## Wednesday, November 29, 2006 ... //

### Choptuik exponent is Regge saturation exponent: maybe

Take two random articles from this weblog:

Such two random articles are expected to have nothing to do with each other. For example, the first one is concerned with a messy classical calculation in GR of a marginal formation of a black hole - essentially some dirty astrophysics - while the second one is some special limit of QCD relevant for some messy nuclear physics. Neither of them seems close to the fundamental equations and they moreover probe very different parts of physics at very different scales. And one of them is classical while the other one is quantum. So they can't be related. Or can they? ;-)

Using the BFKL pomeron exchange in a gauge theory, one can try to calculate some scattering amplitudes in the Regge limit (high-energies, small angle). A linear approximation of this calculation breaks down for some rapidity "y" that is related to the ratio of the sizes of certain two three-dimensional momenta by a scaling law
• exp(y) = (k1 / k2)^{gammaBFKL}
where one can calculate that the exponent is numerically
• gammaBFKL = 0.409552...
This breakdown essentially calculates the ratio of the sizes of the two nucleons as seen by the pomeron exchange.

Now take a completely different physical system: scalars coupled to gravity. Consider a line in the scalar configuration space of initial conditions of the scalars parameterized by a single parameter "p" such that for "p=0", the system evolves into an empty Minkowski space in the future while for large values of "p", a black hole is formed. So there must be a critical value "p0" above which the black hole is formed. How heavy the black hole will be: what is the mass "M"? It depends how closely you get to "p0". Choptuik's relation is
• M = M0 x (p-p0)^{gammaBH}
That's nice, another critical exponent. Note that both of them were denoted by "gamma". Tonight,
argue that it is no coincidence, using the AdS5 / CFT4 correspondence, and both of these exponents are actually equal. If true, that's a rather fascinating relation between quantum phenomena in a gauge theory living in the flat space and a complicated non-linear but classical calculation in general relativity in the context of the birth of a black hole. I actually believe that the relation could be correct even though the Choptuik exponent for five-dimensional gravity is only known - also numerically - as
• gamma = 0.408 +- 2 percent
It would be a good idea to try to calculate this one and perhaps also the other one more accurately or even analytically.