False Doppler

False Doppler refers to the actual Doppler effect that occurs when the distance between emitter and receiver is increasing but the incoming signal is perpendicular to the receiver's path with respect to the receiver's rest frame. The term is not used anywhere except in this Wikipedia article.

In all theories in which a signal (e.g., light or sound) propagates at a finite speed, the angle of incidence of the signal depends on the system of reference. The variation in angle as a function of the system of reference is called aberration. One result of aberration is that when a ray is received at 90 degrees with respect to the co-moving frame of the receiver, the receiver is not necessarily moving purely transversely to the transmitter. The same applies to the angle of the signal emerging from the transmitter.

For example, if a passenger in a moving car shoots a machine gun out the window perpendicular to the path of the car, the bullets will be moving slightly forward so they are not moving perpendicular to the car's path relative to the ground's frame of reference. In order to have the bullets actually moving perpendicular to the car's path with respect to the ground's frame, the passenger must aim the gun slightly to the rear. As a result, the source is actually receeding from the stationary target, so the sequence of bullets will be slightly "red-shifted" compared with the frequency of firing. (Each bullet has to move a slightly longer distance as the car moves further forward of the target).

This should not be confused with the transverse Doppler effect in special relativity, which occurs when the distance between the emitter and receiver is actually not changing, i.e., it is a truly transverse effect, due to time dilation. This is a uniquely relativistic effect, which has nothing to do with combined effects of classical Doppler and aberration discussed in this article.

Example

The shift in frequency referred to as "false Doppler" is a simple consequence of the ordinary Doppler effect and aberration. Suppose a transmitter is at rest with respect to a reference frame in terms of which the signal speed |c| is the same in all directions, and suppose a receiver is moving with velocity v such that the vector c - v is perpendicular to v, where c is the velocity vector of the incident signal at the receiver. Thus we have the dot product (c-v)*v = 0, which implies c*v = |v|^2, and therefore |c||v|cos(alpha) = |v|^2 where alpha is the angle between the vectors c and v. Consequently, cos(alpha) = |v|/|c|. The classical formula for the Doppler shift is $f'/f=1-(|v|/|c|)\cos(\alpha ),\,$

so in this condition we have f'/f = 1 - (v/c)^2.

Comparison with special relativity

According to special relativity the Doppler shift in the situation described above (assuming the signal is light in a vacuum) is:

[1 - (v/c) cos(alpha)]/sqrt(1 - (v/c)^2) 

where alpha is evaluated with respect to the rest frame of the emitter, or equivalently it is

sqrt(1 - (v/c)^2)/[1 + (v/c)cos(alpha)] where alpha is evaluated with respect to the receiver.

In the first case we have cos(alpha) = v/c, so the Doppler shift is simply sqrt(1 - (v/c)^2), and in the second case we have cos(alpha)=0 and again the Doppler shift is sqrt(1 - (v/c)^2). This result can also be computed by dividing the classical Doppler shift by the time dilation factor, which again gives sqrt(1 - (v/c)^2).

Of course, in this condition the receiver is not moving purely transversely to the emitter in terms of the emitter's rest frame. In the emitter's rest frame the true transverse condition occurs when the angle between c and v is 90 degrees with respect to the emitter's frame. In this case the pre-relativistic theory predicts no Doppler effect at all, whereas special relativity predicts the transverse Doppler effect equal to f'/f = 1/sqrt(1 - (v/c)^2), which is the inverse of the effect in the receiver's transverse condition.

Different theories

The classical Doppler effect, and the classical aberration effect, apply to all theories of signal propagation, provided only that the speed of signal propagation is finite. In special relativity there is another factor, called transverse Doppler, which arises due to the effect of time dilation, even when the distance between the source and the receiver is not changing. For example, a receiver moving in a circle around the source would not expect to observe any Doppler shift according to pre-relativistic theories, but there would be a frequency shift in special relativity due to the time dilation of the orbiting receiver. However, it's worth noting that, in classical as well as relativistic theories, the perceived angle of incidence would undergo aberration. This is why (for example) the Sun's apparent position in the sky is not the same as it's actual position, because the Earth intercepts the Sun's radial rays with a large tangential velocity.

Transverse redshifts: uniqueness to special relativity

In modern physics, the concept of "aberration redshift" is unhelpful, misleading and unnecessarily confusing, because it is nothing other than the normal Doppler redshift of a signal that is also undergoing normal aberration. There is no non-classical effect involved. It only requires treating the vectors of the transmitter, receiver, and signal consistently. In contrast, the theory of special relativity introduced a different effect, called transverse Doppler. This is a red-shift that exists even when the distance between the source and the receiver is not changing. It is a uniquely relativistic effect, due to time dilation (and has nothing to do with the subject of this article).