Higher dimensional black holes

TL;DR

This paper surveys higher-dimensional vacuum GR black holes, emphasizing two main families—Kaluza-Klein configurations (including black strings) and asymptotically flat solutions such as Myers-Perry black holes and black rings. It synthesizes exact solutions, perturbative constructions (notably the blackfold approach), and stability analyses, highlighting phenomena like the Gregory-Laflamme instability and ultraspinning instabilities that challenge simple uniqueness. The discussion outlines how topology and symmetry constraints shape possible solutions, and it identifies potential bifurcations to new branches of black holes with reduced symmetry. Overall, the work clarifies the rich landscape of higher-dimensional black holes and lays groundwork for understanding GR in more than four dimensions, with implications for string theory and holography.

Abstract

This article reviews black hole solutions of higher-dimensional General Relativity. The focus is on stationary vacuum solutions and recent work on instabilities of such solutions.

Figures (5)

• Figure 1: Schematic plot of horizon area against mass for Kaluza-Klein black holes/strings with $D=5,6$ based on results of Refs Kudoh:2004hsHeadrick:2009pv. The dashed curve is the uniform black string branch, the solid curve the non-uniform string branch and the dotted curve the localized black hole branch. It seems likely that this will be the qualitative behaviour for all $D \le 11$. (Plot reproduced from Ref. Figueras:2012xj.)
• Figure 2: Schematic plot of horizon area against mass for Kaluza-Klein black holes/strings with $D=12,13$ based on results of Ref. Figueras:2012xj. The localized black hole curve is conjectural. (Plot reproduced from Ref. Figueras:2012xj.)
• Figure 3: Schematic plot of horizon area against mass for Kaluza-Klein black holes/strings with $D>13$ based on results of Ref. Figueras:2012xj. The localized black hole curve is conjectural. (Plot reproduced from Ref. Figueras:2012xj.)
• Figure 4: Parameter space of Myers-Perry black holes for (a) $D=5$ and (b) $D=6$. The axes are dimensionless angular moment $j_I \sim J_I M^{-(D-2)/(D-3)}$. Non-extreme black holes correspond to the shaded region. The boundary of this region corresponds to extreme black holes, except for the vertices of the square, which describe singular solutions. (Plot reproduced from Ref. Emparan:2008eg.)
• Figure 5: Phase space of $D=5$ Myers-Perry black holes and black rings. See Fig. \ref{['figure:MPphasespace']} for notation. For each point of the light grey regions there exists a "thin" black ring. For each point of the mid-grey region there exists a MP black hole. For each point of the dark grey region there exists a MP black hole, a fat black ring and a thin black ring. (Plot reproduced from Ref. Emparan:2008eg.)