Chaos in relativity and cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we disc uss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a so lution depending on 6 arbitrary constants) of Painleve type, but there is a second general solution which is not Painleve. Thus the system does not pa ss the Painleve test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (alpha + beta + gamma) has on ly one maximum for tau = tau(m) and decreases continuously for larger and f or smaller tau. The various Kasner periods increase exponentially for large tau. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite ti me LCN" is positive for finite tau and tends to zero when tau --> infinity. Chaos is introduced mainly near the maximum of (alpha + beta + gamma). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, tau, the Mixmaster mode l has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M-1 and M-2 the orbits of photons are separa ted into three types: orbits falling into M-1 (type I), or M-2 (type II), o r escaping to infinity (type III). Chaos appears because between any two or bits of different types there are orbits of the third type. This is a typic al chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcat ion ratio is the same as in classical conservative systems.