On the Residual Symmetries of the Gravitational Field

TL;DR

This paper introduces a geometric criterion to systematically identify residual symmetries in gravitational Ansätze, applying it to various spacetimes including axisymmetric, noncircular, and AdS waves, revealing complex infinite-dimensional symmetry structures.

Contribution

It provides a new systematic method for finding residual symmetries in gravitational solutions, clarifying their algebraic structures and effects of coordinate choices.

Findings

Reproduced known residual symmetries for the Collinson Ansatz.

Identified supertranslations and conformal symmetry breaking in noncircular cases.

Derived the residual symmetry algebra for AdS waves across dimensions.

Abstract

We develop a geometric criterion that unambiguously characterizes and also provides a systematic and effective computational procedure for finding all the residual symmetries of any gravitational Ansatz . We apply the criterion to several examples starting with the Collinson Ansatz for circular stationary axisymmetric spacetimes. We reproduce the residual symmetries already known for this Ansatz including their conformal symmetry, for which we identify the corresponding infinite generators spanning the two related copies of the Witt algebra. We also consider the noncircular generalization of this Ansatz and show how the noncircular contributions on the one hand break the conformal invariance and on the other hand enhance the standard translations symmetries of the circular Killing vectors to supertranslations depending on the direction along which the circularity is lost. As another application of the method, the well-known relation defining conjugate gravitational potentials introduced by Chandrasekhar, that makes possible the derivation of the Kerr black hole from a trivial solution of the Ernst equations, is deduced as a special point of the general residual symmetry of the Papapetrou Ansatz. We emphasize how the election of Weyl coordinates, which determines the Papapetrou Ansatz, breaks also the conformal freedom of the stationary axisymmetric spacetimes. Additionally, we study AdS waves for any dimension generalizing the residual symmetries already known for lower dimensions and exhibiting a very complex infinite-dimensional Lie algebra containing three families: two of them span the semidirect sum of the Witt algebra and scalar supertranslations and the third generates vector supertranslations. We manage to comprehend that the infinite connected group is the precise diffeomorphisms subgroup allowing to locally deform the AdS background into AdS waves.