Brown-York charges with mixed boundary conditions

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include York's mixed boundary conditions defined by holding fixed the conformal induced metric and the trace of the extrinsic curvature. We show that for all boundary conditions considered, canonical methods give the same answer as covariant phase space methods improved by a boundary Lagrangian, a prescription recently developed in the literature and thus supported by our results. The procedure also suggests a new integrable charge for the Einstein-Hilbert Lagrangian, different from the Komar charge for non-Killing and non-tangential diffeomorphisms. We study how the energy depends on the choice of boundary conditions, showing that both the quasi-local and the asymptotic expressions are affected. Additionally, we study the generalization of the covariant phase space formula for the surface charges to the case of a time-like boundary with non-orthogonal corners. We find that a contribution from the corner Lagrangian must be included and treated as a boundary term for the 3d boundary Lagrangian, if a match with the canonical methods is to be achieved.