(Anti)-de Sitter with leaky boundaries and corners

TL;DR

The paper develops a covariant phase space framework for four-dimensional gravity with a nonzero cosmological constant in a partial Bondi gauge, enabling leaky timelike boundaries and corners. By decomposing the presymplectic potential into boundary and corner contributions, it yields manifestly finite charges and reveals new corner-associated charges, including a Weyl charge under conformal boundary conditions. It analyzes residual symmetries, transformation laws, and the resulting Brown-York boundary charges while allowing partially on-shell field configurations, thereby linking boundary evolution to the yet-unenforced Einstein equations. The authors also construct a Dirichlet variational principle for AdS$_4$ with a corner, showing how surface and corner terms restore finiteness and how Ward identities reproduce mass and angular momentum evolution, with implications for holography and BCFT-like setups.

Abstract

We construct charges for four-dimensional spacetimes with a non-vanishing cosmological constant, including charges that are not conserved because of a leaky boundary and charges associated with corner terms in the symplectic current. The construction leads to manifestly finite charges for any choice of boundary conditions, and reveals new charges in partial Bondi gauge for an enlarged class of field configurations which are not fully on-shell, including a Weyl charge for conformal boundary conditions. In addition, we generalize the variational principle for AdS$_4$ gravity with Dirichlet boundary conditions to a wedge of spacetime where the conformal boundary includes a corner. The boundary data is completely general, with no conditions restricting time dependence or the determinant of the metric. The Ward identities associated with boundary diffeomorphisms are shown to reproduce the evolution equations for the mass and angular momentum of fully on-shell field configurations.