# Fractal Black Holes on Strings

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.

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