Thermodynamic Geodesics in Bardeen Regular Black Hole: Conventional vs. Modified Geometrothermodynamics Metrics

TL;DR

This work analyzes the thermodynamic geometry of the Bardeen regular black hole using Geometrothermodynamics (GTD) and three standard GTD metrics, identifying key boundaries such as the spinodal (Davies) line and the temperature-vanishing line. It shows that conventional GTD metrics fail to consistently encode all thermodynamic features, motivating two modified GTD metrics that retain Legendre invariance while restricting to the entropy–temperature pair in the metric structure. The modified metrics produce thermodynamic geodesics confined to the physical region and exhibit curvature divergences precisely at the thermodynamic boundaries, providing a more faithful geometric reflection of black hole thermodynamics. The results suggest that appropriately tailored GTD metrics may generalize to other black hole systems and ensemble considerations, enhancing insights into microscopic thermodynamic behavior.

Abstract

Thermodynamic geometry allow us to study the microscopic behavior of black hole system by defining a metric structure in thermodynamic phase space. Among the various thermodynamic metric structures, metrics defined by geometrothermodynamics (GTD) are extensively used to study the various thermodynamic system due to its Legendre invariant nature. In this work we investigate the behavior of thermodynamic geodesic of Bardeen regular black hole in thermodynamic space defined by three different GTD metrics. Based on the behavior of thermodynamic geodesic as well as thermodynamic curvature we argued that conventional GTD metric need some modifications to reflect all the thermodynamical properties of a system. We also modified the conventional GTD metrics and explore the behavior of thermodynamic geodesic defined by the modified metrics. Our study shows that the modified GTD metrics contain most of the information about the thermodynamical boundaries such as temperature vanishing line, spinodal line etc. of a black hole system. Based on the property of geodesic and Ricci scalar defined by the modified metrics we argued that the modified version of GTD metric are most suitable metric structures for studying the underlying thermodynamic behavior of a black hole system.

Figures (6)

• Figure 1: Locus of Davis points and temperature vanishing points of Bardeen regular black hole in $s-g$ plane. Green dashed line represents the spinodal line (Devis points) which separate positive specific heat region from negative specific heat region, while blue dashed line represent the temperature vanishing line, which separate positive temperature region from negative temperature region. Blue colored region represents the negative temperature region, red colored region represents the negative specific heat region, while green colored region represents the Physical region (PR); the region with positive specific heat and positive temperature.
• Figure 2: Thermodynamic geodesic of Bardeen regular black hole in thermodynamic $s-g$ space defined by the metric $\mathcal{G}^I$.
• Figure 3: Thermodynamic geodesic of Bardeen regular black hole in thermodynamic $s-g$ space defined by the metric $\mathcal{G}^{II}$.
• Figure 4: Thermodynamic geodesic of Bardeen regular black hole in thermodynamic $s-g$ space defined by the metric $\mathcal{G}^{III}$.
• Figure 5: Thermodynamic geodesic of Bardeen regular black hole in thermodynamic $s-g$ space defined by the metric $\mathcal{G}^{II}_{mod}$