Annus MirabilisEinstein in 1905
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special relativity

The equation explained

cos⁡φ′=−vV.\cos \varphi' = - \frac{v}{V} .
In wordsFor a ray at right angles to the motion in the stationary system, the cosine of its angle in the moving system is minus the speed over the velocity of light.OverviewThe simplest case of aberration: a ray that crosses the motion at right angles in the stationary system meets the moving observer at an angle whose cosine is −vV-\frac{v}{V}.Full explanationWith φ=π2\varphi = \frac{\pi}{2} the cosine of φ\varphi is zero, and the general law leaves cos⁡φ′=−vV\cos \varphi' = -\frac{v}{V}. For small vv the ray is tipped from the right angle by about vV\frac{v}{V} radians, the classical aberration angle. At v=0.6 Vv = 0.6 \, V the cosine is −0.6-0.6 and φ′\varphi' is about 126.87 degrees.Every stepcos⁡φ′=cos⁡φ−vV1−vVcos⁡φ\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi}The general law of aberration, the previous display.cos⁡π2=0\cos \frac{\pi}{2} = 0A ray at right angles to the motion in KK.cos⁡φ′=−vV\cos \varphi' = -\frac{v}{V}The numerator becomes −vV-\frac{v}{V} and the denominator 1.Also

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