to Invent General Relativity by J. Schwichtenberg

- 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.

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

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

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}

equations/einstein_equation.txt · Last modified: 2018/12/19 10:00 by jakobadmin

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