Multiple Killing Horizons

TL;DR

This paper rigorously studies multiple Killing horizons, revealing their algebraic structure, uniqueness of surface gravity, and classifying all types in maximally symmetric spacetimes, with implications for near horizon geometries.

Contribution

It provides a rigorous definition, algebraic classification, and explicit examples of multiple Killing horizons, including their relation to near horizon geometries.

Findings

The set of Killing vectors sharing a horizon forms a Lie algebra of bounded dimension.

Multiple Killing horizons have a unique non-zero surface gravity or none.

Complete classification of multiple Killing horizons in maximally symmetric spacetimes.

Abstract

Killing horizons which can be such for two or more linearly independent Killing vectors are studied. We provide a rigorous definition and then show that the set of Killing vectors sharing a Killing horizon is a Lie algebra $\mathcal{A}_{\mathcal{H}}$ of dimension at most the dimension of the spacetime. We prove that one cannot attach different surface gravities to such multiple Killing horizons, as they have an essentially unique non-zero surface gravity (or none). $\mathcal{A}_{\mathcal{H}}$ always contains an Abelian (sub)-algebra ---whose elements all have vanishing surface gravity--- of dimension equal to or one less than dim $\mathcal{A}_{\mathcal{H}}$. There arise only two inequivalent possibilities, depending on whether or not there exists the non-zero surface gravity. We show the connection with Near Horizon geometries, and also present a linear system of PDEs, the master equation, for the proportionality function on the horizon between two Killing vectors of a multiple Killing horizon, with its integrability conditions. We provide explicit examples of all possible types of multiple Killing horizons, as well as a full classification of them in maximally symmetric spacetimes.