Conventional vs. modified GTD metrics: Survival of modified GTD metrics in AdS spacetime and thermodynamic ensembles

Abstract

Thermodynamic geometry provides a powerful framework for probing the microscopic structure of thermodynamic systems. Among its formulations, Geometrothermodynamics (GTD) has been widely applied to black hole thermodynamics, owing to its Legendre-invariant construction. However, recent work has shown that conventional GTD metrics fail to encode essential physical boundaries of thermodynamic phase space. By modifying the conventional metric structure, three new GTD metrics were previously introduced, which successfully capture these boundaries in regular spacetime. Whether such modified metrics remain viable in different spacetime backgrounds and under changes of thermodynamic ensemble has remained an open question. In this work, I address this issue by investigating the behavior of modified GTD metrics in AdS spacetime and across different thermodynamic ensembles in the framework of Bardeen AdS black hole. An analysis of thermodynamic geodesics demonstrates that the modified GTD metrics consistently respect physical boundaries of the phase space, in contrast to conventional GTD metrics. This behavior is preserved in AdS spacetime and under Legendre transformations, establishing the robustness and universality of the modified GTD framework.

Figures (7)

• Figure 1: $s$--$g$ thermodynamic phase space of the Bardeen AdS black hole. The green dotted curve denotes the spinodal line separating regions of positive and negative specific heat, while the blue dotted curve represents the temperature-vanishing line. The green shaded area corresponds to the physical region with positive temperature and positive specific heat; red and blue shaded regions indicate negative specific heat and negative temperature, respectively.
• Figure 2: Thermodynamic geodesics of the Bardeen AdS black hole in the canonical ensemble, defined by the $\mathcal{G}^{I}$ and $\mathcal{G}^{I}_{\mathrm{mod}}$ metrics. The $\mathcal{G}^{I}$ geodesic is shown by the red dashed line, while the $\mathcal{G}^{I}_{\mathrm{mod}}$ geodesic is shown by the green solid line. The spinodal and temperature-vanishing curves are indicated by the cyan and blue lines, respectively.
• Figure 3: Thermodynamic geodesics of the Bardeen AdS black hole in the canonical ensemble, defined by the $\mathcal{G}^{II}$ and $\mathcal{G}^{II}_{\mathrm{mod}}$ metrics. The $\mathcal{G}^{II}$ geodesic is shown by the red dashed line, while the $\mathcal{G}^{II}_{\mathrm{mod}}$ geodesic is shown by the green solid line. The spinodal and temperature-vanishing curves are indicated by the cyan and blue lines, respectively.
• Figure 4: Thermodynamic geodesics of the Bardeen AdS black hole in the canonical ensemble, defined by the $\mathcal{G}^{III}$ and $\mathcal{G}^{III}_{\mathrm{mod}}$ metrics. The $\mathcal{G}^{III}$ geodesic is shown by the red dashed line, while the $\mathcal{G}^{III}_{\mathrm{mod}}$ geodesic is shown by the green solid line. The spinodal and temperature-vanishing curves are indicated by the cyan and blue lines, respectively.
• Figure 5: $s$--$\phi$ thermodynamic phase space of the Bardeen AdS black hole. The green dotted curve denotes the spinodal line separating regions of positive and negative specific heat, while the blue dotted curve represents the temperature-vanishing line. The green shaded area corresponds to the physical region with positive temperature and positive specific heat; red and blue shaded regions indicate negative specific heat and negative temperature, respectively.