Robust Areal Thermodynamics of the Schwarzschild Black Hole with Robin Boundary Conditions and Weyl Asymptotics
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2018
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| author | Schürmann, Thomas |
| author_facet | Schürmann, Thomas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1807_09128 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Robust Areal Thermodynamics of the Schwarzschild Black Hole with Robin Boundary Conditions and Weyl Asymptotics Schürmann, Thomas General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Quantum Physics 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. |
| title | Robust Areal Thermodynamics of the Schwarzschild Black Hole with Robin Boundary Conditions and Weyl Asymptotics |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/1807.09128 |