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Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions

Shi-Hao Zhang, Zi-Qiang Zhao, Zi-Yuan Li, Jing-Fei Zhang, Xin Zhang

Abstract

First-order phase transitions of black holes have been extensively studied within thermodynamic frameworks, yet the corresponding evolution of spacetime geometric properties remains unclear. This paper establishes a purely differential geometric framework to probe such phase transitions by analyzing the curvature of unstable null orbits. Using the geodesic curvature of the optical metric to locate the light ring, we demonstrate that the corresponding Gaussian curvature $K$ serves as a direct geometric signature of the phase transition. During a first-order phase transition, the curve $K$ versus temperature $T$ exhibits a multivalued structure within the spinodal region, precisely mirroring the swallowtail behavior of the free energy. Numerical analysis of Hayward-Letelier-AdS black holes confirms the effectiveness of this geometric signature. Our work demonstrates that the intrinsic geometric quantities of spacetime encode the information of black hole phase transitions. These quantities serve simultaneously as geometric probes and order parameters for black hole phase transitions. This result provides a purely geometric foundation for understanding the correspondence between thermodynamics and spacetime curvature in the null case.

Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions

Abstract

First-order phase transitions of black holes have been extensively studied within thermodynamic frameworks, yet the corresponding evolution of spacetime geometric properties remains unclear. This paper establishes a purely differential geometric framework to probe such phase transitions by analyzing the curvature of unstable null orbits. Using the geodesic curvature of the optical metric to locate the light ring, we demonstrate that the corresponding Gaussian curvature serves as a direct geometric signature of the phase transition. During a first-order phase transition, the curve versus temperature exhibits a multivalued structure within the spinodal region, precisely mirroring the swallowtail behavior of the free energy. Numerical analysis of Hayward-Letelier-AdS black holes confirms the effectiveness of this geometric signature. Our work demonstrates that the intrinsic geometric quantities of spacetime encode the information of black hole phase transitions. These quantities serve simultaneously as geometric probes and order parameters for black hole phase transitions. This result provides a purely geometric foundation for understanding the correspondence between thermodynamics and spacetime curvature in the null case.

Paper Structure

This paper contains 9 sections, 32 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Thermodynamics and phase structure of spherically symmetric black holes
  4. Gaussian curvature of two-dimensional manifolds
  5. Lyapunov exponent for unstable orbits
  6. Numerical Results and Analysis
  7. Critical Exponent from Lyapunov Exponents and Gaussian Curvature
  8. Discussion
  9. Acknowledgements

Figures (3)

  • Figure 1: Thermodynamic, geometric and chaos signatures of the first-order phase transition in Hayward-Letelier-AdS space-times. (a) $\tilde{F}_{HL}$, (b) $K_{HL}$ of unstable null orbits and (c-d) $\lambda_{HL}$ of unstable null/timelike orbits versus temperature $\tilde{T}_{HL}$ for Hayward-Letelier-AdS black holes, and $\tilde{g}=0.0615,~\tilde{g}_c=0.1042,~a=0.6,~\tilde{g}<\tilde{g}_c,~L=20\ell$. Swallowtail structures in $\tilde{F}_{HL}$ and multivalued $K_{HL}$ at $\tilde{T}_{p}$ evidence geometric degeneracy during phase transitions. $\lambda_{HL}$ at $\tilde{T}_{p}$ also exhibits such multivaluedness.
  • Figure 2: Thermodynamic and geometric signatures of the absence of a first-order phase transition in Hayward-Letelier-AdS space-times. (a) Free energy $\tilde{F}_{HL}$ and (b) Gaussian curvature $K_{HL}$ versus temperature $\tilde{T}_{HL}$ for a Hayward-Letelier-AdS black hole without phase transition ($\tilde{g}=0.0615,~\tilde{g}_c=0.1042,~a=0.6,~\tilde{g}>\tilde{g}_c$). Both quantities exhibit monotonic behavior, confirming the absence of thermodynamic criticality.
  • Figure 3: The reduced null Lyapunov exponents and Gaussian curvatures for the coexistence small and large black holes. (a) Null Lyapunov exponents and (b) Gaussian curvature $K_{HL}$ versus reduced temperature $t$ for a Hayward-Letelier-AdS black hole.