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

TL;DR

This study investigates chaos in Schwarzschild-de Sitter spacetime by analyzing particle motion, revealing how horizon proximity influences chaotic dynamics and confirming the chaos bounds through Lyapunov exponents.

Contribution

It provides new insights into how black hole and cosmological horizons affect chaos, especially highlighting the dominant role of the event horizon and the transition to Nariai spacetime.

Findings

Chaos increases as horizons approach each other.

Lyapunov exponents grow with horizon closeness.

Chaos bounds are satisfied in the analyzed parameter range.

Abstract

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. Poincar\'e 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 Poincar\'e sections fully contract and vanish, marking the system's transition to Nariai spacetime. \textit{Our analysis also suggests that, within the parameter range explored, the event horizon exerts a comparatively more substantial chaotic influence on the system, primarily due to its consistent 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.