Charges and fluxes on (perturbed) non-expanding horizons

Abstract

Abstract In a companion paper [1] we showed that the symmetry group G $$ \mathfrak{G} $$ of non-expanding horizons (NEHs) is a 1-dimensional extension of the Bondi-Metzner-Sachs group B $$ \mathfrak{B} $$ at I $$ \mathcal{I} $$ +. For each infinitesimal generator of G $$ \mathfrak{G} $$ , 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 N $$ \mathcal{N} $$ along the lines of [2–6]. However, N $$ \mathcal{N} $$ is required to be an NEH or a perturbed NEH. Consequently, charges and fluxes associated with generators of G $$ \mathfrak{G} $$ are free of physically unsatisfactory features that can arise if N $$ \mathcal{N} $$ is allowed to be a general null boundary. In particular, all fluxes vanish if N $$ \mathcal{N} $$ 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], I $$ \mathcal{I} $$ ± are NEHs in the conformally completed space-time but with an extra structure that reduces G $$ \mathfrak{G} $$ to B $$ \mathfrak{B} $$ . The flux expressions at N $$ \mathcal{N} $$ reflect this synergy between NEHs and I $$ \mathcal{I} $$ +. In a forthcoming paper, this close relation between NEHs and I $$ \mathcal{I} $$ + will be used to develop gravitational wave tomography, enabling one to deduce horizon dynamics directly from the waveforms at I $$ \mathcal{I} $$ +.

Keywords

• Black Holes
• Space-Time Symmetries