Thermodynamic geometric analysis of D-dimensional RN black hole

TL;DR

This work analyzes the thermodynamic geometry of $D$-dimensional Reissner–Nordström black holes through the Ruppeiner curvature $R$, highlighting strong ensemble dependence. In the fully fluctuating ensemble, the state space is flat ($R=0$) for the dimensions studied, while in the fixed-charge ensemble a second-order phase transition appears for $D>4$, marked by a divergence of both the heat capacity $C_Q$ and $R$ at the critical condition $r_+^{D-3}=(2D-5) r_-^{D-3}$. The authors prove the $D=6$ case yields $R=0$ in the $(M,Q)$ representation and connect the divergence to the Euclidean path integral's Hessian via a conformal factor supplied by the inverse temperature $eta$, establishing a precise geometric–Euclidean dictionary. Overall, the paper clarifies how thermodynamic geometry reveals microscopic interactions and phase structure only when the appropriate ensemble and variables are chosen, with implications for black hole microstructure and extended phase space studies.

Abstract

This paper studies the thermodynamics and Ruppeiner geometry of D-dimensional RN black holes. We analyze the thermodynamic curvature scalar $R$ in various thermodynamic ensembles. It is found that in an ensemble of fixed charge (canonical ensemble), the Ruppeiner curvature is curved and diverges at a critical point, indicating the existence of a phase transition for $D > 4$. In contrast, when all extensive variables are allowed to fluctuate (for example, in a grand-canonical ensemble or with pressure fixed), the Ruppeiner geometry can appear flat. We also demonstrate that the thermodynamic geometric metric has a one-to-one correspondence with the periodicity of the Euclidean path integral method. In particular, the inverse temperature (the Euclidean time period) serves as a bridge connecting the thermodynamic geometry and the Euclidean action approach.