CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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Counterexample: a past-directed parametrization of a timelike line is not a worldline

Statement refuted

False claim: if a $C^{1}$ curve parametrizes a straight timelike line, then it is a future timelike worldline of the model and the proper-time construction gives the clock reading of its history.

The claim is refuted by orienting the parametrization backwards: the same geometric line with the opposite parametrization has past timelike tangents, is not a worldline of the model, and yet the proper-time integrand takes exactly the same values as for the future-directed parametrization. This is why the parametrization, and not merely the image of the curve, is part of the definition of a worldline.

Facts & Assumptions

Given: A future timelike displacement $d$ and an event $p \in A$ , and the two parametrizations $z (s) : = p - d + s d$ and $x (s) : = p - s d$ , $s \in [0, 1]$ , of the same straight timelike segment; $z (s) = x (1 - s)$ .

[L1]

Causal classification (Timelike, null, spacelike and causal vectors of a Lorentz vector space): $v$ is timelike when $η (v, v) < 0$ ; the sign of $η$ does not distinguish future from past.

[L2]

Time orientation (Time orientation, future and past timelike/null/causal vectors, and the closed cones): the timelike set is the disjoint union of the future component $C^{+}$ and the past component $C^{-}$ , and a timelike vector is future or past but not both; $- v \in C^{-}$ whenever $v \in C^{+}$ .

[L3]

Worldlines (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): a future timelike worldline is a $C^{1}$ curve whose tangent at every parameter value is future timelike; the parametrization is part of the curve.

[L4]

Proper time (Proper time along a future timelike worldline): the proper-time integrand is $\sqrt{- η (x^{'} (s), x^{'} (s))}$ , which depends on the tangent through the quadratic form $η$ only.

Counterexample

technique · direct

1.1L1L2

In an inertial chart with $η_{0} = diag (- 1, 1, 1, 1)$ let $d : = (5, 3, 0, 0)$ : then $- η (d, d) = - (- 25 + 9) = 16 > 0$ , so $d$ is timelike, and choosing $d$ in the future component $C^{+}$ makes it a future timelike displacement; then $- d$ belongs to $C^{-}$ by [L2].

2.1step 1.1L3

The curve $x (s) : = p - s d$ on $[0, 1]$ has difference quotients identically $- d$ , so $x^{'} (s) = - d$ is past timelike for every $s$ ; by [L3] the curve is not a future timelike worldline of the model, The curve $z (s) : = p - d + s d = x (1 - s)$ for $s \in [0, 1]$ has tangent $d$ , is a future timelike worldline and has exactly the same image, since $s \mapsto 1 - s$ is a bijection of $[0, 1]$ .

3.1step 1.1step 2.1L4∎

The proper-time integrand of both curves agrees pointwise: $\sqrt{- η (- d, - d)} = \sqrt{- η (d, d)} = 4$ by bilinearity of $η$ , so the proper-time integral alone does not distinguish the future-directed from the past-directed parametrization: by [L4] the proper-time construction is attached to the parametrized history, together with its future timelike hypothesis, and not to the bare image of the curve.

Remarks

Orientation is data, not geometry of the image. The same set of events carries two opposite worldline parametrizations with identical proper-time densities; only the future-directed one is a worldline of the model, and the past-directed one is its orientation reversal.

Counterexample: a past-directed parametrization of a timelike line is not a worldline

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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