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The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equationsthrough the stationary-action principle. With the (− + + +) metric signature, the gravitational part of the action is given as[1]

{\displaystyle S={1 \over 2\kappa }\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x,}

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 the Einstein gravitational constant (
{\displaystyle G}
is the gravitational constant and
{\displaystyle c}
is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. 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. The action was proposed[2] by David Hilbert in 1915 as part of his application of the variational principle to a combination of gravity and electromagnetism.[3]: 119
 
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wishIwasSalludon said:
The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equationsthrough the stationary-action principle. With the (− + + +) metric signature, the gravitational part of the action is given as[1]

{\displaystyle S={1 \over 2\kappa }\int R{\sqrt {-g}}\,\mathrm {d} ^{4}x,}

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 the Einstein gravitational constant (
{\displaystyle G}
is the gravitational constant and
{\displaystyle c}
is the speed of light in vacuum). If it converges, the integral is taken over the whole spacetime. 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. The action was proposed[2] by David Hilbert in 1915 as part of his application of the variational principle to a combination of gravity and electromagnetism.[3]: 119
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