Generalised Couch-Torrence Symmetry for Rotating Extremal Black Holes in Maximal Supergravity

TL;DR

This paper extends the concept of conformal inversion symmetry to rotating extremal black holes in maximal supergravity, revealing mode-dependent and charge-transformed symmetries in the radial equations of these black holes.

Contribution

It introduces a generalized inversion symmetry for rotating extremal STU supergravity black holes, including those with multiple electric and magnetic charges, expanding previous static and non-rotating results.

Findings

Mode-dependent inversion symmetry in rotating black holes

Radial equations map to those with transformed charges

Generalization to black holes with electric and magnetic charges

Abstract

The extremal Reissner-Nordstr\"om black hole admits a conformal inversion symmetry, in which the metric is mapped into itself under an inversion of the radial coordinate combined with a conformal rescaling. In the rotating generalisation, Couch and Torrence showed that the Kerr-Newman metric no longer exhibits a conformal inversion symmetry, but the radial equation arising in the separation of the massless Klein-Gordon equation admits a mode-dependent inversion symmetry, where the radius of inversion depends upon the energy and azimuthal angular momentum of the mode. It was more recently shown that the static 4-charge extremal black holes of STU supergravity admit a generalisation of the conformal inversion symmetry, in which the conformally-inverted metric is a member of the same 4-charge black hole family but with transformed charges. In this paper we study further generalisations of these inversion symmetries, within the general class of extremal STU supergravity black holes. For the rotating black holes, where again the massless Klein-Gordon equation is separable, we show that examples with four electric charges exhibit a generalisation of the Couch-Torrence symmetry of the radial equation. Now, as in the conformal inversion of the static specialisations, the inversion of the radial equation maps it to the radial equation for a rotating black hole with transformed electric charges. We also study the inversion transformations for the general case of extremal BPS STU black holes carrying eight charges (4 electric plus 4 magnetic), and argue that analogous generalisations of the inversion symmetries exist both for the static and the rotating cases.