Chaos in the Case of Two Fixed Black Holes

Abstract

We study the orbits of photons (null geodesics) and particles (time-like geodesics) in the field of two fixed black holes. The orbits of photons terminate either at the black holes M 1 and M 2 (types (I) and (II)), or at infinity (type (III)). The limits of these three types of orbits are Cantor sets, defined by the unstable periodic orbits, that form a set of measure zero. In the case of particles with elliptic energy there are some stable periodic orbits that trap around them both quasiperiodic orbits (type (IV)) and chaotic orbits that do not reach the black holes (type (V)). The stable periodic orbits may become unstable, producing an infinity of period doubling bifurcations. The main periodic orbits are two almost circular satellite orbits around each black hole, M 1 and M 2, and almost elliptical orbits surrounding both black holes. Orbits crossing one of the satellite orbits inwards fall into the corresponding black hole. For certain values of the parameters one or both satellite orbits may not exist. Then the separation of the type (I) and/or type (II) orbits is done by some almost hyperbolic orbits. In the Newtonian case the satellite orbits around M 1 and M 2 never exist (except if M 1 = 0, or M 2 = 0 exactly). However we have many families of higher order periodic orbits that exist both in the relativistic and the Newtonian case. The differences between the corresponding orbits are large close to the black holes.