## Topology of black holes

I've asked this question of some physicist friends of mine and I've never gotten a satisfactory answer: What is topologically possible for a neighborhood of a black hole? To clarify, I'm curious about the topology as a 4-manifold, although I'd also be interested to hear about time-like and space-like slices as well. I've heard that a time-like slice of the event horizon can be a torus or sphere, but this isn't really what I'd like to know, although I imagine that there is a close connection between the topology of the event horizon (as a 3-manifold) and the topology of a neighborhood of the black hole. Please ask if any clarifications are needed.

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 What do you want to use as a definition of a black hole? For example if your space time is a compact 3-manifold x a real line, wouldn't the whole of space-time satisfy Hawking and Ellis's definition? – Ryan Budney Jan 19 2011 at 2:36

Hawking's Theorem of Black Hole topology asserts that the in case of $4$d asymptotically flat stationary black holes satisfying the suitable energy condition (dominant energy condition), the cross sections of the evernt horizon are spherical.

Galloway and Schoen extended this theorem to higher dimensions; they showed that the cross sections of event horizon (stationary case) and the outer (apparent) horizon (general case) are of positive Yamabe type. This paper can be found at Galloway's webpage www.math.miami.edu/~galloway/papers/220_2006_19_OnlinePDF.pdf

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[While I was writing, J. Verma answered the question more succinctly but I already wrote most of my answer so figured I would post it]

First of all I am not a physicist nor do I work in GR, but I have taken a number of courses over the year so I will attempt to give some idea of an answer (I'm sure an expert can expand).

Basically one wants to restrict attention to asymptotically flat spacetimes (in 4d) i.e complete spacelike slices of the spacetime are asymptotically euclidean (i.e. in a neighborhood of infinity the metric looks like the euclidean metric in a definite way). This physically models an isolated gravitating system. I think one probably also wants a global non-vanishing timelike vectorfield (so no timelike closed loops) and also that the the dominant energy condition holds (this is too technical to state but physically means no energy enters the past light cone of a point from outside the past light cone).

Given this setup the event horizon is still not an easy concept to grasp (it depends on the global hyperbolic structure and so requires one to understand the entire spacetime all at once... I'm not sure what can be said about it honestly other than for highly symmetric spacetimes). An easier thing to grab hold of are MOTS (marginally outer trapped surfaces). To find these one takes a space-like complete slice and looks for surfaces whose area is unchanged under the flow by null vectors (this is marginally trapped) and that there is no surface strictly outside of this surface with this property (this is outermost). The existence of one of these implies the formation of a singularity (due to Penrose?) so are natural stand-ins for black holes (I'm not sure how much more is known about the full relationship between MOTS and black holes though I believe understanding it completely is about as hard as showing cosmic censorship). Anyway, it can be shown that these guys (since they are "stable") must be either spheres or tori (this is analogous to theorems about minimal surfaces) and the latter has a rigidity result attached to it which I believe makes it unphysical (due to Galloway and Schoen).

I hope this makes sense and I haven't made any glaring errors.

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 The chronology assumption is not necessary for the topology theorem of Hawking/Galloway-Schoen. The main argument works essentially on an initial data set satisfying the constraint equations (there will be constraints if you take a 1+3 hyperbolic-elliptic decomposition of the Einstein equations). You should think of it as a statement about a 3 manifold and its embedding into a 4 manifold: you only need Einstein's equation to be satisfied on a small neighborhood the embedded 3 manifold (in fact, I think just pointwise along the 3 manifold may be enough). – Willie Wong Jan 19 2011 at 13:15 Likewise, the asymptotic flatness assumption is also unnecessary, except for physical interpretation to attach meaning to the words "inside the horizon" and "outside the horizon". The result is essentially local. Of course, these facts are connected to what you wrote in the second to last paragraph: the statement of Galloway-Schoen is a statement about apparent horizons (MOTS), and not a statement about event horizons (which is connected to the global 4-dimensional geometry of the space-time). MOTS can be defined purely locally, and hence can be studied without regard to global structures. – Willie Wong Jan 19 2011 at 13:20 Thanks for the comments! I see I was being too conservative in my assumptions. – Rbega Jan 19 2011 at 15:46

The previous answers dealt with the physically relevant case of $d=4$ spacetime dimensions. One of the surprising discoveries in recent years is that in higher dimensions the possible topologies are much richer. I believe this started with the discovery by Emparan and Reall of a black hole in $d=5$ with horizon topology $S^2 \times S^1$ (hep-th/0110260). The recent paper arXiv:1002.0490 by Hollands et. al. surveys the situation and discusses restrictions on the possible topologies of the horizon for $d=5$ black holes.

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thanks for pointing this.The review article "Black Holes in Higher dimensions" by Reall and Emparan in Living Reviews also has a section on Black hole topology. – J Verma Jan 19 2011 at 15:05
Actually, in $d=5$, the most relevant question is "which of the ones allowed by Galloway-Schoen is attained?" Since in 5 space-time dimensions, the horizon cross section is 3 dimensional, and all compact 3 manifolds in positive Yamabe class is known, the question becomes "which of those actually corresponds to stationary asymptotically flat solutions". A futher interesting question is the consideration of black-saturn or di-ring type horizons: cases with more than one connected component of the horizon. Such stationary solutions are not expected to exist in 4d. – Willie Wong Jan 19 2011 at 15:18
Willie, thanks for the clarification. Black-saturn solutions in $d=5$ can be found in arXiv.org/pdf/hep-th/0701035. – Jeff Harvey Jan 19 2011 at 15:33
Some di-ring solutions are also known. The first example that Google returns is arxiv.org/abs/hep-th/0511007 ; I mentioned those two by name because there are ones for which I know have been more extensively studied, and they are the known examples of the more general cases of disconnected horizons for which the field is mostly wide open. – Willie Wong Jan 19 2011 at 16:56

So far J Verma and RBega provided two succint descriptions on topology of the apparent horizon itself (for any space-time admitted trapped regions), and so by association, the topology of the event horizon in a stationary asymptotically flat black-hole space-time.

I'll try to provide an answer to an interpretation of the question you asked for: namely that of the topology of a neighborhood of the event horizon. That I have to interpret the question is because you have not actually provide a description of what you mean by a neighborhood. Using the Hawking topology theorem you can easily manage that the topology of the event horizon is $\mathbb{S}^2\times \mathbb{R}$, and is an embedded null hypersurface. So a tubular neighborhood of it necessarily has the topology $\mathbb{S}^2\times \mathbb{R}^2$, which is, for one thing, simply connected. But there is absolutely nothing to prevent you from choosing a neighborhood to be some arbitrary open set containing the event horizon with complicated topology. Indeed, you can easily imagine removing a four dimensional tube disjoint from the event horizon from the $\mathbb{S}^2\times\mathbb{R}^2$ set to get something that is not simply connected.

Because of this freedom to choose subsets, the naive reading of you question leads to the answer that "pretty much as bad as you want".

But that answer is rather physically useless: it doesn't capture anything essential about black holes. In fact, the answer given above is identical to the answer to the following question: let $U$ be an open connected subset of $\mathbb{R}^4$, what kinds of topology can $U$ admit?

A more useful question to ask is: given an isolated gravitating body (such as a black hole), what is the topology of the space-time outside of it? And that question is one admitting a good answer. The content is the topological censorship theorem. In physicist's language, to quote Friedman, Schleich, and Witt,

general relativity does not allow an observer to probe the topology of spacetime: Any topological structure collapses too quickly to allow light to traverse it.

An early version of this is due to Gannon, who showed that Cauchy hypersurface with non-trivial topology will necessarily generate a development which is null geodesically incomplete. The FSW paper showed that under some restrictions, all the non-trivial topology must be hidden behind the event horizon.

A stronger generalisation of the topological censorship theorem is due to Galloway, who later, with Schleich, Witt, and Woolgar, extended the result from asymptotically flat space-times to also asymptotically anti-deSitter ones. One interesting crucial assumption of these theorems is the requirement for null or time-like "Scri".

A somewhat related paper is this one by Schleich and Witt which I didn't read in detail so cannot say more about.

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 great answer.+1 – J Verma Jan 19 2011 at 16:42