Robust Areal Thermodynamics of the Schwarzschild Black Hole with Robin Boundary Conditions and Weyl Asymptotics

TL;DR

This paper develops a thermodynamic framework for Schwarzschild black holes based on horizon area, using quantum spectral analysis and Weyl asymptotics to derive universal equations of state and entropy relations.

Contribution

It introduces a novel areal thermodynamics approach for Schwarzschild black holes utilizing Robin boundary conditions and spectral geometry techniques.

Findings

Derived universal equations of state from Weyl volume coefficients.

Established a finite matter entropy scaling as A^{3/4}.

Reproduced the 4D Weyl law via Matsubara factorization.

Abstract

We formulate an areal thermodynamics for the Schwarzschild black hole that takes the horizon area as the sole macroscopic variable. Quantizing ultrarelativistic interior modes on a regular spacelike slice with a Robin boundary at a stretched horizon leads to a self-adjoint Laplace-Beltrami problem with Heun-type quantization. A maximum-entropy area ensemble introduces an intensive areal temperature $T_A$, and Weyl/heat-kernel asymptotics control the resulting statistical mechanics. The leading equations of state follow universally from the spatial Weyl volume coefficient: in a canonical ensemble of $N$ ultrarelativistic bosons one finds $A = 3 N k_B T_A$ up to a boundary-dependent constant, while in the massless grand-canonical sector $A \propto T_A^{4}$ with a generalized Planck spectrum and a Wien displacement relation. These scaling exponents are insensitive to Dirichlet/Neumann/Robin data and to the foliation; only numerical prefactors vary. Embedding the construction into a static four-dimensional background via Matsubara factorization reproduces the 4D Weyl law and yields a finite matter entropy $S_{\mathrm{rad}} \propto A^{3/4}$, parametrically subleading to the Bekenstein-Hawking term after standard renormalization. The framework provides a concise, mathematically controlled bridge between interior spectral data and macroscopic area relations, clarifying the scope and limitations of areal thermodynamics.