Lyapunov Exponents and Phase Transitions in Four-Dimensional AdS Black Holes with a Nonlinear Electrodynamics Source

TL;DR

The study analyzes four-dimensional AdS black holes with a nonlinear Power-Maxwell source ($p=\tfrac{3}{4}$) and demonstrates that the Lyapunov exponent $\lambda$, which measures the instability of unstable circular geodesics, encodes the full thermodynamic phase structure in the canonical ensemble. The authors show multivalued $\lambda(T)$ and a finite jump $\Delta\lambda$ across the small/large black-hole coexistence line for $Q<Q_c$, with $\Delta\lambda \propto (\tilde{T}_p/\tilde{T}_c-1)^{1/2}$ near the critical point, indicating a second-order endpoint with universal mean-field exponent $1/2$; the same critical behavior is observed for the critical impact parameter $b_c$ in massless probes, while massive probes exhibit energy-dependent but analogous signatures. They link spinodal divergences of the heat capacity $C_Q$ to singularities in $\lambda$, establishing a direct dynamical-thermodynamic correspondence and reinforcing $\lambda$ as a robust diagnostic for both first-order coexistence and second-order criticality. The work highlights a unified dynamical framework connecting geodesic instability, photon-sphere diagnostics, and black-hole thermodynamics in AdS, with potential extensions to rotating and higher-curvature backgrounds.

Abstract

We investigate the relationship between dynamical instability and thermodynamic phase transitions in four-dimensional Anti--de Sitter black holes in Einstein gravity coupled to a nonlinear power-law electromagnetic field with exponent $p = 3/4$. In the canonical ensemble, we identify a critical electric charge $Q_c$ separating a regime exhibiting a first-order small/large black-hole (SBH/LBH) phase transition from a regime with a single thermodynamically stable phase. For both massless and massive probes, the thermal profile of the Lyapunov exponent $λ(T)$ becomes multivalued in the SBH/LBH coexistence region and exhibits a finite discontinuity at the transition temperature. This jump vanishes continuously as $Q \to Q_c$, signaling the termination of the first-order transition at a second-order critical point. Near criticality, the Lyapunov discontinuity obeys a universal mean-field scaling law with critical exponent $1/2$. For massless probes, we further analyze the critical impact parameter $b_c$, which displays the same multivalued structure and critical behavior as the Lyapunov exponent. We also demonstrate that the spinodal temperatures, defined by the extrema of the $T(r_h)$ curve where the heat capacity at fixed charge diverges, coincide with singular features in the Lyapunov exponent. Our results identify the Lyapunov exponent as a unified dynamical probe capable of capturing both first-order phase coexistence and second-order critical behavior in black-hole thermodynamics.

Figures (13)

• Figure 1: Hawking temperature as a function of the horizon radius for $\tilde{\eta}=1$ and for different values of $\tilde{Q}$ below (blue and black) and above (black dotted) the critical point $\tilde{Q}_{c}$.
• Figure 2: The free energy as a function of $\tilde{T}$ for $\tilde{\eta}=1.0$ and different values of $Q$. The top panel corresponds to the case $\tilde{Q}=0.1<\tilde{Q}_c$, while the bottom panel corresponds to $\tilde{Q}=0.25>\tilde{Q}_c$.
• Figure 3: Lyapunov exponent $\lambda$ of the massless particle as a function of temperature $\tilde{T}$ for $\tilde{\eta}=1.0$ and different values of $\tilde{Q}$. Top panel corresponds to $\tilde{Q}=0.1<\tilde{Q}_c$ and the bottom panel to $\tilde{Q}=0.25>\tilde{Q}_c$.
• Figure 4: Effective potential $V_{\text{eff}}(\tilde{r})$ of timelike geodesics as a function of $\tilde{r}$ with $\tilde{Q}=0.1$; $\tilde{\eta}=1.0$. The black dots represent the maxima of the effective potentials, corresponding to unstable time-like circular geodesics. For $\tilde{r}_h=0.8$, the effective potential has no maximum.
• Figure 5: Lyapunov exponents $\lambda$ of massive particles as a function of the temperature $\tilde{T}$. Top panel for $\tilde{Q}=0.1<\tilde{Q}_c$ and bottom panel for $\tilde{Q}=0.25>\tilde{Q}_c$.