Geometry as Thermodynamics:Entropy Stationarity and Horizon Residues
Abstract
Thermodynamic descriptions of gravity involve both variational conditions for spacetime dynamics and analytic structures associated with horizons. We examine their relation while keeping their assumptions distinct. First, we present an explicit derivation of the Einstein equation from the established null-vector entropy functional, including the null constraint, the matter term, and the integration constant associated with the cosmological constant. Second, for an analytic static spherical geometry with a nondegenerate Killing horizon, we define a meromorphic radial one-form whose residue is the inverse of twice the surface gravity. Euclidean regularity then fixes the Hawking temperature. Combining this residue with the Einstein--Hilbert Noether charge gives a normalized contour representation of the Wald entropy. The construction reproduces the Schwarzschild temperature, entropy, and Smarr relation, but does not constitute an independent microscopic derivation of the area law. We establish the limits of a stronger identification between pole structure and dynamics: an asymptotically flat family can retain the Schwarzschild horizon residue, area, surface gravity, and mass while violating the vacuum Einstein equation, and an extremal charged solution possesses a higher-order pole. We also show why exponentiating an unspecified entropy functional does not produce a universal entropy residue. The resulting framework separates entropy stationarity, horizon analyticity, and charge normalization, providing explicit consistency tests for further thermodynamic interpretations of gravitational singularities.
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BibTeX
@misc{chen2026geometry,
title = {{Geometry as Thermodynamics:Entropy Stationarity and Horizon Residues}},
author = {Wen-Xiang Chen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30660},
eprint = {2609.30660},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30660},
}
RIS
TY - UNPB
TI - Geometry as Thermodynamics:Entropy Stationarity and Horizon Residues
AU - Wen-Xiang Chen
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30660
UR - https://arxiv.org/abs/2609.30660
AB - Thermodynamic descriptions of gravity involve both variational conditions for spacetime dynamics and analytic structures associated with horizons. We examine their relation while keeping their assumptions distinct. First, we present an explicit derivation of the Einstein equation from the established null-vector entropy functional, including the null constraint, the matter term, and the integration constant associated with the cosmological constant. Second, for an analytic static spherical geometry with a nondegenerate Killing horizon, we define a meromorphic radial one-form whose residue is the inverse of twice the surface gravity. Euclidean regularity then fixes the Hawking temperature. Combining this residue with the Einstein--Hilbert Noether charge gives a normalized contour representation of the Wald entropy. The construction reproduces the Schwarzschild temperature, entropy, and Smarr relation, but does not constitute an independent microscopic derivation of the area law. We establish the limits of a stronger identification between pole structure and dynamics: an asymptotically flat family can retain the Schwarzschild horizon residue, area, surface gravity, and mass while violating the vacuum Einstein equation, and an extremal charged solution possesses a higher-order pole. We also show why exponentiating an unspecified entropy functional does not produce a universal entropy residue. The resulting framework separates entropy stationarity, horizon analyticity, and charge normalization, providing explicit consistency tests for further thermodynamic interpretations of gravitational singularities.
ER -