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CarlB

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**Title: An analytical computation of asymptotic Schwarzschild quasinormal frequencies**

Lubos Motl, Adv.Theor.Math.Phys. 6 (2003) 1135-1162

*Recently it has been proposed that a strange logarithmic expression for the so-called Barbero-Immirzi parameter, which is one of the ingredients that are necessary for Loop Quantum Gravity (LQG) to predict the correct black hole entropy, is not another sign of the inconsistency of this approach to quantization of General Relativity, but is rather a meaningful number that can be independently justified in classical GR. The alternative justification involves the knowledge of the real part of the frequencies of black hole quasinormal states whose imaginary part blows up. In this paper we present an analytical derivation of the states with frequencies approaching a large imaginary number plus ln 3 / 8 pi M; this constant has been only known numerically so far. We discuss the structure of the quasinormal states for perturbations of various spin. Possible implications of these states for thermal physics of black holes and quantum gravity are mentioned and interpreted in a new way. A general conjecture about the asymptotic states is stated. Although our main result lends some credibility to LQG, we also review some of its claims in a critical fashion and speculate about its possible future relevance for Quantum Gravity.*

http://www.arxiv.org/abs/gr-qc/0212096

In the above paper, (equations 51-54) Lubos obtains a transmission amplitude for the spin-j energy emitted by a black hole. The spin-1/2 and spin-1 amplitudes follow the expected Fermi and Bose statistics:

[tex]T(\omega) = 1/(e^{\beta_{\textrm{Hawking}}\;\omega} \; \mp 1)[/tex]

where the + sign is the Fermi and the - sign is the Bose statistics, and

[tex]\beta_{\textrm{Hawking}} \; = 1/T_{\textrm{Hawking} }\;.[/tex]

What's strange is that the spin-2 amplitude follows the following rule:

[tex]T(\omega) = 1/(e^{\beta_{\textrm{Hawking}}\;\omega} \;+ 3)[/tex]

Lubos calls this "tripled Pauli statistics", writing:

Why do we fail to obtain the same Bose-Einstein factor as we did for odd j? Instead, we calculated a result more similar to the half-integer case, i.e. Fermi-Dirac statistics with the number 3 replacing the usual number 1; let us call it Tripled Pauli statistics. Such an occupation number (51) can be derived for objects that satisfy the Pauli’s principle, but if such an object does appear (only one of them can be present in a given state), it can appear in three different forms. Does it mean that scalar quanta and gravitons near the black hole become (or interact with) J = 1 links (triplets) in a spin network that happen to follow the Pauli’s principle?

I would like to suggest that the above form can be put into the standard form by making two assumptions. First, there are degrees of freedom present in the system that are not apparent in the usual quantum mechanics. For each Pauli state, one finds three states at high temperature, which can be thought of as three colors. Second, the colored states are of extreme high energy (on the order of the Planck energy), and consequently are suppressed, leaving only the colorless state to contribute to the exponential.

Any comments?