The Doppler Shift

20. The Doppler Shift

Now we turn to another important property of light being the Doppler effect. The derivation is based simply on the constant speed of light. It is derived analogously to the Doppler effect in sound, except for the second order sine term which is a result of the photon's limited direction of propagation.

Imagine a point source of light moving at absolute velocity v_s. The front of the photon moves outward a distance l=ct. The source has moved a distance v_st and at this point emits the back of the photon. This situation is shown in the following diagram.

Fig.8: Doppler shifted wave length of photons.

Using the law of cosines we can write:

(1)

With the fact that t=l/c we can solve this for l' and obtain

(2)

This is the wavelength as seen from the absolute rest frame. This then yields

(3)

This is the frequency shift based on the velocity of the source. Traditionally the radical is not included as the wave front is coming from the retarded position of the particle. However, since we are considering photons, the direction is limited and the direction that the photon is aligned on completion of its emission is what is significant¹. It is thus in respect to the retarded position plus the position of the source at the completion of one wavelength of radiation. This yields the radical. The shift based on the velocity of the observer would be given by:

(4)

where q_o is the angle from directly away from the source, that is, if this were related to plane wave fronts as with sound waves. Here, however, it relates to photons. The length of the photon corresponds to the wave length of the emitted light. It turns out that there is a corresponding transverse shift for the observer. Imagine a photon being absorbed by a receiver which is moving transverse to the photon's path. The front of the photon makes contact, but then the receiver moves a distance transverse where the back of the photon is actually further away as the hypotenuse of a right triangle. So it takes longer for the photon to be absorbed, which equates to the frequency, thus the shift based on the velocity of the observer (for photons) is

(5)

We apply both of these factors to obtain the total Doppler shift:

(6)

When the observer is moving away or toward the source which is moving away or toward the observer (q_s=0,p/2; q_o=0,p/2), then

(7)

Note that if v_o=v_s, then f'=f.

1. This is not arbitrary. First, it is clear that a simple plane wave model of a point source cannot be used; Second, the photon is not formed instantaneously, but requires time to form, during which the source has moved to a new location; Third, due to the required symmetry (see Section 15), the direction of transport must coincide with the alignment of the front and back of the photon. A rigorous explanation would require the dynamics of the photon emission, which is discussed some in Section 23, but the complete process is not understood. I assume that in the formation, there is some asymmetry, but by some non-linear processes (the same or similar as those mentioned in Section 3), the photon transforms to a stable soliton immediately after emission.

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