Diagnosis inspired by the thermodynamic geometry for different thermodynamic schemes of the charged BTZ black hole
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Regular Article - Theoretical Physics
Diagnosis inspired by the thermodynamic geometry for different thermodynamic schemes of the charged BTZ black hole Zhen-Ming Xu1,2,3,4,a , Bin Wu1,2,3,4,b , Wen-Li Yang1,2,3,4,c 1
Institute of Modern Physics, Northwest University, Xi’an 710127, China School of Physics, Northwest University, Xi’an 710127, China 3 Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi’an 710127, China 4 Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China
2
Received: 12 May 2020 / Accepted: 18 October 2020 © The Author(s) 2020
Abstract Due to the asymptotic structure of the black hole solution, there are two different thermodynamic schemes for the charged Banados–Teitelboim–Zanelli (BTZ) black hole. In one scheme, the charged BTZ black hole is super-entropic, while in the other, it is not (the reverse isoperimetric inequality is saturated). In this paper, we investigate the thermodynamic curvature of the charged BTZ black hole in different coordinate spaces. We find that in both schemes, the thermodynamic curvature is always positive, which may be related to the information of repulsive interaction between black hole molecules for the charged BTZ black hole if we accept an empirical relationship between the thermodynamic curvature and interaction of a system. More importantly, we provide a diagnosis for the discrimination of the two schemes from the point of view of the thermodynamics geometry. For the charged BTZ black hole, when the reverse isoperimetric inequality is saturated, the thermodynamic curvature of an extreme black hole tends to be infinity, while when the reverse isoperimetric inequality is violated, the thermodynamic curvature of the extreme black hole goes to a finite value.
1 Introduction At present, black hole physics is generally considered as one of the best and effective ways to explore quantum gravity. Especially with the pioneering discovery of Hawking and Bekenstein about the temperature and entropy in the black hole [1–4], the general relativity, quantum mechanics and statistical physics are closely linked together to make it posa e-mail:
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sible for us to glimpse the tip of the iceberg of quantum gravity. Thermodynamics theory, which has been rested on general principles spanning a wide range of pure fluid physical systems, is now applied to black hole systems extensively and successfully [5]. One of the most prominent is the introduction of the extended phase space [6–8], which makes the charged AdS black hole and the van der Waals fluid have a close relationship [9,10]. Black holes exhibit abundant phase transitions and critical behaviors in such an extended phase space [11–13]. Recently, the theory of the thermodynamics geometry [14–17] is widely applied to the thermodynamics system of black holes, which provides a new and more traceable perspective for studying the micro-mechanism of black holes from the axioms of thermodyn
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