Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Local massive motion in smooth orthogonal prescribed forces

Physical scope

Conditional local dynamics of one massive test body with constant primitive mass on a prescribed smooth time-oriented Lorentzian background, under a separately adopted smooth external force orthogonal to its future unit velocity. It proves local motion and a chart-local compact-continuation criterion only; it makes no universal-force, global-completeness, self-force, or empirical claim.

Statement

Let $(M, g, N)$ be a prescribed smooth time-oriented Lorentzian four-manifold and let $m, c > 0$ . Let $V \subset T M$ be an open set of future timelike vectors, and let $f$ be a smooth vector field along the bundle projection $V \to M$ , with $f (p, V) \in T_{p} M$ . For the named massive test body, adopt the constant-mass force model of Prescribed constant-mass force law: $P = m U$ , $g (U, U) = - c^{2}$ , $D_{τ} P = f (x, U)$ , and $g (U, f (x, U)) = 0$ . If $τ$ is measured in seconds, then $P$ has SI units kilogram metres per second and $f$ has units newtons.

Fix an initial proper-time value $τ_{0} \in R$ . For each initial datum $(p_{0}, U_{0}) \in V$ with $g_{p_{0}} (U_{0}, U_{0}) = - c^{2}$ , there is a unique local $C^{2}$ future-timelike worldline $x : J \to M$ on an open interval $J ∋ τ_{0}$ , with $C^{1}$ tangent $U = d x / d τ$ , $x (τ_{0}) = p_{0}$ , and $U (τ_{0}) = U_{0}$ , satisfying the adopted equation. It remains on the future unit timelike shell and has a unique maximal solution on an open interval containing $τ_{0}$ . In any chart-state domain where the coordinate force vector field is smooth, if a finite endpoint is approached while the state remains in a compact subset of that domain, the solution extends past the endpoint. No global lifetime or spacetime completeness is asserted.

Facts & Assumptions

Given: The prescribed smooth spacetime and force field, $m, c > 0$ , a specified initial proper-time value $τ_{0}$ , initial position $p_{0}$ , future unit timelike initial velocity $U_{0}$ , and the equalities $x (τ_{0}) = p_{0}$ , $U (τ_{0}) = U_{0}$ , and the adopted constant-mass force model.

[A1]

The physical model is an optional adoption for a named massive test body with constant primitive $m > 0$ : on an open neighborhood of the future unit timelike shell it specifies a smooth force $f$ , the equation $D_{τ} P = f (x, U)$ with $P = m U$ , and the compatibility condition $g (U, f (x, U)) = 0$ (Prescribed constant-mass force law). This is not a universal force principle; that postulate is a current in-run draft and is used here at its inspected statement.

[F1]

The RPP background is a prescribed smooth time-oriented Lorentzian four-manifold, and massive worldlines are future timelike curves with a specified parameter; observer, congruence, adapted chart and tetrad are distinct objects (Particle spacetime, observers and worldlines).

[F2]

In the adopted massive sector, the mechanical momentum is $P = m U$ for constant $m > 0$ and $g (U, U) = - c^{2}$ (Mechanical momentum and observer energy).

[L1]

For a smooth force on an open future-timelike velocity domain with $g (U, f) = 0$ , the system $d x / d s = U$ , $D_{s} U = f (x, U) / m$ has a unique local solution, preserves $g (U, U) = - c^{2}$ and future direction, and has a unique maximal open interval (Smooth shell-tangent force initial-value problem). The actual supplier is a draft from in-run batch 22, so the use here is conditional on its inspected interface and proof status.

[L2]

A smooth solution of that shell-force equation which remains in a compact subset of its open chart-state domain cannot have a finite maximal endpoint (Orthogonal shell force local IVP and compact continuation).

Proof

technique · identify the adopted force equation with the shell-tangent ODE and apply its local and continuation results

1.1givenA1F1F2L1algebra

The prescribed smooth background and future timelike worldline use the conventions of [F1]. Since $m$ is constant, [F2] gives $D_{τ} P = m D_{τ} U$ . Substituting [A1] yields $D_{τ} U = f (x, U) / m$ , with the force smooth on the open velocity domain and orthogonal to $U$ . These are exactly the hypotheses of [L1], so apply [L1] with $s_{0} = τ_{0}$ and $s = τ$ , so for each stated initial datum there is a unique local solution with $x (τ_{0}) = p_{0}$ and $U (τ_{0}) = U_{0}$ ; [L1] also proves preservation of the future unit timelike shell.

2.1step 1.1L1L2∎

The unique local solutions patch by uniqueness to the maximal open-interval solution supplied by [L1]. At either finite endpoint, if the coordinate state remained in a compact subset of a chart-state domain on which the force vector field is smooth, [L2] would extend that solution, contradicting maximality. Therefore every finite maximal endpoint leaves every such compact subset.

Remarks

This is a model deduction conditional on the separately adopted force law. Orthogonality is the compatibility condition for constant-mass shell preservation; it does not select a physical interaction. The result does not prove that the model describes a measured body.

Depends on

Dependency tree · 2 levels

Sources

Gautam Iyer, Brief Notes on Introduction to Differential Equations, complete ODE notes