Horizon Hair from Inversion Symmetry

TL;DR

This paper investigates the symmetry properties of the extreme Reissner-Nordstr"om black hole, revealing how the Couch-Torrence inversion relates gauge parameters and conserved quantities at the horizon and null infinity.

Contribution

It introduces a gauge fixing compatible with inversion symmetry and relates gauge parameters and conserved charges at different regions of the black hole spacetime.

Findings

Constructed Newman-Penrose and Aretakis conserved quantities.

Related horizon and null infinity charges via inversion symmetry.

Provided a gauge fixing linking gauge parameters at horizon and infinity.

Abstract

The extreme Reissner-Nordstr\"om solution has a discrete conformal isometry that maps the future event horizon to future null infinity and vice versa, the Couch-Torrence inversion isometry. We study the dynamics of a probe Maxwell field on the extreme Reissner-Nordstr\"om solution in light of this symmetry. We present a gauge fixing that is compatible with the inversion symmetry. The gauge fixing allows us to relate the gauge parameter at the future horizon to future null infinity, which further allows us to study global charges for large gauge symmetries in the exterior of the extreme Reissner-Nordstr\"om black hole. Along the way, we construct Newman-Penrose and Aretakis like conserved quantities along future null infinity and the future event horizon, respectively, and relate them via the Couch-Torrence inversion symmetry.