Minimal Length Effects on Motion of a Particle in Rindler Space

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 Poincaré 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.

Near horizon local instability and quantum thermality

“…considering the (t−r) sector of metric (5), one can readily show that time and radial components of l a are given by those given in (7). Therefore, one again finds the same radial behaviour as obtained in (14). Also as here Θ vanishes and the determinant of the metric is g = −1, the definition for expansion parameter (A7) reduces to Eq.…”

“…( 16). Hence one finds (14) again and so the existence of the instability in the particle motion in the near horizon region persists in this case as well. This indicates that the present instability is completely liable for the influence of horizon in the spacetime, not a specific feature to number of spacetime dimensions.…”

“…r * → −∞) implies t → −∞, one obtains C = r H . Therefore we have the same solution (14). We now make a comment for the same in (1 + 1) dimension static black hole case.…”

“…Therefore, the aim of our next approach is to identify the observer and the corresponding vacuum state corresponding this thermality. One such popular approach is investigating through the detector response of a two level atomic detector which can give us the clear idea to identify our observer and the vacuum.The choice of the observer here is the one which is following the path (14) in the near horizon regime. The vacuum is chosen to be Boulware vacua which is defined with respect to the static observer in Schwarzschild coordinates.…”

“…Let us consider a two-level atomic detector (say a is the excited level and b is the ground state) is moving along the geodesic (14). We consider the massless scalar field Φ under this background and its modes are denoted by u ν with frequency ν.…”