{"title":"The Smarr formula within the Geroch-Held-Penrose formalism","authors":"Alberto Guilabert and Pedro Bargueño","doi":"10.1088/1361-6382/adc180","DOIUrl":null,"url":null,"abstract":"The connection between classical thermodynamics and black hole horizons is a fundamental topic in gravitational physics, offering a potential pathway to understanding quantum aspects of gravity. However, while black hole mechanics exhibits well-known thermodynamic parallels, a rigorous geometric interpretation of thermodynamic variables directly from the field equations warrants further research. In this manuscript, we present a thermodynamic formulation of the field equations through the decomposition of the Riemann tensor, employing the Geroch-Held-Penrose (GHP) formalism, to clarify a strong correspondence between black hole thermodynamic variables and geometrical quantities derived from horizon geometry. Our analysis reveals an intrinsic connection between the Penrose and Rindler K-curvature and the Smarr relation, motivating a revised definition of both trapping gravity and black hole internal energy. Additionally, we derive through this GHP formalism the Smarr formula for the Reissner–Nördstrom black hole cointained in an AdS spacetime and we explore the implications of this relationship for black holes with exotic topologies and in the context of extended theories, exemplified by f(R) gravity. We also briefly analyze this relation in more general configurations, such as Taub-NUT spacetimes. These findings suggest a deeper geometrical basis for black hole thermodynamics, potentially advancing our understanding of gravitational energy, horizon entropy, and their significance within quantum gravity frameworks.","PeriodicalId":10282,"journal":{"name":"Classical and Quantum Gravity","volume":"30 1","pages":""},"PeriodicalIF":3.6000,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Classical and Quantum Gravity","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1088/1361-6382/adc180","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
引用次数: 0
Abstract
The connection between classical thermodynamics and black hole horizons is a fundamental topic in gravitational physics, offering a potential pathway to understanding quantum aspects of gravity. However, while black hole mechanics exhibits well-known thermodynamic parallels, a rigorous geometric interpretation of thermodynamic variables directly from the field equations warrants further research. In this manuscript, we present a thermodynamic formulation of the field equations through the decomposition of the Riemann tensor, employing the Geroch-Held-Penrose (GHP) formalism, to clarify a strong correspondence between black hole thermodynamic variables and geometrical quantities derived from horizon geometry. Our analysis reveals an intrinsic connection between the Penrose and Rindler K-curvature and the Smarr relation, motivating a revised definition of both trapping gravity and black hole internal energy. Additionally, we derive through this GHP formalism the Smarr formula for the Reissner–Nördstrom black hole cointained in an AdS spacetime and we explore the implications of this relationship for black holes with exotic topologies and in the context of extended theories, exemplified by f(R) gravity. We also briefly analyze this relation in more general configurations, such as Taub-NUT spacetimes. These findings suggest a deeper geometrical basis for black hole thermodynamics, potentially advancing our understanding of gravitational energy, horizon entropy, and their significance within quantum gravity frameworks.
期刊介绍:
Classical and Quantum Gravity is an established journal for physicists, mathematicians and cosmologists in the fields of gravitation and the theory of spacetime. The journal is now the acknowledged world leader in classical relativity and all areas of quantum gravity.