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Right-wedge accelerated clocks and rays — Examples

1 · Prerequisites

2 · Summary

These symbolic examples specialize the page's conditional results: a pair of stationary Rindler clocks and a radial ray; a particle released from rest in a uniform electric field; a planar circular orbit in a uniform magnetic field; and the exact constant drift solution in subcritical crossed fields. Parameters are chosen for calculation and do not represent reported measurements. The instantaneous Rindler frequency ratio uses the theorem's additional adopted local phase-reception rule and retains its local ray-regime and finite-apparatus qualifications.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

ExampleConstruction: AI-adaptedVerification: 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 →

Two stationary clocks and one radial Rindler ray

Physical scope

A symbolic specialization of the conditional right-wedge Rindler theorem. The numerical parameters are chosen model values, not measured quantities; the instantaneous received-frequency reading is conditional on the theorem's additional adopted local phase-reception model, with its local ray-regime and finite-apparatus qualifications and no numerical accuracy or empirical validation claim.

Example

Specialize Right-wedge Rindler clocks, radial rays and frequency ratio to $a = 2 m s^{- 2}$ , emission radius $ρ_{E} = 2 m$ , reception radius $ρ_{R} = 4 m$ , and a future outward radial ray. These are exact model parameters. The one-way coordinate travel time is $(c / 2 m s^{- 2}) \log 2$ , and over a shared Rindler-time interval the ideal-clock ratio is $Δ τ_{R} / Δ τ_{E} = 2$ . The instantaneous received-frequency ratio is $ω_{R} / ω_{E} = 1 / 2$ under the theorem's additional adopted local phase-reception rule, with its local ray-regime and short-observation qualifications for finite apparatus and no numerical error bound or empirical validation. No experiment is described.

Facts & Assumptions

Given: The parameter values above and the right-wedge definitions and model readings of the theorem.

[F1]

Right-wedge Rindler clocks, radial rays and frequency ratio: radial travel has $Δ T = (c / a) ∣ \log (ρ_{R} / ρ_{E}) ∣$ , stationary ideal-clock increments scale as $ρ$ , and the wave-covector contraction functional scales as $1 / ρ$ ; the instantaneous reception interpretation adopts the theorem's [A2] and retains all its qualifications.

Verification

1.1givenF1algebra

Substitute $a = 2 m s^{- 2}$ , $ρ_{R} / ρ_{E} = 2$ into the theorem's travel-time formula. Since both radii are positive and the ray is outward, $Δ T = (c / a) \log 2 = (c / 2 m s^{- 2}) \log 2$ , which is finite.

2.1step 1.1F1algebra∎

The clock-gradient formula gives $Δ τ_{R} / Δ τ_{E} = ρ_{R} / ρ_{E} = 4 m / (2 m) = 2$ . The contraction functional gives $ω_{R} / ω_{E} = ρ_{E} / ρ_{R} = 1 / 2$ . By [F1], the latter is the instantaneous received-frequency ratio within the additional adopted local phase-reception model, with its local ray-regime and finite-apparatus limits; no measured accuracy is asserted.

ExampleConstruction: AI-adaptedVerification: 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 →

Uniform electric field from rest

Physical scope

A symbolic specialization of the conditional constant-electric-field test-particle theorem with chosen SI model parameters. It is not a measurement or empirical claim.

Example

Choose $m = 1 k g$ , $q = 1 C$ , $E_{0} = 2 N C^{- 1}$ , $B = 0$ , and initial data $x_{0} = 0 m$ , $p_{0} = 0 k g m s^{- 1}$ at $t = 0$ in Exact relativistic orbit in a uniform electric field. Coordinate time $t$ is in seconds. Then $F_{0} = 2 N$ , $p_{1} (t) = F_{0} t$ , and for every finite $t$ the model orbit is

$x^{1} (t) = \frac{c}{F_{0}} (\sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}} - m c), x^{2} (t) = x^{3} (t) = 0,$

with $v_{1} (t) = c F_{0} t / \sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}}$ . This is a symbolic model calculation with chosen parameters, not a report of measured field or particle data.

Facts & Assumptions

Given: The chosen $m, q, E_{0}$ , zero initial position and momentum, and the adopted model assumptions in the theorem.

[F1]

Exact relativistic orbit in a uniform electric field: for $F_{0} = q E_{0} \neq 0$ , $Π = \sqrt{m^{2} c^{2} + ∣ p_{⊥} ∣^{2}}$ , $p_{1} = p_{01} + F_{0} t$ , and the exact position and velocity are the formulas stated there.

Verification

1.1givenF1algebra

Here $F_{0} = q E_{0} = (1 C) (2 N C^{- 1}) = 2 N \neq 0$ , $p_{01} = 0$ , $p_{⊥} = 0$ , and $Π = m c$ . Inserting these values into [F1] gives $p_{1} (t) = F_{0} t$ , zero transverse coordinates, and the displayed $x^{1} (t)$ .

2.1step 1.1F1algebra∎

Substitution in the velocity formula of [F1] gives $v_{1} = c F_{0} t / \sqrt{m^{2} c^{2} + F_{0}^{2} t^{2}}$ and $v_{2} = v_{3} = 0$ . At $t = 0$ , $x^{1} = 0$ and $v_{1} = 0$ ; for finite $t$ , $∣ v_{1} ∣ < c$ because the denominator is strictly larger than $∣ F_{0} t ∣$ .

ExampleConstruction: AI-adaptedVerification: 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 →

A circular magnetic orbit with zero axial momentum

Physical scope

A symbolic specialization of the conditional constant-magnetic-field test-particle theorem with chosen model parameters. It is not a measurement or empirical claim.

Example

In Exact relativistic helix in a uniform magnetic field, choose $q B_{0} > 0$ , $p_{01} = p_{*} > 0$ with units kilogram metre per second, $p_{02} = p_{03} = 0$ , and arbitrary $m > 0$ kilograms. Then $S_{0} = \sqrt{m^{2} c^{2} + p_{*}^{2}}$ has momentum units, $γ_{0} = S_{0} / (m c)$ is dimensionless and $Ω = q B_{0} / (γ_{0} m) > 0$ has units inverse seconds. The orbit is the circle

$x^{1} = x_{0}^{1} + \frac{c p_{*}}{S_{0} Ω} \sin (Ω t), x^{2} = x_{0}^{2} - \frac{c p_{*}}{S_{0} Ω} (1 - \cos (Ω t)), x^{3} = x_{0}^{3},$

of radius $p_{*} / (q B_{0})$ , with constant energy $c S_{0}$ . At $t = π / (2 Ω)$ its momentum is $(0, - p_{*}, 0)$ . All values are model parameters, not observations.

Facts & Assumptions

Given: The parameter restrictions above and the adopted magnetic test-particle model in the theorem.

[F1]

Exact relativistic helix in a uniform magnetic field: the momentum and position formulas have signed angular frequency $Ω = q B_{0} / (γ_{0} m)$ , constant energy $c S_{0}$ , and transverse radius $∣ p_{⊥} ∣ / ∣ q B_{0} ∣$ .

Verification

1.1givenF1algebra

With $p_{02} = p_{03} = 0$ and $Ω > 0$ , substitution into [F1] gives $p_{1} = p_{*} \cos (Ω t)$ , $p_{2} = - p_{*} \sin (Ω t)$ and $p_{3} = 0$ . Substitution into the position formulas gives the displayed coordinates and $x^{3} = x_{0}^{3}$ .

2.1step 1.1F1algebra∎

The radius clause of [F1] gives $∣ p_{⊥} ∣ / ∣ q B_{0} ∣ = p_{*} / (q B_{0})$ , and the energy is $c S_{0}$ . At $Ω t = π / 2$ , sine is $1$ and cosine is $0$ , so $p = (0, - p_{*}, 0)$ . The orbit is planar because $p_{03} = 0$ makes $x^{3}$ constant.

ExampleConstruction: AI-adaptedVerification: 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 →

The exact constant drift solution in subcritical crossed fields

Physical scope

A symbolic specialization of the subcritical crossed-field theorem. The chosen field parameters describe an exact model solution and are not measured values; general initial data still give inverse-boosted magnetic orbits.

Example

Let $B_{0} > 0$ be in tesla and set $E_{0} = c B_{0} / 2$ in newtons per coulomb, so $E = E_{0} e_{1}$ , $B = B_{0} e_{3}$ is subcritical. In Subcritical crossed fields and exact drift-frame orbits, the drift velocity and boost factor are $v_{d} = - (c / 2) e_{2}$ in metres per second and $γ_{d} = 2 / \sqrt{3}$ . The constant-velocity orbit $x (t) = x_{0} + v_{d} t$ in a chart with $x^{0} = c t$ solves the adopted model for any $m > 0$ and constant charge $q$ : its Lorentz force vanishes. This is one exact solution; other initial data yield boosted magnetic orbits.

Facts & Assumptions

Given: The chosen fields, positive $B_{0}$ , $m > 0$ , constant $q$ , and the subcritical external-field model of the theorem.

[F1]

Subcritical crossed fields and exact drift-frame orbits: for $∣ E_{0} ∣ < c ∣ B_{0} ∣$ , $v_{d} = - (E_{0} / B_{0}) e_{2}$ , $γ_{d} = (1 - ∣ v_{d} ∣^{2} / c^{2})^{- 1 / 2}$ , and $E + v_{d} \times B = 0$ ; general solutions are inverse boosts of magnetic orbits.

Verification

1.1givenF1algebra

The ratio $E_{0} / (c B_{0}) = 1 / 2$ verifies the strict subcritical hypothesis. Substituting $E_{0} = c B_{0} / 2$ in [F1] gives $v_{d} = - (c / 2) e_{2}$ and $γ_{d} = (1 - 1 / 4)^{- 1 / 2} = 2 / \sqrt{3}$ .

2.1step 1.1F1algebra∎

Directly, $v_{d} \times B = - (c B_{0} / 2) (e_{2} \times e_{3}) = - (c B_{0} / 2) e_{1} = - E$ . Thus [F1] makes the Lorentz force zero, momentum stays constant, and the straight worldline is an exact orbit. The theorem's inverse-boost construction applies to general initial data, so this example does not assert that every orbit has this constant velocity.

Right-wedge accelerated clocks and rays — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Two stationary clocks and one radial Rindler ray

Example

Facts & Assumptions

Verification

Uniform electric field from rest

Example

Facts & Assumptions

Verification

A circular magnetic orbit with zero axial momentum

Example

Facts & Assumptions

Verification

The exact constant drift solution in subcritical crossed fields

Example

Facts & Assumptions

Verification

Sources