Particle Motion Around Perturbed Black Holes: The Onset of Chaos

Abstract

I describe an example of relativistic chaos that arises in the motion of a relativistic particle around a black hole. In the case of a static, Schwarzschild black hole, for which geodesic motion is integrable, there are, in addition to the stable circular orbits present in Newtonian gravity, unstable circular orbits near r = 6M. These give rise to other orbits, which approach the unstable ones for t → ±∞ and wind outwards up to some maximum radius in between. The latter are examples of the so-called homoclinic orbits in the phase space of an integrable system, which are often seeds of chaos when a perturbation is added to the system. By using the Melnikov method for detecting homoclinic chaos in near integrable systems, we conclude that, as one adds a time-periodic perturbation to the static black hole, a region of chaotic motion replaces the homoclinic orbit.