Black Rings

Black rings are higher-dimensional black holes whose horizon contains a noncontractible circle. The neutral singly spinning five-dimensional ring is an exact asymptotically flat vacuum solution with horizon topology $S^1\times S^2$. Rotation can balance it only on a specific conical-regularity locus. Balanced rings furnish an explicit failure of uniqueness by mass and angular momentum alone, but they are not observed objects and their stability is a separate question.

Five-dimensional black-ring geometry

Property	Value
Object class	Exact solution family
Theory	Five-dimensional vacuum general relativity for the baseline ring
Parameters	scale $R$ and shape parameters $(\lambda,\nu)$
Asymptotics	Asymptotically flat with two rotational symmetries
Physical status	Exact higher-dimensional solution; not an observed object class

Use ring coordinates

$$-1\le x\le1, \qquad y\le-1, \qquad 0\lt\nu\le\lambda\lt1,$$

and define

$$F(\xi)=1+\lambda\xi, \qquad G(\xi)=(1-\xi^2)(1+\nu\xi).$$

One singly spinning metric is

$$\begin{aligned} ds^2={}&-\frac{F(y)}{F(x)} \left(dt-CR\frac{1+y}{F(y)}d\psi\right)^2\\ &+\frac{R^2F(x)}{(x-y)^2} \left[-\frac{G(y)}{F(y)}d\psi^2-\frac{dy^2}{G(y)} +\frac{dx^2}{G(x)}+\frac{G(x)}{F(x)}d\phi^2\right], \end{aligned}$$

where

$$C=\sqrt{\lambda(\lambda-\nu)\frac{1+\lambda}{1-\lambda}}.$$

Spatial infinity is the corner $x,y\to-1$. The horizon is at $y=-1/\nu$, while $F(y)=0$ locates the stationary-limit surface. Horizon-regular coordinates reveal the $S^1$ generated by $\psi$ and the $S^2$ described by $(x,\phi)$. The solution was discovered by Emparan and Reall 2002, complete article, printed pp. 101101-1–101101-4, eqs. (1)–(17).

In the coordinates displayed, regularity at the asymptotic axes $x=y=-1$ requires the common identification

$$\Delta\phi=\Delta\psi =2\pi\frac{\sqrt{1-\lambda}}{1-\nu}.$$

Canonical $2\pi$ angles are $\widetilde\phi=(1-\nu)\phi/\sqrt{1-\lambda}$ and $\widetilde\psi=(1-\nu)\psi/\sqrt{1-\lambda}$. The charge formulas below refer to the corresponding canonically normalized asymptotic rotation generators.

Balance and global regularity

Axis regularity fixes angular periods. After removing conical singularities at the asymptotic axes, the inner axis is regular only if

$$\lambda=\frac{2\nu}{1+\nu^2}.$$

Without this balance condition, the metric is vacuum away from the axis but contains a conical disk or strut supplying external force. Local Ricci flatness is therefore not enough to define an isolated ring. The allowed coordinate cell, common angular normalization, and every axis must be checked together.

The parameters outside $0\lt\nu\lt1$ or away from the balance locus do not form a conventional “superextremal ring” branch. They can change signature, destroy the regular horizon, or introduce a defect. Unlike Kerr, the neutral singly spinning ring is not organized by one charge discriminant with a universal extremal continuation.

Charges, branches, and nonuniqueness

With asymptotic azimuths normalized to period $2\pi$,

$$M=\frac{3\pi R^2}{4G_5}\frac{\lambda}{1-\nu}, \qquad J_\psi=\frac{\pi R^3}{2G_5} \frac{\sqrt{\lambda(\lambda-\nu)(1+\lambda)}}{(1-\nu)^2}.$$

Define the scale-free spin

$$j^2=\frac{27\pi J_\psi^2}{32G_5M^3}.$$

On the balanced branch,

$$j^2(\nu)=\frac{(1+\nu)^3}{8\nu}.$$

It reaches a minimum $27/32$ at $\nu=1/2$. For $27/32\lt j^2\lt1$, a thin and a fat balanced ring have the same $(M,J_\psi)$, and a singly spinning Myers–Perry black hole also exists with those charges. This is exact stationary nonuniqueness, not merely a perturbative degeneracy. The broader phase structure is organized in Emparan and Reall 2008, § 5.1.1.

Charges and dipole parameters in supergravity rings require further care. A dipole flux through the $S^2$ need not be conserved at infinity, yet it can enter local mechanics. Such rings solve a different action from the neutral metric above; they should not be merged into one parameter table without stating gauge fields and charge definitions.

Mechanics, limits, and dynamics

A balanced neutral ring obeys

$$dM=\frac{\kappa}{8\pi G_5}\,dA_H+\Omega_H\,dJ_\psi,$$ $$2M=3\left(\frac{\kappa A_H}{8\pi G_5}+\Omega_HJ_\psi\right).$$

An unbalanced ring needs an additional defect work term. Applying the balanced first law before removing the conical defect silently drops its external support.

Thin-ring limit. Along balance, $\nu\to0$ and $\lambda\sim2\nu$. A local segment approaches a boosted black string while the global $S^1$ closes the object and makes total charges finite.

Cusp. At $\nu=1/2$, thin and fat branches meet at the minimum dimensionless spin. This is a turning point in the family, not an extremal zero-surface-gravity horizon.

Fat endpoint. As $\nu\to1$, the singly spinning balanced branch approaches a singular endpoint. It does not become a regular Myers–Perry black hole by direct substitution; coexistence at equal charges is not a continuous regular limit within this family.

Exactness and balance still do not imply stability. Santos and Way 2015, complete article, printed pp. 221101-1–221101-5 computed a nonaxisymmetric $m=2$ instability extending from thin rings into part of the fat branch. Independently, Figueras, Murata, and Reall 2011, §§ 5.1–5.3 established an axisymmetric instability for fat rings using local Penrose inequalities. Formation and nonlinear endpoints remain separate dynamical problems.

Domain and comparison diagnostics

The coordinate boundaries have different roles: $x=\pm1$ are poles of the $S^2$, an exterior $y$ boundary is an axis of the ring circle, $y=-1/\nu$ is the horizon, and the corner $x,y\to-1$ is infinity. A common angular period must regularize all intended axes. Checking curvature only away from an axis can miss a distributional conical source, so balance and invariant regularity are complementary tests.

Nonuniqueness by $(M,J)$ does not mean the geometries are observationally or dynamically indistinguishable. Horizon topology, area, angular velocity, multipole data, and response spectra can separate the branches. Nor does the existence of three stationary solutions at the same charges determine which one forms or is stable; that requires initial data, perturbations, and nonlinear evolution.

The $S^1\times S^2$ section satisfies higher-dimensional positive-curvature restrictions because the sphere supplies positive scalar curvature while the circle is flat. This explains why the topology is allowed without implying that arbitrary products or higher-genus sections are regular asymptotically flat vacuum horizons.

Connections to the detailed theory

The coordinate domain, metric, periods, and charge normalization are developed in Five-Dimensional Black Rings. The balance condition, mechanics, and exact failure of charge uniqueness are organized in Black Rings and Horizon Nonuniqueness. These exact results neither classify every ring in every theory nor give direct astrophysical evidence for extra dimensions.

References

• Emparan, Roberto, and Harvey S. Reall. “A Rotating Black Ring Solution in Five Dimensions.” Physical Review Letters 88 (2002): 101101. DOI.
• Emparan, Roberto, and Harvey S. Reall. “Black Holes in Higher Dimensions.” Living Reviews in Relativity 11 (2008): 6. DOI.
• Figueras, Pau, Keiju Murata, and Harvey S. Reall. “Black Hole Instabilities and Local Penrose Inequalities.” Classical and Quantum Gravity 28 (2011): 225030. arXiv.
• Santos, Jorge E., and Benson Way. “The Black Ring Is Unstable.” Physical Review Letters 114 (2015): 221101. DOI.