paper

Schwarzschild Black Hole Quantum Statistics from Z(2) Orientation Degrees of Freedom and its Relations to Ising Droplet Nucleation

arXiv:gr-qc/9906104 · doi:10.1002/1521-3889(200007)9:7<503::AID-ANDP503>3.0.CO;2-L

Abstract

Generalizing previous quantum gravity results for Schwarzschild black holes from 4 to D > 3 space-time dimensions yields an energy spectrum E_n = alpha n^{(D-3)/(D-2)} E_P, n=1,2,..., alpha = O(1), where E_P is the Planck energy in that space-time. This spectrum means that the quantized area A_{D-2}(n) of the D-2 dimensional horizon has universally the form A_{D-2} = n a_{D-2}, where a_{D-2} is essentially the (D-2)th power of the D-dimensional Planck length. Assuming that the basic area quantum has a Z(2)-degeneracy according to its two possible orientation degrees of freedom implies a degeneracy d_n = 2^n for the n-th level. The energy spectrum with such a degeneracy leads to a quantum canonical partition function which is the same as the classical grand canonical partition function of a primitive Ising droplet nucleation model for 1st-order phase transitions in D-2 spatial dimensions. The analogy to this model suggests that E_n represents the surface energy of a "droplet" of n horizon quanta. Exploiting the well-known properties of the so-called critical droplets of that model immediately leads to the Hawking temperature and the Bekenstein-Hawking entropy of Schwarzschild black holes. The values of temperature and entropy appear closely related to the imaginary part of the partition function which describes metastable states

29 pages, Latex; version to appear in Annalen der Physik (Leipzig) (Refs. added, a few text improvements, no changes of results)

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