Ruppeiner geometry and the fluctuation of the RN-AdS black hole in framework of the extensive thermodynamics

Abstract

In this study, the Ruppeiner geometry and fluctuations of a four-dimensional charged Anti-de Sitter (AdS) black hole within the context of extensive thermodynamics have been investigated. By fixing the AdS radius in Vissers' construction, an extensive thermodynamics description of the RN-AdS black hole is established, with the central charge playing the role of the particle number, introducing the concept of a subsystem into black hole thermodynamics. By employing the density of thermodynamic variables, the phase transition of the black hole is reexamined, and this result holds for black holes with different AdS backgrounds. Furthermore, the authors delve into a detailed analysis of the Ruppeiner geometry using fluctuation theory. Interestingly, an increase in the particle number of the smaller subsystem is found to induce a reduction in the thermodynamic curvature scalar.