# Torus shaped event horizon

Is there a solution to GR field equations for a rotating black hole that has a torus shaped event horizon? If so, when a craft flies through the torus, it can pass through a ring singularity while remaining outside of the event horizon. Would it enter a strange region of repulsive gravity and closed timelike curves? How far from the black hole would this region extend?

There is a result in four-dimensional General Relativity that forbids such situations. Namely, the Hawking's Theorem on the Topology of Black Holes. It is given, e.g., on the classic book by Hawking and Ellis, The Large Scale Structure of Space-Time, which I quote:

Proposition 9.3.2

Each connected component in $$J^+(\mathscr{I}^{-},\overline{\mathscr{M}})$$ of the horizon $$\partial \mathscr{B}(\tau)$$ in a stationary regular predictable space is homeomorphic to a two-sphere.

Hence, at least under this proposition's conditions, you can't have a toroidal black hole. Notice that we are assuming the black hole is stationary, for example, and, implicitly, that we are working with four-dimensional GR. The dimension of spacetime is extremely relevant: in five-dimensions, you do get black rings.

If you're interested, I believe the original reference due to Hawking is Comm. Math. Phys. 25, 152–166 (1972). DOI: 10.1007/BF01877517.

• As a side comment, while searching for the statement of the theorem (I recalled it existed, but not the details), I found arXiv: gr-qc/0509107, which apparently generalizes the results to higher dimensions. In it, the authors claim on a footnote that "Actually the torus $T^2$ arises as a borderline case in Hawking’s argument, but can occur only under special circumstances." I didn't read the proof on Hawking & Ellis with much attention, but it does seem to discuss the case of a torus and discard it. Perhaps someone who understands these results + Oct 1, 2022 at 2:51
• better than I do might add to the discussion later and explain whether this is or not a "loophole". Oct 1, 2022 at 2:51
• The more exotic the symbols, the more advanced the math, I suppose? ;-) Oct 1, 2022 at 12:00

The proposition cited in the answer by Nickolas Alves applies to “stationary” and “asymptotically flat” spacetimes, i.e. in four dimensional GR there are no stationary torus-shaped event horizons of asymptotically flat black hole. If we consider highly dynamical situations such as black hole formation or black hole coalescence, then it is possible to observe horizons for which constant time slices have toroidal topology.

For example here is a figure 10 from the paper:

It depicts an instantaneous configuration of event horizon during the merger of two black holes one of which is rapidly rotating while the other is much heavier (so called extreme mass ratio). We can see (in the zoomed in inset) that there is a small and stretched toroidal “hole” in the “pinch” connecting two black holes. The cited paper also provides references to other works where toroidal transient event horizons have been observed and discussed.

… when a craft flies through the torus, it can pass through a ring singularity while remaining outside of the event horizon.

No. First, ring singularity of the Kerr metric is mostly an artifact of analytic solution of GR, realistic black holes are not expected to have such a feature due to instabilities developing around inner horizon. Second, in scenarios where temporary toroidal event horizons form they do not contain ring inside them, for example in the figure above, the ring singularity would be fully inside the gray spheroid, so spacecraft passing through the hole in a torus would not pass through the singularity. Such a spacecraft (if it stays outside event horizon) would not experienced any “strangeness”.

For completeness, let us also mention that if one drops “asymptotic flatness” as a requirement than it is possible to construct stationary/static “black holes” with horizons of unusual topology even in four dimensions. But such topological black holes are unrealistic, since in such cases event horizon merely inherits the topology from underlying background and away from the black hole spacetime does not approaches to a flat one.

• It is also worth noting that the Topological Censorship Theorem implies that one cannot fly in from infinity, pass through the hole, and then escape to infinity, at least not in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition. Whether that applies here (Kerr is not globally hyperbolic), or some generalisation of the theorem applies, can be debated. Oct 1, 2022 at 11:11
• @AndersSandberg “Kerr is not globally hyperbolic” - It is hyperbolic everywhere outside the inner horizon. Oct 6, 2022 at 7:40
• @safesphere - Which is why it is somewhat unclear whether the theorem applies or not. Oct 6, 2022 at 16:38