Probing thermodynamic phase transitions by dynamics of timelike particle around a magnetic AdS black hole

TL;DR

The paper addresses how thermodynamic phase transitions of a nonminimal coupled magnetic AdS black hole manifest in the dynamics of orbiting timelike particles. It studies this connection by computing the Lyapunov exponent of unstable circular orbits and by deriving QPO frequencies within the relativistic precession model, using the exact Wu-Yang magnetic SU(2) solution and the extended BH thermodynamics where the cosmological constant is treated as pressure. The results show that both the Lyapunov exponent and the QPO spectra exhibit signatures of Van der Waals-like phase transitions, including multivalued behavior for subcritical charges/couplings and branch-merging at criticality, consistent with the Gibbs free energy and Hawking-temperature diagram. These dynamical indicators thus provide a potentially observable window into black hole thermodynamics in AdS-like spacetimes and motivate extensions to rotating or other modified-gravity BHs.

Abstract

In this paper, we investigate the phase structure of a nonminimal coupled magnetic AdS black hole by connecting its thermodynamic properties with the dynamical behavior of orbiting particles, within the framework of Lyapunov exponent and quasi-periodic oscillation. The analysis of the free energy as a function of the Hawking temperature reveals a Van der Waals-like phase transition, characterized by the coexistence of small, intermediate, and large black hole phases. By examining timelike geodesics corresponding to unstable circular orbits, we evaluate the Lyapunov exponent of the test particles crossing the phase transition and explore its role as dynamical indicator of stability. Furthermore, by perturbing the unstable circular orbit and employing the relativistic precession model, we explore the associated upper and lower quasi-periodic oscillation frequencies. Our findings show that the occurrence of thermodynamic phase transitions induces marked changes in both Lyapunov exponent and quasi-periodic oscillation spectra, indicating that these dynamical quantities can serve as sensitive probes of the underlying thermodynamic phase structure of the black hole.

Figures (13)

• Figure 1: The typical phase diagram of the parameter space $Q_m-\xi$, illustrating regions with BH solutions and horizonless configurations, thereby characterizing the valid domain for nonminimal coupled magnetic AdS BHs.
• Figure 2: The typical plot showing the dependence of the horizon radius on the magnetic charge of the magnetic BH, for various values of the coupling parameter. The solid curves are for the radius of event horizon ($r_+$) while the dashed curves are for Cauchy horizon ($r_-$), and their intersection describes the extremal case.
• Figure 3: The critical quantities are shown in the plots against the magnetic charged parameter and nonminimal coupling parameter. In the left plot, the critical values are plotted as a function of $Q_m$, showing the critical horizon radius (blue curve), the critical nonminimal coupling parameter (green curve), and the critical BH temperature (red curve). In the right plot, the graph displays the corresponding critical values as a function of $\xi$.
• Figure 4: The Hawking temperature as a function of event horizon. In the left plot, we fix $\xi = 0.30$, and the curves are for different $\tilde{Q}_{m}$ as follows: $\tilde{Q}_{m} < \tilde{Q}_{mc}$ (blue curve), $\tilde{Q}_{m} = \tilde{Q}_{mc}$ (red curve), and $\tilde{Q}_{m} > \tilde{Q}_{mc}$ (green curve). The right plot, with a fixed value of $Q_m = 0.03$, presents the temperature for different $\xi$: $\xi < \xi_c$ (blue curve), $\xi = \xi_c$ (red curve), and $\xi > \xi_c$ (red curve), respectively.
• Figure 5: The typical phase transition plots show the free energy as a function of the Hawking temperature for the magnetic AdS BH, which includes two distinct nonminimal coupling parameter regimes with (left) and without (right) phase transition. Here, we take a fixed value of $Q_m = 0.03$ as an example. In the left plot, with a fixed value of $\xi = 0.15$ ($\xi < \xi_c$), the phase transition curves show the existence of three BH solutions and indicate first-order phase transitions between small-large BHs. In the right plot, with a fixed value of $\xi = 0.88$ ($\xi > \xi_c$), no phase transition occurs, indicating the existence of one BH solution and no transition.