Annus MirabilisEinstein in 1905
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special relativity

The equation explained

ω′=ωβ(1−avV),\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right) ,
In wordsThe angular frequency in the moving system is the angular frequency in the stationary system times the factor times one minus the direction cosine of the wave along the motion times the speed over the velocity of light.OverviewHow fast the wave oscillates in the moving system. It depends on how the wave's direction lies against the motion: lower for a wave running the same way as kk, higher for a wave running against it.Full explanationThe factor ω′\omega' is what stands before τ\tau when the phase is written in kk's time and coordinates. The cosine aa says how the wave's direction lies against the motion: a=1a = 1 for a wave travelling along +x+x, the direction kk moves, a=−1a = -1 for one travelling against it, and a=0a = 0 for one crossing the motion at right angles. For a=1a = 1 and v=0.6 Vv = 0.6 \, V, ω′=ω⋅1.25⋅0.4=0.5 ω\omega' = \omega \cdot 1.25 \cdot 0.4 = 0.5 \, \omega; for a=−1a = -1, ω′=2 ω\omega' = 2 \, \omega. Even at a=0a = 0 the frequency changes, to βω\beta \omega, a change of second order in vV\frac{v}{V} that the first-order formula ω′=ω(1−avV)\omega' = \omega \left( 1 - a \frac{v}{V} \right) does not show. The paper draws the consequences for the frequency an observer perceives in the next paragraph.Every stepω(t−axV)=ωβ(1−avV)τ−ωVβ(a−vV)ξ\omega \left( t - \frac{a x}{V} \right) = \omega \beta \left( 1 - a \frac{v}{V} \right) \tau - \frac{\omega}{V} \beta \left( a - \frac{v}{V} \right) \xiThe phase of the wave written in kk's time and coordinates (the working of the previous display). The time appears only in this part of it.ω′=ωβ(1−avV)\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right)The coefficient of τ\tau is how fast the phase advances in kk's time, the angular frequency in kk.a=1:ω′=ωβ(1−vV)a = 1: \quad \omega' = \omega \beta \left( 1 - \frac{v}{V} \right)For a wave running the same way as kk: at v=0.6 Vv = 0.6 \, V, β=1.25\beta = 1.25 and 1−0.6=0.41 - 0.6 = 0.4, so ω′=0.5 ω\omega' = 0.5 \, \omega.Historian's marginModern texts write this as the relativistic Doppler factor, with the letter gamma for the paper's β\beta and the letter beta for the ratio of the speed to the velocity of light (the notation concordance; the paper, p. 911).Also

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