Annus MirabilisEinstein in 1905
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special relativity

The equation explained

b′=bβ(1−avV),b' = \frac{b}{\beta \left( 1 - a \frac{v}{V} \right)} ,
In wordsThe direction cosine along the second axis in the moving system is the stationary one divided by the factor and by one minus the direction cosine along the motion times the speed over the velocity of light.OverviewThe second new direction cosine, for the axis η\eta across the motion. The motion does not act along this axis directly, but the cosine changes so that the three new cosines still describe a direction.Full explanationIt comes from the coefficient of η\eta in the phase written in kk's coordinates, divided by ω′V\frac{\omega'}{V}. The η\eta term keeps its coefficient, ωVb\frac{\omega}{V} b, but ω′\omega' is not ω\omega, so b′=bβ(1−avV)b' = \frac{b}{\beta \left( 1 - a \frac{v}{V} \right)}. For the wave crossing at right angles with b=1b = 1 and v=0.6 Vv = 0.6 \, V, b′=11.25=0.8b' = \frac{1}{1.25} = 0.8, and with a′=−0.6a' = -0.6 this gives 0.36+0.64=10.36 + 0.64 = 1, a direction again.Every stepω′=ωβ(1−avV)\omega' = \omega \beta \left( 1 - a \frac{v}{V} \right)The angular frequency in kk. In the phase written in kk's coordinates, η\eta appears only in the term −ωVbη-\frac{\omega}{V} b \eta, since y=ηy = \eta.ωVb=ω′Vb′\frac{\omega}{V} b = \frac{\omega'}{V} b'To write the phase as ω′\omega' times τ\tau less the new cosines' terms over VV, the coefficient of η\eta must equal ω′Vb′\frac{\omega'}{V} b'.b′=bβ(1−avV)b' = \frac{b}{\beta \left( 1 - a \frac{v}{V} \right)}Divide by ω′V\frac{\omega'}{V} and put in ω′\omega'.Also

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