Conformal Symmetries of the Einstein-Hilbert Action on Horizons of Stationary and Axisymmetric Black Holes

TL;DR

This paper explores conformal symmetries on black hole horizons by reducing the Einstein-Hilbert action, revealing infinite-dimensional conformal symmetries related to angular velocities in stationary, axisymmetric black holes.

Contribution

It introduces a novel approach using Kaluza-Klein reduction to identify conformal symmetries on black hole horizons, connecting angular velocities to infinite-dimensional algebras.

Findings

Identifies $k$-copies of conformal symmetries on horizons

Relates angular velocities to $SL(2,R)$ subgroups and Witt algebra

Shows classical Einstein-Hilbert action admits these symmetries

Abstract

We suggest a way to study possible conformal symmetries on black hole horizons. We do this by carrying out a Kaluza-Klein like reduction of the Einstein-Hilbert action along the ignorable coordinates of stationary and axisymmetric black holes. Rigid diffeomorphism invariance of the $m$-ignorable coordinates then becomes a global $SL(m,R)$ gauge symmetry of the reduced action. Related to each non-vanishing angular velocity there is a particular $SL(2,R)$ subgroup, which can be extended to the Witt algebra on the black hole horizons. The classical Einstein-Hilbert action thus has $k$-copies of infinite dimensional conformal symmetries on a given black hole horizon, with $k$ being the number of non-vanishing angular velocities of the black hole.