Minimal length effects on chaotic motion of particles around black hole horizon

Abstract

Recently, it was conjectured that the Lyapunov exponent of chaotic motion of a particle in a black hole is universally bounded from above by the surface gravity of the black hole. On the other hand, the minimal length appears in various theories of quantum gravity and leads to the deformed canonical position-momentum commutation relation. In this paper, we use the Hamilton-Jacobi method to study effects of the minimal length on the motion of a massive particle perturbed away from an unstable equilibrium near the black hole horizon. We find that the minimal length effects make the particle move faster away from the equilibrium, and hence the corresponding Lyapunov exponent is greater than that in the usual case with the absence of the minimal length. It therefore shows that if the minimal length effects are taken into account, the conjectured universal bound on the Lyapunov exponent could be violated.

Chaotic motion around a black hole under minimal length effects

“…So it would be interesting to use other chaos indicators, e.g., the Poincaré surfaces of section, the Lyapunov characteristic exponents and the method of fractal basin boundaries, to detect chaotic behavior in systems perturbed by the minimal length effects. In [51], we calculated the minimal length effects on the Lyapunov exponent of a massive particle perturbed away from an unstable equilibrium near the black hole horizon and found that the classical trajectory in black holes becomes more chaotic, which is consistent with the chaotic behavior found in this paper. Finally, the minimal length effects on the dual conformal field theory was analyzed in [60].…”

“…In the classical limith → 0, the effects of the minimal length can be studied in the classical context. For example, the minimal length effects have been analyzed for the observational tests of general relativity [37][38][39][40][41][42][43][44] , classical harmonic oscillator [45,46], equivalence principle [47], Newtonian potential [48], the Schrödinger-Newton equation [49], the weak cosmic censorship conjecture [50] and motion of particles near a black hole horizon [51,52]. Moreover, the minimal length corrections to the Hawking temperature were also obtained using the Hamilton-Jacobi method in [53][54][55][56].…”

“…In [51], we considered the minimal length effects on motion of a massive particle near the black hole horizon under some external potential, which was introduced to put the particle at the unstable equilibrium outside the horizon. It was found that the minimal length effects could make the classical trajectory in black holes more chaotic, which motivates us to study the minimal length effects on geodesic motion in black holes.…”

Modified Brans-Dicke cosmology with Minimum Length Uncertainty

“…In such a case the dynamical system of our consideration is reduced to a four-dimensional system. Let P = (x (P ) , y (P ) , z (P ) , µ (P )) be a stationary point for the system composed of the equations ( 29), ( 30), (31) and (34). Then by definition at the stationary point, P , the rhs of equations ( 29), ( 30), ( 31) and ( 34) are zero.…”

“…The modification of the Uncertainty Principle for the quantum observables leads to the definition of a deformed Heisenberg algebra and consequently to the modification of the Poisson Brackets on the classical limits. The effect of GUP in Hamiltonian Mechanics and in General Relativity was the subject of study of a series of works by various authors, see for instance [28][29][30][31][32][33][34] and references therein. There are various applications of GUP in cosmological studies.…”