Introduction

A black ring is a D-dimensional black hole with horizon topology S1 × SD-3. There is a simple heuristic way of understanding why such solutions might exist. Consider a black string, the product of the (D - 1)-dimensional Schwarzschild solution with a flat direction, with horizon topology ℝ × SD-3. Imagine bending the string into a loop, so the topology is now S1 × SD-3. This loop would tend to contract owing to its tension and gravitational self-attraction. However, if the loop rotates then it might be possible to balance these forces with centrifugal repulsion. That this is indeed possible is proved by the existence of an explicit black ring solution of the five-dimensional vacuum Einstein equation [1, 2]. The existence of analogous solutions for D > 5 dimensions (or when a cosmological constant is included) is strongly suggested by the perturbative methods reviewed in Chapter 8.

The discovery of black rings revealed that higher-dimensional black holes can exhibit properties very different from those of four-dimensional holes. First, higherdimensional black holes need not be topologically spherical. Second, higherdimensional black holes are not uniquely parameterized by mass and angular momenta: there exist distinct black ring solutions with the same mass and angular momenta. Furthermore, there exist black ring and Myers–Perry solutions [3] with the same mass and angular momenta. Thus the topology and uniqueness theorems that underpin our understanding of four-dimensional black holes do not extend to higher dimensions in an obvious way.