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Regular black holes and reductions of thermodynamic phase spaces

Meng-Sen Ma, Huai-Fan Li, Jian-Hua Shi

TL;DR

This work addresses thermodynamic inconsistencies observed in regular black holes by attributing them to an extra constraint among black hole parameters that reduces the thermodynamic phase space. The authors advocate computing thermodynamic quantities in the full phase space of the underlying singular black holes and only after deriving these quantities applying the constraint to obtain consistent regular-hole thermodynamics. They illustrate the mechanism with the Bardeen black hole, showing how a reduced-temperature $T^{(II)}$ can diverge from the geometric temperature $T_g$ when the constraint modifies the first law, and propose a correct procedure: define in the higher-dimensional space (e.g., $(S,Q,\alpha)$) and then reduce. This framework yields physically meaningful quantities such as heat capacity and free energy and can be extended to AdS settings and to other black holes with similar constraints, providing a universal method for constrained thermodynamics in gravitational systems.

Abstract

The thermodynamic inconsistency observed in regular black holes is resolved through the framework of reduced thermodynamic phase spaces. We demonstrate that regular black holes are essentially induced from singular black holes by adding an extra requirement, which imposes a constraint among black hole parameters. This constraint reduces the thermodynamic phase space, rendering the standard form of the first law of black hole thermodynamics inapplicable. Accordingly, we propose a novel methodology to study the thermodynamic properties of regular black holes. Thermodynamic quantities must be defined in the full, unconstrained thermodynamic phase space of the underlying singular black holes, only afterward is the constraint imposed to derive the consistent and meaningful thermodynamic quantities of the regular black holes. Crucially, this framework extends beyond regular black holes and applies universally to any black hole with this kind of constraint.

Regular black holes and reductions of thermodynamic phase spaces

TL;DR

This work addresses thermodynamic inconsistencies observed in regular black holes by attributing them to an extra constraint among black hole parameters that reduces the thermodynamic phase space. The authors advocate computing thermodynamic quantities in the full phase space of the underlying singular black holes and only after deriving these quantities applying the constraint to obtain consistent regular-hole thermodynamics. They illustrate the mechanism with the Bardeen black hole, showing how a reduced-temperature can diverge from the geometric temperature when the constraint modifies the first law, and propose a correct procedure: define in the higher-dimensional space (e.g., ) and then reduce. This framework yields physically meaningful quantities such as heat capacity and free energy and can be extended to AdS settings and to other black holes with similar constraints, providing a universal method for constrained thermodynamics in gravitational systems.

Abstract

The thermodynamic inconsistency observed in regular black holes is resolved through the framework of reduced thermodynamic phase spaces. We demonstrate that regular black holes are essentially induced from singular black holes by adding an extra requirement, which imposes a constraint among black hole parameters. This constraint reduces the thermodynamic phase space, rendering the standard form of the first law of black hole thermodynamics inapplicable. Accordingly, we propose a novel methodology to study the thermodynamic properties of regular black holes. Thermodynamic quantities must be defined in the full, unconstrained thermodynamic phase space of the underlying singular black holes, only afterward is the constraint imposed to derive the consistent and meaningful thermodynamic quantities of the regular black holes. Crucially, this framework extends beyond regular black holes and applies universally to any black hole with this kind of constraint.

Paper Structure

This paper contains 5 sections, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reductions of thermodynamic phase spaces
  3. Temperatures of regular black holes
  4. Thermodynamics of regular black holes
  5. Final remarks

Figures (1)

  • Figure 1: The behaviors of the temperature, the heat capacity and the free energy for the Bardeen black hole. We set $Q=0.5$. The maximum of the temperature lies at $r_{+}=1.907$ and the divergent point of the heat capacity lies at $r_{+}=2.189$. The dot-dashed line in the temperature curve corresponds to $T=0.026$.