Is the Transverse Doppler Effect Dependent on General Relativity?

[arildno]

I saw in a chapter on special relativity a derivation of the transverse Doppler effect, which seemed okay, but I have a question concerning this:

If the motion of a source is always perpendicular to the position vector connecting the observer to the moving source (i.e. the distance remains constant), that is the source rotates about the obs., does not
(technically, at least) the validity of the result (transv. D. eff.) depend on general relativity, rather than special relativity, since the source's rest frame is in non-uniform motion relative to the observer's rest frame?

 

[Doc Al]

As I am sure you realize, the derivation of the transverse Doppler effect does not require that the motion of the source be in pure transverse motion. As long as there is a transverse component of motion, there will be a transverse Doppler effect due to time dilation. In any case, even if the light source does purely circle around the observer, the observing frame is still inertial, so special relativity should be sufficient.

Or have I missed your point?

 

[DW]

I saw in a chapter on special relativity a derivation of the transverse Doppler effect, which seemed okay, but I have a question concerning this:

If the motion of a source is always perpendicular to the position vector connecting the observer to the moving source (i.e. the distance remains constant), that is the source rotates about the obs., does not
(technically, at least) the validity of the result (transv. D. eff.) depend on general relativity, rather than special relativity, since the source's rest frame is in non-uniform motion relative to the observer's rest frame?

The special relativistic formula for transverse Doppler shift will still be valid for describing how the light is seen according to the inertial frame observer that was emmited from the circling object unless you are also considering a gravitational source such as a planet about which it orbits for example. Then, one must use general relativity to determine the total relativistic Doppler shift.

 

[arildno]

Thx for the response!
Doc Al, yes, I was aware of that the derivation of the transverse Doppler effect did not at all depend upon a pure transverse motion; in fact, the result followed from the decomposition of a uniform motion (in accordance with spec. rel.) into a radial and transverse component,
the radial comp. accounting for the classical Doppler effect, whereas the transverse effect enters as a result of the time dilation factor (since the transverse velocity component is part of the total velocity).

Hence, I have no trouble in recognizing the presence of a transverse Doppler effect in the case of a general, uniform motion, and that this is derivable from the postulates/conditions for the validity of special relativity.

My point only concerned the limiting case (radial comp->0),
i.e. when the motion can no longer be regarded as strictly uniform, since the source will, in fact, rotate around the observer.

However, from your own answer, and DW's, I think I have gained a satisfactory response.