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Fixed-endpoint curved charge variation — Examples

1 · Prerequisites

2 · Summary

The pure-gauge example uses the polynomial potential $A = d χ$ in flat spacetime to separate kinetic momentum from the gauge-dependent canonical covector. Its field strength and Lorentz force vanish, and its fixed-endpoint stationary curves remain free massive straight timelike curves. A position-dependent canonical component can nevertheless vary along a curve. All conclusions are conditional on the adopted test-body action and its exact mathematical hypotheses; no self-field, radiation reaction, force measurement or energy measurement is inferred.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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

Pure-gauge potential and free massive charged motion

Physical scope

Symbolic flat-Minkowski example of the adopted external-field massive charged test-body action, on finite fixed-endpoint timelike segments and under the inherited countable-choice hypothesis. The pure-gauge potential changes a local canonical covector without changing the kinetic momentum or producing a Lorentz force. No self-field, radiation reaction, measured energy, empirical finding or interaction mechanism is asserted.

Example

Use flat affine Minkowski spacetime $(A, V, η, C^{+})$ with signature $(-, +, +, +)$ in one named inertial chart $(x^{0}, x^{1}, x^{2}, x^{3})$ , $x^{0} = c t$ , with coordinates in metres. Adopt the external charged test-body action of External-field charged-particle action, the model quantities of Charged and spinning test-body quantities, and the countable-choice hypothesis ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) required by the final action-equivalence supplier; no full Axiom of Choice is assumed. Fix $m > 0$ kilograms, $q \in R$ coulombs and $c > 0$ metres per second. For a chosen constant $b \in R$ in tesla, put

$χ (x) = \frac{b}{2} (x^{1})^{2}, A = d χ = b x^{1} d x^{1} .$

Here $χ$ has units tesla metre squared and the covector components $A_{a}$ have units tesla metres. Then $F = d A = 0$ . On every finite $C^{2}$ future-timelike segment $z : [τ_{0}, τ_{1}] \to A$ with $τ_{0} < τ_{1}$ , proper-time parameter in seconds and fixed endpoint events $X_{0}, X_{1}$ , the action differs from its zero-potential value by

$S_{A} [z] - S_{0} [z] = q (χ (X_{1}) - χ (X_{0})) .$

Thus its stationary curves are precisely the free massive straight timelike curves on these segments. Their kinetic momentum $P = m U$ , $U = d z / d τ$ , is constant. With $p = m U^{♭}$ and the local canonical covector $π_{A} = p + q A$ of Gauge boundary term and kinetic/canonical momentum,

$π_{A, 1} = m U_{1} + q b x^{1}, \frac{d π_{A, 1}}{d τ} = q b U^{1} .$

The raised and lowered spatial components agree in this chart. This canonical component can vary when $q b U^{1} \neq 0$ , while the Lorentz force and kinetic-momentum derivative vanish. If any factor vanishes, the canonical component is constant too. The shift is a gauge-dependent covector calculation, not a force or an energy measurement.

Facts & Assumptions

Given: The flat inertial chart, $m, c > 0$ , constants $q, b$ , the displayed polynomial potential, finite timelike segments parameterized by proper time, fixed endpoint events and the adopted action model.

[A1]

External-field charged-particle action adopts stationarity of $S_{A} = - m c \int \sqrt{- g (z^{'}, z^{'})} d λ + q \int A (z^{'}) d λ$ under finite fixed-endpoint timelike variations on a local potential domain. The displayed potential is smooth globally, so every stated finite segment lies in such a domain.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only ${A C}_{ω}$ for the canonical tangent-bundle smooth structure used by the action-equivalence supplier.

[F1]

Oriented time-oriented affine Minkowski space (the spacetime primitives) and Inertial frames and inertial charts of the affine Minkowski model supply the constant flat metric matrix $diag (- 1, 1, 1, 1)$ and the inertial chart. Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields supplies the Levi–Civita coefficient formula; the constant metric makes every coefficient zero, so covariant differentiation in this chart is ordinary component differentiation.

[F2]

Gauge boundary term and kinetic/canonical momentum: replacing zero potential by $d χ$ leaves the field strength zero, adds $q [χ (X_{1}) - χ (X_{0})]$ to the finite fixed-endpoint action, and changes the canonical covector to $m U^{♭} + q d χ$ while leaving the kinetic covector unchanged.

[F3]

Stationary charged action and Lorentz force: under [A2] and the finite proper-time/potential-domain conditions, stationarity is equivalent to $m \nabla_{U} U = q F^{♯} (U)$ .

[F4]

Charged and spinning test-body quantities: $U$ is future timelike with $g (U, U) = - c^{2}$ , $P = m U$ is kinetic momentum, and $m, q$ are constant model parameters.

[F5]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: a differentiable real function on a connected interval with zero derivative is constant.

Verification

1.1givenA1F2algebra

The polynomial $χ = b (x^{1})^{2} / 2$ is smooth and $d χ = b x^{1} d x^{1}$ , so its only nonzero potential component is $A_{1} = b x^{1}$ . The alternating coordinate derivatives give $F_{a b} = \partial_{a} A_{b} - \partial_{b} A_{a} = 0$ : all off-diagonal partial derivatives vanish and the only diagonal derivative cancels itself. Applying [F2] with initial potential zero gives the displayed action difference. It is fixed under the endpoint-preserving variations of [A1], so stationarity is unchanged.

2.1givenA1A2F1F3F4F5step 1.1algebra

By [A2] and [F3], stationarity is equivalent to $m \nabla_{U} U = 0$ . Since $m > 0$ and the metric matrix is constant, [F1] gives $Γ^{a}_{b c} = \frac{1}{2} η^{a d} (\partial_{b} η_{d c} + \partial_{c} η_{d b} - \partial_{d} η_{b c}) = 0$ ; thus the equation is $d U^{a} / d τ = 0$ in the inertial chart. Apply [F5] to each component on the interval to obtain constant $U = U_{0}$ ; applying it to $z^{a} (τ) - U_{0}^{a} τ$ gives $z (τ) = X_{0} + (τ - τ_{0}) U_{0}$ . Conversely a future-timelike proper-time straight curve has $d U / d τ = 0$ and is stationary by the reverse implication of [F3]. No compatible endpoint curve is asserted to exist for arbitrary endpoint events or arbitrary proper-time duration.

3.1givenF1F2F4step 1.1step 2.1algebra∎

By [F4], $P = m U_{0}$ is constant, and [F2] gives $π_{A, 1} = m U_{1} + q b x^{1}$ . Differentiating along the curve from step 2.1 yields $d π_{A, 1} / d τ = q b U^{1}$ , with $U_{1} = U^{1}$ by [F1]. Hence the canonical component may vary exactly as stated while $q F^{♯} (U) = 0$ and $d P / d τ = 0$ . The action shift has units joule-seconds and the canonical covector components have momentum units; neither is an empirical measurement.