Ambiguity resolution for integrable gravitational charges

TL;DR

This paper clarifies the ambiguity in defining gravitational charges in extended phase spaces by resolving covariant phase space ambiguities, leading to more precise charges and understanding of integrability issues.

Contribution

It demonstrates how to obtain unambiguous gravitational charges in extended phase spaces by using the variational principle, resolving previous ambiguities and clarifying the nature of integrability obstructions.

Findings

Ambiguity-free charges can be derived using the variational principle.

Corrections to diffeomorphism charges are identified.

Obstructions to integrability are clarified, linking to Wald-Zoupas charges.

Abstract

Recently, Ciambelli, Leigh, and Pai (CLP) [arXiv:2111.13181] have shown that nonzero charges integrating Hamilton's equation can be defined for all diffeomorphisms acting near the boundary of a subregion in a gravitational theory. This is done by extending the phase space to include a set of embedding fields that parameterize the location of the boundary. Because their construction differs from previous works on extended phase spaces by a covariant phase space ambiguity, the question arises as to whether the resulting charges are unambiguously defined. Here, we demonstrate that ambiguity-free charges can be obtained by appealing to the variational principle for the subregion, following recent developments on dealing with boundaries in the covariant phase space. Resolving the ambiguity produces corrections to the diffeomorphism charges, and also generates additional obstructions to integrability of Hamilton's equation. We emphasize the fact that the CLP extended phase space produces nonzero diffeomorphism charges distinguishes it from previous constructions in which diffeomorphisms are pure gauge, since the embedding fields can always be eliminated from the latter by a choice of unitary gauge. Finally, we show that Wald-Zoupas charges, with their characteristic obstruction to integrability, are associated with a modified transformation in the extended phase space, clarifying the reason behind integrability of Hamilton's equation for standard diffeomorphisms.