[Paper Review] Thermodynamic curvature and isoperimetric inequality for the charged BTZ black hole

Zhen-Ming Xu, Bin Wu+1 more|arXiv (Cornell University)|Feb 1, 2020

Black Holes and Theoretical Physics1 citations

TL;DR

This paper investigates the thermodynamic curvature of the charged BTZ black hole in two phase space schemes, finding consistently positive curvature across both, suggesting repulsive interactions between black hole molecules. It establishes equivalence of curvature in different phase spaces and conjectures that extreme black holes with saturated reverse isoperimetric inequality exhibit infinite curvature, while super-entropic ones yield finite curvature.

ABSTRACT

In this paper, we investigate the thermodynamic curvature of the charged Banados-Teitelboim-Zanelli (BTZ) black hole and the relationship between thermodynamic curvature and reverse isoperimetric inequality. There are two schemes for analyzing the thermodynamic behavior of the charged BTZ black hole. In one scheme, the charged BTZ black hole is super-entropic, while in the another, it is not (the reverse isoperimetric inequality is saturated). 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. Furthermore, we find that for different phase spaces, the obtained thermodynamic curvatures in both schemes are equivalent respectively, which provides a clue to establish the coordinate-free definition of the thermodynamic geometry. Meanwhile, we give a conjecture that {\em when the reverse isoperimetric inequality is saturated, the thermodynamic curvature of extreme black hole tends to be infinity and for super-entropic black holes, the thermodynamic curvature of extreme black hole will go to a finite value}.

Motivation & Objective

• To analyze the thermodynamic curvature of the charged BTZ black hole in different phase space frameworks.
• To examine the relationship between thermodynamic curvature and the reverse isoperimetric inequality.
• To investigate whether curvature remains invariant under different phase space choices, supporting a coordinate-free definition of thermodynamic geometry.
• To explore the behavior of thermodynamic curvature at the extreme black hole limit under varying isoperimetric conditions.

Proposed method

• Analysis of the charged BTZ black hole's thermodynamic properties using two distinct phase space schemes.
• Computation of the thermodynamic curvature via the Ruppeiner metric in both phase space frameworks.
• Comparison of curvature values across the two schemes to assess invariance and consistency.
• Application of the reverse isoperimetric inequality to classify black holes as super-entropic or non-super-entropic.
• Use of geometric thermodynamics to interpret curvature as a measure of microscopic interactions.
• Formulation of a conjecture on extreme black hole curvature behavior based on isoperimetric saturation.

Experimental results

Research questions

• RQ1 How does the thermodynamic curvature of the charged BTZ black hole behave in different phase space schemes?
• RQ2 What is the relationship between the reverse isoperimetric inequality and the sign or magnitude of thermodynamic curvature?
• RQ3 Are the thermodynamic curvatures obtained in different phase spaces equivalent, and what does this imply for a coordinate-free thermodynamic geometry?
• RQ4 What happens to the thermodynamic curvature of an extreme charged BTZ black hole when the reverse isoperimetric inequality is saturated?
• RQ5 Can a general pattern be established for extreme black hole curvature based on isoperimetric properties?

Key findings

• The thermodynamic curvature of the charged BTZ black hole is consistently positive across both phase space schemes, indicating repulsive interactions between black hole molecules.
• In both schemes, the curvature remains finite and positive, supporting the presence of microscopic repulsive forces.
• The thermodynamic curvatures derived in the two phase space frameworks are equivalent, suggesting a potential foundation for a coordinate-free definition of thermodynamic geometry.
• When the reverse isoperimetric inequality is saturated, the curvature of the extreme black hole is conjectured to diverge to infinity.
• For super-entropic black holes, the curvature of the extreme state is conjectured to remain finite, indicating a distinct thermodynamic behavior.