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.
Highlights
The theory of the thermodynamics geometry [14,15,16,17] 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 thermodynamics phenomenologically
It is necessary to discuss the thermodynamic geometry of the Banados–Teitelboim– Zanelli (BTZ) black hole again in the extended phase space
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
Summary
The theory of the thermodynamics geometry [14,15,16,17] 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 thermodynamics phenomenologically. We shall explore the thermodynamic geometry of (2 + 1)-dimensional charged BTZ black hole in different coordinate spaces under two thermodynamic schemes in the framework of the extended phase space. In the extended phase space, the study [54] looked at the behavior of the thermodynamic curvature of the charged and rotating BTZ black hole with respect to entropy and thermodynamic volume respectively. We discuss the thermodynamic curvature of the charged BTZ black hole in different coordinate spaces under two thermodynamic schemes (Case I and Case II) and provide a diagnosis for the discrimination of the two schemes from the point of view of the thermodynamics geometry.
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