Probing phase transitions of regular black holes in anti-de Sitter space with Lyapunov exponent

TL;DR

This work analyzes charged regular AdS black holes in five- and seven-dimensional quasi-topological gravity, linking thermodynamic phase transitions in extended space to the dynamical stability of particle orbits. By deriving the Lyapunov exponents for massless (and where relevant, massive) particles in these spacetimes, the authors show oscillatory behavior of the Lyapunov exponent across first-order transitions and a discontinuous-to-continuous evolution along the coexistence curve at the critical point. The key finding is that the difference in Lyapunov exponents between small and large black holes, $\Delta\lambda$, vanishes at the critical point with a critical exponent of $1/2$, indicating that $\Delta\lambda$ can serve as an order parameter for black hole phase transitions. The results suggest a dynamical diagnostic for black hole thermodynamics in quasi-topological gravity and motivate further exploration of connections to photon spheres and quasinormal modes.

Abstract

We investigate the relationship between thermodynamic phase transitions and the Lyapunov exponent of charged regular anti-de Sitter black holes in quasi-topological gravity. Our results show that the Lyapunov exponent displays oscillatory behavior during phase transitions. Moreover, along the coexistence curve the Lyapunov exponent changes discontinously and continuously at the critical point. Near the critical point, the Lyapunov exponent follows a power-law behavior with a critical exponent of 1/2, suggesting its role as an order parameter and encodes information on black hole phase transitions.

Figures (9)

• Figure 1: (a). The oscillatory behavior in the $P-v$ plane for five-dimensional charged regular AdS black holes in quasi-topological gravity; (b). The swallowtail structure in the $F-P$ plane for five-dimensional charged regular AdS black holes in quasi-topological gravity. Parameters are chosen as $q=1/2$, $b=4/5~(\alpha = 9/25)$.
• Figure 2: (a). The oscillatory behavior in the $P-v$ plane for seven-dimensional charged regular AdS black holes in quasi-topological gravity; (b). The swallowtail structure in the $F-P$ plane for seven-dimensional charged regular AdS black holes in quasi-topological gravity. Parameters are chosen as $q=1/2$, $b=4/5~(\alpha = 1)$.
• Figure 3: (a). The swallowtail structure in the $F-T$ plane for five-dimensional charged regular AdS black holes in quasi-topological gravity. (b). The coexistence curve of phase transition for the five-dimensional charged regular AdS black holes in quasi-topological gravity. Parameters are chosen as $q=1/2$, $b=4/5~(\alpha = 9/25)$.
• Figure 4: (a). The swallowtail structure in the $F-T$ plane for seven-dimensional charged regular AdS black holes in quasi-topological gravity. (b). The coexistence curve of phase transition for the seven-dimensional charged regular AdS black holes in quasi-topological gravity. Parameters are chosen as $q=1/2$, $b=4/5~(\alpha = 1)$.
• Figure 5: Effective potential $V_{\text{eff}}$ of massive particles with varying angular momentum $L$ as a function of the radial coordinate $r$ in the large black hole phase at $T = 0.75 T_{\text{c}}$. The parameters are set to $q=1/2$ and $b=4/5$. (a) five-dimensional case; (b) seven-dimensional case.