Black ring

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Author: Dr. Roberto Emparan, University of Barcelona and ICREA, Spain
Author: Dr. Harvey Reall, DAMTP, Cambridge University

A black ring is a black hole with an event horizon with topology $S^1 \times S^p$ . Black rings can exist only in spacetimes with five or more dimensions. Exact black ring solutions of General Relativity are known only in five dimensions, but approximate solutions for thin black rings (with the radius of $S^1$ much larger than the radius of $S^p$ ) have been constructed in spacetimes with more than five dimensions. The existence of black ring solutions shows that higher-dimensional black holes can have non-spherical topology and are not uniquely specified by their conserved charges.

1 Background
2 Heuristic construction of black rings
3 Non-uniqueness
4 Black rings with dipoles

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Background

In four dimensional space-time, the black hole uniqueness theorem asserts that the Kerr solution is the unique black hole solution of the vacuum Einstein equation that is time-independent and asymptotically flat (i.e. approaches Minkowski spacetime far from the hole). This solution is topologically spherical and is uniquely labelled by its mass $M$ and angular momentum $J$ . This result proves that all multipole moments of the gravitational field of a time-independent black hole are uniquely determined by the lowest two, namely $M$ and $J$ .

In 2001, the discovery [1] ([2]) of an exact black ring solution of the five-dimensional vacuum Einstein equation proved that these simple topological and uniqueness properties of 4d black holes do not extend to higher dimensions.

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Heuristic construction of black rings

A heuristic argument that suggests the possible existence of black rings is the following. Take a neutral black string in five dimensions, constructed as the direct product of the Schwarzschild solution and a line, so the geometry of the horizon is $\mathbf{R}\times S^2$ . Imagine bending this string to form a circle, so the topology is now $S^1\times S^2$ . In principle this circular string tends to contract, decreasing the radius of the $S^1$ , due to its tension and gravitational self-attraction. However, if the string can be made to rotate along the $S^1$ then Newtonian arguments suggest that these forces could be balanced by centrifugal repulsion. The result is a rotating black ring: a black hole with an event horizon of topology $S^1 \times S^2$ .

Ref.[3] ([4]) obtained an explicit solution of five-dimensional vacuum general relativity describing a black ring that rotates along its circle. It provided the first example of non-spherical horizon topology and of black hole non-uniqueness in vacuum gravity. Ref.[5] found a generalization of this solution in which the black ring rotates also along the $S^2$ , i.e., a doubly-spinning black ring.

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Non-uniqueness

The black ring with a single angular momentum (along its circle) illustrates the main novelties of the solution more clearly than the doubly-spinning ring. The absence of uniqueness is illustrated in a plot the area of the horizon as a function of angular momentum for fixed mass. Contrary to what happens for rotating black holes in four dimensions, and for the MP black hole in five dimensions, the angular momentum of the black ring (for fixed mass) is bounded below, but not above. Non-uniqueness is reflected in the fact that there is a narrow range of angular momenta for which there exist one MP black hole and two black rings all with the same values of the mass and the spin. Since the latter are the only conserved quantities carried by these objects, there is an explicit violation of black hole uniqueness.

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Black rings with dipoles

For topologically spherical black holes (\eg the Kerr-Newman solution), the combination of electric charges and rotation gives rise to associated magnetic dipoles, which do not violate uniqueness since they do not provide parameters independent of the conserved charges.

Black rings can carry conserved gauge charges, but more remarkably, they can also support gauge dipoles that are independent of all conserved charges, in fact they can be present even in the absence of any gauge charge. The possible presence of dipoles on a black ring is most easily understood by regarding the ring as a circular string that can source an electric two-form potential $B_{t\psi}$ , whose magnetic dual $A_\phi$ is sourced by a circular distribution of magnetic monopoles. The topology of a black ring allows to define dipole charges $q$ as one would do for a magnetic charge,

$q = \frac{1}{2 \pi} \int_{S^2} F\,,$

by performing the integral on a surface $S^2$ that links the ring once. However, even if there is a local distribution of charge, the total magnetic charge is zero: in order to compute the magnetic charge in five dimensions, one has to specify a two-sphere that encloses a point of the ring, and a vector tangent to the string. So there are opposite magnetic charges at diametrically opposite points of the ring, which justifies the analogy to a dipole.

Dipole rings are therefore specified by the independent physical parameters $(M,J,q)$ . The dipole $q$ is a non-conserved, classically continuous parameter and it implies continuous violations of non-uniqueness in five dimensions.