Black Rings and Horizon Nonuniqueness

A neutral five-dimensional black ring is a stationary, asymptotically flat vacuum black hole with horizon topology $S^1\times S^2$. Rotation along the noncontractible circle can balance the ring’s tendency to contract, but only when a conical-regularity condition is satisfied. The balanced family provides an exact and especially sharp failure of four-dimensional uniqueness: within a finite interval of dimensionless spin, two inequivalent rings and one spherical Myers–Perry black hole have the same ADM mass and angular momentum.

Helpful background. Exact Solutions of Einstein’s Equations supplies the full ring metric and coordinate domains; Global Structure supplies horizon topology and regularity; the Einstein equation fixes the vacuum equation; Black-Hole Perturbation Theory supplies the separate stability question; and Black-Hole Geometry and Mechanics supplies the charge and horizon-generator conventions.

Ring geometry and balance

Use ring coordinates with

$$-1\leq x\leq1, \qquad y\leq-1,$$

and dimensionless parameters satisfying

$$0\lt\nu\leq\lambda\lt1.$$

The scale $R$ sets the overall length. The coordinate $\psi$ follows the ring’s $S^1$, while $(x,\phi)$ describe the $S^2$. In a standard parameterization, the event horizon is at $y=-1/\nu$ and the stationary-limit surface is at $y=-1/\lambda$. Horizon-regular coordinates reveal a smooth $S^1\times S^2$ section; these coordinate locations alone are not a regularity proof.

There are three relevant axes. Choosing the angular periods to remove conical singularities at spatial infinity and one pole of the sphere leaves a possible defect at the inner axis. Requiring that axis to have circumference-to-radius ratio $2\pi$ gives

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

This is the balance condition of the exact ring discovered in Emparan and Reall 2002, pp. 101101-1–101101-4. Without it, the metric is vacuum away from the axis but contains a conical disk or strut that supplies an external force. A locally Ricci-flat expression is therefore not yet an isolated balanced black hole.

The condition has an intuitive interpretation: centrifugal support along the $S^1$ offsets self-attraction. It is nevertheless a fully relativistic axis-regularity equation, not a Newtonian force balance. The same distinction matters for more complicated multi-horizon solutions, where regularity conditions can couple several rods and angular velocities.

Charges and a scale-free comparison

With asymptotic azimuths normalized to period $2\pi$, one conventional parameterization gives

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

The formulas must be used with the same angular normalization as the metric. Changing a coordinate period without changing the charge formula creates a spurious factor. To compare geometries independently of their overall scale, define

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

After imposing balance,

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

This function decreases from infinity, reaches $27/32$ at $\nu=1/2$, and then rises to $1$ as $\nu$ approaches the singular fat-ring endpoint. The interval $0\lt\nu\lt1/2$ is called the thin branch; $1/2\lt\nu\lt1$ is the fat branch.

Exact nonuniqueness

For

$$\frac{27}{32}\lt j^2\lt1,$$

the equation above has two balanced-ring solutions: one thin and one fat. A singly spinning five-dimensional Myers–Perry hole also exists throughout $j^2\lt1$. Hence three stationary, asymptotically flat, regular vacuum black holes share the same $(M,J)$ in that interval while differing in horizon topology and geometry. The broader phase structure and its limits are reviewed in Emparan and Reall 2008, §§5.1–5.2.

This result refutes uniqueness by asymptotic monopole charges alone; it does not say that no classification is possible. Additional data—horizon topology, interval structure, multiple angular momenta, dipole charges in matter theories, or sufficiently rich multipoles—can distinguish solutions. Nor does exact coexistence imply equal stability. The fat ring and sufficiently thin or rapidly rotating branches have distinct perturbative questions that require an evolution or spectral problem.

In the thin limit $\nu\to0$, balance gives $\lambda\sim2\nu$ and the dimensionless spin grows without bound. A short segment of the ring approaches a boosted black string. This local limit explains why strings are useful approximations, while the global closure into an $S^1$ supplies the finite total mass and angular momentum.

Mechanics and regularity checks

A regular ring obeys the same vacuum first law and Smarr relation as any connected five-dimensional stationary vacuum horizon,

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

An unbalanced ring requires an additional work term associated with the conical defect. This supplies an independent diagnostic: using the balanced first law before removing the defect silently omits the external support.

The topology also passes the higher-dimensional curvature restriction. A product metric on $S^1\times S^2$ has positive scalar curvature because the sphere contributes positively and the circle contributes zero. The ring is therefore compatible with the positive-Yamabe horizon theorem even though it is not spherical.

Check your understanding

Locate the cusp of the balanced-ring curve and explain why the interval just above it contains two rings.

Solution

Differentiate the logarithm of the scale-free spin:

$$\frac{d}{d\nu}\log j^2 =\frac{3}{1+\nu}-\frac1\nu =\frac{2\nu-1}{\nu(1+\nu)}.$$

The derivative changes from negative to positive at $\nu=1/2$, so this is a minimum. Its value is

$$j^2_{\min} =\frac{(3/2)^3}{8(1/2)} =\frac{27}{32}.$$

Every horizontal line with $27/32\lt j^2\lt1$ intersects the curve once at $\nu\lt1/2$ and once at $\nu\gt1/2$, producing one thin and one fat balanced ring. In the same interval it also intersects the Myers–Perry branch.

The next page opens the ring locally and studies horizons extended along compact or noncompact directions: Black Strings, Branes, and Kaluza–Klein Horizons.

References

• Emparan, Roberto, and Harvey S. Reall, “A Rotating Black Ring Solution in Five Dimensions,” Physical Review Letters 88, 101101 (2002). DOI. Open PDF.
• Emparan, Roberto, and Harvey S. Reall, “Black Holes in Higher Dimensions,” Living Reviews in Relativity 11, 6 (2008). DOI. Open PDF.