Circular orbits and chaos bound in slow-rotating curved acoustic black holes

Abstract Acoustic black holes, analogs of gravitational black holes created in fluid systems, have recently been embedded within Schwarzschild spacetime using the Gross–Pitaevskii theory, leading to configurations with both event and acoustic horizons. This study examines the motion of vortices, modeled as unit-mass relativistic test particles, around a slow-rotating curved acoustic black hole. We analyse the stability of circular orbits, identifying the innermost stable circular orbit (ISCO), and investigate the chaotic dynamics of vortices perturbed from unstable circular orbits near the acoustic horizon. Using the Lyapunov exponent to quantify this chaos, we assess whether it satisfies the Maldacena–Shenker–Stanford bound $$(\lambda \le 2 \pi T_H)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , a limit established for gravitational black holes in general relativity. Our results show that, in non-extremal cases $$(\xi &gt; 4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the Lyapunov exponent respects the bound near the horizon, while in extremal cases $$(\xi = 4)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , it is violated due to vanishing surface gravity. These findings highlight similarities between acoustic and gravitational black holes, advancing the analogy in the context of chaos and orbital dynamics.