Phases of Five-Dimensional Black Holes

TL;DR

The paper shows that in five-dimensional vacuum gravity, configurations maximizing entropy for fixed mass and angular momentum are black Saturns, comprising a central near-static black hole plus a very thin surrounding ring, which yields an effectively infinite-dimensional phase space from multi-ring arrangements. It develops a first-law formalism for multi-black-hole systems and analyzes how entropy scales with ring thickness, revealing that entropy-maximizing states are not in thermodynamic equilibrium unless radiative effects are included. Imposing thermodynamic equilibrium collapses the phase space to a small set of phases (MP black holes, black rings, and single-ring Saturns). The work exposes rich, non-unique phase structure beyond four-dimensional uniqueness and discusses dynamical stability and the impact of quantum effects on the entropy bounds.

Abstract

We argue that the configurations that approach maximal entropy in five-dimensional asymptotically flat vacuum gravity, for fixed mass and angular momentum, are `black Saturns' with a central, close to static, black hole and a very thin black ring around it. For any value of the angular momentum, the upper bound on the entropy is equal to the entropy of a static black hole of the same total mass. For fixed mass, spin and area there are families of multi-ring solutions with an arbitrarily large number of continuous parameters, so the total phase space is infinite-dimensional. Somewhat surprisingly, the phases of highest entropy are not in thermal equilibrium. Imposing thermodynamical equilibrium drastically reduces the phase space to a finite, small number of different phases.

Figures (2)

• Figure 1: Phases of five-dimensional black holes including black Saturns. We fix total mass $M=1$ and plot total area vs. spin. The solid curves correspond to single MP black holes and black rings. The semi-infinite shaded strip, spanning $0\leq \tilde{J}<\infty$, $0<\tilde{\mathcal{}}{A}<2\sqrt{2}$, is covered by black Saturns. Each point in the strip actually corresponds to a one-parameter family of Saturn solutions. The top end $\tilde{\mathcal{}}{A}=2\sqrt{2}$ with $\tilde{J}\neq 0$ is reached only asymptotically for black Saturns with infinitely long rings. Solutions at the bottom $\tilde{\mathcal{}}{A}=0$ are naked singularities. For a fixed value of $\tilde{J}$ we can move from the top of the strip to the bottom by varying the spin of the central black hole $\tilde{J}_{h}$ from 0 to 1. For fixed area we can move horizontally by having $\tilde{J}_{h}<0$ and varying the spin of the ring between $|\tilde{J}_{h}|$ and $\infty$.
• Figure 2: Phases in thermodynamical equilibrium. Besides the gray curves for MP black holes and black rings, we have the solid black curve of single-ring black Saturns for which the central black hole and the black ring have equal temperature and equal angular velocity. There is only one black Saturn (and not a continuous family of them) for each point in this curve. The minimum $\tilde{J}$ along the curve is $\approx 0.9245$, slightly above $\sqrt{27/32}$, and the maximum total area is $\tilde{\mathcal{A}} \approx 0.81$. Note that black Saturns are never entropically dominant among the phases in thermodynamical equilibrium. In the text we argue that it is unlikely that multi-ring black Saturns exist as states in thermodynamical equilibrium. Barring more exotic possibilities, this diagram may then contain all thermodynamical equilibrium states of 5D black holes.