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Thermodynamic geometric analysis of RN black holes under f(R) gravity

Wen-Xiang Chen, Yao-Guang Zheng

TL;DR

This work probes the thermodynamic geometry of Reissner–Nordström black holes in f(R) gravity, using GTD and related geometric formalisms to link fluctuations, curvature scalars, and phase structure. By examining multiple f(R) models, horizon geometries, and extended-thermodynamics (pressure-volume) in both constant and non-constant initial curvature scenarios, the authors show that RN black holes can mimic an ideal gas when the initial curvature is constant, and exhibit van der Waals–like phase behavior when the curvature is non-constant with negative exponent terms. The study computes modified entropies, Hawking temperatures, Gibbs free energies, and thermodynamic curvatures $R(S)$ across a suite of models, identifying conditions under which GTD curvature divergences signal phase transitions and how swallowtail $G$–$T$ diagrams arise. These results illuminate how modified gravity terms reshape black hole microstructure in thermodynamic terms and suggest pathways to connect macroscopic thermodynamics with underlying quantum gravity phenomena.

Abstract

In this article, we explore the RN black hole under f(R) gravity and its thermodynamic properties. We begin by examining the small fluctuations around the equilibrium state and summarizing the expression for the modified thermodynamic entropy of this black hole. Additionally, we delve into the geometric thermodynamics (GTD) of black holes and investigate the suitability of the curvature scalar of the GTD method for the phase transition point of the black hole. Moreover, we investigate the effects of modified parameters on the thermodynamic behavior of black holes.Within the framework of $f(R)$ modified gravity theory, we discovered that several RN black holes demonstrate thermodynamic properties resembling those of an ideal gas when the initial curvature scalar of the black hole remains constant. However, if the initial curvature scalar is non-constant and the cosmological constant term possesses a negative exponent, the Reissner-Nordström (RN) black holes could exhibit characteristics akin to those of a van der Waals gas.We separately list the general solutions for the case of non-negative powers and the special solutions for the case of negative powers. We observe that, under certain conditions, the phase transition analogous to the Van der Waals gas exists for charged black holes under f(R) gravity.

Thermodynamic geometric analysis of RN black holes under f(R) gravity

TL;DR

This work probes the thermodynamic geometry of Reissner–Nordström black holes in f(R) gravity, using GTD and related geometric formalisms to link fluctuations, curvature scalars, and phase structure. By examining multiple f(R) models, horizon geometries, and extended-thermodynamics (pressure-volume) in both constant and non-constant initial curvature scenarios, the authors show that RN black holes can mimic an ideal gas when the initial curvature is constant, and exhibit van der Waals–like phase behavior when the curvature is non-constant with negative exponent terms. The study computes modified entropies, Hawking temperatures, Gibbs free energies, and thermodynamic curvatures across a suite of models, identifying conditions under which GTD curvature divergences signal phase transitions and how swallowtail – diagrams arise. These results illuminate how modified gravity terms reshape black hole microstructure in thermodynamic terms and suggest pathways to connect macroscopic thermodynamics with underlying quantum gravity phenomena.

Abstract

In this article, we explore the RN black hole under f(R) gravity and its thermodynamic properties. We begin by examining the small fluctuations around the equilibrium state and summarizing the expression for the modified thermodynamic entropy of this black hole. Additionally, we delve into the geometric thermodynamics (GTD) of black holes and investigate the suitability of the curvature scalar of the GTD method for the phase transition point of the black hole. Moreover, we investigate the effects of modified parameters on the thermodynamic behavior of black holes.Within the framework of modified gravity theory, we discovered that several RN black holes demonstrate thermodynamic properties resembling those of an ideal gas when the initial curvature scalar of the black hole remains constant. However, if the initial curvature scalar is non-constant and the cosmological constant term possesses a negative exponent, the Reissner-Nordström (RN) black holes could exhibit characteristics akin to those of a van der Waals gas.We separately list the general solutions for the case of non-negative powers and the special solutions for the case of negative powers. We observe that, under certain conditions, the phase transition analogous to the Van der Waals gas exists for charged black holes under f(R) gravity.
Paper Structure (13 sections, 122 equations, 32 figures)

This paper contains 13 sections, 122 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Thermodynamic parameters and thermodynamic entropy correction in f(R) theory
  3. Thermodynamic of RN black holes in $f(R)$ gravity background
  4. Black Hole in the Form of f(R) Theory: $f(R)=R-\alpha R^{n}$
  5. Black hole in the form of f(R) theory: $f(R)=R-\lambda \exp (-\xi R)+\kappa R^{n}+\eta \ln (R)$
  6. When d=4, ${k_1}$=1
  7. When d=4, ${k_1}$=0
  8. When d=3, ${k_1}$=1
  9. When d=3, ${k_1}$=-1
  10. When d=3, ${k_1}$=0
  11. Black hole in the form of f(R) theory: $f(R)=-2 \eta M \ln (6 \Lambda_0+R)+R_{0}$
  12. Black hole in the form of f(R) theory:$f(R)=2 a \sqrt{R-\alpha}$
  13. Summary and Discussion

Figures (32)

  • Figure 1: At this time, q=1, b=0.2, G is the horizontal axis, and T is the vertical axis
  • Figure 2: At this time, q=1, b=0.4, G is the horizontal axis, and T is the vertical axis
  • Figure 3: At this time, $Q$=1, P=0.4,$G_1$ is the horizontal axis, $T_1$ is the vertical axis
  • Figure 4: At this time, $Q$=1, P=0.5, $G_1$ is the horizontal axis, $T_1$ is the vertical axis
  • Figure 5: Here(section 3.2.1-FIG.5) is the plot of the function $G-T$ where $G_1$ is the Gibbs free energy, $T_1$ is the temperature, $Q=1, r_{+}$ranges from 1 to 14 , and $P$ is an algebraic number ranging from -10 to 10 .
  • ...and 27 more figures