Black Holes in Higher Dimensions

TL;DR

The paper surveys the landscape of black holes in higher dimensions, focusing on vacuum and gauged-supergravity contexts. It develops a framework for conserved charges and horizon properties, reviews Myers–Perry solutions and the novel five-dimensional phenomena (notably black rings and the Weyl/rod formalisms), and extends to higher dimensions with approximate and solution-generating approaches. It also analyzes stability, phase structure, topology, and AdS/cosmological-constant contexts, highlighting open problems such as horizon rigidity, uniqueness, and the full classification of high-dimensional black holes. Overall, the work illuminates how extra dimensions enable rich horizon geometries, non-uniqueness, and intricate phase diagrams with potential implications for string theory, gravity, and holography.

Abstract

We review black hole solutions of higher-dimensional vacuum gravity, and of higher-dimensional supergravity theories. The discussion of vacuum gravity is pedagogical, with detailed reviews of Myers-Perry solutions, black rings, and solution-generating techniques. We discuss black hole solutions of maximal supergravity theories, including black holes in anti-de Sitter space. General results and open problems are discussed throughout.

Figures (14)

• Figure 1: Horizon area vs. angular momentum for Myers-Perry black holes with a single spin in $d=5$ (black), $d=6$ (dark gray), $d=10$ (light gray).
• Figure 2: Phase space of (a) five-dimensional and (b) six-dimensional MP rotating black holes: black holes exist for parameters within the shaded regions. The boundaries of the phase space correspond to extremal black holes with regular horizons, except at the corners of the square in five dimensions where they become naked singularities. The six-dimensional phase space extends along the axes to arbitrarily large values of each of the two angular momenta (ultra-spinning regimes).
• Figure 3: Phase space of (a) seven-dimensional, and (b) eight-dimensional MP rotating black holes (in a representative quadrant $j_i\geq 0$). The surfaces for extremal black holes are represented: black holes exist in the region bounded by these surfaces. (a) $d=7$: the hyperbolas where the surface intersects the planes $j_i=0$ (which are $j_k j_l=1/\sqrt{6}$, i.e., $a_k a_l=\sqrt{\mu}$, and $r_0=0$) correspond to naked singularities with zero area, otherwise the extremal solutions are non-singular. The three prongs extend to infinity: these are the ultra-spinning regimes in which one spin is much larger than the other two. The prong along $j_i$ becomes asymptotically of the form $|j_k|+|j_l|\leq f(j_i)$, i.e., the same shape as the five-dimensional diagram fig. \ref{['figure:phasespace']}(a). (b) $d=8$: ultra-spinning regimes exist in which two spins are much larger than the third one. The sections at large constant $j_i$ approach asymptotically the same shape as the six-dimensional phase space fig. \ref{['figure:phasespace']}(b).
• Figure 4: Horizon area $a_H(j_1,j_2)$ of five-dimensional MP black holes. We only display a representative quadrant $j_1,j_2\geq 0$ of the full phase space of figure \ref{['figure:phasespace']}(a), the rest of the surface being obtained by reflection along the planes $j_1=0$ and $j_2=0$.
• Figure 5: Curve $a_H(j)$ of horizon area vs. spin for five-dimensional black rings rotating along their $S^1$ (solid). The dashed curve corresponds to five-dimensional MP black holes (see figure \ref{['figure:aHj']}). The solid curve for black rings has two branches that meet at a regular, non-extremal minimally rotating black ring at $j=\sqrt{27/32}$: an upper branch of thin black rings, and a lower branch of fat black rings. Fat black rings always have smaller area than MP black holes. Their curves meet at the same zero-area naked singularity at $j=1$.