[Paper Review] First-order corrected thermodynamic geometry of a static black hole in $f (R)$ gravity

Sudhaker Upadhyay, Saheb Soroushfar+1 more|arXiv (Cornell University)|Jan 26, 2018

Black Holes and Theoretical Physics14 references3 citations

TL;DR

This paper investigates the first-order corrected thermodynamic geometry of a static black hole in f(R) gravity using geometrothermodynamics (GTD). It shows that the curvature scalar of the Ruppeiner metric singularities precisely coincide with heat capacity zeros, indicating phase transition points; however, this correspondence weakens as the α parameter increases.

ABSTRACT

In this paper, we consider a static black hole in $f(R)$ gravity. We recapitulate the expression for corrected thermodynamic entropy of this black hole due to small fluctuations around equilibrium. Also, we study the geometrothermodynamics (GTD) of this black hole and investigate the adaptability of the curvature scalar of geothermodynamic methods with phase transition points of this black hole. Moreover, we point out the effect of $ \alpha $ parameter on thermodynamic behavior of this black hole. As a result, we see that, the singular point of the curvature scalar of Ruppeiner metric is completely coincided with zero point of the heat capacity and by increasing $ \alpha $, this match will be less.

Motivation & Objective

• To analyze the first-order corrected thermodynamic entropy of a static black hole in f(R) gravity due to small thermal fluctuations.
• To examine the applicability of geometrothermodynamics (GTD) in capturing phase transitions in f(R) gravity black holes.
• To investigate how the α parameter in f(R) gravity influences the thermodynamic behavior and geometric structure of the black hole.
• To determine the relationship between the curvature scalar of the Ruppeiner metric and thermodynamic phase transition points.

Proposed method

• Derives the first-order corrected entropy expression for a static black hole in f(R) gravity using statistical mechanical fluctuations around equilibrium.
• Applies the Ruppeiner geometric approach to model thermodynamic systems via information geometry, using the Hessian of the entropy.
• Computes the curvature scalar of the Ruppeiner metric to detect thermodynamic instabilities and phase transitions.
• Analyzes the behavior of the curvature scalar in relation to the heat capacity, identifying singularities as indicators of phase transitions.
• Varying the α parameter in f(R) gravity to assess its influence on the correspondence between curvature singularities and heat capacity zeros.
• Compares the location of curvature singularities with the zeros of the heat capacity to evaluate consistency in phase transition detection.

Experimental results

Research questions

• RQ1 How does first-order thermal fluctuation correction affect the thermodynamic entropy of a static black hole in f(R) gravity?
• RQ2 Can the curvature scalar of the Ruppeiner metric accurately signal thermodynamic phase transitions in this black hole system?
• RQ3 What is the role of the α parameter in modifying the thermodynamic geometry and phase structure of the black hole?
• RQ4 To what extent does the singularity of the Ruppeiner curvature scalar align with the zero of the heat capacity?
• RQ5 How does increasing α affect the agreement between geometric curvature singularities and thermodynamic instability points?

Key findings

• The curvature scalar of the Ruppeiner metric exhibits singularities that exactly coincide with the points where the heat capacity vanishes, indicating a strong geometric signature of phase transitions.
• The correspondence between curvature singularities and heat capacity zeros is most precise at low values of the α parameter.
• As the α parameter increases, the agreement between the curvature scalar singularity and the heat capacity zero becomes progressively weaker.
• The first-order corrected entropy expression captures quantum fluctuations around equilibrium, providing a refined thermodynamic description.
• The study confirms that GTD via the Ruppeiner metric effectively detects phase transitions in f(R) gravity black holes when α is small.
• The α parameter acts as a control parameter that modulates the strength and detectability of thermodynamic instabilities through geometric means.