Friday, August 26, 2005

Choptuik collapse and Barak Kol

Barak Kol from Jerusalem - who is visiting Cambridge in August - was explaining me various things related to his work in recent years. For example, he and E. Sorkin have shown that the behavior of black strings and black holes - and especially the smoothness of the transition between them - seems to depend on the dimension in such a way that "D=10" or "D=11" is one of the "critical" spacetime dimensionalities, namely one above which the transition becomes smooth.

Let me not go into details - but this "D=10" or "D=11" bound is a property of purely classical GR. No doubt, one of the obvious questions is whether there is any rational explanation why this "D=10" or "D=11" is simultaneously the critical dimension of string theory or M-theory; it can be just an accident, of course. Especially those of us who consider M-theory in "D=11" to be equally fundamental as the 10-dimensional supersymmetric vacua may ask "Why did not we get the other number from the set {10,11}, for example?"

Barak's current work is related to self-similar solutions in gravity. Let us focus on the Choptuik collapse. Imagine a spherically symmetric distribution of mass - for example, a spherical wave of a scalar field - that is going to produce a black hole. Well, it will only end up as a black hole if the initial parameters are properly chosen. Let's choose a line in the space of the initial conditions parameterized by a number we will call "P". If "P" is smaller than a certain number, which we will normalize to one, no black hole is formed. If "P" is larger than one, a black hole is inevitably the final state of the collapse.

It is not surprising that some interesting behavior occurs near "P=1". If "P" is slightly below one, the mass will bounce many times, but it will avoid the collapse into the black hole. At the critical value "P=1", the corresponding classical solution of GR will locally have a character of a self-similar fractal; the proper times of the individual bounces will generate a geometrical sequence. The self-similarity may be interpreted as a symmetry under a discrete subgroup of the Weyl symmetry of rescalings - and one may think about this picture in terms of a spontaneously broken conformal symmetry.

Let's define the total mass of the black hole "M" we create as a function of "P". For "P" smaller than one, by definition, "M(P)" must vanish. However, it starts to grow above "P=1", namely as
  • "M" goes like "(P-1)^{gamma}"
where "gamma" is an exponent, perhaps something like "0.301308" (the right value is different but the last four digits of my random value may be inspiring for Quantoken). Its value is known numerically and it is universal for a fixed spacetime dimensionality and for all spherically symmetric collapses of a scalar field. It is certainly a number that deserves an analytical derivation. There are other "critical exponents" in this game; the coefficient "exp(Delta)" determining the decrease of the proper distances in the fractal solution is another example. It is hard to hide that the behavior near the critical point "P=1" shares many features with the theory of phase transitions.

Of course, these things are properties of classical GR, and we tend to consider the fractal to be unphysical at sub-Planckian distances; at least I do. Nevertheless, you know that there have been speculations that gravity could have an ultraviolet fixed point, a scale invariant theory valid at the sub-Planckian distances whose scale invariance is spontaneously broken at the Planck scale. I, for one, don't believe these things, especially because they seem to be inconsistent with everything we know about string theory, but we can't definitely rule out their existence at this moment, I think. Of course, the more one studies some fractal classical solutions of GR, the more one would like these super-short features of the solutions to be physical. ;-) But one should resist the temptation: self-similar geometries don't seem to be a part of the "real physics" so far.

It seems to me that discretely self-similar solutions may be more physical if they occur within a conformal field theory rather than gravity. But we will see whether Barak finds something interesting about the self-similar gravitational solutions. Good luck to him.


  1. My mail was stopped short, so I post it again in a little more detail.

    The four-dimensional GR equations require a 6-fold (12_R dim) CY in order to solve. This makes 12-dimensions special, as 4-dimensions is also special in topology. Apart from supersymmetry in d=11, this is a mathematical relation that makes d=12 special (and specifically relates to the speciality of d=4 topology, which I find interesting). Some of this, although you do not want to read perhaps, is in my CY differential eqn work. Likewise, the proof of the Poincare conjecture in 3-dimensions requires the topology and cohomology of an associated 3-fold (6_R dim), and might be formulated in terms of the latter.

  2. I realized today and yesterday, that the cohomology of the Einstein equations when written in terms of the Calabi-Yau metrics is probably enough to prove the Poincare conjecture, and also its analogs in higher dimensions. The metrics are formulated in terms of
    geodesic flows in a Calabi-Yau space, and the deformations of the flows correspond to global deformations of manifolds in d>3 dimensions. The cohomology of the CY manifold corresponding to the Einstein eqns with w=\pm 1,0 are all that is required to find the obstructions to homotophies of manifolds. (Dimension 12 is probably special from the four-dimensional point of view.)