Annus MirabilisEinstein in 1905
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special relativity

The equation explained

U2=(dxdt)2+(dydt)2,w2=wξ2+wη2\begin{aligned} U^2 &= \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2 , \\ w^2 &= w_\xi^2 + w_\eta^2 \end{aligned}
In wordsThe square of the resulting speed is the square of how fast the place along the motion changes with the stationary time, plus the square of how fast the place across it does; and the square of the speed in the moving system is the sum of the squares of its component along the motion and its component across.OverviewTwo names for what is about to be compared: the speed of the point as the stationary system reckons it, and the speed of the same point as the moving system reckons it. Each is built from its own components in the ordinary way.Full explanationThese are definitions, not results. UU is the speed of the point in the stationary system, got from the two rates at which its coordinates there change with the stationary time, and ww is the speed of the point in the moving system, got from the two constants of the first display. Each is formed by the ordinary rule for combining components at right angles, and each uses the coordinates and the time of one system only. That is the care the section takes: a quantity measured partly in one system and partly in another would mean nothing. The angle of the next display completes the description of ww, and then the two can be compared.Every stepU2=(dxdt)2+(dydt)2U^2 = \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2The speed in the stationary system, from the rates at which the two coordinates of that system change with its own time. The printed definition.w2=wξ2+wη2w^2 = w_\xi^2 + w_\eta^2The speed in the moving system, from the two constants of the equations of motion there. The second printed definition.dxdt=wξ+v1+vwξV2,dydt=1−(vV)21+vwξV2wη\frac{d x}{d t} = \frac{w_\xi + v}{1 + \frac{v w_\xi}{V^2}}, \quad \frac{d y}{d t} = \frac{\sqrt{1 - \left( \frac{v}{V} \right)^2}}{1 + \frac{v w_\xi}{V^2}} w_\etaThe two rates, read off the display before: both coordinates are constants times tt, so each rate is that constant.Also

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