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

TL;DR

This work extends restricted phase space thermodynamics (RPST) to charged static and rotating black holes in $f(R)$ gravity, keeping the cosmological constant fixed while introducing a central charge $C$ and its conjugate chemical potential $μ$ so mass acts as internal energy. Analytic results are obtained for the static case, with $M(S,\tilde{Q},C)$ exhibiting first-order homogeneity and clear $T$–$S$ and $F$–$T$ signatures of Van der Waals-like transitions, plus a second-order transition at the critical point; the rotating case requires a numerical approach, introducing $\sigma=J/\tilde{Q}$ and yielding scaling forms for the critical entropy and central charge. The thermodynamic geometry of these black holes is analyzed via geometrothermodynamics (GTD), showing Legendre-invariant curvature divergences that coincide with specific-heat instabilities for both static and rotating solutions, thereby geometrically diagnosing phase transitions in modified gravity. Overall, the results support the universality of RPST beyond Einstein gravity and highlight GTD as a robust tool for connecting thermodynamics and geometry in $f(R)$ black hole spacetimes, with potential extensions to other modified theories and holographic interpretations of the central charge.

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 $μ$, 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.

Figures (9)

• Figure 1: $T-S$ and $F-T$ curves for charged static black hole in the iso-$e$-charge processes for $C=53$ and $b=2$.
• Figure 2: $\mu-C$ curves for charged static black hole in the iso-$e$-charge processes for $C=53$ and $b=2$.
• Figure 3: $f(\sigma)$ and $g(\sigma)$ curves in the iso-$e$-charge processes for charged rotating black hole.
• Figure 4: $T-S$ and $F-T$ curves for charged rotating black hole in the iso-$e$-charge processes for $C=30$ and $b=2$.
• Figure 5: $\mu-C$ curves for the charged rotating in the iso-$e$-charge processes for $C=30$ and $b=2$.