Time machines and the Principle of Self-Consistency as a consequence of the Principle of Stationary Action (II): the Cauchy problem for a self-interacting relativistic particle

TL;DR

This paper demonstrates that in a relativistic setting with wormholes, the principle of self-consistency naturally emerges from the principle of stationary action, showing that only globally self-consistent trajectories are stationary solutions.

Contribution

It extends the principle of self-consistency as a consequence of the stationary action principle to relativistic particles in wormhole spacetimes, analyzing the Cauchy problem and solution multiplicity.

Findings

Stationary trajectories are globally self-consistent.

Number of solutions is finite or infinite depending on initial data constraints.

Confirms the classical ill-posedness of the Cauchy problem in wormhole spacetimes.

Abstract

We consider the action principle to derive the classical, relativistic motion of a self-interacting particle in a 4-D Lorentzian spacetime containing a wormhole and which allows the existence of closed time-like curves. In particular, we study the case of a pointlike particle subject to a `hard-sphere' self-interaction potential and which can traverse the wormhole an arbitrary number of times, and show that the only possible trajectories for which the classical action is stationary are those which are globally self-consistent. Generically, the multiplicity of these trajectories (defined as the number of self-consistent solutions to the equations of motion beginning with given Cauchy data) is finite, and it becomes infinite if certain constraints on the same initial data are satisfied. This confirms the previous conclusions (for a non-relativistic model) by Echeverria, Klinkhammer and Thorne that the Cauchy initial value problem in the presence of a wormhole `time machine' is classically `ill-posed' (far too many solutions). Our results further extend the recent claim by Novikov et al. that the `Principle of self-consistency' is a natural consequence of the `Principle of minimal action.'