3-DIMENSIONAL BILLIARDS WITH TIME MACHINE

Self-collision of a nonrelativistic classical point-like body, or part icle, in the spacetime containing closed time-like curves (time-machin e spacetime) is considered. A point-like body (particle) is an idealiz ation of a small ideal elastic billiard ball. The known model of a tim e machine is used containing a wormhole leading to the past. If the bo dy enters one of the mouths of the wormhole, it emerges from another m outh in an earlier time so that both the particle and its ''incarnatio n'' coexist during some time and may collide. Such self-collisions are considered in the case when the size of the body is much less than th e radius of the mouth, and the latter is much less than the distance b etween the mouths. Three-dimensional configurations of trajectories wi th a self-collision are presented. Their dynamics is investigated in d etail. Configurations corresponding to multiple wormhole traversals ar e discussed. It is shown that, for each world line describing self-col lision of a particle, dynamically equivalent configurations exist in w hich the particle collides not with itself but with an identical parti cle having a closed trajectory (Jinnee of Time Machine).