✎ Einstein Equation [The Physics Travel Guide]

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====== Einstein Equation: $ \quad G_{\mu \nu} = 8 \pi G T_{\mu \nu}$ ======

//see also [[theories:general_relativity]] //

* [[http://jakobschwichtenberg.com/how-to-invent-general-relativity/|How to Invent General Relativity]] by J. Schwichtenberg 
  
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* [[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.

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The static limit of the Einstein equation is known as [[equations:newtons_law|Newton's law]].

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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}

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