Black Rings

TL;DR

This review surveys five-dimensional black rings, highlighting their horizon topology $S^1\times S^2$, non-uniqueness, and the interplay between gravity and string theory. It develops ring coordinates and analyzes neutral, charged, and supersymmetric rings, including their microscopic descriptions in M-theory via IR circular strings and UV supertubes. The work discusses multi-ring solutions, small rings, non-supersymmetric generalizations, and the fuzzball program, connecting macroscopic black hole physics to microscopic brane dynamics. It emphasizes stability issues, potential generalizations to higher dimensions, and the broader implications for black hole entropy and microstate counting.

Abstract

A black ring is a five-dimensional black hole with an event horizon of topology S1 x S2. We provide an introduction to the description of black rings in general relativity and string theory. Novel aspects of the presentation include a new approach to constructing black ring coordinates and a critical review of black ring microscopics.

Figures (3)

• Figure 1: Ring coordinates for flat four-dimensional space, on a section at constant $\phi$ and $\psi$ (and $\phi+\pi$, $\psi+\pi$). Dashed circles correspond to spheres at constant $|x|\in [0,1]$, solid circles to spheres at constant $y\in [-\infty,-1]$. Spheres at constant $y$ collapse to zero size at the location of the ring of radius $R$, $y=-\infty$. The disk bounded by the ring is an axis for $\partial_\phi$ at $x=+1$.
• Figure 2: Horizon area $a_H$ vs (spin)$^2$$j^2$, for given mass, for the neutral rotating black ring (solid) and MP black hole (dotted). There are two branches of black rings, which branch off from the cusp at $(j^2,a_H) = (27/32,1)$, and which are dubbed "thin" and "fat" according to their shape. When $27/32 < j^2<1$ we find the Holey Trinity: three different solutions ---two black rings and one MP black hole--- with the same dimensionless parameter $j$ ( i.e., with the same mass and spin). The minimally spinning ring, with $j^2=27/32$, has a regular non-degenerate horizon, so it is not an extremal solution. Other interesting features are: At $j^2={a_H}^2=8/9$ the curves intersect and we find a MP black hole and a (thin) black ring both with the same mass, spin and area. The limiting solution at $(j^2, a_H)=(1,0)$ is a naked singularity. Rapidly spinning black rings, $j^2\to\infty$, become thinner and their area decreases as ${a_H}\sim 1/(j\sqrt{2})$.
• Figure 3: The dipole $q_i$ measured from the magnetic flux of $F^i$ across an $S^2$ that encloses a section of the string. An azimuthal angle has been suppressed in the picture, so the $S^2$ is represented as a circle.