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.
Highlights
Chaos is one of the most important ideas to understand various nonlinear phenomena in general relativity
The existence of simple zeros of the Melnikov function leads to the Smale horseshoes structure in phase space, which implies that the dynamical system is chaotic
We used the Melnikov method to investigate the chaotic behavior in geodesic motion on a Schwarzschild metric perturbed by the minimal length effects
Summary
Chaos is one of the most important ideas to understand various nonlinear phenomena in general relativity. The geodesic motion of a ring string instead of a point particle has been shown to exhibit chaotic behavior in a Schwarzschild black hole [9]. 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].
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