Billiard-Ball Time Loop (2D): A Self-Consistent Paradox

Setup

Launch speed v₀ 5.0

Impact-normal angle δ 30°

Wormhole twist ψ 35°

Solver damping α 0.42

Animation speed 1.0×

Self-Consistency Solver

Show velocity vectors

Show residual history

Not yet converged — press Run.

Iteration n

Residual |Δv|

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Deflection of ball 1

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Struck-ball speed

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How it works

A ball rolls at velocity v₁ toward a wormhole mouth A, in top-down view. Shortly before it arrives, it is grazed by a ball that just emerged from mouth B, exit velocity v₂ — but that second ball is the first ball's future self, having already fallen through the wormhole and come out earlier. This is Polchinski's billiard-ball paradox for a Morris–Thorne traversable wormhole.

The grazing collision is a standard equal-mass 2D elastic collision: for a hard-sphere collision along impact normal n̂, the normal velocity component is exchanged between the two balls and each ball keeps its own tangential component:

v₁' = (v₂·n̂)n̂ + [v₁ − (v₁·n̂)n̂]
v₂' = (v₁·n̂)n̂ + [v₂ − (v₂·n̂)n̂]

Note: the original 3D article's source code carried a typo in this second line — it wrote the tangential term as [v₂ − (v₁·n̂)n̂] instead of [v₂ − (v₂·n̂)n̂]. It never affected the simulation, because the solver only ever needs v₁' (below) to close the loop and v₂' is not evaluated — but this 2D build documents the corrected formula.

Ball 1 then carries v₁' into mouth A. If the two mouths are not perfectly co-aligned (a fixed relative twist ψ), it emerges from B rotated by ψ. Self-consistency (the Novikov / Deutsch principle) demands that this emergent velocity equal the v₂ we assumed:

v₂ = R(ψ) · v₁'(v₂)   — a fixed-point equation in v₂

The solver applies damped (relaxed) fixed-point iteration, v₂ ← v₂ + α·[R(ψ)v₁'(v₂) − v₂], until the residual vanishes. Echeverria, Klinkhammer & Thorne (1991) showed this equation generally has a whole continuum of self-consistent solutions rather than none — time travel need not forbid a paradox, it just over-determines the trajectory. Changing δ or ψ moves the solver to a different member of that family; changing v₀ rescales it; the damping α here is exposed as its own slider — turn it up for faster convergence or down to watch the iteration spiral in more gradually.

Ball 1 (present self) Ball 2 (future self, exits B) Wormhole mouths A / B

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top: velocity-space view of v₁, v₂ and the fixed-point target · bottom: residual |Δv| per iteration (log scale)