Generalized First Law and Smarr-type Relation: Beyond Additivity and Extensivity
Abstract
In this work, we develop a general framework for deriving the first law of black hole thermodynamics and the associated Smarr-type relation for 4-dimensional static spherically symmetric spacetimes within a broad class of generalized entropy models. In addition, by employing the entropy representation for the Reissner--Nordström (RN) black hole, we determine a concise curvature functional that depends only on this representation and its derivatives. A general flatness condition is derived, demonstrating that when the entropy Hessian metric is regular and nondegenerate, both homogeneous power-law transformations and a separate logarithmic branch leave the RN black hole's Ruppeiner geometry flat. On the other hand, Rényi, Sharma-Mittal, Kaniadakis, and LQG-inspired entropy deformations typically give rise to nonvanishing curvature in the regular thermodynamic regime, demonstrating that the extremality of the RN black hole does not by itself require a curvature divergence.
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BibTeX
@misc{zafar2026generalized,
title = {{Generalized First Law and Smarr-type Relation: Beyond Additivity and Extensivity}},
author = {Usman Zafar and Krishnakanta Bhattacharya and Kazuharu Bamba},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.00381},
eprint = {2605.00381},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00381},
}
RIS
TY - UNPB
TI - Generalized First Law and Smarr-type Relation: Beyond Additivity and Extensivity
AU - Usman Zafar
AU - Krishnakanta Bhattacharya
AU - Kazuharu Bamba
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2605.00381
UR - https://arxiv.org/abs/2605.00381
AB - In this work, we develop a general framework for deriving the first law of black hole thermodynamics and the associated Smarr-type relation for 4-dimensional static spherically symmetric spacetimes within a broad class of generalized entropy models. In addition, by employing the entropy representation for the Reissner--Nordström (RN) black hole, we determine a concise curvature functional that depends only on this representation and its derivatives. A general flatness condition is derived, demonstrating that when the entropy Hessian metric is regular and nondegenerate, both homogeneous power-law transformations and a separate logarithmic branch leave the RN black hole's Ruppeiner geometry flat. On the other hand, Rényi, Sharma-Mittal, Kaniadakis, and LQG-inspired entropy deformations typically give rise to nonvanishing curvature in the regular thermodynamic regime, demonstrating that the extremality of the RN black hole does not by itself require a curvature divergence.
ER -