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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Subcritical crossed fields and exact drift-frame orbits

Physical scope

Exact conditional flat-space statement for constant perpendicular electric and magnetic fields in the adopted external charged test-particle model. When $∣ E ∣ < c ∣ B ∣$ , a subluminal passive boost removes the electric field and every solution is the inverse boost of a uniform-magnetic-field orbit. A constant drift-velocity solution is singled out, but the theorem does not claim that all particle velocities equal the drift velocity. Null and supercritical crossed fields are excluded; no self-field, radiation reaction, measurement, or empirical claim is made.

Statement

In the flat SI charged-particle model with adopted action, conditional force equation, flat massive-particle momentum model, and ${A C}_{ω}$ assumption required by the current local RPM carriers External-field charged-particle action, Stationary charged action and Lorentz force, Charged test-body mass shell and laboratory work, Adopted massive-particle momentum, observer measurement and isolated-collision conservation, and The Axiom of Countable Choice ( ${A C}_{ω}$ ), let $m > 0$ be in kilograms, $q$ in coulombs, $c$ in metres per second, $E_{0}$ in newtons per coulomb, and $B_{0}$ in tesla. Use an inertial chart with $x^{0} = c t$ and all coordinates in metres, so $t$ is in seconds. Set $E = E_{0} e_{1}$ , $B = B_{0} e_{3}$ , $B_{0} \neq 0$ , and $∣ E_{0} ∣ < c ∣ B_{0} ∣$ . Use the smooth combined potential $ϕ = - E_{0} x^{1}$ , $A = (- B_{0} x^{2} / 2, B_{0} x^{1} / 2, 0)$ . Define

$v_{d} = \frac{E \times B}{∣ B ∣^{2}} = - \frac{E_{0}}{B_{0}} e_{2}, γ_{d} = (1 - ∣ v_{d} ∣^{2} / c^{2})^{- 1 / 2} .$

The drift velocity has units of metres per second and $γ_{d}$ is dimensionless.

The passive Lorentz boost of velocity $v_{d}$ transforms these fields to $E^{'} = 0$ , $B^{'} = B / γ_{d}$ . Consequently every flat-model solution in the laboratory frame is the inverse boost of the uniform-magnetic-field orbit of Exact relativistic helix in a uniform magnetic field with the boosted initial data. The laboratory coordinate time increases strictly and ranges over all real values along that inverse-boosted orbit. In particular, the constant velocity $v = v_{d}$ is an exact solution because $E + v_{d} \times B = 0$ . Other initial data give boosted helical orbits; the theorem does not identify every velocity with $v_{d}$ . The cases $∣ E_{0} ∣ = c ∣ B_{0} ∣$ and $∣ E_{0} ∣ > c ∣ B_{0} ∣$ are outside the statement.

Facts & Assumptions

Given: The adopted external-field test-particle and flat massive-particle models, the named inertial SI chart, $m > 0$ , constant $q$ , $c > 0$ , the constant perpendicular fields and their smooth combined potential, and ${A C}_{ω}$ for the canonical tangent-bundle smooth structure.

[F1]

Charged and spinning test-body quantities: the model has constant $m > 0, q$ , $P = m U$ , and the flat SI field components use $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ .

[A1]

External-field charged-particle action: model trajectories are stationary points of the external-charge action on the local potential domain.

[F2]

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

[F3]

Charged test-body mass shell and laboratory work: in an inertial flat chart the coordinate force is $d p / d t = q (E + v \times B)$ , and its local solutions are unique for given future-timelike initial data.

[A2]

Adopted massive-particle momentum, observer measurement and isolated-collision conservation: the flat 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]

Covariance, boosts and the two field invariants: under the passive boost of velocity $v$ , $E^{'} = γ_{v} (E + v \times B) - \frac{γ_{v}^{2}}{γ_{v} + 1} v (v \cdot E) / c^{2}$ and $B^{'} = γ_{v} (B - v \times E / c^{2}) - \frac{γ_{v}^{2}}{γ_{v} + 1} v (v \cdot B) / c^{2}$ .

[F5]

Exact relativistic helix in a uniform magnetic field: for a uniform magnetic field and zero electric field, initial position and momentum specified at coordinate time zero determine the stated global future-timelike helix (or its free degeneracy), unique on common connected time intervals.

[F6]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition: Lorentz boosts preserve the Minkowski timelike cone, and a future timelike chart velocity has magnitude below $c$ .

[F7]

The vector triple product identity $u \times (v \times w) = ⟨ u, w ⟩ v - ⟨ u, v ⟩ w$ : the vector triple product is $u \times (v \times w) = v (u \cdot w) - w (u \cdot v)$ .

[F8]

Adopted inertial covariance of the model equations: an inertial change of description is an affine isometry of Minkowski spacetime preserving the metric and future cone.

[F9]

The passive Lorentz boost with parameter beta: a passive boost has $x^{' 0} = γ (x^{0} - β \cdot x)$ and inverse $x^{0} = γ (x^{' 0} + β \cdot x^{'})$ .

[F10]

The cross product in $R^{3}$ : in the standard right-handed basis, $e_{2} \times e_{3} = e_{1}$ and $e_{1} \times e_{3} = - e_{2}$ .

[F11]

The passive boost is a proper orthochronous Lorentz map with inverse B(-beta): the passive Lorentz boost $B (β)$ is a proper orthochronous Lorentz map with inverse $B (- β)$ .

[F12]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : the standard spatial basis is orthonormal, so $e_{i} \cdot e_{j} = δ_{i j}$ .

Proof

1.1givenF1A1A2A3F2F3algebra

The scalar and vector potentials are smooth. By [F1], their tensor components are $F_{01} = - E_{0} / c$ , $F_{12} = B_{0}$ , and the remaining independent components vanish, so they give precisely the displayed fields. Thus [A1] applies and, under the explicit ${A C}_{ω}$ premise [A3], [F2] yields the covariant force equation. Under the flat massive-particle adoption [A2] and the same countable-choice condition, [F3] gives the coordinate Lorentz law in the named inertial chart.

2.1step 1.1F4F7F10F12algebra

The component definition [F10] gives $v_{d} = - E_{0} e_{2} / B_{0}$ , so $∣ v_{d} ∣ / c = ∣ E_{0} ∣ / (c ∣ B_{0} ∣) < 1$ . Orthogonality from [F12] gives $v_{d} \cdot E = v_{d} \cdot B = 0$ and, by the triple-product identity [F7], $v_{d} \times B = - E$ . Substitution in [F4] makes the electric parallel correction terms zero and gives $E^{'} = γ_{d} (E + v_{d} \times B) = 0$ . The same component definition gives $v_{d} \times E = (E_{0}^{2} / B_{0}) e_{3}$ , so the magnetic formula gives $B^{'} = γ_{d} (B_{0} - E_{0}^{2} / (B_{0} c^{2})) e_{3} = B / γ_{d}$ , using $1 - ∣ v_{d} ∣^{2} / c^{2} = γ_{d}^{- 2}$ .

3.1step 2.1A1A2A3F2F3F5F8algebra

Boost any laboratory initial event and future-timelike four-velocity into the primed chart; metric and future-cone preservation keep the transformed data future timelike by [F8]. Let the boosted initial event have coordinate time $t'_{0}$ and spatial position $x'_{0}$ , with spatial momentum $p'_{0}$ . The primed field is uniform magnetic by step 2.1. By [A2] and [F3], its coordinate equations are $d x' / d t' = c p' / \sqrt{m^{2} c^{2} + ∣ p' ∣^{2}}$ and $d p' / d t' = q (d x' / d t') \times B'$ ; their right sides depend on momentum and the constant field, not on time. Put $s = t' - t'_{0}$ . Apply [F5] at $s = 0$ to position $x'_{0}$ , momentum $p'_{0}$ and field $B'$ , obtaining its global orbit $(ξ (s), π (s))$ . Set $x' (t') = ξ (t' - t'_{0})$ and $p' (t') = π (t' - t'_{0})$ . The chain rule leaves those autonomous equations unchanged and gives precisely the required initial data at $t' = t'_{0}$ ; future timelikeness is unchanged. Conversely, translating any primed solution by the same constant time shift gives a solution with those data at $s = 0$ , so [F5] gives uniqueness on every common connected interval. The action in [A1] is unchanged by the boost: [F8] preserves the metric term, and pulling the one-form potential back by the coordinate change preserves its line integral. Thus stationary curves map to stationary curves, and under [A3], [F2] gives the covariant force equation in either chart. Applying the inverse Lorentz map therefore gives a laboratory worldline with the original initial data. Conversely, any laboratory solution transforms to a primed solution and coincides with the unique [F5] orbit; hence all laboratory solutions are exactly the inverse boosts of those magnetic orbits.

4.1step 3.1F5F6F9F11algebra

With $β = v_{d} / c$ , the inverse $B (- β)$ from [F11] applied to the time-coordinate formula [F9] gives $t = γ_{d} (t^{'} + v_{d} \cdot x^{'} / c^{2})$ . Differentiating along the magnetic-frame orbit gives $d t / d t^{'} = γ_{d} (1 + v_{d} \cdot v^{'} / c^{2})$ . Since its speed satisfies $∣ v^{'} ∣ < c$ by [F5, F6], this derivative is at least $γ_{d} (1 - ∣ v_{d} ∣ / c) > 0$ . The positive lower bound makes $t$ strictly increasing and unbounded in both directions as $t^{'}$ ranges over $R$ ; thus the inverse-boosted curve is defined as a laboratory-time orbit for all real $t$ .

5.1step 2.1step 3.1A3F3algebra∎

Finally, $v_{d} \times B = - E$ from step 2.1, so the laboratory force in [F3] under [A3] vanishes at constant $v = v_{d}$ . The corresponding momentum is constant and its speed is subluminal by step 2.1. This exhibits one exact drift solution; step 3.1 shows that general initial data instead produce inverse-boosted magnetic orbits. No assertion is made about null or supercritical fields.

Subcritical crossed fields and exact drift-frame orbits

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources