Brown-York charges with mixed boundary conditions

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.

v2: Revised discussion of the case with non-orthogonal corners, the matching between covariant and canonical formulas is now obtained thanks to the Legendre transform at the corner. Further amendments. Matches published version

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• Wald-Zoupas prescription with (soft) anomalies
• Dual Charges for AdS Spacetimes and the First Law of Black Hole Mechanics
• Covariant canonical formulations of classical field theories
• Freelance Holography, Part I: Setting Boundary Conditions Free in Gauge/Gravity Correspondence
• Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms