Transverse Doppler effect

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

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

The transverse Doppler effect is a direct consequence of the relativistic Doppler effect

:$f\_o\; =\; frac\{f\_s\}\{gammaleft(1+frac\{vcos\; heta\_o\}\{c\}\; ight)\}$

In the particular case when $heta\_o=pi/2$, one obtains the transverse Doppler effect

:$f\_o=frac\; \{f\_s\}\; \{gamma\}\; ,$

For receding or approaching objects, the redshift factor $frac\{1\}\{gamma\}$ 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 from "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 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.

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 in terms of 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,Fact|date=February 2007 many other longitudinal tests have been performed since with much higher precision [http://* 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).

ee also

*Doppler effect
*Relativistic Doppler effect
*Ives-Stilwell experiment
*Time dilation

References

* A. Einstein (1907), "Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips", Annalen der Physik SER.4, no.23
* J. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999).