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

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Counterexample: the temporal order of a spacelike-separated pair is not the same in every inertial frame

Statement refuted

False claim. For any two distinct events $p, q$ of the affine Minkowski model, the sign of the time-coordinate difference $x^{0} (q) - x^{0} (p)$ is independent of the inertial frame (Inertial frames and inertial charts of the affine Minkowski model).

Facts & Assumptions

Given: An inertial frame with chart $x$ , the events $p = x^{- 1} (0, 0, 0, 0)$ and $q = x^{- 1} (11, 13, 0, 0)$ , and the displacement $D = q - p$ with coordinates $(11, 13, 0, 0)$ .

[L1]

For a spacelike displacement $∣ D^{0} ∣ < ∣ D_{s} ∣$ holds in every inertial chart, and for every $∣ b ∣ < 1$ the chart $x_{b}$ of the inertial frame with basis $B (b e_{1})^{- 1} e_{a}$ (same origin) satisfies $x_{b} = B (b e_{1}) \circ x$ and $x_{b}^{0} (q) - x_{b}^{0} (p) = γ_{b} (D^{0} - b ∣ D_{s} ∣)$ with $γ_{b} = (1 - b^{2})^{- 1 / 2}$ (Spacelike-separated events: the spacelike coordinate inequality in every inertial chart, and an explicit family of inertial frames realizing every time-order sign).

[L2]

For $∣ b ∣ < 1$ the passive boost acts in the $(0, 1)$ -plane by $x^{' 0} = γ (x^{0} - b x^{1})$ , $x^{' 1} = γ (x^{1} - b x^{0})$ and is a proper orthochronous Lorentz map (The passive Lorentz boost with parameter beta, The passive boost is a proper orthochronous Lorentz map with inverse B(-beta)).

[L3]

Diagonal form and classification: $η (D, D) = - (D^{0})^{2} + ∣ D_{s} ∣^{2}$ ; a nonzero vector with positive squared norm is spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L4]

Time monotonicity: for every piecewise $C^{1}$ future causal path that is nonconstant on $[a^{'}, b^{'}]$ , every inertial chart gives $(x^{0} \circ y) (b^{'}) > (x^{0} \circ y) (a^{'})$ ; and for $q \neq p$ with $q \in J^{+} (p)$ the straight path from $p$ to $q$ is a future causal path (Future causal reachability, absence of causal loops, time monotonicity and the straight-worldline proper-time maximum).

[L5]

The false claim of the statement refuted section.

Counterexample

technique · direct

1.1givenL1L3algebra

The displacement has coordinates $D = (11, 13, 0, 0)$ , so $η (D, D) = - 121 + 169 = 48 > 0$ and $∣ D^{0} ∣ = 11 < 13 = ∣ D_{s} ∣$ ; by [L3] the displacement is spacelike, so [L1] applies to the pair.

2.1step 1.1L1L2algebra

With $b = 3 / 5$ , admissible since $∣ b ∣ < 1$ , one has $γ_{b} = (1 - 9 / 25)^{- 1 / 2} = 5 / 4$ , so the chart $x_{3 / 5}$ of the inertial frame of [L1] satisfies $x_{3 / 5}^{0} (q) - x_{3 / 5}^{0} (p) = γ_{b} (11 - 13 b) = (5 / 4) (11 - 39 / 5) = (5 / 4) (16 / 5) = 4 > 0$ .

2.2step 1.1L1L2algebra

With $b = 12 / 13$ , admissible since $∣ b ∣ < 1$ , one has $γ_{b} = (1 - 144 / 169)^{- 1 / 2} = 13 / 5$ , so $x_{12 / 13}^{0} (q) - x_{12 / 13}^{0} (p) = γ_{b} (11 - 13 b) = (13 / 5) (11 - 12) = - 13 / 5 < 0$ .

3.1step 1.1step 2.1step 2.2L1L4L5∎

The charts $x_{3 / 5}$ and $x_{12 / 13}$ are charts of inertial frames of the model by [L1] and [L2], and they assign to the same pair of events $(p, q)$ time-coordinate differences of opposite sign by steps 2.1 and 2.2; the claim of [L5] is therefore false. The failure is confined to spacelike-separated pairs: if $q \neq p$ with $q - p$ future causal, then [L4] gives the straight path from $p$ to $q$ as a nonconstant future causal path, so every inertial chart gives $x^{0} (q) - x^{0} (p) > 0$ ; no choice principle is used.

Counterexample: the temporal order of a spacelike-separated pair is not the same in every inertial frame

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources