Annus MirabilisEinstein in 1905
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special relativity

The equation explained

τ=φ(v)β(t−vV2x),ξ=φ(v)β(x−vt),η=φ(v)y,ζ=φ(v)z,\begin{aligned} \tau &= \varphi(v) \beta \left( t - \frac{v}{V^2} x \right) , \\ \xi &= \varphi(v) \beta (x - v t), \\ \eta &= \varphi(v) y, \\ \zeta &= \varphi(v) z, \end{aligned}
In wordsThe time of the moving system and its three coordinates, each as a function of the speed still unknown times what the transformation gives: for the time and the first coordinate, the factor times the time and the first coordinate of the stationary system taken together with the speed of the moving system and the velocity of light; across the motion, the coordinate unchanged.OverviewThe transformation, assembled. Each line carries one function of the speed that is still unknown; apart from it, these are the equations the section is looking for.Full explanationThe four lines before this one are written in the auxiliary coordinate the section introduced for the distance from the origin of the moving system, whose value is x−vtx - v t. Substituting that value turns them into equations in the coordinates and the time of the stationary system, and the algebra collects the coefficients into the factor β\beta defined just below. In the same step the paper replaces the coefficient aa, which was still unknown, by a function that is also still unknown: φ(v)\varphi(v) is aβa \beta, and the paper calls it a provisionally unknown function of vv. Nothing has been assumed about it. The shape of the equations is now fixed, and only that one factor, common to all four lines, is left to determine; § 3 does that on p. 902 and finds it to be 1.Every stepτ=a(t−vV2−v2(x−vt))=aV2V2−v2(t−vV2x)\tau = a \left( t - \frac{v}{V^2 - v^2} (x - v t) \right) = a \frac{V^2}{V^2 - v^2} \left( t - \frac{v}{V^2} x \right)The paper's instruction is to put in for the auxiliary coordinate its value, which is x−vtx - v t. Do that and collect. The terms in tt give t(1+v2V2−v2)=tV2V2−v2t \left( 1 + \frac{v^2}{V^2 - v^2} \right) = t \frac{V^2}{V^2 - v^2}, and the term in xx is −vxV2−v2- \frac{v x}{V^2 - v^2}; taking V2V2−v2\frac{V^2}{V^2 - v^2} out in front leaves t−vV2xt - \frac{v}{V^2} x.V2V2−v2=11−(vV)2=β2,VV2−v2=β\frac{V^2}{V^2 - v^2} = \frac{1}{1 - \left( \frac{v}{V} \right)^2} = \beta^2 , \qquad \frac{V}{\sqrt{V^2 - v^2}} = \betaDivide numerator and denominator by V2V^2. The coefficient in the first two lines is β2\beta^2 and the one in the transverse lines is β\beta, with β\beta as the display below defines it.τ=aβ2(t−vV2x),ξ=aβ2(x−vt),η=aβy,ζ=aβz\tau = a \beta^2 \left( t - \frac{v}{V^2} x \right) , \quad \xi = a \beta^2 (x - v t) , \quad \eta = a \beta y , \quad \zeta = a \beta zThe four lines with the coefficients written in β\beta. The first two carry β2\beta^2 and the transverse two carry β\beta.φ(v)=aβ⟹aβ2=φ(v)β,aβ=φ(v)\varphi(v) = a \beta \quad \Longrightarrow \quad a \beta^2 = \varphi(v) \beta , \quad a \beta = \varphi(v)The paper writes the one unknown that is left as φ(v)\varphi(v). With that name the first two lines read φ(v)β\varphi(v) \beta and the transverse two read φ(v)\varphi(v), which is the display as printed.Historian's marginThe paper adds a qualification under this display that is easy to pass over: if no assumption at all is made about the initial position of the moving system and about the zero of τ\tau, an additive constant is to be added on the right of each of these equations (p. 900). The equations as printed already carry the choice that the origins meet and that τ\tau and tt are zero together. What is still open at this point is the common factor, and § 3 spends the next two pages on it.Also

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