Minimal Length Effects on Motion of a Particle in Rindler Space

TL;DR

This paper investigates how a minimal measurable length, predicted by quantum gravity theories, influences particle motion in Rindler space, revealing increased chaos, faster horizon crossing, and violations of chaos bounds near black holes.

Contribution

It demonstrates that minimal length effects enhance chaos and reduce scrambling time in Rindler space, challenging existing bounds on Lyapunov exponents near black hole horizons.

Findings

Chaos increases with minimal length effects.

Particle crossing Rindler horizon faster.

Lyapunov exponents can exceed surface gravity.

Abstract

Various quantum theories of gravity predict the existence of a minimal measurable length. In this paper, we study effects of the minimal length on the motion of a particle in the Rindler space under a harmonic potential. This toy model captures key features of particle dynamics near a black hole horizon, and allows us to make three observations. First, we find that the chaotic behavior is stronger with the increases of the minimal length effects, which manifests that the maximum Lyapunov characteristic exponents mostly grow, and the KAM curves on Poincar\'{e} surfaces of section tend to disintegrate into chaotic layers. Second, in the presence of the minimal length effects, it can take a finite amount of Rindler time for a particle to cross the Rindler horizon, which implies a shorter scrambling time of black holes. Finally, it shows that some Lyapunov characteristic exponents can be greater than the surface gravity of the horizon, violating the recently conjectured universal upper bound. In short, our results reveal that quantum gravity effects may make black holes prone to more chaos and faster scrambling.