Condensation of area quanta ensembles with quantum statistics in Schwarzschild spacetimes

TL;DR

This work builds a statistical model of Schwarzschild geometry using indistinguishable area quanta from loop quantum gravity and the near-horizon energy $E_g = g A/(8\pi)$. By treating tiles as spin-determined bosons or fermions and employing a fixed effective Boltzmann weight $e^{-\beta \epsilon}=e^{-2\pi \gamma}$, the authors analyze large-area, high-$g$ limits and find a highly entropic-behavior for the fermions but a condensed, low-entropy bosonic state dominated by the $j=1$ tile. In this regime the entropy scales logarithmically with area, $S_b \simeq 3 \ln(\langle A \rangle/(3 a_1))$, and the energy-entropy balance yields a microscopic origin for near-horizon geometric fluctuations without invoking holographic degeneracy. The results point to an observer-dependent condensate as a framework for geometric quantization, with the Barbero-Immirzi parameter playing a crucial role, and motivate the companion paper’s focus on fluctuations and the quantum atmosphere of black holes.

Abstract

As is well known, near-horizon (equivalently high acceleration) observers in spherically symmetric black hole spacetimes have a particularly simple form of the quasi-local energy. Using this energy and indistinguishable area quanta satisfying quantum statistics a statistical mechanical description of the Schwarzschild black hole geometry for uniformly accelerating observers is developed. The resulting model has several phases including one with highly excited states, Bose-Einstein condensates, condensates distinct from the usual Bose gas, and degenerate Fermi gases. In the large area limit, relevant for comparison to the Bekenstein-Hawking entropy, the new condensed state is favored over Bose-Einstein condensation and the degenerate Fermi gas. The entropies of the phases, and the entropy of mixing, are computed. The resulting low-entropic condensed state, where the quanta are essentially all in the lowest Bose energy state, provides the framework for the quantization of near-horizon geometric fluctuations, which is explored in a companion paper.

Figures (1)

• Figure 1: The entropy in the high-$g$, large area limit as a function of area. In (a) the numerically computed bosonic entropy $S_b$ is shown with the red filled circles. The integral approximation of the bosonic entropy, $3 \ln (A/3a_1)+7$, is shown with the blue solid line. Similarly in (b) the numerically computed fermionic entropy $S_f$ is shown with the integral approximation with the same choice of filled dots for the numerical approximation and the solid line for the integral approximation.