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

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 latter ones being able to describe radiative processes taking place on the horizon. As a consistency check we repeat the analysis for Bondi–Metzner–Sachs (BMS) supertranslations defined on null infinity, . 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 . The full analysis is made in exact, non-linear, general relativity.