Annus MirabilisEinstein in 1905
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special relativity

The equation explained

A′′′=A′′1+vVcos⁡φ′′1−(vV)2=A1−2vVcos⁡φ+(vV)21−(vV)2,A''' = A'' \frac{1 + \frac{v}{V} \cos \varphi''}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} = A \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{1 - \left( \frac{v}{V} \right)^2} ,
In wordsThe amplitude of the reflected light in the stationary system is its amplitude in the mirror's system times one plus the speed over the velocity of light times the cosine of its angle there, divided by the square root of one minus the square of that ratio; in terms of the incident amplitude and the angle of incidence, this is one minus twice the ratio times the cosine plus its square, over one minus its square.OverviewThe third move: the reflected light's amplitude transformed back to the stationary system, and then written in terms of the incident light alone.Full explanationGoing back from kk to KK is the transformation with velocity −v-v, so § 7's amplitude formula applies with −v-v in place of vv and the angle measured in kk. Putting in the first two moves gives the reflected amplitude in terms of the incident amplitude and angle. A mirror receding from light that falls straight on it (φ=0\varphi = 0) returns light of amplitude A1−v/V1+v/VA \frac{1 - v/V}{1 + v/V}: at v=0.6 Vv = 0.6 \, V, a quarter of AA.Every stepA′′′=A′′1+vVcos⁡φ′′1−(vV)2A''' = A'' \frac{1 + \frac{v}{V} \cos \varphi''}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Back to KK: § 7's amplitude formula with −v-v for vv, and φ′′\varphi'' for the angle.1+vVcos⁡φ′′=1−vVcos⁡φ−vV1−vVcos⁡φ1 + \frac{v}{V} \cos \varphi'' = 1 - \frac{v}{V} \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi}Reflection gave cos⁡φ′′=−cos⁡φ′\cos \varphi'' = -\cos \varphi', and aberration gave cos⁡φ′\cos \varphi' in terms of cos⁡φ\cos \varphi.1+vVcos⁡φ′′=1−2vVcos⁡φ+(vV)21−vVcos⁡φ1 + \frac{v}{V} \cos \varphi'' = \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{1 - \frac{v}{V} \cos \varphi}Over the common denominator the numerator is 1−vVcos⁡φ−vVcos⁡φ+(vV)21 - \frac{v}{V} \cos \varphi - \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2.A′′′=A1−2vVcos⁡φ+(vV)21−(vV)2A''' = A \frac{1 - 2 \frac{v}{V} \cos \varphi + \left( \frac{v}{V} \right)^2}{1 - \left( \frac{v}{V} \right)^2}With A′′=A′=A1−vVcos⁡φ1−(vV)2A'' = A' = A \frac{1 - \frac{v}{V} \cos \varphi}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}, the factor 1−vVcos⁡φ1 - \frac{v}{V} \cos \varphi cancels and the two square roots multiply to 1−(vV)21 - \left( \frac{v}{V} \right)^2.Also

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