Supertranslations: redundancies of horizon data and global symmetries at null infinity

Abstract

Abstract:We characterise the geometrical nature of smooth supertranslations defined on a generic non-expanding horizon (NEH) embedded in vacuum. To this end we consider the constraints imposed by the vacuum Einstein's equations on the NEH structure, and discuss the transformation properties of their solutions under supertranslations. We present a freely specifiable data set which is both necessary and sufficient to reconstruct the full horizon geometry, and is composed of objects which are invariant under supertranslations. We conclude that smooth supertranslations do not transform the geometry of the NEH, and that they should be regarded as pure gauge. Our results apply both to stationary, and non-stationary states of a NEH, the later ones being able to describe radiative processes taking place on the horizon. As a consistency check we repeat the analysis for BMS supertranslations defined on null infinity, I. Using the same framework as for the NEH we recover the well known result that BMS supertranslations act non-trivially on the free data on I. The full analysis is done in exact, non-linear, general relativity.

On Super-Translation transition between Quasi-local Black holes

“…The discussion closely follows [34] but our argument is essentially in the reversed order. In recent times there has been discussions on symmetries of intrinsic geometry of non-expanding horizons [35], null shells [36,37] and general null surfaces [38]. While, we recover some (appropriate to our choice of boundary conditions) of these results, our argument differs from these.…”

“…In the current context one might be able to discuss the transition completely on the phase space of non-expanding horizon that includes radiative solutions as in [35]. The phase space of boundary data on a non-expanding horizon (in this case ∆ 1 ∪ H ∪ ∆ 2 ) therefore needs to be constructed with the inclusion of matter as opposed to gravitational data alone.…”

Area law unification and the holographic event horizon

“…Given a Cauchy surface formed by light sheets, the characteristic initial data formalism implies that a spacetime exists for self-consistent initial data satisfying the constraint equations. For the null portion of a Cauchy surface in the k direction, the constraint equations are [29,30,[60][61][62][63][64]…”

Inducing supertranslations: The membrane picture

“…The main aim then was to reproduce the Bekenstein -Hawking entropy formula [7,8] by counting the states in a representation of an effective conformal symmetry on the horizon [9,10]. In recent times, however, the general interest has been to find BM S like symmetries near the horizon [12][13][14][15][16][17][18][19][20][21][22][23] or null surfaces in general [24,25]. The main aim of this paper wont be along these lines but an attempt to study dynamical black hole spacetime in relation to these symmetries.…”