Horizon phase spaces in general relativity

Abstract

We derive a prescription for the phase space of general relativity on two intersecting null surfaces. The boundary symmetry group is the semidirect product of the group of arbitrary diffeomorphisms of each null boundary which coincide at the corner, with a group of reparameterizations of the null generators. The phase space can be extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces, and it then includes the relative boost angle between the null surfaces. We compute Wald-Zoupas gravitational charges and fluxes associated with the boundary symmetries. There is a two-parameter freedom in the charges corresponding to different choices of polarization on the phase space, which cannot be eliminated using the Wald-Zoupas stationarity criterion. We also derive the symmetry groups and charges for a phase space subspace obtained by fixing the direction of the normal vectors, and for another subspace obtained by fixing their direction and normalization. The latter group consists of Carrollian diffeomorphisms. Specializing to perturbations about stationary event horizons, we determine the independent dynamical degrees of freedom by solving the constraint equations along the horizons. We mod out by the degeneracy directions of the presymplectic form, and apply a similar procedure for weak non-degeneracies, to obtain the horizon edge modes and the Poisson structure. We show that the area operator of the black hole generates a shift in the relative boost angle under the Poisson bracket.