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

TL;DR

This paper derives a quantum statistical model for Schwarzschild black holes in higher dimensions, linking horizon quantization, degeneracy from orientation degrees of freedom, and phase transition analogies to explain black hole thermodynamics.

Contribution

It introduces a novel quantum spectrum with Z(2) degeneracy for black hole horizons and connects it to Ising droplet nucleation models, providing new insights into black hole entropy and temperature.

Findings

Derived energy spectrum for higher-dimensional black holes.

Linked horizon area quantization to a Z(2) degeneracy.

Connected black hole thermodynamics to phase transition models.

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