Conformal symmetries of the Einstein–Hilbert action on horizons of stationary and axisymmetric black holes

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.