Multiplicity of self-consistent solutions for a billiard ball in a wormhole time-machine spacetime: existence and classification

Echeverria, Klinkhammer, and Thorne [Phys. Rev. D 44, 1077 (1991)] asked whether the Cauchy problem is well posed for the simplest self-interacting system in a spacetime with closed timelike curves: a nonrelativistic, perfectly elastic billiard ball moving in a wormhole time machine whose two mouths, at rest a unit distance apart with a backward time shift of order one, return the ball to its own past. Defining the multiplicity of an initial trajectory as the number of self-consistent classical solutions that begin with it, they found that 'dangerous' near-coplanar trajectories — those whose collision-free continuation makes the ball collide with its past self — have infinite multiplicity rather than the naively expected zero, that a ball initially at rest far away has multiplicity one, and that an extensive perturbative and numerical search for zero-multiplicity (genuinely paradoxical) initial data comes up empty. However, they could not prove the nonexistence of such data: the analysis covered only single-self-collision solutions, near-coplanar data within a perturbative decoupling neighborhood, and most of the 'dangerous triangle' of collision parameters, and solutions with multiple self-collisions were only cursorily searched. The announced quantum companion paper, which was to show that a sum-over-histories formulation assigns unique probability distributions to the multiple classical solutions, was never published. The open problem is to settle the well-posedness of this system: prove or disprove the existence of zero-multiplicity initial data throughout the model's parameter space, and determine the full multiplicity structure, including multi-self-collision solutions and generic noncoplanar data.

Background

General relativity permits spacetimes containing closed timelike curves (CTCs) — worldlines of massive objects that return to their own past. Morris, Thorne, and Yurtsever (1988) showed that a traversable wormhole (a tunnel joining two distant regions of spacetime, held open by matter violating the averaged null energy condition) can be turned into a time machine: by inducing a time shift between the two ends (the wormhole 'mouths'), an object entering one mouth can exit the other earlier than it entered, and can thereby meet — and collide with — its younger self. In the 'twin-paradox' configuration used by the source paper, the two mouths are at rest a distance $D$ apart (in the paper's units $D=1$), with the second mouth's interior shifted by a time $T_0$ of order one into the past relative to the first.

An object that can influence its own past raises the grandfather paradox. The precise mathematical form of the worry is the Cauchy problem: whether initial data posed in a region free of CTCs determine a unique evolution. In ordinary spacetimes they do, and the evolution is called well posed; in a spacetime with a time machine there is a Cauchy horizon beyond which the initial data no longer uniquely determine what happens. Novikov's self-consistency principle, articulated by the consortium of Friedman, Morris, Novikov, Echeverria, Klinkhammer, Thorne, and Yurtsever (1990), responds that the local physical laws hold unchanged and only globally self-consistent solutions — those compatible with the already-fixed past — are admissible. For linear systems this works: the Cauchy problem for the scalar wave equation is well defined on a class of spacetimes with CTCs (Friedman and Morris, 1991). For self-interacting systems the question is nonlinear and genuinely open.

Echeverria, Klinkhammer, and Thorne (EKT) posed it for the simplest self-interacting system: a single nonrelativistic, perfectly elastic hard-sphere billiard ball of radius $R$ with no internal degrees of freedom, moving in the twin-paradox wormhole spacetime with mouth radius $B$, in the hierarchy $R \ll B \ll D$; the paper presumes the ball small enough compared to the mouth ($R/B$ sufficiently small) that the impulsive tidal force at the throat can be ignored (it assumes that no collision occurs while the ball straddles the throat, that the ball's center moves through the wormhole as an arbitrarily small ball would, and that tidal distortions heal arbitrarily fast after a traversal); the ball is treated as a test object that does not move the mouths. Between encounters the ball moves freely; on self-collision it obeys the elastic hard-sphere law for equal masses; and a wormhole traversal preserves the ball's speed as measured in the exterior while changing the angle of its velocity by twice the angular location of the traversal point on the throat: $v_{\rm out} = v_{\rm in}$ and $\psi_{\rm out} = \psi_{\rm in} - 2\phi$, where $\psi$ is the angle of the velocity from the mouths' line of centers and $\phi$ is the angular location of the traversal point on the throat (EKT Eqs. (2.1); equivalently, straight-line, constant-speed motion of the ball's center as measured at the throat). The multiplicity of an initial trajectory is the number of self-consistent solutions of the equations of motion that begin with that trajectory. The Cauchy problem is well posed only if all initial trajectories have multiplicity one.

EKT found the opposite of the science-fiction expectation. A trajectory is dangerous if, followed assuming no collisions, it takes the ball through the wormhole and into collision with itself — the billiard-ball analogue of going back in time and killing one's younger self — and the naive expectation is multiplicity zero ('no solutions'). Instead, all dangerous trajectories in a wide class (nearly coplanar with the line of centers of the mouths) have infinite multiplicity: two of the solutions are slight perturbations of the collision-free motion, and the infinitely many others are qualitatively different, built from repeated wormhole traversals. A ball initially at rest far from the wormhole has multiplicity one, and probably also one that is almost at rest. The multiplicity structure was mapped by solving a set of highly nonlinear, coupled algebraic self-consistency equations with perturbation theory and numerics: the parameters of single-traversal dangerous data fill a 'dangerous triangle' in the plane of transverse offset and incoming angle, whose 'white' regions (left corner and lower-left edge) yielded only numerical spot solutions, not analytic control. For initial data displaced out of the plane by a distance $z_1$, the perturbative proof decouples in-plane from out-of-plane motion only in a small neighborhood of coplanarity, far smaller than the full dangerous range, so EKT stated they could 'say nothing about the existence of solutions over the full range' and could only conjecture that all initial trajectories have at least one self-consistent solution. Solutions involving sequences of multiple self-collisions and multiple traversals were searched only cursorily (suspected absent for nearly-at-rest data, without proof).

EKT closed by announcing a companion 'paper II' (Klinkhammer and Thorne, in preparation) in which a sum-over-histories formulation of quantum mechanics — the path-integral prescription in which probabilities arise from summing amplitudes over all histories — would restore well posedness: the many classical solutions would be assigned a unique probability distribution for the outcomes of all measurements. That paper was never published, and no later work has completed it for the wormhole spacetime itself. This announced sequel is historical context; it is not part of the problem stated below, whose deliverables are classical.

Problem Statement

For the model of Echeverria, Klinkhammer, and Thorne — a single nonrelativistic, perfectly elastic hard-sphere billiard ball of radius $R$, with no internal degrees of freedom, moving in the twin-paradox wormhole spacetime whose two mouths, each of radius $B$, are at rest a distance $D=1$ apart with a backward time shift $T_0$ of order one per traversal, in the regime $R \ll B \ll D$ (with the ball treated as a test object that does not move the mouths), with the local laws fixed as Newtonian free flight, elastic hard-sphere self-collision, and the EKT wormhole traversal rules — $v_{\rm out} = v_{\rm in}$ and $\psi_{\rm out} = \psi_{\rm in} - 2\phi$, where $\psi$ is the angle of the ball's velocity from the mouths' line of centers and $\phi$ is the angular location of the traversal point on the throat (EKT Eqs. (2.1)) — determine the multiplicity of every initial trajectory, where multiplicity is the number of self-consistent classical solutions beginning with that initial trajectory and initial data are posed before the Cauchy horizon. This comprises two linked deliverables that jointly settle the source paper's central question of whether the Cauchy problem is well posed:

(1) Existence. Prove, or disprove by explicit construction, that every initial trajectory admits at least one self-consistent classical solution — equivalently, that zero-multiplicity ('paradoxical') initial data do not exist. The regions left unsettled by the source analysis are the full noncoplanar range of dangerous trajectories (out-of-plane initial displacements beyond the paper's perturbative decoupling neighborhood of the coplanar limit), the white regions of the dangerous triangle (its left corner and lower-left edge, where the perturbative expansion fails), and solutions built from sequences of multiple self-collisions and multiple wormhole traversals.

(2) Multiplicity classification. Determine the multiplicity structure over the same range of initial data left unsettled in (1), beyond the source's nearly coplanar results: whether 'peculiar' multi-self-collision, multi-traversal solutions exist (conjectured absent for nearly-at-rest initial data in the source, without proof), and whether the coplanar structure — infinite multiplicity for dangerous data, unit multiplicity for data at rest far from the wormhole — persists for noncoplanar data.

References

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