Multiple Killing Horizons and Near Horizon Geometries

TL;DR

This paper classifies near horizon geometries with multiple degenerate Killing horizons, showing they all derive from hypersurface-orthogonal Killing vectors and are uniquely determined up to isometry.

Contribution

It provides a complete classification of near horizon geometries with multiple degenerate Killing horizons, revealing their structure and uniqueness properties.

Findings

All near horizon geometries from a given multiple degenerate Killing horizon are isometric.

Cuts of the horizon are warped products with maximally symmetric fibers.

The structure of the metric is explicitly determined for horizons of order m ≥ 3.

Abstract

Near Horizon Geometries with multiply degenerate Killing horizons $\mathcal{H}$ are considered, and their degenerate Killing vector fields identified. We prove that they all arise from hypersurface-orthogonal Killing vectors of any cut of $\mathcal{H}$ with the inherited metric -- cuts are spacelike co-dimension two submanifolds contained in $\mathcal{H}$. For each of these Killing vectors on a given cut, there are three different possibilities for the Near Horizon metric which are presented explicitly. The structure of the metric for Near Horizon Geometries with multiple Killing horizons of order $m\geq 3$ is thereby completely determined, and in particular we prove that the cuts on $\mathcal{H}$ must be warped products with maximally symmetric fibers (ergo of constant curvature). The question whether multiple degenerate Killing horizons may lead to inequivalent Near Horizon Geometries by using different degenerate Killings is addressed, and answered on the negative: all Near Horizon geometries built from a given multiple degenerate Killing horizon (using different degenerate Killings) are isometric.