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

Generalizing previous quantum gravity results for Schwarzschild black holes from 4 to D ≥ 4 space-time dimensions yields an energy spectrum En = αn (D−3)/(D−2) EP,D , n = 1, 2, . . . , α = O(1) , where EP,D is the Planck energy in that space-time. This energy spectrum means that the quantized area AD−2(n) of the D−2 dimensional horizon has universally the form AD−2 = n aD−2, where aD−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 dn = 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 potential of a primitive Ising droplet nucleation model for 1st-order phase transitions in D−2 spatial dimensions. The analogy to this model suggests that En represents the surface energy of a ”bubble” of n horizon area 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.