Gibbons–Hawking–York boundary term

Summary

The Gibbons–Hawking–York boundary term is an integral over the boundary of a spacetime region that is added to the Einstein–Hilbert action. Its integrand is the trace of the extrinsic curvature of the boundary, that is, its mean curvature. The scalar curvature contains second derivatives of the metric. So without the term, the Einstein–Hilbert action has no well-posed variational principle when the metric is fixed on the boundary. The term matters in the Hamiltonian formulation of gravity and in Euclidean quantum gravity, where it contributes to black hole entropy.