Euclidean Thermodynamics and Lyapunov Exponents of Einstein-Power-Yang-Mills AdS Black Holes

Abstract

We study the thermodynamics of Einstein-Power-Yang-Mills AdS black holes via the Euclidean path integral method, incorporating appropriate boundary and counterterms. By analyzing unstable timelike and null circular geodesics, we demonstrate that their Lyapunov exponents reflect the thermodynamic phase structure obtained from the Euclidean action. Specifically, the small-large black hole phase transition, analogous to a van der Waals fluid, is signaled by a discontinuity in the Lyapunov exponent. Treating this discontinuity as an order parameter, we observe a universal critical exponent of $1/2$, consistent with mean-field theory. These results extend previous insights from black hole spacetimes with Abelian charges to scenarios involving nonlinear, non-Abelian gauge fields, highlighting the interplay between black hole thermodynamics and chaotic dynamics.

Figures (11)

• Figure 1: Hawking temperature vs. horizon radius depicted for two distinct charge regimes: $\tilde{Q} = 0.11 < \tilde{Q}_c$ (left) and $\tilde{Q} = 0.16 > \tilde{Q}_c$ (right), and $\gamma =3/2$.
• Figure 2: Left: Free energy $\tilde{F}$ as a function of temperature $\tilde{T}$ for three distinct charge regimes: $\tilde{Q}=0.11<\tilde{Q}_c$, $\tilde{Q}= \tilde{Q}_c$ and $\tilde{Q}=0.2>\tilde{Q}_c$, and $\gamma =3/2$. Right: For temperatures in the range $\tilde{T}_1<\tilde{T}<\tilde{T}_2$, three black hole solutions coexist, and a first-order phase transition between small and large black holes occurs at the temperature $\tilde{T}_p$.
• Figure 3: Effective potential $V_\text{eff}$ governing the motion of massive particles around an EPYM AdS black hole, plotted for parameters $\gamma=3/2$, $\tilde{Q}=0.11$, angular momentum $L=20\ell$, and different horizon radii $\tilde{r}_{\text{h}}=0.2$, $0.4$, and $0.6$. The vertical dashed lines represent the event horizon radius, and brown dots indicate unstable circular geodesics. Unstable circular orbits disappear as $\tilde{r}_{\text{h}}$ increases, shown explicitly for $\tilde{r}_{\text{h}}=0.6$.
• Figure 4: Three-dimensional plot of $\log_{100}(\lambda+1)$ as a function of $\tilde{r}_{\text{h}}$ and $\tilde{Q}$ for a massive particle with angular momentum $L = 20\ell$. Here we chose $\gamma=3/2$.
• Figure 5: (a) Lyapunov exponent $\lambda$ for massive particles with angular momentum $L=20\ell$ on unstable timelike circular orbits, plotted as a function of temperature $\tilde{T}$ for $\tilde{Q}=0.11<\tilde{Q}_c$ ($\gamma = 3/2$ case). Three black hole solutions coexist in the temperature range $\tilde{T}_1<\tilde{T}<\tilde{T}_2$, with the phase transition between small and large black holes occurring at $\tilde{T}=\tilde{T}_p$. The inset shows the behavior of $\lambda$ near $\tilde{T}=0$. (b) Lyapunov exponents $\lambda$ of massive particles on unstable timelike circular orbit as a function of temperature $\tilde{T}$ for $\tilde{Q}=0.11<\tilde{Q}_c$ with varying $\gamma$ values.