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

Thomas Schürmann

TL;DR

This work formulates an areal thermodynamics for Schwarzschild black holes by treating the horizon area $A$ as the sole macroscopic constraint and deriving universal relations from the Weyl density of interior Laplace–Beltrami modes with Robin boundary conditions at a stretched horizon. Using a maximum-entropy approach, it introduces an areal temperature $T_A$ conjugate to $A$ and shows that, in the ultrarelativistic (massless) limit, the canonical sector yields $A = 3 N k_B T_A$ (plus boundary offsets) while the radiation sector gives $A = ilde{ 4pha} T_A^4$ with a Planck-like spectrum and Wien displacement; these exponents depend only on the spatial Weyl coefficient $C_0$ and are robust to boundary data and foliation. Embedding the 3D spatial construction into a static 4D spacetime via Matsubara factorization reproduces the 4D Weyl law and yields a finite areal matter entropy $S_{ ext{rad}} \,\propto A^{3/4}$, subleading to the Bekenstein–Hawking term after renormalization within the generalized entropy framework $S_{ ext{gen}}=S_{ ext{BH}}+S^{ ext{ren}}_{ ext{out}}$. The framework thus provides a concise, mathematically controlled bridge between interior spectral data and macroscopic area relations, clarifying the scope of areal thermodynamics and its compatibility with the generalized second law, while highlighting the universal role of Weyl geometry in setting the leading exponents.

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.

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

TL;DR

This work formulates an areal thermodynamics for Schwarzschild black holes by treating the horizon area as the sole macroscopic constraint and deriving universal relations from the Weyl density of interior Laplace–Beltrami modes with Robin boundary conditions at a stretched horizon. Using a maximum-entropy approach, it introduces an areal temperature conjugate to and shows that, in the ultrarelativistic (massless) limit, the canonical sector yields (plus boundary offsets) while the radiation sector gives with a Planck-like spectrum and Wien displacement; these exponents depend only on the spatial Weyl coefficient and are robust to boundary data and foliation. Embedding the 3D spatial construction into a static 4D spacetime via Matsubara factorization reproduces the 4D Weyl law and yields a finite areal matter entropy , subleading to the Bekenstein–Hawking term after renormalization within the generalized entropy framework . The framework thus provides a concise, mathematically controlled bridge between interior spectral data and macroscopic area relations, clarifying the scope of areal thermodynamics and its compatibility with the generalized second law, while highlighting the universal role of Weyl geometry in setting the leading exponents.

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 , 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 ultrarelativistic bosons one finds up to a boundary-dependent constant, while in the massless grand-canonical sector 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 , 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.

Paper Structure

This paper contains 43 sections, 78 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Guide to the paper.
  3. Terminology (areal temperature).
  4. Notation and units.
  5. Geometric and spectral setup
  6. Spacelike interior slice and Laplace--Beltrami operator
  7. Robin boundary condition at a stretched horizon
  8. Normal derivative and stretched--horizon regulator.
  9. Remark (Physical interpretation and scale of $h$).
  10. Area operator and areal thermodynamics
  11. On the identification $E=\langle \widehat{H} \rangle$ with the ADM energy
  12. What is being identified.
  13. Maximum entropy principle and areal temperature
  14. Spectral asymptotics and single--particle partition function
  15. Explicit $C_0$ and geometry of $(B_{r_{\mathrm{s}}},h)$.
  16. ...and 28 more sections

Figures (1)

  • Figure 1: Illustration of the canonical laws $A=3Nk_{\mathrm{B}} T_A+\delta A_{\mathrm{b.c.}}$ with the boundary shifts from \ref{['eq:deltaA-explicit']} (solid lines) and the grand-canonical radiation law $A=\sigma T_A^4$ mapped to the dimensionless variables $\tilde{A} := A/(N a)$ and $\tilde{T} := k_{\mathrm{B}} T_A / a$ (dashed). For the radiation curve we used $\sigma$ from \ref{['eq:sigma']} with $C_0=5/24$, cf. \ref{['eq:C0-value']}, and set $N=100$ for visual comparison. This explains the prefactor $(\pi^4/24)/N$ shown in the legend.