What kind of image would a transcient toroidial black hole merger produce from it's gravitalional lensing effect?


What does a torus shaped lens actually look like?

Are there any other hypothesized transcient shapes for gravitational lenses that could arise from a merger or collapse?


2 Answers 2


Not even light can pass through a toroidal black hole before the hole closes up, so the question is moot.

To set the stage for why toroidal black holes are interesting, I'll start with Hawking's 1972 result that "a stationary black hole must have topologically spherical boundary." That is, once everything has settled down and your metric is no longer changing in time, any event horizons must be spherical (in the topological sense). Black holes do not come in very many shapes at all.

But what if we don't wait to settle into some steady state? Well, Gannon (1976) asked the same question and had this to say:

Let $(M,g)$ be strongly future asymptotically predictable from a partial Cauchy surface regular near infinity. If $(M,g)$ satisfies the strong energy condition ($T^{ab}V_aV_b \geq 0$, $T^{ab}V_a$ timelike and future directed for every timelike and future directed vector $V$) then the topology of a smooth black hole in $S$ is either that of a sphere or a torus.

While the result is subject to its energy condition caveat (and note that the strong energy condition excludes e.g. dark energy), this early result helped lay the groundwork for proving that black holes are either spherical (i.e. boring) or toroidal and not anything much more exotic, even if you freeze them at a moment in time before they have settled down.

But tori are interesting: One can imagine going through the hole in the center, and indeed this speculation was raised. However, Friedman et al. (1993) put an end to that with their topological censorship theorem:

If an asymptotically flat, globally hyperbolic spacetime $(M,g_{ab})$ satisfies the averaged null energy condition, then every causal curve from $\mathscr{I}^-$ to $\mathscr{I}^+$ is deformable to $\gamma_0 \mathbin{\mathrm{rel}} \mathscr{I}$.

In other words, trajectories for all objects, both massive and massless, running from past infinity to future infinity must be topologically equivalent. Light passing through a toroidal black hole could not have its path smoothly deformed (even mathematically) to match that of a ray not passing through the hole in the torus, and thus the setup violates the theorem. Note also that this theorem holds under even broader conditions than the last: The null energy condition is satisfied by every form of bulk stress-energy observed, and even allows for some exotic stuff (negative density, so long as density and pressure sum to a nonnegative value) not observed. As the paper says in the abstract, "general relativity does not allow an observer to probe the topology of spacetime."

Even if you do "probe the topology of spacetime" mathematically, you will find that toroidal black holes are rare beasts indeed. For example, Bowdy & Galloway (1995) showed that event horizons must be spherical under only mild assumptions, including the null energy condition, strongly future asymptotic predictability, and the assumption that new generators do not enter the horizon, but not relying on an entirely stationary metric as Hawking did. In some sense, the technical limit on generators was all the "not-evolving-in-time"-ness needed for the proof. Around the same time, Jacobson & Venkataramani (1995) proved horizons must be topologically spherical even if the restriction on generators is lifted, so long as you wait for about a light crossing time after something happens to the black hole (e.g. a merger with another). So even in theory it is rather difficult to contrive a scenario where a toroidal black hole will be seen.

Transient toroidal black holes have been seen, though, at least in simulations. This does not conflict with topological censorship, because in a simulation you can ask questions ("What is the topology of this particular surface?") that an observer living in that spacetime could not answer. To illustrate this more definitively, Shapiro et al. (1995) analyzed toroidal event horizons both analytically and within their simulation that produced them. They found, as expected, that light cannot pass through the torus:

Thus the "hole" in the torus closes up faster than the speed of light. Consequently, no causal signal can link through the torus and escape back to the exterior spacetime region to provide a violation of topological censorship.

  • $\begingroup$ Hi, what is the import of the recent refutation of strong cosmic censorship by Dafermos and Luk for the topological censorship part of this answer? Does their result show that the global hyperbolicity holds less often than we thought and possibly therefore also toplogical censorship? Thanks $\endgroup$ May 21, 2018 at 15:36

I'm sure the first order effects at distances large from the observational axis are just that of a point source with the mass of the sum, discussed in this question. The strong field effects near the observational axis will probably have some interesting features, but I'm not familiar with them. You might try perusing the pulsar timing array (a gravitational wave detection mechanism) literature, since they are measuring a similar effect to the lensing of light.


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