Probing Chaos in Schwarzschild-de Sitter Spacetime: The Role of Black Hole and Cosmological Horizons

In this paper, we study the motion of a massless, chargeless particle in Schwarzschild-de Sitter spacetime, revealing exponential radial growth and potential chaos in an integrable system. Poincaré sections show regular Kolmogorov-Arnold-Moser (KAM) tori when black hole and cosmological horizons are distant, but distortions and chaos emerge as they converge. As the horizons coincide, the Poincaré sections fully contract and vanish, marking the system’s transition to Nariai spacetime. $\mathit{\text{Our}}\mathit{\text{analysis}}\mathit{\text{also}}\mathit{\text{suggests}}\mathit{\text{that,}}\mathit{\text{within}}\mathit{\text{the}}\mathit{\text{parameter}}\mathit{\text{range}}\mathit{\text{explored,}}\mathit{\text{the}}\mathit{\text{event}}\mathit{\text{horizon}}\mathit{\text{exerts}}\mathit{\text{a}}\mathit{\text{comparatively}}\mathit{\text{more}}\mathit{\text{substantial}}\mathit{\text{chaotic}}\mathit{\text{influence}}\mathit{\text{on}}\mathit{\text{the}}\mathit{\text{system,}}\mathit{\text{primarily}}\mathit{\text{due}}\mathit{\text{to}}\mathit{\text{its}}\mathit{\text{consistent}}\mathit{\text{proximity.}}$ Additionally, we analyze the Lyapunov exponents to quantify the degree of chaos in the system. Our findings indicate that as the closeness of the two horizons increases, the most prominent Lyapunov exponent also increases, signifying a rise in chaotic behavior. By examining the long-term saturation values of the Lyapunov exponents, we confirm that they consistently comply with the Maldacena-Shenker-Stanford (MSS) bound.