Gia's Hairy Black Holes

Now that the hope for a quick LHC start-up has literally vaporized, I have six more months to play with particle theory without worrying about experimental constraints. This is a good moment to return to the workshop on black holes that is still trickling here at CERN. Last Friday, Gia Dvali was presenting his ideas concerning gravity theories with a large number of particle species. The subject is more than one year old and I already wrote about it in this blog. Gia is trying to derive general features of field theory coupled to gravity, without making specific assumptions about the underlying quantum gravity theory. To this end he produces gedanken black holes (at the LHC, or anywhere in the Universe) and employs unitarity plus general properties of semi-classical black holes to obtain some interesting conclusions.

In particular, Gia argued one year ago that in a theory with N species of particles there is a bound on the fundamental scale $M_*$ where the gravitational interactions become strong. Namely, the bound reads $M_* \leq M_P/\sqrt{N}$ where $M_P \sim 10^{19}$ GeV is the Planck scale as we know. This opens a possibility to solve the hierarchy problem by postulating $N = 10^{32}$ new particle species at TeV. One can immediately see the analogy to the ADD (as in Arkani-Hamed, Dimopoulos, Dvali) large extra dimensions. In ADD, the relation between the fundamental scale and the Planck scale is $M_P^2 = M_*^2 (M_* R)^n$, where n and R are the number and the radius of the extra dimensions. This relation is in fact equivalent to $M_* = M_P/\sqrt{N}$ because in ADD $(M_* R)^n$ is just the number of graviton KK modes below the cutoff of the theory. Gia argues that the ADD solution to the hierarchy problem is just one example in a larger class: gravity must become strong well below the Planck scale whenever there is a large number of particles species, regardless whether extra dimensions exist or not. The fact that in ADD the multitude of particles species are Kaluza-Klein modes of a higher dimensional graviton is just a red herring.

After the initial proposal Gia has posted several papers further developing this idea. On Friday Gia talked mostly about the consequences for black hole physics summarized in this paper. It turns out that micro black holes in theories with a large number of species have peculiar properties that make them quite distinct from ordinary black holes in Einsteinian gravity. First of all, small black holes with sizes of the order of $M_*^{-1}$ are hairy. This Freudian feature can be argued as follows. Consider a collision of a particle and an anti-particle of one species at center-of mass energies of order $M_*$. The production rate of micro black holes in that collision must also be of order $M_*$ because it is the only scale available. By unitarity, the decay into the same species must also proceed at the rate $M_*$. However, this cannot be true for the decay in the remaining N-1 species: the decay rate can be at most $\sim M_*/N$, while assuming a faster decay rate leads to a contradiction. For example, black hole exchange would lead to a too fast growth of scattering amplitudes that would be at odds with the assumption that the cut-off of the theory is at $M_*$. Thus, the micro black holes in a theory with N species are highly non-democratic, and they need to carry a memory of the process in which they were produced.

As black holes grow heavier and older they should start losing their hair. The scale where the Einsteinian behavior is recovered cannot however be smaller than $M_* N$. Thus, black holes in the mass range $M_P/\sqrt{N} < M_{BH} < M_P \sqrt{N}$ must be non-standard, hairy and undemocratic. The proper black hole behavior, which entails democratic decay to all available species via Hawking radiation, can be expected only for heavier black holes. Again, these properties can be readily understood in the specific example of ADD large extra dimensions as a consequence of the geometric structure of the extra dimensions (for example, the crossover scale to the Einsteinian behavior is related to the radius of the extra dimension). Gia's arguments just generalize it to any theory with a large number of particle species. It seems that some kind of "emergent geometry" and "localization" must be a feature of any consistent low scale quantum gravity theory.

Of course there is a lot of assumptions that enter in that game (no black hole remnants, for example), and it is not unthinkable that the true quantum gravity theory may violate some of these assumptions. Nevertheless, I find amusing that such simple hand-and-all-body-waving arguments lead to quite profound consequences.

More details in the paper.