Black ring

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Roberto Emparan and Harvey Reall (2010), Scholarpedia, 5(9):8786. doi:10.4249/scholarpedia.8786 revision #135545 [link to/cite this article]
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Curator: Roberto Emparan

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.



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

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 \(D\geq 5\) dimensions, constructed as the direct product of the \(D-1\)-dimensional Schwarzschild-Tangherlini solution and a line, so the geometry of the horizon is \( \mathbf{R}\times S^{D-3} \ .\) Imagine bending this string to form a circle, so the topology is now \(S^1\times S^{D-3} \ .\) 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 \( S^1 \times S^{D-3} \ .\)

Ref.[3] ([4]) presented 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.


The five-dimensional 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 of the area of the horizon as a function of angular momentum for fixed mass (see Figure 1).
Figure 1: Area \(a_H\)vs. spin \(j\) for fixed mass for five-dimensional Myers-Perry (MP) black holes and black rings (see text for normalization of \(a_H\) and \(j\))

In this plot the horizon area \(A_H\) and the spin \(J\) have been conveniently normalized to dimensionless magnitudes \(a_H=\sqrt{\frac{27}{256\pi}}\frac{A_H}{(GM)^{3/2}}\) and\(j=\sqrt{\frac{27\pi}{32G}}\frac{J}{M^{3/2}}\) (for ease of visualization the horizontal axis plots \(j^2\) instead of \(j\)).

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. Observe also that above this minimum angular momentum there exist two branches of black rings: the one with higher area is referred to as the branch of thin black rings, and the other as fat black rings. (See Figure 2.)
Figure 2: Thin and fat black rings

Non-uniqueness is reflected in the fact that there is a narrow range of angular momenta, \(\sqrt{27/32}\leq j<1\ ,\) for which there exist one MP black hole and two black rings (a fat and a thin one) 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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See Also

Exact solutions of Einstein's equations, General relativity.

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