Multiple Killing Horizons

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.

arXiv comment41 pages

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• General matching across Killing horizons of zero order
• Spacetimes foliated by non-expanding null surfaces in the presence of a cosmological constant
• On static and rotating decoupled black holes without inner horizons
• Multiple Killing Horizons: the initial value formulation for $\Lambda$-vacuum
• Locally Homogeneous Kundt Triples and CSI Metrics