Black rings in 4D gravity

Physics Asked on May 8, 2021

In Introduction to Black Hole Physics by Valeri P. Frolov, Andrei Zelnikov there is a discussion of gravity solutions. They present some examples of solutions with non-spherical horizon topology:

But such solutions are known in 5D.

About the possibility of such solutions in 4D authors say:

In 4D it is believed that there are no stationary solutions for vacuum black holes with horizons consisting of several disconnected components.

What about black ring solutions in 4D?

Is it impossible due to "no hair theorem"? If not, why?

Or maybe exist some other theorems which prohibit such solutions?

One Answer

Yes, in 4D the topology censorship theorem seems to rule them out:

Every causal curve extending from past null infinity to future null infinity can be continuously deformed to a curve near infinity.

Roughly speaking, this says that an observer, whose trip begins and ends near infinity, and who thus remains outside all black holes, is unable to probe any nontrivial topological structures.

That is, if you somehow made a ring you cannot detect that it is a ring by shining light through it; the photons will either miss it or be absorbed, and you cannot shoot one through the ring.

A real shame.

Answered by Anders Sandberg on May 8, 2021

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