Theory of Doppler’s Principle and of Aberration

In the system K, very far from the origin of co-ordinates, let there be a source of electrodynamic waves, which in a part of space containing the origin of co-ordinates may be represented to a sufficient degree of approximation by the equations

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Here (X0 , Y0 , Z0 ) and (L0 , M0 , N0 ) are the vectors defining the amplitude of the wave-train, and l, m, n the direction-cosines of the wave-normals. We wish to know the constitution of these waves, when they are examined by an observer at rest in the moving system k. Applying the equations of transformation found in § 6 for electric and mag- netic forces, and those found in § 3 for the co-ordinates and the time, we obtain directly

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15 From the equation for ω 0 it follows that if an observer is moving with velocity v relatively to an infinitely distant source of light of frequency ν, in such a way that the connecting line “source-observer” makes the angle φ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency ν 0 of the light perceived by the observer is given by the equation

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This is Doppler’s principle for any velocities whatever. When φ = 0 the equation assumes the perspicuous form

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We see that, in contrast with the customary view, when v = −c, ν 0 = ∞. If we call the angle between the wave-normal (direction of the ray) in the moving system and the connecting line “source-observer” φ0 , the equation for l0 assumes the form

cos φ0 = cos φ − v/c 1 − cos φ · v/c

This equation expresses the law of aberration in its most general form. If φ = .. the equation becomes simply cos φ0 = −v/c.

We still have to find the amplitude of the waves, as it appears in the moving system. If we call the amplitude of the electric or magnetic force A or A0 respectively, accordingly as it is measured in the stationary system or in the moving system, we obtain

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which equation, if φ = 0, simplifies into

..

It follows from these results that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.