Chaos in Relativity and Cosmology |
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| Authors: | G Contopoulos N Voglis C Efthymiopoulos |
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| Institution: | (1) Research Center for Astronomy, Academy of Athens, Greece;(2) Department of Astronomy, University of Athens, Greece |
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| Abstract: | Chaos appears in various problems of Relativity and Cosmology. Here we discuss (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 solution depending on 6
arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does
not pass the Painlevé 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 (α + β + γ) has only one maximum for τ = τm and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus
the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ
→ ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner
periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics
of a chaotic scattering problem. (b) In the case of two fixed black holes M1 and M2 the orbits of photons are separated into three types: orbits falling into M1 (type I), or M2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are
orbits of the third type. This is a typical 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 bifurcation ratio is the same as in classical conservative systems.
This revised version was published online in July 2006 with corrections to the Cover Date. |
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