Transverse Doppler effect

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In special relativity, the transverse Doppler effect is the nominal redshift component associated with transverse (i.e. lateral) observation, and is important both theoretically and experimentally.

Overview

If the predictions of special relativity are compared to those of a simple flat nonrelativistic light medium that is stationary in the observer’s frame (“classical theory”), SR’s physical predictions of what an observer sees are always “redder”, by the Lorentz factor

\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}\,}}}.

For receding or approaching objects, this additional redshift term modifies the redshift or blueshift predictions of "classical theory". Where the two effects act against each other, the propagation-based effects are stronger. But for the case of an object passing directly across the observer’s line of sight, special relativity’s predictions are qualitatively different to "classical theory" – a redshift where the “classical theory” reference model would have predicted no shift effect at all for the case that the observer is at rest in the aether.

Because of this, the transverse Doppler effect is sometimes held up as one of the main new predictions of the special theory. As Einstein^[1] put it in 1907: according to special relativity the moving object's emitted frequency is reduced by the Lorentz factor, so that - in addition to the classical Doppler effect - the received frequency is reduced by the same factor.

The formula for the transverse Doppler effect is

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

where f ' refers to the Doppler shifted frequency of oscillation.

The above consideration by Einstein applies to the case of a detectorin rest. For a moving detector with an emitter "in rest", the situation is different; this will be discussed next.

Reciprocity

Sometimes the question arises as to how the transverse Doppler effect can lead to a redshift as seen by the "observer" whilst another observer moving with the emitter would also see a redshift of light sent (perhaps accidentally) from the receiver.

It is essential to understand that the concept "transverse" is not reciprocal. Each participant understands that when the light reaches her/him transversely as measured that person's rest frame, the other had emitted the light aftward as measured in the other person's rest frame. In addition, each participant measures the other's frequency as reduced ("time dilation"). These effects combined make the observations fully reciprocal, thus obeying the principle of relativity.

Experimental verification

In practice, experimental verification of the transverse effect usually involves looking at the longitudinal changes in frequency or wavelength due to motion for approach and recession: by comparing these two ratios together we can rule out the relationships of "classical theory" and prove that the real relationships are "redder" than those predictions.

longitudinal tests

The first of these experiments was carried out by Ives and Stilwell in (1938) and although the accuracy of this experiment has since been questioned,^{[citation needed]} many other longitudinal tests have been performed since. These usually claim greater certainty than Ives-Stilwell, but also tend to be more complicated.

Herbert E. Ives and G.R. Stilwell, “An experimental study of the rate of a moving clock”

J. Opt. Soc. Am 28 215-226 (1938) and part II. J. Opt. Soc. Am. 31, 369-374 (1941)

transverse tests

To date, only one inertial experiment seems to have verified the redshift effect for a detector actually aimed at 90 degrees to the object.

D. Hasselkamp, E. Mondry, and A. Scharmann, "Direct Observation of the Transversal Doppler-Shift"

Z. Physik A 289, 151-155 (1979).

Classical transverse Doppler effect

Because of the predicted Doppler effect as resulting from 'time dilation', in numerous textbooks on special relativity the incorrect assertion may be found that the transverse Doppler effect^{[citation needed]} was not known in classical physics. Although in fact little attention had been focused on this effect of second order (it had not yet been given the name under which it is known today), it was already implicated in classical optics of moving bodies, where it follows immediately from the law of velocity aberration (Bradley's theorem) ^[2].

Examples of textbooks that don't mention the classical transverse Doppler effect:

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

"… 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."

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

"… transverse Doppler shift … this is a purely relativistic effect …"

Robert Resnick, Introduction to Special Relativity, New York 1968, p. 90.

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

"… 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."

In other textbooks a 'theoretical proof' of the asserted absence of the transverse Doppler effect in classical physics is attempted by resorting to the plane wave approximation of a spherical wave ^[3]. As a matter of fact, if an observer moves in parallel to the arriving wave fronts of a plane wave, no Doppler shift is observed, because in the next moment he measures the same value of instantaneous phase, which he would measure if he were at rest. The same argument applies for a point source emitting spherical waves, when the observer moves at a fixed radial distance from the point source in parallel to the spherical wave fronts, i.e. if his (circular) line of motion is concentric with respect to the spherical wave. 'Disproofs' of this kind of classical physics have to be rejected, however, since the observer does not receive the wavefronts under a straight angle while the transverse Doppler effect is defined as such.

Measuring the classical transverse Doppler effect

The classical transverse Doppler effect is formally the same as the transverse Doppler effect predicted by special relativity^{[citation needed]}. In acoustics it can be easily measured by observing an acoustic point source being at rest in the acoustic medium of propagation by means of a direction sensitive microphone installed on a moving platform. The transverse Doppler effect is the Doppler shift which is observed in the short moment of tracking the point source, when the axis of the microphone is at a right angle with respect to the line of motion of the platform. It is a second-order shift toward lower frequencies of oscillatiom.

References

^ Albert Einstein (1907) "Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips", Annalen der Physik, Ser. 4, 23, 197 - 198 (1907).
^ R. Emde (1926) "Aberration und Relativitätstheorie", Die Naturwissenschaften 14, 327 - 335.
^ J. D. Jackson, Classical Electrodynamics, 3rd edition, New York 1999, Chapter 11.