Higher-dimensional Einstein gravity

Under influence from the string theory juggernaut, much attention has been paid recently to generalizing to higher dimensions various results concerning the well established theory of standard four-dimensional Einstein gravity (that is, general relativity). At present, this work can probably be most fairly described as well-motivated and extended theoretical speculation. Currently it apparently has no direct observational and experimental support, in contrast to four dimensional general relativity.

Exact solutions

The higher dimensional generalization of the Kerr metric was discovered by Myers and Perry. Like the Kerr metric, the Myers-Perry metric has spherical horizon topology. The construction involves making a Kerr-Schild ansatz; by a similar method, the solution has been generalized to include a cosmological constant.

The black ring is a solution of five dimensional general relativity. It inherits its name from the fact that its event horizon is topologically S^1 x S^2. This is in constract to other known black hole solutions in five dimensions which have horizon topology S^3. See also black string

Black hole uniqueness

In four dimensions, Hawking proved that the topology of the event horizon of a black hole must be spherical. Because the proof uses the Gauss-Bonnet theorem, it does not generalize to higher dimensions. The discovery of black ring solutions in five dimensions shows that other topologies are allowed in higher dimensions, but it is unclear precisely which topologies are allowed. It has been shown that the horizon must be of positive Yamabe type, that is it must admit a metric of positive scalar curvature.