Einstein-Hilbert Functional

The Einstein-Hilbert functional of a Riemannian metric on a closed manifold is the integral of its scalar curvature,

Its first variation under a covariant metric variation is the integral pairing with , the negative Einstein tensor. In dimension at least three, its critical metrics with a fixed-volume constraint are Einstein metrics. On a noncompact manifold, the scalar-curvature integral and boundary terms need not converge separately, so a renormalized version requires specified asymptotic data.