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Published August 15, 1991 | public
Journal Article Open

Billiard balls in wormhole spacetimes with closed timelike curves: Classical theory


The effects of self-interaction in classical physics, in the presence of closed timelike curves, are probed by means of a simple model problem: The motion and self-collisions of a nonrelativistic, classical billiard ball in a space endowed with a wormhole that takes the ball backward in time. The central question asked is whether the Cauchy problem is well posed for this model problem, in the following sense: We define the multiplicity of an initial trajectory for the ball to be the number of self-consistent solutions of the ball's equations of motion, which begin with that trajectory. For the Cauchy problem to be well posed, all initial trajectories must have multiplicity one. A simple analog of the science-fiction scenario of going back in time and killing oneself is an initial trajectory which is dangerous in this sense: When followed assuming no collisions, the trajectory takes the ball through the wormhole and thereby back in time, and then sends the ball into collision with itself. In contrast with one's naive expectation that dangerous trajectories might have multiplicity zero and thereby make the Cauchy problem ill posed ("no solutions"), it is shown that all dangerous initial trajectories in a wide class have infinite multiplicity and thereby make the Cauchy problem ill posed in an unexpected way: "far too many solutions." The wide class of infinite-multiplicity, dangerous trajectories includes all those that are nearly coplanar with the line of centers between the wormhole mouths, and a ball and wormhole restricted by (ball radius)≪(wormhole radius)≪(separation between wormhole mouths). Two of the infinity of solutions are slight perturbations of the self-inconsistent, collision-free motion, and all the others are strongly different from it. Not all initial trajectories have infinite multiplicity: trajectories where the ball is initially at rest far from the wormhole have multiplicity one, as also, probably, do those where it is almost at rest. A search is made for initial trajectories with zero multiplicity, and none are found. The search entails constructing a set of highly nonlinear, coupled, algebraic equations that embody all the ball's laws of motion, collision, and wormhole traversal, and then constructing perturbation theory and numerical solutions of the equations. A future paper (paper II) will show that, when one takes account of the effects of quantum mechanics, the classically ill-posed Cauchy problem ("too many classical solutions") becomes quantum-mechanically well posed in the sense of producing unique probability distributions for the outcomes of all measurements.

Additional Information

©1991 The American Physical Society. Received 27 February 1991. We thank Joseph Polchinski for motivating this research by asking, in a letter to one of us, how the laws of physics could deal with paradoxical situations of the sort embodied in Fig. 3(a). For helpful discussions we thank John Friedman, Mike Morris, Nicolas Papastamatiou, Leonard Parker, and Ulvi Yurtsever. This paper was supported in part by National Science Foundation Grant No. AST-8817792.


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