TIME MACHINES AND THE PRINCIPLE OF SELF-CONSISTENCY AS A CONSEQUENCE OF THE PRINCIPLE OF STATIONARY ACTION .2. THE CAUCHY-PROBLEM FOR A SELF-INTERACTING RELATIVISTIC PARTICLE

We consider the action principle to derive the classical, relativistic motion of a self interacting particle in a 4D Lorentzian spacetime co ntaining a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subj ect to a ''hard-sphere'' self-interaction potential and which can trav erse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary ar e those which are globally self-consistent. Generically, the multiplic ity of these trajectories (defined as the number of self-consistent so lutions to the equations of motion beginning with given Cauchy data) i s finite, and it becomes infinite if certain constraints on the same i nitial data are satisfied. This confirms the previous conclusions (for a nonrelativistic model) by Echeverria, Klinkhammer and Thorne that t he Cauchy initial value problem in the presence of a wormhole ''time m achine'' is classically ''ill-posed'' (far too many solutions). Our re sults further extend the recent claim by Novikov et al. that the ''pri nciple of self-consistency'' is a natural consequence of the ''princip le of minimal action.''