Chaotic Motion around a Black Hole under Minimal Length Effects

Abstract

We use the Melnikov method to identify chaotic behavior in geodesic motion perturbed by the minimal length effects around a Schwarzschild black hole. Unlike the integrable unperturbed geodesic motion, our results show that the perturbed homoclinic orbit, which is a geodesic joining the unstable circular orbit to itself, becomes chaotic in the sense that Smale horseshoes chaotic structure is present in phase space.

Minimal Length Effects on Motion of a Particle in Rindler Space

“…In light of our numerical results, we come to the conclusion that chaotic behavior is more likely to happen in the presence of the minimal length effects. This is in agreement with earlier observations and generic arguments for a massive particle perturbed away from an unstable equilibrium near the black hole horizon [62], as well as recent findings for the geodesic motion perturbed by the minimal length effects around a Schwarzschild black hole [70]. In addition, black hole horizons have been conjectured to be fastest scramblers in nature [81] with the scrambling time t s ∼ T −1 ln S, where T and S are the temperature and entropy of the black hole, respectively.…”

“…Here, we numerically study the motion of a particle and the corresponding chaos indicators in the Rindler space. Our numerical results not only support the findings of [62,70], but also signal a shorter scrambling time, a notion that is connected to chaos.…”

“…This study is a follow-up of our previous works [62,70], which demonstrated that the minimal length effects tend to increase chaos. Analytical approaches, i.e., the perturbation method and Melnikov method, were employed to investigate the chaotic motion of a particle around a black hole in [62,70]. Here, we numerically study the motion of a particle and the corresponding chaos indicators in the Rindler space.…”

“…In this paper, we discuss the minimal length effects on the motion of a particle under a harmonic potential in the Rindler space, which is the approximation of the near-horizon region. This study is a follow-up of our previous works [62,70], which demonstrated that the minimal length effects tend to increase chaos. Analytical approaches, i.e., the perturbation method and Melnikov method, were employed to investigate the chaotic motion of a particle around a black hole in [62,70].…”

Lyapunov exponents and phase structure of Lifshitz and hyperscaling violating black holes

“…If λ < 0, the particle trajectory will be asymptotically stable, resulting in the nearby trajectories tending to overlap. Extensive research has been devoted to explore the chaotic motion of particles within various black hole spacetimes [88][89][90][91][92][93][94][95][96][97][98][99][100]. Specifically, investigations into particle motion near black hole horizons have revealed that the Lyapunov exponent adheres to a universal upper bound proposed within the gauge/gravity duality framework 5 [102,103].…”