Annus MirabilisEinstein in 1905
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special relativity

The equation explained

(x−Vat)2+(y−Vbt)2+(z−Vct)2=R2(x - V a t)^2 + (y - V b t)^2 + (z - V c t)^2 = R^2
In wordsThe points whose coordinates lie at a fixed distance, the radius, from a centre that moves with the velocity of light along the direction of the wave as the time runs: a sphere that travels with the light.OverviewA sphere that moves along with the light. No energy crosses its surface, so it always encloses the same portion of light, which the paper calls a light complex.Full explanationThe paper wants to compare the energy of a given portion of light in the two systems, and first needs to say which portion. The centre of this sphere is at (Vat,Vbt,Vct)(V a t, V b t, V c t): it moves with the velocity of light along the wave normal, whose direction cosines are aa, bb, cc, and every point of the surface moves with it. For the plane waves of § 7, whose energy also moves along the wave normal at the velocity of light, no energy passes through any element of such a surface, so it encloses the same light at every moment. The paper then asks how much energy this surface encloses when it is regarded from the moving system.Every step(Vat,Vbt,Vct)(V a t, V b t, V c t)The centre of the sphere at the time tt: it has travelled the distance VtV t along the direction with cosines aa, bb, cc, the direction the light travels.(x−Vat)2+(y−Vbt)2+(z−Vct)2=R2(x - V a t)^2 + (y - V b t)^2 + (z - V c t)^2 = R^2The points at the distance RR from that centre, by Pythagoras in three dimensions. As tt runs the whole sphere slides along with the light, keeping its radius.Also

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