# Einstein Equations

Let $G$ denote Newton's constant. The Einstein equations are

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

where $G_{\mu\nu}$ is the Einstein tensor and $T_{\mu\nu}$ is the stress-energy-momentum tensor. These equations extremize the action

$S[g_{\mu\nu},\Phi] = \frac{1}{16\pi G} \int d^4x \sqrt{-g} R + S^{\rm matter}[g_{\mu\nu},\Phi]$

where $g$ is the determinant of the spacetime metric $g_{\mu\nu}$ and $R$ is the spacetime curvature scalar. The matter fields are denoted collectively by $\Phi$. The stress-energy-momentum tensor is obtained from the matter action by

$T^{\mu\nu} \equiv \frac{2}{\sqrt{-g}} \frac{\delta S^{\rm matter}}{\delta g_{\mu\nu}}$

Formulations of the Einstein equations include: