Fractal Black Holes on Strings

By Sean Carroll | March 4, 2011 9:18 am

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.

CATEGORIZED UNDER: arxiv, Science, Top Posts
  • ossicle

    Gah, I get depressed when a post goes _completely_ over my head. From this onward, I lack any comprehension whatsoever:

    “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?”


  • Sean

    Sorry about that; post written in haste. I added a little figure, which probably won’t help but who knows. (Note that it’s just a generic “extra dimensions” figure, without any black holes illustrated.)

    The question is whether or not the diameter of the black hole is smaller or larger than the size of the circle around the cylinder, representing the extra dimension. If it’s larger, it can’t fit inside; instead, it stretches across the extra dimension. Then (confusingly), the black hole is *itself* a cylinder — one that is perpendicular in orientation to the cylinder shown in the first picture.

  • Sean Matthews

    Minor philosophical quibble to an otherwise interesting post.

    I wince when I read something like ‘Nature has the capacity to surprise us…’ in this sort of context. This has nothing to do with nature, until we confirm that we live in a 5d manifold, etc. where this sort of thing happens. Until then it is an interesting observation about a formal model that has no identifiable real-world correlate.

  • Sam Gralla

    Haha, I think this title is a bit misleading, conjuring up images of networks of D-branes to make fractals. It’s cool work of course though.

  • Left_Wing_Fox

    Sean @3: I guess I read that phrase a little differently than you. My take was more that as odd as this prediction is, we might actually see evidence that supports the hypothetical models, even if the results are unusual or counterintuitive.

    Best example I can think of in physics would be Quantum Entanglement. Einstein and his co-writers did the math, noted that Quantum Mechanics theory predicted “spooky action at a distance”, which was absurd, and therefore QM was flawed. What ended up happening was that we actually observed said “Spooky Action”, which supported the theory.

  • Steven Colyer

    Sean, I love you not just because you look like you could be my younger (slightly less good looking but still awesome) brother, or because of your awesome book which I bought as soon as it came out, or your great appearance on Colbert, but, really man, did you write:

    But of course, there could be a fifth dimension of space that is compactified on a tiny circle. (Of course.)

    My objection, possibly wrong, is with the phrase “of course”, that you used twice. Does stating an opinion twice make it more of a fact, or less of a speculation? My point is, who the heck says the 5th dimension (i.e., a fourth dimension of space) HAS to be compactified?

    Couldn’t it be WAY larger than the others, such that our whole observable Universe is a point in a 4th spacial dimension? My point being, and again possibly wrong, is that somewhere between Kaluza and Klein something was lost. Possibly. If I throw a frisbee, then the world on the frisbee thinks the world is just rotating, and its world is, but what it doesn’t know and what we know looking at the big picture is that it’s also moving in a direction besides that rotation. Am I making any sense?

    Ah, speculation, what utter fun. We will always be able to find mathematics to back up any idea, crazy or not, but reality always reveals itself, in time, to use only very specific mathematics.

    In any event, great work, and thanks for the heads up.

  • James

    Ossicle @1… I sympathise. In my PhD interview, Ruth Gregory asked me what would happen if you took a 5-dimensional black hole and squeezed it or added mass to it, or something. My eyes widened and I uttered some rambling response, which was apparently “wrong, but [she] liked the way I thought”. I’ve no idea what I possibly could have said to have made any sense, but when I came across the Gregory-Laflamme instability some time later, I guessed it must have something to do with that.

  • Lievo


    I’ve been said that naked singularities would be such a strange thing that it seems safe to postulate it can’t exist. Of course who knows, but is it correct it would be really surprising?

    If yes, then can we interpret the result of Lehner and Pretorius as an evidence that BHs cannot be smaller than a compactified dimension, i.e. that any compactified dimension is likely to be as small as it is physically possible to be small?


  • randommuser

    What does this imply about the validity of the cosmic-censorship conjecture, in five and four dimensions? If the counterexample is a fractal, maybe that is why it is so hard to prove/disprove? If the conjecture is false, what will happen to causality???

    On a side note, if space-time is discrete at the Planck scale, the fractal pattern must eventually break down at some point, when some unknown physics becomes important. So maybe the numerical simulation is not accurate after all?

    Very exciting stuff indeed!

  • Jordan

    I have no idea what you just said, but that is the coolest headline I’ve ever seen.

  • Sean

    Not every post can be a gem of explanatory transparency. Sometimes we must content ourselves with the dazzle.

  • ossicle

    Thanks, Sean@2 and James @7! Sean: (1) just to super-duper-clarify, I wasn’t even remotely complaining. I never want to be the guy who whines about the flavor of free ice cream people are giving him. And (2), I finally bought your book last weekend, look forward to starting it tonight or tomorrow!

  • SleazyCarl

    Is it me or does all this talk of naked singularities, black G(ravity)-strings and what look like adult beads kind of kinky?

    Wishful thinking? Mmmmaybe. But not on my part.

  • ossicle

    Stephen @6, you misread that part of Sean’s post. His “(Of course.)” was a totally hilarious acknowledgment of how non-self-evident this stuff is.

  • Low Math, Meekly Interacting

    Doesn’t the Kerr solution give one a “black string” of sorts in 4 dimensions? I imagine that could wiggle around, too. Would entropy be maximized if a Kerr ring started pinching off segments? If so, could this be a general phenomenon of extended black holes, regardless if spatial dimension, i.e. would this ever happen in a universe we can describe empirically, vs. a toy universe? I’ve read in a couple places that the spin of a one or more stellar black holes has been measured to be very close to the the maximum velocity allowed to preserve cosmic censorship. Maybe we might see one that exceeds that speed limit. Maybe something like this might allow it? Seems possible to at least measure rotations of real black holes to see if any naked singularities might exist. Maybe something like this is an observable phenomenon.

    Just speculating wildly.

  • Sean

    LMMI– in the Kerr solution, the singularity is a ring, but the event horizon is still a sphere. Here we’re really talking about a cylindrical black hole.

    The Kerr singularity is a funny thing — there are some lurking instabilities, but I don’t think anyone thinks it will break off into pieces. The phenomenon is known as “mass inflation”:

  • Nex

    Haha, I’ve read the “of course” part as a funny joke, though maybe it was meant to be serious…

  • GregH

    Thanks Sean Carroll for tweeting this article and excellent blog. The graphic looks spookily like saliva stretched between two fingers with the little blobs that form from surface tension.

  • Evan Keane

    I’m with Sean Matthews (comment 3) on this. This is, although cool, just a calculation done for a model which has no known connection with reality. It may represent something real but it seems to be more Maths than Physics.

    Ok, now to continue the quest for a pulsar – black hole binary, wherein the Cosmic Sensorship Conjecture could actually be tested …

  • Low Math, Meekly Interacting

    Hm. OK, thanks, Sean. Seems like you’re saying the cylindrical horizon is a necessary component, and the overall phenomenon isn’t so easily generalized.

    I guess it still is interesting (for me, anyway) to wonder if the instability of the Kerr singularity has a thermodynamic origin, or, at least, can be thought of as an inevitability due to the 2nd Law.

  • Lab Lemming

    So do the little ones get pinched off into black holes small enough to evaporate?

  • Anders Tell

    Can someone help me out and explain how to understand 5 dimensions “exactly”
    I can see that a two dimensional plane is as a surface without extension in the direction of the third, -and all other dimensions above the plane must be felt as time if “something” -say a “being” restricted into perceiving only that plane. “Our” third direction don’t “exist” in the two of the plane. Third dimension repeated infinitely in the direction that is not in two, makes a form in same way as the first.. a line repeated infinitely makes two dimensions.
    Likewise – fourth dimensions is for us the “direction” of time, -together with the dimensions existing above felt as same for us, in “three”. Yes? And same with sixth, five repeated infinitely?
    In same way as three dimensions behave on a smaller or bigger scale then our size. As electrons is smaller then 0 in our scale and dimensions – but with theirs extension in repeating infinitely, their fifth and sixth dimensions -make traces in “our” and form for us the mirage of three.
    Or ..-do you mean something else with “dimensions”?

  • Mitchell Porter

    Lievo @8, these are all purely classical (non-quantum) calculations. A quantum theory of gravity should explain what happens where they break down.

  • Pingback: 5 March 2011 am « blueollie()

  • Lievo

    @Mitchell 23: I screwed my question: I should have say that the compactified dimension should be expected to be at least as large, not smaller, than the tiniests BHs. So it’s an evidence against the space-time being continuous up to any desired precision. As you said this is to be expected anyway. Thx for your insight. 😉

  • TimG

    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).

  • beginner

    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?

  • Sean

    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.

  • Lila Sovietskaya

    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


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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