# Transverse Doppler shift

MOTION

#### Transverse Doppler shift

The Transverse Doppler effect under special relativity is a Lorentz factor redshift seen in signals coming form objects moving at 90 degrees to the the observer's viewing angle (where angles are defined in the observer's frame).

$\nu' = \nu × \sqrt{ 1 - \frac{v^2}{c^2}}$

A great deal of attention is given to the transverse Doppler effect in experimental testing, as it represents the difference between the simple Doppler predictions for lightspeed fixed in the observer's frame, and those of special relativity.

The effect is traditionally interpreted as being due to time dilation, and is considered to be present for a moving object regardless of its direction of motion.

## The transverse component in the longitudinal equations

We can also choose to write write the longitudinal Doppler relationship for special relativity as

$\nu' = \nu × \frac{c}{c+v} × \sqrt{ 1 - \frac{v^2}{c^2}}$

, where the (c/(c+v)) part represents the propagation shift associated with globally constant c for the observer, and the Lorentz term represents the "transverse Doppler component".

## Comparison with pre-SR theory

Although transverse redshifts used to be claimed to be unique to SR (and identifying a transverse component treted as a validation of the special theory), if we look at C19th Newtonian theory, we find corresponding predictions of

$\nu' = \nu × ({ 1 - \frac{v^2}{c^2})}$

and

$\nu' = \nu × \frac{c}{c+v} × ({ 1 - \frac{v^2}{c^2}}) = \frac{c-v}{c}$

, so not only is the effect not unique to special relativity, the SR version of the effect is nominally weaker than the Lorentz-squared redshift component that appears under older Newtonian optics.

In general, a relativistic argument or interpretation based on the existence of transverse redshifts that works under special relativity (such as the physical time-dilation of a circling clock) can also be attempted under Newtonian theory. Because of this, and experiment that successfully identifies a Lorentzlike time-dilation effect is not necessarily a significant validation of special relativity unless we can also (a) show that the SR result is physically different to the NM result, and (b) show that the SR prediction is superior to the NM prediction.

## SR testing

Much C20th Sr testing was concerned with verifying the existence of a Transverse component in Doppler shifts that coud be said not to be any weker than the SR prediction – the question of redshifts stronger than SR was not widely addressed, due to shortcomings in the test theory, which assumed that redder shifts didn't correspond to any known theory.

Most of these tests were carried out using longitudinally or mostly-longitudinally-moving particles, the exception being a test in 19xxx which trained the detector actually at 90 degrees to a particle beam. This experiment actually found roughtly twice the SR prediction, before correction.