Annus MirabilisEinstein in 1905
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Read · Special relativity: from clock operations to electrodynamics · Results

§7 · Doppler shift and aberration

Clocks, rods and light first, then electricity and magnetism: follow the relativity paper through its introduction and all ten sections, including the later ones most retellings skip.

The paper's results

Pages 911, 912

Doppler's principle and aberration for any velocity

An observer who moves with the velocity vv relative to a very distant source of frequency ν\nu, at the angle φ\varphi between the line from source to observer and the observer's velocity, perceives the frequency ν′=ν(1−cos⁡φ vV)/1−(v/V)2\nu'=\nu(1-\cos\varphi\,\frac{v}{V})/\sqrt{1-(v/V)^2} and finds the ray at the angle φ′\varphi' given by cos⁡φ′=(cos⁡φ−vV)/(1−vVcos⁡φ)\cos\varphi'=(\cos\varphi-\frac{v}{V})/(1-\frac{v}{V}\cos\varphi): Doppler's principle and the law of aberration for arbitrary velocities.

As printed

Aus der Gleichung für ω′\omega' folgt: Ist ein Beobachter relativ zu einer unendlich fernen Lichtquelle von der Frequenz ν\nu mit der Geschwindigkeit vv derart bewegt, daß die Verbindungslinie „Lichtquelle–Beobachter“ mit der auf ein relativ zur Lichtquelle ruhendes Koordinatensystem bezogenen Geschwindigkeit des Beobachters den Winkel φ\varphi bildet, so ist die von dem Beobachter wahrgenommene Frequenz ν′\nu' des Lichtes durch die Gleichung gegeben:

ν′=ν1−cos⁡φvV1−(vV)2.\nu' = \nu \frac{1 - \cos \varphi \frac{v}{V}}{\sqrt{1 - \left( \frac{v}{V} \right)^2}} .Frequency, moving systemFrequency, stationary systemAngle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation

Dies ist das Doppelersche Prinzip für beliebige Geschwindigkeiten. Für φ=0\varphi = 0 nimmt die Gleichung die übersichtliche Form an:

ν′=ν1−vV1+vV.\nu' = \nu \sqrt{\frac{1 - \frac{v}{V}}{1 + \frac{v}{V}}} .Frequency, moving systemFrequency, stationary systemSpeed of the moving systemSpeed of lightExplain this equation

Man sieht, daß — im Gegensatz zu der üblichen Auffassung — für v=−∞v = -\infty, ν=∞\nu = \infty ist.

cos⁡φ′=cos⁡φ−vV1−vVcos⁡φ.\cos \varphi' = \frac{\cos \varphi - \frac{v}{V}}{1 - \frac{v}{V} \cos \varphi} .Angle to the motion, moving systemAngle to the motion, stationary systemSpeed of the moving systemSpeed of lightExplain this equation

Diese Gleichung drückt das Aberrationsgesetz in seiner allgemeinsten Form aus.

Text on page 911 of the German source, § 7, paragraph 3 · Display on page 911 of the German source, § 7, display 7 · Text on pages 911–912 of the German source, § 7, paragraph 3 · Display on page 912 of the German source, § 7, display 8 · Text on page 912 of the German source, § 7, paragraph 3 · Display on page 912 of the German source, § 7, display 9 · Text on page 912 of the German source, § 7, paragraph 5

In modern notation

q:=ν′νq := \frac{\nu'}{\nu}
q=γ (1−vc cos⁡(φ))q = \gamma\,\left(1 - \frac{v}{c}\,\cos\left(\varphi\right)\right)
cos⁡(φ′)=cos⁡(φ)−vc1−vc cos⁡(φ)\cos\left(\varphi'\right) = \frac{\cos\left(\varphi\right) - \frac{v}{c}}{1 - \frac{v}{c}\,\cos\left(\varphi\right)}

Shown in modern letters; Einstein's are in §7 of the German source face.

  • Premise: The source is so distant that near the observer its light is represented, to a sufficient approximation, by a plane wave, as § 7 sets it up; the frequency and the direction follow from transforming that wave's phase and fields.
  • A comparison, not a premise: Stellar aberration had been observed since Bradley's report of 1729, and Ives and Stilwell measured the second-order Doppler shift in canal rays in 1938, reading it within Lorentz's theory. Both are measurements of the world, separate from this calculation.

Decoder

ν\nu, ν′\nu'
The frequency of the light in the system of the source, and the frequency the moving observer perceives.
φ\varphi
The angle between the line from source to observer and the observer's velocity, measured in the system at rest relative to the source. At φ=0\varphi=0 the observer moves away from the source along that line.
φ′\varphi'
The angle between the ray and the same line, as the moving observer measures it.
v=−∞v=-\infty, ν=∞\nu=\infty
As printed. The equation above it grows without bound as vv approaches −V-V, an observer approaching the source at nearly the speed of light, and has no real value beyond; the older relation ν′=ν(1−v/V)\nu'=\nu(1-v/V) grows without bound only as vv goes to −∞-\infty, which is the usual view the sentence sets itself against. So the sentence reads as v=−Vv=-V, and its ν\nu most likely as the perceived ν′\nu'. Every face keeps the printed reading; the correction is offered here only.

Try it

SR-09: Approach the source along the line of sight at 0.6 of the speed of light: by what factor does the perceived frequency grow? Choose the preset “Approaching light along axis at 0.6c” there.

SR-09: A ray at 90 degrees to the motion in the source's system: what frequency and what angle does the observer find? Choose the preset “Transverse ray at 90 deg, 0.6c” there.

SR-09: At the earth's speed in its orbit, how far is a star at right angles to the motion displaced? Choose the preset “Earth orbit aberration” there.

Where this stops: Light that crosses the motion at right angles in one frame does not, in general, do so in the other. No observer can move at the speed of light, so the formulas’ limit at v = c describes no frame. Read the argument

Argument
A consequence of the transformation of a plane wave's phase and fields
Model
Exact within the paper's electrodynamics, for a plane wave from a very distant source
Evidence
A derivation, not evidence
History
Doppler's principle and aberration were known before, and the paper names the first after Doppler; their forms for arbitrary velocities are derived in this paper

§7 · Doppler shift and aberration

One phase fixes frequency and direction

Rewriting the phase of a light wave in the moving observer’s coordinates gives the new frequency and the new direction in one step. At 0.6c, light travelling the same way as the observer has half its frequency in the new frame; light that crosses the motion at right angles in the original frame has 1.25 times its frequency.

Read the argument