Here’s a fascinating new result about black holes in five dimensions — actually from last October, but I missed it when it came out. I just noticed it this week because of a write-up by Gary Horowitz in Matters of Gravity, the newsletter of the gravity group of the American Physical Society. (I obviously missed David Berenstein’s post as well.)
You might be thinking that black holes in five dimensions can’t be that interesting, since they are probably pretty similar to black holes in four dimensions, and after all we don’t live in five dimensions. But of course, there could be a fifth dimension of space that is compactified on a tiny circle. (Of course.) So then you have to consider two different regimes: the size of the circle is much larger than the size of the black hole — in which the fact that it’s compact doesn’t really matter, and you just have a regular black hole in five dimensions — or the size of the circle is smaller than the black hole — in which case, what?
The answer is that you get a black string — a cylindrical configuration that stretches across the extra dimension. This was figured out a long time ago by Ruth Gregory and Raymond LaFlamme. But they were also clever enough to ask — what if you had that kind of cylindrical black hole, but it stretched across a relatively large extra dimension? That sounds like a configuration you can make, but it might be unstable — wiggles in the string could grow, leading it to pinch off into a set of distinct black holes. One way of seeing that something like that is likely is to calculate the entropy of each configuration; for long enough black strings, the entropy is lower than a collection of black holes with the same mass, and entropy tends to grow. Indeed, Gregory and LaFlamme showed that long black strings are unstable. However, it wasn’t clear what exactly would ultimately happen to them.
The problem is this: there is a singularity at the center of the black string. If the string simply divides into multiple black holes, that singularity should (at least for a moment) become “naked” to the outside world, violating cosmic censorship. Cosmic censorship is a conjecture, not a theorem, so maybe it is violated, but that would certainly be interesting.
What Lehner and Pretorius have done is to numerically follow the decay of an unstable black string, much further than anyone had ever done before. They find that yes, it does decay into multiple black holes, and the strings connecting them seem to shrink to zero size in a finite time. The implication seems strongly to be that cosmic censorship is indeed violated, although the numerical simulations aren’t enough to establish that for sure.
The cool part is the way in which the strings decay into black holes. They form a self-similar pattern along the way — a fractal configuration of black holes of every size, from the largest on down. Here’s a result from their simulations.
Beautiful, isn’t it? As the string shrinks in radius, it keeps beading off smaller and smaller black holes. Eventually we would expect them all just to bump into each other and make one big black hole, but the intermediate configuration is complex and elegant. And cosmic censorship is apparently violated when the strings finally shrink to zero radius. So it goes.
Little chance we’ll be observing any of this in an experiment any time soon. But Nature has the capacity to surprise us even if we’re just solving equations that are many decades old.
Sean, I found your first comment helpful in clarifying things. If I might suggest an illustration: Taking your first picture (where the large dimensions are represented as a single dimension), you could make a round dot on the surface of the cylinder (to represent the black hole which is smaller than the extra dimension). Then reproduce the picture again with the size of the dot increased so that it becomes a band which wraps all the way around the cylinder (representing the black hole which is larger than the extra dimension).
Sean, when you say that cosmic censorship is a conjecture, what do you mean exactly? The Wikipedia page says there are spacetimes for which it is known to be true and others for which it is known to be violated; is there some succinct way of characterizing the class of spacetimes for which it is unknown? Is there a specific unknown case that is particularly important?
The thing about conjectures is, they are attempts to capture something true without necessarily being able to state that true thing precisely. Cosmic censorship is roughly “there are no naked singularities.” But it’s easy to get naked singularities, so you have to say something like “unless they are there to begin, or if you allow exotic forms of energy.” Then Choptuik and collaborators proved that you could have exquisitely finely-tuned initial conditions that gave rise to naked singularities; so the caveat “arising from generic initial conditions” was added. Now it looks like naked singularities arise from generic initial conditions in five dimensions, so “in four dimensions” needs to be added. There’s still not a proof that even that is true.
Great math, but does it reflect reality. Could a really well skilled physicist explain this to an illiterate Bedouin? or to a high school dropout? or to somebody who has a BS or to liberal Arts PHD? Trying to write for the average tax payer to convince that research in this issue is worth the money is as challenging as to proof that this cylindrical black hole can or cannot extend into a sixth spatial dimension or beyond