Annus MirabilisEinstein in 1905
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special relativity

The equation explained

ν′=ν1−vV1+vV.\nu' = \nu \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .
In wordsThe frequency perceived by an observer moving straight away from the source is the source's frequency times the square root of this quotient: one minus the speed over the velocity of light, divided by one plus it.OverviewDoppler's principle for an observer moving along the line from the source, the case φ=0\varphi = 0: the frequency falls by a square-root factor as the observer recedes, and rises as it approaches.Full explanationWith φ=0\varphi = 0 the observer moves directly away from the source. Then 1−cos⁡φvV1 - \cos \varphi \frac{v}{V} is 1−vV1 - \frac{v}{V}, and dividing it by (1−vV)(1+vV)\sqrt{\left( 1 - \frac{v}{V} \right) \left( 1 + \frac{v}{V} \right)} leaves the perspicuous form, as the paper calls it. At v=0.6 Vv = 0.6 \, V the frequency is halved: 0.4/1.6=0.5\sqrt{0.4 / 1.6} = 0.5. With vv negative the observer approaches, and the frequency rises without bound as vv approaches −V-V. The paper points this out as contrary to the usual view.Every stepν′=ν1−cos⁡φvV1−(vV)2\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Doppler's principle for any angle, the previous display.ν′=ν1−vV1−(vV)2\nu' = \nu \frac{1 - \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}}Put φ=0\varphi = 0, so cos⁡φ=1\cos \varphi = 1.ν′=ν1−vV(1−vV)(1+vV)\nu' = \nu \frac{1 - \frac{v}{V}}{\sqrt{\left( 1 - \frac{v}{V} \right) \left( 1 + \frac{v}{V} \right)}}Factor the difference of two squares under the root.ν′=ν1−vV1+vV\nu' = \nu \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}}Cancel one square root of 1−vV1 - \frac{v}{V} against the numerator.Historian's marginThe sentence after this display is printed: for v=−∞v = -\infty, ν=∞\nu = \infty (p. 912). The source receipt records two suspected misprints in it: the equation gives an infinite perceived frequency as vv approaches −V-V, not −∞-\infty, since for vv beyond −V-V its root is not real (err-typo-p912-2); and the frequency that grows is the perceived ν′\nu', not the source's ν\nu (err-typo-p912-1). The faces keep the printed reading.Also

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