{"title":"Black Hole Thermodynamic Free Energy as A-discriminants","authors":"Mounir Nisse, Yen-Kheng Lim, Linus Chang","doi":"10.1007/s10773-024-05711-x","DOIUrl":null,"url":null,"abstract":"<p>We show that the free energy <span>\\(\\mathcal {F}\\)</span> and temperature <i>T</i> of black holes, considered as a thermodynamic system, can be viewed as an <i>A</i>-discriminant of an appropriately-defined polynomial. As such, mathematical results about <i>A</i>-discriminants may lead to implications about black hole thermodynamics. In particular, for static spacetimes with spherical, planar, or hyperbolic symmetry, the number of distinct thermodynamic phases depend on the number of distinct terms in the metric component <span>\\(g_{tt}\\)</span>. We prove that if <span>\\(g_{tt}\\)</span> consists of <span>\\(N_{f}\\)</span> distinct terms, then the <span>\\(\\mathcal {F}\\)</span>-<i>T</i> curve consists of <span>\\(N_f-2\\)</span> cusps, which in turn leads to <span>\\(N_f-1\\)</span> distinct thermodynamic phases. This result is applied to explicit examples of the Schwarzschild-AdS, Reissner–Nordström, power-law Maxwell, and Euler–Heisenberg black holes.</p>","PeriodicalId":597,"journal":{"name":"International Journal of Theoretical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Journal of Theoretical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10773-024-05711-x","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We show that the free energy \(\mathcal {F}\) and temperature T of black holes, considered as a thermodynamic system, can be viewed as an A-discriminant of an appropriately-defined polynomial. As such, mathematical results about A-discriminants may lead to implications about black hole thermodynamics. In particular, for static spacetimes with spherical, planar, or hyperbolic symmetry, the number of distinct thermodynamic phases depend on the number of distinct terms in the metric component \(g_{tt}\). We prove that if \(g_{tt}\) consists of \(N_{f}\) distinct terms, then the \(\mathcal {F}\)-T curve consists of \(N_f-2\) cusps, which in turn leads to \(N_f-1\) distinct thermodynamic phases. This result is applied to explicit examples of the Schwarzschild-AdS, Reissner–Nordström, power-law Maxwell, and Euler–Heisenberg black holes.
期刊介绍:
International Journal of Theoretical Physics publishes original research and reviews in theoretical physics and neighboring fields. Dedicated to the unification of the latest physics research, this journal seeks to map the direction of future research by original work in traditional physics like general relativity, quantum theory with relativistic quantum field theory,as used in particle physics, and by fresh inquiry into quantum measurement theory, and other similarly fundamental areas, e.g. quantum geometry and quantum logic, etc.