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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Exact relativistic orbit in a uniform electric field

Physical scope

Exact global coordinate-time solutions of the adopted flat external-field charged test-particle model for a uniform constant electric field, with fixed SI conventions and a smooth explicit potential. The force law is conditional on the local RPM action postulate and its draft equivalence and mass-shell suppliers. No self-field, radiation reaction, measurement, or empirical claim is included.

Statement

In the flat Minkowski specialization of Charged and spinning test-body quantities, adopt the fixed-endpoint external-charge action External-field charged-particle action, its conditional stationary-action equivalence Stationary charged action and Lorentz force, the flat massive-particle momentum model Adopted massive-particle momentum, observer measurement and isolated-collision conservation, and ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the canonical tangent-bundle smooth structure required by the current local IVP and action-equivalence carriers. Let $m > 0$ in kilograms, $q \in R$ in coulombs, $c > 0$ in metres per second, and $E_{0} \in R$ in newtons per coulomb; let the uniform field in one named inertial SI chart be $E = E_{0} e_{1}$ , $B = 0$ . A smooth global potential is $ϕ = - E_{0} x^{1}$ , $A = 0$ , so the local-potential hypothesis is satisfied. Spatial coordinates are in metres, coordinate time $t$ is in seconds, and initial momenta are in kilograms metres per second. Set $F_{0} = q E_{0}$ (newtons), take initial event $(0, x_{0})$ and initial momentum $p_{0} = (p_{01}, p_{⊥})$ at $t = 0$ , and define $Π$ (momentum units), $E$ (joules) and $v$ (metres per second) by

$p_{1} (t) = p_{01} + F_{0} t, Π = \sqrt{m^{2} c^{2} + ∣ p_{⊥} ∣^{2}} > 0, E (t) = c \sqrt{Π^{2} + p_{1} (t)^{2}}, v (t) = \frac{c (p_{1} (t) e_{1} + p_{⊥})}{\sqrt{Π^{2} + p_{1} (t)^{2}}} .$

If $F_{0} \neq 0$ , the exact position for every $t \in R$ is

$x^{1} (t) = x_{0}^{1} + \frac{c}{F_{0}} (\sqrt{Π^{2} + p_{1} (t)^{2}} - \sqrt{Π^{2} + p_{01}^{2}}),$ $x_{⊥} (t) = x_{⊥ 0} + \frac{c p_{⊥}}{F_{0}} \log ⁣ (\frac{p_{1} (t) + \sqrt{Π^{2} + p_{1} (t)^{2}}}{p_{01} + \sqrt{Π^{2} + p_{01}^{2}}}) .$

If $F_{0} = 0$ , the field exerts no force on the particle and the orbit is free: $p (t) = p_{0}$ , $E (t) = E (0)$ and $x (t) = x_{0} + c^{2} p_{0} t / E (0)$ . These formulas define future-timelike worldlines for all finite coordinate times; any model solution with the same data agrees with them on every common connected time interval. This is a conditional model deduction, not evidence that the adopted action describes a measured particle.

Facts & Assumptions

Given: A named inertial SI chart $x^{0} = c t$ , prescribed $m > 0$ , charge $q$ , constant $c > 0$ , initial momentum and position, the uniform field and its displayed smooth potential, the adopted local action and flat massive-particle models, and ${A C}_{ω}$ for the canonical tangent-bundle smooth structure.

[F1]

Charged and spinning test-body quantities: $m > 0$ and $q$ are constant parameters, $P = m U$ , the field convention is $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ , and a local $C^{2}$ potential may represent $F = d A$ .

[A1]

External-field charged-particle action: model trajectories are stationary points of the fixed-endpoint charged-particle action in a local potential domain.

[F2]

Stationary charged action and Lorentz force: under the local action hypotheses and ${A C}_{ω}$ , stationarity is equivalent to $m \nabla_{U} U = q F^{♯} (U)$ .

[F3]

Charged test-body mass shell and laboratory work: in the flat inertial SI specialization, the equation gives $d p / d t = q (E + v \times B)$ and $d E / d t = q E \cdot v$ .

[A2]

Adopted massive-particle momentum, observer measurement and isolated-collision conservation: a flat-model massive particle has constant $m > 0$ and kinetic four-momentum $P = m U$ .

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure, as stated by the current RPM suppliers.

[F4]

Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds: in a named inertial frame, the massive particle has $E = γ m c^{2}$ , $p = γ m v$ , and $E^{2} = m^{2} c^{4} + c^{2} ∣ p ∣^{2}$ .

[F5]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition: future timelike chart motion has $d t / d τ = γ > 0$ and $∣ v ∣ < c$ .

[F6]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: on positive arguments, $d (\log s) / d s = 1 / s$ and the logarithmic integral rule applies.

[F7]

Continuity and derivatives of positive-base real powers: for positive $s$ , $d (s^{1 / 2}) / d s = 1 / (2 \sqrt{s})$ .

[F8]

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 scalar function on an interval with zero derivative is constant there.

Proof

1.1givenF1A1A3F2algebra

The displayed scalar and vector potentials are smooth on all of Minkowski space. With the covector convention $A_{0} = - ϕ / c$ and $A_{i} = A_{i}$ , they give $F_{01} = - E_{0} / c$ , $F_{i j} = 0$ , hence exactly the field in [F1]. Thus the action potential-domain hypothesis [A1] is met, and under the exact assumption [A3], [F2] gives its covariant Lorentz-force equation.

2.1step 1.1A2A3F3F4F5algebra

Apply [F3] under [A3] to this field: ${\dot{p}}_{1} = F_{0}$ , ${\dot{p}}_{⊥} = 0$ , and $\dot{E} = F_{0} v_{1}$ . Integration with the initial data gives $p_{1} (t) = p_{01} + F_{0} t$ and constant $p_{⊥}$ . By adopted [A2] and the inertial momentum relations [F4], $E^{2} = m^{2} c^{4} + c^{2} ∣ p ∣^{2}$ ; since $E = γ m c^{2} > 0$ , this is $E = c \sqrt{m^{2} c^{2} + ∣ p ∣^{2}}$ , while $p = γ m v$ gives $v = c^{2} p / E$ . Thus the definitions of $Π, E, v$ in the statement follow. Differentiating the shell identity with $\dot{p} = F_{0} e_{1}$ gives $2 E \dot{E} = 2 c^{2} p_{1} F_{0}$ , hence $\dot{E} = c^{2} p_{1} F_{0} / E = F_{0} v_{1}$ , as required by [F3]. Since $m, c > 0$ , the square-root denominator is at least $m c > 0$ , and [F5] gives subluminal motion.

3.1step 2.1F6F7algebra

Suppose $F_{0} \neq 0$ . Differentiating the proposed $x^{1} (t)$ and using ${\dot{p}}_{1} = F_{0}$ gives ${\dot{x}}^{1} = c p_{1} / \sqrt{Π^{2} + p_{1}^{2}} = v_{1}$ . For the transverse formula, the derivative of $\log (p + \sqrt{Π^{2} + p^{2}})$ is $1 / \sqrt{Π^{2} + p^{2}}$ : the argument is positive since $\sqrt{Π^{2} + p^{2}} > ∣ p ∣$ , and the identity follows from [F6, F7] and the chain rule. Hence ${\dot{x}}_{⊥} = c p_{⊥} / \sqrt{Π^{2} + p_{1}^{2}} = v_{⊥}$ . At $t = 0$ both position increments vanish, so these formulas have the prescribed initial position and velocity.

3.2step 2.1F3F5algebra

If $F_{0} = 0$ , [F3] makes $p$ and $E$ constant, so the displayed free position has derivative $c^{2} p_{0} / E (0) = v$ and the initial position. Its speed is below $c$ by [F5].

4.1step 2.1step 3.1step 3.2F3F8algebra∎

If $F_{0} \neq 0$ , the constructed curve satisfies the force equation and initial data by steps 2.1 and 3.1; if $F_{0} = 0$ , those facts follow from steps 2.1 and 3.2. Any other solution has the same momentum by integrating [F3]; subtracting its position from the displayed one gives a differentiable vector with zero derivative and zero initial value, so each component vanishes by [F8]. Thus it agrees throughout any common connected interval. The formulas have positive square-root arguments and positive logarithm arguments for every finite $t$ , so they extend smoothly over all $R$ .

Exact relativistic orbit in a uniform electric field

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources