False Doppler

False Doppler refers to an effect in some theories, preceding special relativity, whereby a transversely-moving object should appear to be redshifted, even though its distance from the observer is not obviously changing. It may be more usefully referred to as aberration redshift.

Explanation

Aberration can alter the angles of rays of light from a moving object. In physics theories that allow some relationship between emitter speed and signal speed, aberration will normally cause rays leaving a moving object to have a forward deflection.

If we draw a line on a laboratory floor marking the path along which an emitting object will move, draw a second line intersecting it at 90 degrees, and then calibrate a detector to only register light entering the detector along that line, then a lightray that the object emits at 90 degrees, arriving in the lab frame with a forward slant, will not enter this detector. The ray that will enter the detector will have originally been aimed slightly more to the rear, and will contain a certain amount of recession redshift as a result of that original recession velocity component.

Worked example (emission theory)

The effect is probably easiest to calculate for the case of a presumed light-medium stationary in the emitter’s frame (or for simple emission theory, which has the same relationships). To calculate the visible redshift at our detector, we can (a) calculate the aberration angle for a ray received at 90 degrees in the lab frame, (b) calculate the longitudinal velocity component for the ray at its original angle, and then (c) apply the appropriate longitudinal Doppler relationship to that component.

(a) Working in the emitter’s frame: if the speed of light is c meters per second, and the detector moves at v m/s while intercepting a ray that deviates from 90 degrees by the angle A, then if light happens to take one second to reach our moving detector, the diagonal path of the received ray is a hypoteneuse of length c, the baseline of the triangle parallel to the direction of motion has length v (the distance that the detector moved in that time), and we get the relationship:

$\sin(A)=v/c.\,$

(b) The lightsignal takes a finite amount of time to reach the detector, and during that time the detector is described as having moved on from its “true” transverse position, so that it now has a slight recession component with respect to the emitter, with this recession speed w being:

$w=\sin(A).\,$

Replacing sin(A) with v/c then gives

$w=v\cdot v/s\,$

$w=v^{2}/s.\,$

(c) Finally, applying the appropriate Doppler relationship for the model to calculate the amount of recession redshift that the emitter thinks the detector ought to see, gives a predicted frequency change in the ray of

$f'/f=(c-w)/c=1-w/c\,$

$f'/f=1-v^{2}/c^{2}.\,$

"Sanity check": Comparison with special relativity

Where special relativity says that a detector aimed at 90 degrees to an object’s path in the lab frame will report a Lorentz redshift, of $f'/f={\sqrt {1-v^{2}/c^{2}}}$, the previous calculations say that the corresponding emission theory prediction is stronger, a Lorentz-squared redshift. This result already exists as part of the structure of special relativity: since the special theory allows us to calculate an agreed final physical result by assuming that light “really” propagates in any legal inertial frame, we must be able to calculate the usual SR transverse redshift result by declaring that light “really” propagates in the emitter’s frame, and that the “moving” observer sees light from the emitter to be Lorentz blueshifted due to their own time dilation. So, if the emitter-frame propagation effects predict an (unspecified) change in visible frequency, of ratio f’/f = X, and multiplying this with a Lorentz blueshift must give the appropriate SR “Lorentz redshift” outcome, then X must be a prediction of a Lorentz-squared redshift, as calculated.

Different theories

Most theories of light that incorporate at least some sort of partial source dependency will predict some sort of aberration redshift effect, typically somewhere between zero and Lorentz-squared.

The effect also arises in acoustics - a highly directional microphone pointed at right angles to its direction of motion and moving through still air should receive signals from stationary objects with a frequency shift of f ’/f = 1 − v²/s², where s is the speed of sound.

Transverse redshifts: uniqueness to special relativity

In modern physics, the concept of aberration redshift is likely to be considered unhelpful, misleading and unnecessarily confusing: since the concept of a transverse redshift effect is considered unique to SR (and equivalent theories), it is considered desirable to avoid methods that produce descriptions of “apparent” transverse redshifts in other theories.

This is normally achieved by specifying that for each theory being considered, angles should only be specified and calibrated in the frame in which a signal is presumed to propagate (if there is one). In emission theory, a detector tilted to intercept a ray aimed at 90 degrees to the direction of relative motion should report no redshift.

Applying this protocol, although a range of older theories of light will agree that a detector aimed at “90 degrees” in the lab frame will report a redshift, it is only under special relativity that we would consider the detector to be truly aimed “transversely”.

Quotes

“ … Doppler effect caused by motion of the observer is … a case of common aberration.”

Oliver Lodge, “Aberration Problems,” Phil.Trans.Roy.Soc. (1893) sections 56-57.

“ … a spurious or apparent Doppler effect.”

Oliver Lodge, The Ether of Space (Harper & Brothers London 1909) chapter X.

“… According to the theory of special relativity, if a beam of atoms which is emitting light is observed in a direction which according to the observer is at right angles to the direction of relative motion, then the frequency of the light should differ from the frequency the light would have if the source were at rest relative to the observer. This is the transverse Doppler effect. According to the classical ether theories there should be no change in frequency in this case.”

W.G.V. Rosser, An Introduction to the Theory of Relativity (Butterworths, London, 1964) section 4.4.7 pp.160.

“… transverse Doppler shift … this is a purely relativistic effect …”

Ray d’Inverno, Introducing Einstein’s Relativity (OUP, Oxford, 1992) pp.40.

“… transverse Doppler effect. This is a relativistic effect, for classically one would not expect a frequency shift from a source that moves by right angles.”

Richard A. Mould, Basic Relativity, (Springer-Verlag, NY, 1994) pp.80.