Brown-York charges with mixed boundary conditions

TL;DR

This paper computes Hamiltonian surface charges in gravity under various boundary conditions, demonstrating consistency between methods and proposing a new integrable charge, with implications for energy definitions and boundary effects.

Contribution

It introduces a unified approach to compute gravitational charges for mixed boundary conditions and proposes a new integrable charge distinct from the Komar charge.

Findings

Canonical and covariant methods agree on charges for all boundary conditions.

A new integrable charge for Einstein-Hilbert Lagrangian is proposed.

Energy expressions depend on boundary condition choices.

Abstract

We compute the Hamiltonian surface charges of gravity for a family of conservative boundary conditions, that include Dirichlet, Neumann, and 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. Finally, we generalize the analysis to non-orthogonal corners, confirm the matching between the covariant and canonical results without any change in the prescription, and discuss the subtleties associated with this case.