Einstein Equations

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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:

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