[Paper Review] All vacuum near horizon geometries in arbitrary dimensions

Stefan Hollands, Akihiro Ishibashi|arXiv (Cornell University)|Sep 18, 2009

Black Holes and Theoretical Physics14 citations

TL;DR

This paper presents a matrix-based method to classify all stationary, non-static, extremal near-horizon geometries in D-dimensional vacuum Einstein gravity with D−3 commuting rotational symmetries. It identifies three families of solutions with horizon topologies S²×T^{D−4}, S³×T^{D−5}, or their quotients, depending on two discrete topological parameters and (D−2)(D−3)/2 continuous parameters—extending prior work in D=4,5 to arbitrary dimensions using a novel matrix formulation of the Einstein equations.

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 $S^2 imes T^{D-4}$, or $S^3 imes T^{D-5}$, 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.

Motivation & Objective

• To extend the classification of extremal near-horizon geometries beyond D=4,5 to arbitrary dimensions.
• To provide a systematic construction of all stationary, non-static, extremal vacuum near-horizon geometries with D−3 commuting rotational isometries.
• To identify the full moduli space of such geometries, including their topological and continuous parameters.
• To determine whether these geometries arise as near-horizon limits of known black hole solutions in D≥6.

Proposed method

• A matrix formulation of the vacuum Einstein equations is developed, enabling a systematic analysis in arbitrary dimensions.
• The method exploits the presence of D−3 commuting rotational symmetries to reduce the field equations to a set of matrix differential equations.
• The ansatz for the metric is constructed to preserve the required isometries and extremality conditions.
• The resulting matrix equations are solved to classify all possible solutions based on topological and continuous parameters.
• The approach avoids the dimensional dependence of earlier methods, such as those relying on explicit coordinate parametrizations.
• Solutions are classified by their horizon topology, with distinct families arising from different topological choices.

Experimental results

Research questions

• RQ1 What is the complete set of stationary, non-static, extremal near-horizon geometries in D-dimensional vacuum gravity with D−3 rotational symmetries?
• RQ2 How do the solutions depend on discrete topological parameters and continuous moduli in arbitrary D?
• RQ3 Do all solutions in D≥6 arise as near-horizon limits of known black hole solutions?
• RQ4 Can a unified method be developed to classify such geometries across all dimensions, independent of dimension-specific techniques?
• RQ5 What are the possible horizon topologies for these extremal geometries beyond S²×T^{D−4} and S³×T^{D−5}?

Key findings

• The paper identifies three distinct families of extremal near-horizon geometries: with horizon topologies S²×T^{D−4}, S³×T^{D−5}, or their quotients.
• Solutions depend on two discrete parameters specifying the topology and (D−2)(D−3)/2 continuous parameters, forming a moduli space of solutions.
• The matrix formulation enables a dimension-independent classification, generalizing prior results limited to D=4,5.
• Not all solutions in D≥6 can be identified as near-horizon limits of known black hole solutions, suggesting new geometric structures.
• The method provides a systematic framework to explore extremal black hole geometries beyond the scope of previous approaches.