Charges and fluxes on (perturbed) non-expanding horizons

Abstract

Abstract In a companion paper [1] we showed that the symmetry group $$ \mathfrak{G} $$ G of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group $$ \mathfrak{B} $$ B at $$ \mathcal{I} $$ I +. For each infinitesimal generator of $$ \mathfrak{G} $$ G , we now define a charge and a flux on NEHs as well as perturbed NEHs. The procedure uses the covariant phase space framework in presence of internal null boundaries $$ \mathcal{N} $$ N along the lines of [2–6]. However, $$ \mathcal{N} $$ N is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of $$ \mathfrak{G} $$ G are free of physically unsatisfactory features that can arise if $$ \mathcal{N} $$ N is allowed to be a general null boundary. In particular, all fluxes vanish if $$ \mathcal{N} $$ N is an NEH, just as one would hope; and fluxes associated with symmetries representing ‘time-translations’ are positive definite on perturbed NEHs. These results hold for zero as well as non-zero cosmological constant. In the asymptotically flat case, as noted in [1], $$ \mathcal{I} $$ I ±are NEHs in the conformally completed space-time but with an extra structure that reduces $$ \mathfrak{G} $$ G to $$ \mathfrak{B} $$ B . The flux expressions at $$ \mathcal{N} $$ N reflect this synergy between NEHs and $$ \mathcal{I} $$ I +. In a forthcoming paper, this close relation between NEHs and $$ \mathcal{I} $$ I + will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at $$ \mathcal{I} $$ I +.