$ G_{\mu \nu} = 8 \pi G T_{\mu \nu}$ ====== Einstein Equation ====== //see also [[models:general_relativity]] // Einstein's equation describes how spacetime gets curved through mass and energy. ----- * [[http://jakobschwichtenberg.com/how-to-invent-general-relativity/|How to Invent General Relativity]] by J. Schwichtenberg * [[http://math.ucr.edu/home/baez/einstein/einstein.pdf|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 [[formulas:newtons_law|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}