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Counterexample: a transverse wave configuration need not have zero Doppler shift

Physical scope

Counterexample to a universal transverse-Doppler claim within the ideal plane-wave model of flat classical special-relativistic spacetime at relative speed (3/5)c. Conditional on the model; no measurement, statistical inference or empirical falsification.

Statement refuted

The following universal claim is refuted: "if the propagation direction of an ideal plane wave is perpendicular to the relative velocity of two inertial frames, then the frequency read by the observer at rest in one frame equals the frequency read by the observer at rest in the other, $ω^{'} = ω$ ."

Facts & Assumptions

Given: Inertial charts $x$ and $x^{'}$ with $x^{'} = B (β) x + b$ , $β = (3 / 5) e_{1}$ , and an ideal plane wave with frame- $x$ data $(ω, s)$ , $ω > 0$ , $∣ s ∣ = 1$ , in the ideal plane-wave model.

[A1]

Adopted ideal plane-wave model: the wave is a dimensionless affine phase with future null wave covector, and the rest observers of $x$ and $x^{'}$ read $ω$ and $ω^{'}$ as defined in the model (Adopted ideal plane-wave and frequency-reading model, Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave).

[L1]

For the chart pair of the Given, the theorem gives $ω^{'} = γ (1 - β \cdot s) ω > 0$ with $γ = (1 - ∣ β ∣^{2})^{- 1 / 2}$ , and in the axis case $β = b e_{1}$ : $s_{1}^{'} = (s_{1} - b) / (1 - b s_{1})$ , $s_{2}^{'} = s_{2} / [γ (1 - b s_{1})]$ , $s_{3}^{'} = s_{3} / [γ (1 - b s_{1})]$ (Doppler shift and aberration for an ideal plane wave between inertial descriptions, clause (i)).

[L2]

Transverse protocols of the theorem: if $β \cdot s = 0$ then $ω^{'} = γ ω$ ; if $β \cdot s = ∣ β ∣^{2}$ , equivalently $s^{'} \cdot β_{h a t} = 0$ for $β \neq 0$ , then $ω^{'} = ω / γ$ ; in both cases $s$ , $s^{'}$ are unit and the shift is not one when $β \neq 0$ (Doppler shift and aberration for an ideal plane wave between inertial descriptions, clause (iv)).

Counterexample

technique · direct

1.1A1L2algebra

First witness. With $β = (3 / 5) e_{1}$ one has $∣ β ∣^{2} = 9 / 25$ and $γ = (1 - 9 / 25)^{- 1 / 2} = (16 / 25)^{- 1 / 2} = 5 / 4$ . Take $s = e_{2}$ ; then $β \cdot s = (3 / 5) (0) = 0$ , so the propagation direction is perpendicular to the relative velocity in the frame $x$ , and [L2] gives $ω^{'} = γ ω = (5 / 4) ω \neq ω$ because $ω > 0$ . Thus the antecedent of the refuted universal claim holds in $x$ while its conclusion fails.

1.2A1L1L2algebra

Second witness. Take $s = (3 / 5, 4 / 5, 0)$ ; then $∣ s ∣^{2} = 9 / 25 + 16 / 25 = 1$ and $β \cdot s = (3 / 5) (3 / 5) = 9 / 25 = ∣ β ∣^{2}$ , so by [L2] $s^{'} \cdot β_{h a t} = 0$ , the propagation direction is perpendicular to the relative velocity in the frame $x^{'}$ , and [L2] gives $ω^{'} = γ (1 - 9 / 25) ω = (5 / 4) (16 / 25) ω = (4 / 5) ω \neq ω$ . Explicitly, the axis formulas of [L1] give $s_{1}^{'} = (3 / 5 - 3 / 5) / (1 - 9 / 25) = 0$ , $s_{2}^{'} = (4 / 5) / [(5 / 4) (16 / 25)] = (4 / 5) / (4 / 5) = 1$ and $s_{3}^{'} = 0$ , so $s^{'} = e_{2}$ .

2.1step 1.1step 1.2A1∎

Conclusion. Each witness satisfies the antecedent of the refuted claim — transversality in $x$ in step 1.1 and transversality in $x^{'}$ in step 1.2 — and yields $ω^{'} \neq ω$ ; the two factors $(5 / 4)$ and $(4 / 5)$ differ and neither equals $1$ for the stated nonzero $β$ , so the universal claim is false in the ideal model, and the direction and size of the shift depend on the protocol (the frame in which transversality is required). The counterexample is conditional on the adopted idealizations, performs no measurement and claims no empirical falsification; no choice principle is used.

Counterexample: a transverse wave configuration need not have zero Doppler shift

Statement refuted

Facts & Assumptions

Counterexample

Depends on

Dependency tree · 2 levels

Sources