21. Lowest Order Shifts

In the last section we found the Doppler in relation to two absolute velocities, but it is not very useful since in an experiment it would be the relative velocity between the source and the frame of reference of the lab, so in this section we will go through the somewhat tedious process of converting to the appropriate coordinates.

To understand the observable consequences of the Doppler shift, it will be useful to view it from the perspective of a lab frame, moving at the observer's velocity v and a source at a relative-to-the-lab velocity u which is at an angle f from the direction of the absolute velocity of the lab. The source will be observed at an angle q from the direction of the absolute velocity of the lab. First it will be useful to start the conversion with rectangular coordinates utilizing the following diagram.



Fig.9: Diagram used to convert the Doppler equation to general rectilinear coordinates.

This is done by the following substitutions:

(1ab)

(2ab)

(3ab)

(4ab)

We then have:

(5)

(6)

(7)

(8)

We now convert the position of observation to polar coordinates based on the angle q with substitutions:

(9)

This yields:

(10)

(11)

(12)

(13)

Now imagine that the observer is at rest with respect to a lab moving in the x-direction at velocity v and the source is moving relative to the lab with velocity u. We thus apply the following substitutions:

(14)

This yields:

(15)

(16)

(17)

(18)

Now we express u, the magnitude of relative velocity of the source in polar coordinates dependent on the angle f from the direction of the lab's velocity. This yields:

(19)

(20)

(21)

(22)

Now these can be substituted into Eq(20-7) to obtain:

(23)

Note that when u=0, f'=f. It will be further useful to expand this to second order. With the binomial expansion (v,u much less than c) this becomes

(24)

This is the equation for the Doppler shift to second order as seen in a moving lab frame with source relative velocity u which is moving an angle f from the direction of the absolute velocity of the lab and is being observed from an angle q from the direction of the absolute velocity of the lab. Thus when f-q=0 the observer is looking along the flight path (head on) of the source. When one is doing an actual measurement, however, the factor from Eq.(14-3) must be used to show the dynamic decrease in frequency. The same transformation can be done here to get

(25)

Then we can expand it to second order yielding:

(26)

Multiplying Eq.(24) by this results in

(27)

The corresponding wavelength is

(28)

This is the total shift that would be observed. Imagine that we are observing the wavelength of a source that is always moving toward the observer in the lab frame (f=q). Then

(29)

Note that a relative second order shift is observed independent of direction and the absolute second order is also independent of direction, so if the shift is measured in relation to a changing relative velocity, the lab velocity would not be observed¹. Imagine now that the photon is being absorbed by the observer. The allowable frequency of absorption is dependent on the observer's absolute velocity in the same way as it is the source's. Therefore dividing by the factor in Eq.(14-2) (and expanding to second order) given the velocity of the lab, Eq.(12) becomes

(30)

Note that the absolute term disappears and there is no transverse Doppler shift². It is rather extraordinary that while the effects are based in the absolute velocity of the observer and the source, with a dynamic frequency shift also based in absolute velocity, that the observable shifts are all essentially relative and directionally isotropic.

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1. A second order shift of this amount based on relative velocity (the third term) was verified by Ives and Stillwell²⁰. A second order shift of this amount based on the absolute velocity of the lab (the fourth term) was verified by Kennedy and Thorndike²¹, which will be further described in the next section.

2. The lack of a second order transverse shift was verified by Champeney and Moon²² using the Mössbauer effect. A second order shift was measured when the absorber and emitter were under different stress on a rotating arm (by Champeney, Isaak and Khan²³) I suspect that stress was the cause in correspondence to temperature dependent frequency shifts as shown by Pound and Rebka²⁴. Note that the stress depends on the rim velocity squared. The temperature effect includes sources moving in all directions, so the lack of transverse shift does not eliminate the effect. It should be noted that the second order shift for frequency is actually opposite to what was expected, but the experimental setups using the Mössbauer effect did not distinguish the sign, only the magnitude. If it is stress that is causing this shift then it can be verified by changing the radius of the arms in the experiment of Champeney, Isaak and Khan without changing the velocity at the rim.

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