Annus MirabilisEinstein in 1905
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special relativity

The equation explained

X′=X0sin⁡Φ′,L′=L0sin⁡Φ′,Y′=β(Y0−vVN0)sin⁡Φ′,M′=β(M0+vVZ0)sin⁡Φ′,Z′=β(Z0+vVM0)sin⁡Φ′,N′=β(N0−vVY0)sin⁡Φ′,Φ′=ω′(τ−a′ξ+b′η+c′ζV),\begin{aligned} X' &= X_0 \sin \Phi' , &\qquad L' &= L_0 \sin \Phi' , \\ Y' &= \beta \left( Y_0 - \frac{v}{V} N_0 \right) \sin \Phi' , &\qquad M' &= \beta \left( M_0 + \frac{v}{V} Z_0 \right) \sin \Phi' , \\ Z' &= \beta \left( Z_0 + \frac{v}{V} M_0 \right) \sin \Phi' , &\qquad N' &= \beta \left( N_0 - \frac{v}{V} Y_0 \right) \sin \Phi' , \\ \Phi' &= \omega' \left( \tau - \frac{a' \xi + b' \eta + c' \zeta}{V} \right) , \end{aligned}
In wordsIn the moving system, each component of the electric force and of the magnetic force is again an amplitude times the sine of one phase: along the motion the stationary system's electric or magnetic amplitude itself; across it, the factor times one amplitude mixed with the other's, weighted by the speed over the velocity of light. The phase advances with the moving system's time at a new angular frequency and travels along its coordinates in a new direction.OverviewThe same wave as the moving system kk sees it: still a plane wave, with amplitudes transformed as § 6 says, and a phase that has its own frequency and its own direction in kk.Full explanationThe paper asks what these waves are like for an observer at rest in kk, and answers by applying two earlier results: § 6 for the forces and § 3 for the coordinates and time. It says the result follows directly, and shows none of the work. The forces transform component by component, and because every component carries the same factor sin⁡Φ\sin \Phi, the transformation acts on the amplitudes alone. The phase is a number attached to each event, the same whichever system computes it; written in kk's time and coordinates it takes the same form as before, ω′(τ−a′ξ+b′η+c′ζV)\omega' \left( \tau - \frac{a' \xi + b' \eta + c' \zeta}{V} \right), with the new frequency and direction cosines of the next four displays. So in kk too the light is a plane wave moving at the velocity VV.Every stepX′=X0sin⁡Φ,Y′=β(Y0−vVN0)sin⁡ΦX' = X_0 \sin \Phi, \quad Y' = \beta \left( Y_0 - \frac{v}{V} N_0 \right) \sin \PhiPut the wave of the first display into the transformation of § 6: X′=XX' = X, Y′=β(Y−vVN)Y' = \beta \left( Y - \frac{v}{V} N \right), and so on. The common factor sin⁡Φ\sin \Phi comes out of every bracket.t=β(τ+vV2ξ),x=β(ξ+vτ),y=η,z=ζt = \beta \left( \tau + \frac{v}{V^2} \xi \right), \quad x = \beta (\xi + v \tau), \quad y = \eta, \quad z = \zetaTo write the phase in kk's coordinates, express KK's time and coordinates through kk's: § 3's transformation run backwards, which is the same transformation with −v-v.ω(t−axV)=ωβ(1−avV)τ−ωVβ(a−vV)ξ\omega \left( t - \frac{a x}{V} \right) = \omega \beta \left( 1 - a \frac{v}{V} \right) \tau - \frac{\omega}{V} \beta \left( a - \frac{v}{V} \right) \xiSubstitute into the part of the phase that holds tt and xx, and collect. The terms in τ\tau are ωβτ−ωaβvVτ\omega \beta \tau - \omega a \beta \frac{v}{V} \tau; the terms in ξ\xi are ωβvV2ξ−ωaβVξ\omega \beta \frac{v}{V^2} \xi - \frac{\omega a \beta}{V} \xi. The other two parts, with byb y and the third cosine times zz, become the same with η\eta and ζ\zeta, because y=ηy = \eta and z=ζz = \zeta.Φ′=ω′(τ−a′ξ+b′η+c′ζV)\Phi' = \omega' \left( \tau - \frac{a' \xi + b' \eta + c' \zeta}{V} \right)Take out the factor ωβ(1−avV)\omega \beta \left( 1 - a \frac{v}{V} \right) and call it ω′\omega'. What stands before ξ\xi, η\eta and ζ\zeta inside the bracket are then the new direction cosines a′a', b′b', c′c' of the next displays. The paper names the phase written this way Φ′\Phi'.a′2+b′2+c′2=1a'^2 + b'^2 + c'^2 = 1A check the paper does not print: since the squares of the three old cosines add to one, the numerator (a−vV)2+(1−a2)(1−v2V2)\left( a - \frac{v}{V} \right)^2 + (1 - a^2) \left( 1 - \frac{v^2}{V^2} \right) equals (1−avV)2\left( 1 - a \frac{v}{V} \right)^2. So a′a', b′b', c′c' describe a direction, and the wave crosses kk at the velocity VV, as the principle of the constancy of the velocity of light requires.Also

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