Einstein–Hilbert action

The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the principle of least action. With the (− + + +) metric signature, the gravitational part of the action is given as

where ${\displaystyle g=\det(g_{\mu \nu })}$ is the determinant of the metric tensor matrix, ${\displaystyle R}$ is the Ricci scalar, and ${\displaystyle \kappa =8\pi Gc^{-4}}$ is Einstein's constant (${\displaystyle G}$ is the gravitational constant and ${\displaystyle c}$ is the speed of light in vacuum). The integral is taken over the whole spacetime if it converges. If it does not converge, ${\displaystyle S}$ is no longer well-defined, but a modified definition where one integrates over arbitrarily large, relatively compact domains, still yields the Einstein equation as the Euler–Lagrange equation of the Einstein–Hilbert action.

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