Restricted Phase Space Thermodynamics of Charged Static and Charged Rotating Black Holes in $f(R)$ Gravity

Abstract

The thermodynamics of black holes provides a profound link between gravity, quantum theory and statistical mechanics. It serves as a useful tool for testing theories beyond Einstein's gravity. In this work of ours, we investigate the newly found restricted phase space thermodynamics (RPST) of charged static and charged rotating black holes in $f(R)$ gravity. Unlike the extended phase space (EPST) approach, RPST keeps the cosmological constant fixed and introduces the central charge $C$ along with its conjugate chemical potential $\mu$, thereby allowing the black hole mass to be consistently interpreted as internal energy. Within this framework, we derive the relevant thermodynamic quantities and analyse the temperature-entropy $(T-S)$ and Helmholtz free energy-temperature $(F-T)$ behaviours. Our results reveal characteristic features of first-order phase transitions through non-monotonic $T-S$ curves along with the swallow-tail structures in $F-T$ plots, while second-order transitions appear at critical points. To further validate these findings, we employ the formalism of geometrothermodynamics (GTD), which provides a Legendre-invariant geometric description of thermodynamic geometry. We demonstrate that the curvature singularities of the GTD scalar curvature coincides exactly with that of the divergences in the specific heat capacity curves, thereby establishing a geometric correspondence for phase transitions. This study facilitates the first systematic exploration of RPST within $f(R)$ gravity and highlights the universality of RPST in capturing black hole criticality in modified gravity theories.

14 pages, 9 figures

References in corpus (23)

• Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models
• Geometrothermodynamics
• Geometrothermodynamics of black holes
• Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions
• Thermodynamics of AdS Black Holes: Central Charge Criticality
• Regular Multi-Horizon Black Holes in Modified Gravity with Non-Linear Electrodynamics
• Restricted phase space thermodynamics for AdS black holes via holography
• Instabilities and Anti-Evaporation of Reissner-Nordström Black Holes in modified $F(R)$ gravity
• Thermodynamics of Kerr-AdS black holes in the restricted phase space
• Spherically symmetric black holes with electric and magnetic charge in extended gravity: Physical properties, causal structure, and stability analysis in Einstein's and Jordan's frames
• 4D-EGB Black Holes in RPS Thermodynamics
• $P-v$ criticalities, phase transitions and geometrothermodynamics of charged AdS black holes from Kaniadakis statistics
• Barrow Entropy and AdS Black Holes in RPS Thermodynamics
• Black hole geometrothermodynamics and critical phenomena: a look from Tsallis entropy-based perspective
• Restricted phase space thermodynamics of charged AdS black holes in conformal gravity
• Thermodynamics and Phase Structure of an Einstein-Maxwell-scalar Model in Extended Phase Space
• Infinitesimal Legendre symmetry in the Geometrothermodynamics programme
• Thermodynamic geometry of pure Lovelock black holes
• Restricted Phase Space Thermodynamics of NED-AdS Black Holes
• Topology of restricted phase space thermodynamics in Kerr-Sen-Ads black holes
• Exploring critical behavior of thermodynamic variables of the Kerr-Newman-AdS black hole in the restricted phase space
• Topological Signatures and Geometrothermodynamics of Critical Phenomena in Regularized Maxwell Black Holes
• Thermodynamics of Flat 4D Einstein-Gauss-Bonnet Black Hole with Rényi Entropy: An RPST-like formalism