Horizon phase spaces in general relativity

Abstract

Abstract We derive a prescription for the phase space of general relativity on two intersecting null surfaces using the null initial value formulation. The phase space allows generic smooth initial data, and the corresponding 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 consistently extended by acting with half-sided boosts that generate Weyl shocks along the initial data surfaces. The extended phase space includes the relative boost angle between the null surfaces as part of the initial data.We then apply the Wald-Zoupas framework to compute gravitational charges and fluxes associated with the boundary symmetries. The non-uniqueness in the charges can be reduced to two free parameters by imposing covariance and invariance under rescalings of the null normals. We show that the Wald-Zoupas stationarity criterion cannot be used to eliminate the non-uniqueness. The different choices of parameters correspond to different choices of polarization on the phase space. We also derive the symmetry groups and charges for two subspaces of the phase space, the first obtained by fixing the direction of the normal vectors, and the second by fixing the direction and normalization of the normal vectors. The second symmetry group consists of Carrollian diffeomorphisms on the two boundaries.Finally we specialize to future event horizons by imposing the condition that the area element be non-decreasing and become constant at late times. For perturbations about stationary backgrounds 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.