CHAOS AROUND A BLACK-HOLE

We apply the Melnikov method for identifying chaos in near integrable systems to relativistic particle motion around a Schwarzschild black hole. We start by giving a self-contained introduction to the Melnikov method together with some relevant background on dynamical systems. Then we show that a relativistic particle has unstable circular orbits around a Schwarzschild black hole, and that each one of these gives rise to a homoclinic orbit in phase space, which tends to the unstable one for t --> +/-infinity. Finally, we use the Melnikov method to conclude that, under most periodic perturbations of the black-hole metric, the homoclinic orbit becomes chaotic.