ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Straight timelike free particle

Physical scope

Direct flat-space solution of the adopted free massive test-particle equation; the displayed trajectory and conserved quantities are model deductions, not reported observations.

Example

In the affine Minkowski model $(A, V, η, C^{+})$ , let $p \in A$ , let $U_{0} \in V$ be future timelike with $η (U_{0}, U_{0}) = - c^{2}$ , and let $m > 0$ . The curve $x : R \to A$ , $x (τ) = p + τ U_{0}$ , is a straight free massive trajectory.

Facts & Assumptions

Given: The flat affine model with background metric $g = η$ , $U_{0}$ and $m$ as above. Choose an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ and its inertial chart; at each event of the curve let $n = e_{0}$ under the canonical identification $T_{q} A ≅ V$ .

[F1]

The free massive equation is $D_{τ} U = 0$ and its solutions are affine timelike geodesics (Free massive and null particle geodesics).

[F2]

The momentum and observer energy are $P = m U$ and $E_{n} (P) = - c g (P, n)$ (Mechanical momentum and observer energy).

[F3]

An inertial frame supplies an inertial chart and a future unit time-axis vector $e_{0}$ ; the affine tangent-space identification makes this the same observer along the curve (Inertial frames and inertial charts of the affine Minkowski model, The tangent space of the affine Minkowski model at any event is canonically V).

[F4]

In the flat relativistic-particle model, the background metric used in the momentum and observer-energy definitions is $g = η$ (Particle spacetime, observers and worldlines).

Verification

1.1givenF1algebra

Differentiating $x (τ) = p + τ U_{0}$ gives $U = U_{0}$ , which is constant in the flat affine connection. It is future timelike by assumption, and $D_{τ} U = 0$ , so it solves the free massive equation of [F1]. The mechanical momentum $P = m U_{0}$ is also constant.

2.1step 1.1F2F3F4algebra∎

In the inertial chart of [F3], write $U_{0} = (U^{0}, U)$ . Since $χ (x (τ)) = χ (p) + τ (U^{0}, U)$ and $t = x^{0} / c$ , the coordinate velocity is $v = c U / U^{0}$ . The norm gives $(U^{0})^{2} = c^{2} + ∣ U ∣^{2}$ , while future direction gives $U^{0} > 0$ , so $∣ v ∣ < c$ . The observer $n = e_{0}$ is fixed under the affine tangent identification; [F2] and [F4] therefore make $E_{n} (P) = - c η (P, e_{0})$ constant because $P$ is constant.

Straight timelike free particle

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources