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Worked aberration and Doppler shift at beta = 3/5 and cos theta = 4/5

Physical scope

Worked numerical aberration and Doppler computation of the ideal plane-wave model at relative speed (3/5)c and source angle cosine 4/5. Conditional on the model; no measurement, statistical inference or empirical claim.

Example

Worked example in the model of sr-light-and-doppler, on the page sr-light-and-doppler-examples. Let $x$ and $x^{'}$ be inertial charts with $x^{'} = B (β) x + b$ , $β = (3 / 5) e_{1}$ and $γ = 5 / 4$ , and let an ideal plane wave have frame- $x$ propagation direction $s = (4 / 5, 3 / 5, 0)$ , so $\cos θ = 4 / 5$ and $\sin θ = 3 / 5$ with $θ \in [0, π]$ the angle between $s$ and the boost axis. Then Doppler shift and aberration for an ideal plane wave between inertial descriptions gives $s^{'} = (5 / 13, 12 / 13, 0)$ , hence $\cos θ^{'} = 5 / 13$ , and the angular frequency ratio $ω^{'} / ω = γ (1 - β s_{1}) = (5 / 4) (1 - 12 / 25) = (5 / 4) (13 / 25) = 13 / 20$ , so $ν^{'} = (13 / 20) ν$ and $λ^{'} = (20 / 13) λ$ . Define the apparent source direction here to mean the incoming unit direction $- s$ (and $- s^{'}$ in $x^{'}$ ); this is a direction convention, not a source or detector model. Since $5 / 13 < 4 / 5$ (as $25 < 52$ ), its axis component moves from $- s_{1} = - 4 / 5$ to $- s_{1}^{'} = - 5 / 13$ measured along the boost axis, that is, toward the direction of motion of the observer's frame: this is the sign of stellar aberration in this configuration. The example records the exact direction transformation and leaves any observational interpretation to the framework's experimental work. Scope: a conditional computation in the ideal plane-wave model; no measurement and no empirical claim.

Facts & Assumptions

Given: Inertial charts $x, x^{'}$ with $x^{'} = B (β) x + b$ , $β = (3 / 5) e_{1}$ ; an ideal plane wave with frame- $x$ direction $s = (4 / 5, 3 / 5, 0)$ and angular frequency $ω > 0$ .

[A1]

Wave data and readings: the rest observers of $x$ and $x^{'}$ read the frequencies $ω$ and $ω^{'}$ , and the linear frequency and wavelength satisfy $ν = ω / (2 π)$ , $λ = c / ν$ (Null wave covectors, frame frequency data and the observer frequency of an ideal plane wave); the direction and frequency transformation used below is the theorem's (Doppler shift and aberration for an ideal plane wave between inertial descriptions).

[L1]

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

[L2]

Aberration clause: for $β \neq 0$ , with $θ, θ^{'}$ the angles between $s, s^{'}$ and the boost axis, $\cos θ^{'} = (\cos θ - ∣ β ∣) / (1 - ∣ β ∣ \cos θ)$ (Doppler shift and aberration for an ideal plane wave between inertial descriptions, clause (v)).

[L3]

$γ = (1 - ∣ β ∣^{2})^{- 1 / 2}$ , so for $β = (3 / 5) e_{1}$ : $γ = (16 / 25)^{- 1 / 2} = 5 / 4$ and $∣ β ∣ = 3 / 5$ ; the boost transformation of the frame data is the one of Boost transformation of a future null wave covector and of its frame data (The passive Lorentz boost with parameter beta).

Verification

technique · direct

1.1A1L1L2L3

The transformed direction. Here $s = (4 / 5, 3 / 5, 0)$ has $∣ s ∣^{2} = 16 / 25 + 9 / 25 = 1$ and $s_{1} = 4 / 5$ ; the axis formulas of [L1] with $b = 3 / 5$ and $γ = 5 / 4$ [L3] give $s_{1}^{'} = (4 / 5 - 3 / 5) / (1 - (3 / 5) (4 / 5)) = (1 / 5) / (13 / 25) = 5 / 13$ , $s_{2}^{'} = (3 / 5) / [(5 / 4) (13 / 25)] = (3 / 5) / (13 / 20) = 12 / 13$ and $s_{3}^{'} = 0$ , so $s^{'} = (5 / 13, 12 / 13, 0)$ with $∣ s^{'} ∣^{2} = 25 / 169 + 144 / 169 = 1$ ; the same value follows from the aberration clause [L2], $\cos θ^{'} = (4 / 5 - 3 / 5) / (1 - (3 / 5) (4 / 5)) = (1 / 5) / (13 / 25) = 5 / 13$ .

1.2A1L1L3

Frequency and wavelength. By [L1] with $β \cdot s = (3 / 5) (4 / 5) = 12 / 25$ : $ω^{'} / ω = γ (1 - 12 / 25) = (5 / 4) (13 / 25) = 13 / 20$ , so $ν^{'} = ω^{'} / (2 π) = (13 / 20) ν$ and $λ^{'} = c / ν^{'} = (20 / 13) λ$ by [A1].

2.1step 1.1step 1.2L2∎

The apparent direction. By the incoming-direction convention in the Example, the apparent source direction is $- s$ in $x$ and $- s^{'}$ in $x^{'}$ ; its components along the boost axis $β_{h a t} = e_{1}$ are $- s_{1} = - 4 / 5$ and $- s_{1}^{'} = - 5 / 13$ by step 1.1. Since $5 / 13 < 4 / 5$ (as $25 < 52$ ), also $- 5 / 13 > - 4 / 5$ : the axis component of the apparent direction increases, so the apparent direction moves toward $+ e_{1}$ , the direction of motion of the frame $x^{'}$ relative to $x$ , which is the sign of stellar aberration in this configuration. The computation is conditional on the ideal plane-wave model, records the exact direction transformation, and makes no measurement or empirical claim; no choice principle is used.

Worked aberration and Doppler shift at beta = 3/5 and cos theta = 4/5

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

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