A rotating black ring in five dimensions

TL;DR

The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1xS2, which describes a rotating "black ring".

Abstract

The vacuum Einstein equations in five dimensions are shown to admit a solution describing an asymptotically flat spacetime regular on and outside an event horizon of topology S^1 x S^2. It describes a rotating ``black ring''. This is the first example of an asymptotically flat vacuum solution with an event horizon of non-spherical topology. There is a range of values for the mass and angular momentum for which there exist two black ring solutions as well as a black hole solution. Therefore the uniqueness theorems valid in four dimensions do not have simple higher dimensional generalizations. It is suggested that increasing the spin of a five dimensional black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.

Figures (3)

• Figure 1: Plots, as functions of $\nu$ at fixed $M$, of the inner ($R_i$) and outer ($R_o$) radii of curvature of the $S^1$, total area $\cal A$ of the ring, surface gravity $\kappa$, and angular velocity at the horizon $\Omega_H$. All quantities are rendered dimensionless by dividing by an appropriate power of $GM$.
• Figure 2: $(27\pi/32G)J^2/M^3$ as a function of $\nu$. Here and in the following graph, the solid line corresponds to the black ring, the dashed line to the black hole. The two dotted lines delimit the values for which a black hole and two black rings with the same mass and spin can exist.
• Figure 3: ${\cal A}/(GM)^{3/2}$ against $\sqrt{27\pi/32G}J/M^{3/2}$, around the regime in which a black hole and two black rings with the same $M$ and $J$ exist. For $\sqrt{27\pi/32G} J/M^{3/2}\approx 0.942$ there exist a black hole and a black ring with the same mass, spin, and area ${\cal A}\approx 5.157 (GM)^{3/2}$.