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Quasi-homogeneous geometrothermodynamics of a noncommutative Reissner-Nordstrom black hole

Alberto Maya, Hernando Quevedo

TL;DR

The paper addresses the thermodynamics of a noncommutative Reissner-Nordström black hole (NCRN BH) by applying geometrothermodynamics (GTD) to a quasi-homogeneous framework in which the noncommutative parameter $\Theta$ is treated as a thermodynamic variable. The fundamental equation $M(S,Q,\Theta)$ is derived from a large-$r$ NC-corrected metric, and three Legendre-invariant GTD metrics on the equilibrium space are constructed to identify phase transitions via Ricci scalar singularities. Two non-trivial phase-transition curves, $Q^{(I)}(S,\Theta)$ and $Q^{(II)}(S,\Theta)$, generalize NC Schwarzschild and classical RN results in appropriate limits, with the divergence of the heat capacity $C_{Q,\Theta}$ matching the curvature singularities. The results show that the NC parameter plays a dynamic thermodynamic role within GTD, offering a unified geometric description of NC BH thermodynamics and guiding future explorations of NC and quantum-gravity effects in black-hole physics.

Abstract

We present the thermodynamic properties of a noncommutative Reissner Nordstrom (NCRN) black hole (BH) modeled with Lorentzian distributions. The analysis is carried out using a Legendre invariant formalism called Geometrothermodynamics (GTD) which is applied to a quasihomogeneous system generated by the NCRN BH. This formalism enables the study of phase transitions by locating Ricci scalar singularities, from which the phase transition points are determined in terms of the thermodynamic variables. We also examine how the noncommutative (NC) parameter can be interpreted as a thermodynamic variable within quasi-homogeneous thermodynamic laws, highlighting its potential role on phase transitions beyond those well-known characterized by divergences in the heat capacity.

Quasi-homogeneous geometrothermodynamics of a noncommutative Reissner-Nordstrom black hole

TL;DR

The paper addresses the thermodynamics of a noncommutative Reissner-Nordström black hole (NCRN BH) by applying geometrothermodynamics (GTD) to a quasi-homogeneous framework in which the noncommutative parameter is treated as a thermodynamic variable. The fundamental equation is derived from a large- NC-corrected metric, and three Legendre-invariant GTD metrics on the equilibrium space are constructed to identify phase transitions via Ricci scalar singularities. Two non-trivial phase-transition curves, and , generalize NC Schwarzschild and classical RN results in appropriate limits, with the divergence of the heat capacity matching the curvature singularities. The results show that the NC parameter plays a dynamic thermodynamic role within GTD, offering a unified geometric description of NC BH thermodynamics and guiding future explorations of NC and quantum-gravity effects in black-hole physics.

Abstract

We present the thermodynamic properties of a noncommutative Reissner Nordstrom (NCRN) black hole (BH) modeled with Lorentzian distributions. The analysis is carried out using a Legendre invariant formalism called Geometrothermodynamics (GTD) which is applied to a quasihomogeneous system generated by the NCRN BH. This formalism enables the study of phase transitions by locating Ricci scalar singularities, from which the phase transition points are determined in terms of the thermodynamic variables. We also examine how the noncommutative (NC) parameter can be interpreted as a thermodynamic variable within quasi-homogeneous thermodynamic laws, highlighting its potential role on phase transitions beyond those well-known characterized by divergences in the heat capacity.
Paper Structure (7 sections, 61 equations, 3 figures)

This paper contains 7 sections, 61 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. The Noncommutative Reissner-Nordström Black Hole
  3. The metric
  4. Thermodynamics
  5. Quasi-Homogeneous Geometrothermodynamics in brief
  6. Phase transitions in NCRN BHs
  7. Conclusions

Figures (3)

  • Figure 1: Electric charge $+Q^{I}$ (blue) and $-Q^{I}$ (red) represented as functions by parts showing the values where phase transitions can occur in a NCRN BH, according to the condition \ref{['eq:54']}. Here, $S=1$.
  • Figure 2: Analogously to Figure 1, this image represents the electric charge $+Q^{II}$ (blue) and $-Q^{II}$ (red) for the possible GTD phase transitions according to the condition \ref{['eq:55']}.
  • Figure 3: The heat capacity $C_{Q,\Theta}=C_{Q,\Theta} (\frac{\Theta}{S},\frac{Q^2}{S})$ of the NCRN BH. Here, $C_{Q,\Theta} < 0$ (red) indicates unstable configurations, while the stable values appear for $C_{Q,\Theta} > 0$ (blue). The phase transition curve \ref{['eq:62']} can be visualized for the positive values of the axis since $\frac{Q^2}{S}> 0$.