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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The causal vectors of a Lorentz vector space are not closed under addition

Statement refuted

Statement refuted: "the set of causal vectors of a Lorentz vector space is a linear subspace of $V$ , in particular closed under addition" (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

Facts & Assumptions

Given: An inertial frame of an oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ with $η = diag (- 1, 1, 1, 1)$ in the frame's coordinates, and the displacements $u = (1, 0, 0, 0)$ and $w = (- 1, 9 / 10, 0, 0)$ .

[L1]

$η (v, v) = - (v^{0})^{2} + (v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2}$ in an $η$ -orthonormal basis (Inertial frames and inertial charts of the affine Minkowski model, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

[L2]

A vector is timelike when $η (v, v) < 0$ , null when $v \neq 0$ and $η (v, v) = 0$ , spacelike when $η (v, v) > 0$ , and causal when timelike or null (Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L3]

The future closed cone is ${v : v^{0} \geq ∣ v_{v e c} ∣}$ and the past closed cone is ${v : v^{0} \leq - ∣ v_{v e c} ∣}$ ; a timelike vector in the first is future timelike and one in the second is past timelike (Time orientation, future and past timelike/null/causal vectors, and the closed cones, The timelike set has exactly two convex components, described in orthonormal coordinates).

Counterexample

1.1L1L2L3algebra

The displacement $u = (1, 0, 0, 0)$ satisfies $η (u, u) = - 1 < 0$ , so $u$ is timelike by [L2]; since $u^{0} = 1 > 0 = ∣ u_{v e c} ∣$ , it is future timelike by [L3].

1.2L1L2L3algebra

The displacement $w = (- 1, 9 / 10, 0, 0)$ satisfies $η (w, w) = - 1 + 81 / 100 = - 19 / 100 < 0$ , so $w$ is timelike by [L2]; since $w^{0} = - 1 \leq - 9 / 10 = - ∣ w_{v e c} ∣$ , it is past timelike by [L3].

2.1step 1.1step 1.2L1L2L3algebra∎

The sum has coordinates $u + w = (0, 9 / 10, 0, 0)$ , so $η (u + w, u + w) = 81 / 100 > 0$ by [L1] and $u + w$ is spacelike by [L2]. Both summands $u, w$ are causal (indeed timelike) by steps 1.1 and 1.2, while their sum is not; hence the set of causal vectors is not closed under addition and is not a linear subspace, refuting the displayed statement. Moreover $- u = (- 1, 0, 0, 0)$ is causal by step 1.1 and past timelike, not future causal, so the future causal set is not closed under negation either.

Remarks

This counterexample refutes the closure of the full causal set under addition. It makes no claim about sums of two future causal vectors: that closure is a different statement, proved with the causality material, and the displayed witnesses involve one future and one past timelike vector.

Depends on

Dependency tree · 2 levels

Sources

B. Crowell, Special Relativity, rev. 2020-12-10, sections 1.3-1.4 pp. 23-31 and Ch. 2 pp. 43-51 (signature (+---) converted to (-+++))