Physics theoremStatement: AI-adaptedProof: AI-adaptedPipeline-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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Charged test-body mass shell and laboratory work

Physical scope

Local solutions of the external-field charged test-body model on the open domain of its prescribed field, with constant $m > 0$ and $q$ , preserve the future timelike four-velocity shell in a fixed curved Lorentzian background. The laboratory three-force and power identities are asserted only in the flat Minkowski specialization with a named inertial SI chart and the compatible EM tensor convention. No global completeness, self-force, radiation reaction, measurement or empirical claim is included.

Statement

Assume the charged test-body quantities of Charged and spinning test-body quantities, the field convention of The electromagnetic two-form, its observer decomposition and the four-current, and the stationary-action equivalence Stationary charged action and Lorentz force, and assume ${A C}_{ω}$ for the canonical tangent-bundle smooth structure. Let $O \subseteq M$ be the open local-potential domain of the model, with prescribed field $F = d A$ there. For every future-timelike initial datum $(x_{0}, U_{0})$ with $x_{0} \in O$ and $g (U_{0}, U_{0}) = - c^{2}$ , the equation of motion has a unique local solution by External Lorentz ODE local well-posedness. Along it the kinetic four-momentum $P = m U$ remains future timelike and satisfies

$g (P, P) = - m^{2} c^{2} .$

In the flat Minkowski specialization, use the massive-particle momentum result Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds under its stated model assumptions, and choose a named inertial chart with $x^{0} = c t$ , $E, B$ defined by The electromagnetic two-form, its observer decomposition and the four-current, and $v$ the Euclidean spatial chart velocity. Set $γ : = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ . Then, with $p = m γ v$ and $E_{p a r t} = γ m c^{2}$ supplied by Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds, the same equation gives

$\frac{d p}{d t} = q (E + v \times B), \frac{d E_{p a r t}}{d t} = q E \cdot v .$

The mass-shell and force statements are conditional deductions within the specified model; they do not establish empirical applicability.

Facts & Assumptions

Given: A fixed smooth time-oriented Lorentzian background and prescribed external $C^{1}$ two-form on the open local-potential domain $O$ ; $m > 0$ and constant $q$ ; future-timelike initial data $(x_{0}, U_{0})$ with $x_{0} \in O$ ; assume ${A C}_{ω}$ for the canonical tangent-bundle smooth structure, and, for the laboratory clause, use the flat Minkowski and SI inertial-chart specialization.

[F1]

Charged and spinning test-body quantities: the kinetic four-momentum is $P = m U$ , the external field is a prescribed alternating two-form, and $g (U, U) = - c^{2}$ is the massive-worldline normalization.

[F2]

Stationary charged action and Lorentz force: trajectories stationary for the adopted local action satisfy $m \nabla_{U} U = q F^{♯} (U)$ .

[F3]

External Lorentz ODE local well-posedness: for future-timelike initial data, the equation has a unique local solution and preserves $g (U, U) = - c^{2}$ and the future direction.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume the axiom of countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure.

[F4]

Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds: in the flat massive-particle model, $P = m U$ has inertial components $(E_{p a r t} / c, p)$ with $E_{p a r t} = γ m c^{2}$ and $p = m γ v$ , where the chart Lorentz factor is $γ = (1 - ∣ v ∣^{2} / c^{2})^{- 1 / 2}$ .

[F5]

The electromagnetic two-form, its observer decomposition and the four-current: in the named flat SI chart, $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , hence $F^{i}_{0} = E_{i} / c$ , $F^{i}_{j} = ϵ_{i j k} B_{k}$ , and $F^{0}_{j} = E_{j} / c$ .

Proof

1.1givenA1F1F2F3algebra

For the given initial data with $x_{0} \in O$ , [F2] and [F3] give a unique local solution of the action model's equation, with $g (U, U) = - c^{2}$ and future direction preserved. From [F1], $P = m U$ ; therefore $g (P, P) = m^{2} g (U, U) = - m^{2} c^{2}$ , and $P$ is future timelike because $m > 0$ .

1.2givenA1F2F4F5algebra

In the flat inertial chart, [F4] gives $U^{0} = γ c$ , $U^{i} = γ v^{i}$ , $P^{i} = p^{i}$ , and $E_{p a r t} = c P^{0}$ , while [F2] gives $m d U^{μ} / d τ = q F^{μ}_{ν} U^{ν}$ . Since $d t / d τ = γ$ , [F5] yields $m d U^{i} / d τ = q γ (E_{i} + (v \times B)_{i})$ and $m d U^{0} / d τ = q γ (E \cdot v) / c$ ; multiplying by $1 / γ$ and, for the temporal equation, by $c / γ$ , gives the two displayed laboratory identities.

2.1step 1.1step 1.2∎

The curved-space shell conclusion is step 1.1, and the flat coordinate force and power conclusions are step 1.2; both inherit exactly the prescribed-field and test-body assumptions stated above. No empirical premise is used.

Charged test-body mass shell and laboratory work

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources