Bigger than the FCC

We’ve been studying rotating black holes in my class this week, which has been fun. We get to apply the techniques we’ve been honing in the context of the Schwarzschild solution (link to recent posts below) to a bit more complicated solution, the Kerr solution, which includes rotation. Some equations follow, although you needn’t be put off by them. Most of this will make sense without really understanding them much. Just so you can see the shape of the things we scribble, I’ll show you the equation that captures this curved spacetime geometry, with no real explanation (sorry):

kerr solution

Here [tex]\Delta\equiv r^2-2Mr+a^2[/tex] and [tex]\rho^2\equiv r^2+a^2\cos^2\theta\ .[/tex] The parameter [tex]a[/tex] is the ratio of the solution’s spin or angular momentum [tex]J[/tex] to its total mass [tex]M[/tex], measured in appropriate units. It’s a very important solution to get to grips with, since it’s not just fantasy physics, but highly relevant for astrophysics since black holes that are “out there” are unlikely to be non-rotating, and in fact, one can expect them to be rotating at quite a clip in many cases. A good many black holes – including some of the supermassive ones that power active galactic nuclei making some of the most spectacularly energetic objects known in the universe – are likely to be rotating almost as much as they can.

As much as they can? A key thing about black holes is that they can only rotate so much. If [tex]a[/tex] exceeds [tex]M[/tex], their horizon (see earlier post), located at [tex]r=M+\sqrt{M^2-a^2}[/tex], disappears and the singularity contained within would be visible for all to see – A Naked Singularity!! As far as we know, this can’t happen in Nature. That this can’t happen is called “Cosmic Censorship”. It is a conjecture. Black hole singularities always seem to come clothed inside an horizon, shielding them from our gaze (and protecting our physics from the inherent unpredictability that seems to arise at these points of infinite spacetime curvature). (There’s a lot more to the issue than this, which you can read about elsewhere, but this’ll do for our purposes.)

Happily, there was the appropriate giggle at this point, and one of the students in the class raised his hand and asked something like “Is it kind of like the FCC?”, to which I replied “It’s bigger than the FCC”.

That was on Tuesday, and today we studied the motion of objects in the geometry around the rotating black hole. A cute and simple computation that supports the Cosmic Censorship conjecture is as follows. You might imagine that you can start with a black hole that has as much angular momentum as it possibly can [tex]a=M[/tex] – the “extremal” case it is called. Now take an object that has the right amount of spin compared to mass and throw it toward the black hole. If it falls into the black hole it could increase [tex]a[/tex] by an amount more than the amount [tex]M[/tex] increases – the horizon would disappear and so the singularity would be naked! This would seem to be a natural way to create naked singularities in Nature. You can imagine the right sort of matter falling into an extremal or near-extremal black hole in a real astrophysics context and making it turn into a naked singularity.

Well, the real test of this is to see from the equations themselves whether it is actually possible to throw such an object into the hole at all, and the analysis actually yields something rather cute – it’s impossible. (In slightly technical terms, the “scattering problem” produces an effective potential whose shape depends upon the angular momentum and mass of the object being thrown in. For just the type of object that can over-spin the hole and make it into a naked singularity, the effective potential has a barrier that reflects the object away from the black hole before it can cross the horizon and merge with it!)

We don’t know why, but Nature seems to naturally want us to be protected from singularities, throwing up horizons to clothe them. Horizons and singularities seem to go together in this way, horizons being the tool by which the cosmic censors express their modesty – In Einstein’s General Relativity, anyway.

While typing this, I just had a thought. If string theory goes where General Relativity cannot go and repairs singularities, supplying new physics that restores predictability (this is still being worked on – see earlier post), then it would seem that horizons are totally unnecessary. The natural conclusion would be that they themselves are simply figments of the limitations of General Relativity. So by this chain of logic (perhaps I’m stretching the word here) they probably are also removed by string theory. Horizons are then, one would conclude, just General Relativity’s way of covering its embarrassment at being incomplete. They are approximations to something else. Ah. Perhaps this is what Sumir Mathur’s conjecture* is all about. Huh. I never thought of that before.



* As you may not know what that is I’ll try to find a good link to further information about it later. Too tired to type more.

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6 Responses to Bigger than the FCC

  1. Tommy says:

    May I suggest,

    and for a somewhat more detailed review:

    The original proposal is also nicely summarized in Samir’s gravity essay from a few years ago:

    Of course, if you wanted more details and modern developments (including the suggested link to Strominger-Vafa state counting) I could link some of my own work, but I’m too humble to do that ;).

  2. Clifford says:

    Excellent! Thanks!


  3. Tommy says:

    Oh, you know I forgot a great little essay Samir wrote about what happens falling into a microstate and how it reproduces black hole like behavior:

  4. stevem says:

    Some people might disagree but I would say that the Kerr solution is the most remarkable (and beautiful) exact solution in all of mathematical physics, and a great triumph of general relativity. Despite its mathematical richness it is of course highly physical too (as you point out) since the universe is no doubt saturated with Kerr black holes, the end result of very massive collapsed spinning stars.
    There is an extensive treatment of the solution in Chandrasekhar’s (now classic) book “The Mathematical Theory of Black Holes”. Beautiful stuff. I may be mistaken but did’nt GS&Witten mention in their string theory book (Vol 1) that the linear Regge slope relation/trajectory between J and M in string theory for massive string states is kind of like the J=aM relation for the black hole? Of course, a lot of work has been done since then on the connections between strings and black holes.

  5. Oliver says:

    Dublin was fortunate enough to have a public lecture by Roy Kerr, himself, last night. He was the speaker at the 9th J. L. Synge memorial lecture, hosted by Trinity College.

    The talk was, to be fair, a little on the ramshackle side, with the speaker being as surprised as anyone when the computer running his presentation told him he had no more slides left! Nevertheless, there were some real gems – not so much about the science itself, which was mentioned only superficially – but more to do with the process of doing good science.

    It was very apparent that physical insight was at the heart of Kerr finding his solution and he really emphasised this. Coming to the problem of finding solutions to Einstein’s equations as a mathematician, he got the impression that physicist thought `he wouldn’t be able to tell physics from a hole in his head’. But he
    was actually most interested in solutions which had physical applicability. Moreover, he knew that the problem could only be solved if it was simplified, and physics provided him with the means to effectively do this.

    Subsequent to his paper being published – which was only 1.5pages long – he expressed considerable annoyance that, as he put it, every other sentence was expanded by others into some unwieldy 50page proof, obscuring, in his opinion, the simple ideas behind what he had done.

    He concluded his talk with his opinions on dark matter and dark energy. For the former, he thinks the evidence – particularly from gravitational lensing – is very strong. But he is very skeptical about dark energy. He says that the conclusion it exists relies too strongly on the assumptions of homogeneity and isotropy. As he went on to say, `the more we look at the universe, the less homogeneous and isotropic is appears to be’.

  6. Clifford says:

    Oliver – Thanks! That was really fascinating to hear about… Always good to hear about these sorts of reflections. I had no idea that Kerr was on the lecture circuit.