Schwarzschild black hole quantum statistics from Z(2) orientation degrees of freedom and its relations to Ising droplet nucleation

Abstract

AbstractGeneralizing previous quantum gravity results for Schwarzschild black holes from 4 to D ≥ 4 space‐time dimensions yields an energy spectrum En = α n(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 Planck length in D dimensions. Assuming that the basic area quantum has a Z(2)‐degeneracy according to its two possible orientation degrees of freedom implies a degeneracy dn = 2n 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.