Lyapunov Exponent Approach to Phase Structure of Schwarzschild AdS Black Holes Surrounded by a Cloud of Strings

Abstract

We investigate Schwarzschild black holes in anti-de Sitter (AdS) spacetimes surrounded by a cloud of strings (BH-AdS-CoS), incorporating both electric- and magnetic-like components of the string bi-vector. Thermodynamically, these systems exhibit small/intermediate/large black hole phases with first- and second-order transitions governed by the string parameter $c_0$. Dynamically, we probe the phase structure using Lyapunov exponents $\hat{I}»$ from unstable circular geodesics. For massless particles ($\hat{I}´=0$), analytical expressions $\hat{I}»$ reveal multivalued behavior in first-order transition regimes ($c_0<c_{\text{cri}}$), with branches mapping to thermodynamic phases ($\hat{I}{»}_{\text{SBH}},\hat{I}{»}_{\text{IBH}},\hat{I}{»}_{\text{LBH}}$). The discontinuity $\hat{I}”\hat{I}»=\hat{I}{»}_{\text{SBH}}-\hat{I}{»}_{\text{LBH}}$ at $T_p$ follows mean-field scaling: $\hat{I}”\hat{I}»/\hat{I}{»}_{\text{cri}}\propto (T_{\text{cri}}-T{)}^{1/2}({\hat{I}}^2=1/2)$. For massive particles ($\hat{I}´=1$), numerical computation of timelike geodesics confirms $\hat{I}»$ as an order parameter, with critical exponent ${\hat{I}}^2=1/2$ universally. Key distinctions emerge: $\hat{I}»\to 1$ asymptotically for photons, while $\hat{I}»\to 0$ in the significant black hole phase for massive particles due to vanishing unstable orbits. The transition of $\hat{I}»$ from multivalued to single-valued at $c_0=c_{\text{cri}}$ establishes it as a universal dynamical probe of black hole criticality. The universal critical exponent of 1/2 for \(Δλ\) further reinforces the analogy with conventional thermodynamic systems. Our results confirm a direct connection between the thermodynamic phase structure of BH-AdS-CoS and the dynamics of test particles, with the Lyapunov exponent emerging as a sensitive diagnostic of black hole criticality.

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