All Vacuum Near-Horizon Geometries in $D$-dimensions with $(D-3)$ Commuting Rotational Symmetries

TL;DR

This paper classifies all extremal near-horizon geometries in D-dimensional vacuum spacetimes with specific symmetries, revealing three families of solutions with diverse topologies and many parameters, some of which are not linked to known black holes.

Contribution

It provides a comprehensive, dimension-independent classification of near-horizon geometries with $(D-3)$ commuting rotational symmetries using a novel matrix formulation of Einstein's equations.

Findings

Identifies three families of near-horizon geometries with distinct topologies.

Finds that these geometries depend on multiple parameters, including topology and continuous variables.

Notes that some solutions in higher dimensions are not associated with known black hole solutions.

Abstract

We explicitly construct all stationary, non-static, extremal near horizon geometries in $D$ dimensions that satisfy the vacuum Einstein equations, and that have $D-3$ commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in $D=4,5$. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology $S^2 \times T^{D-4}$, or $S^3 \times T^{D-5}$, or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as $(D-2)(D-3)/2$ continuous parameters. Not all of our metrics in $D \ge 6$ seem to arise as the near horizon limits of known black hole solutions.