# Relativistic Doppler Effect between rotating bodies

The Wikipedia article "Relativistic Doppler effect - One object in circular motion around the other" confused me.

There they say that when the light source orbits the observer, the observation of the observer would be different than the case where the observer orbits the light source.

Now, what I knew from my basic physics background is that there is no such thing as "orbiting" because despite from the outside the earth seems like it is orbiting around the sun, that is just a misinterpretation. If we fix the sun at the center, the earth will seem like orbiting around the sun and if we fix the earth at the center, the sun will seem like orbiting around the earth. In essence, they both are rotating around their overall center of mass.

So, how is it possible that when I and the sun are rotating around our overall center of mass, I could observe redshift and blueshift at the same time?

## 3 Answers

You don't need circular motion for this paradox. It's enough to consider two objects moving inertially in these two configurations:

Redshift:                Blueshift:

E-->                     E

R                     <--R


E stands for emitter and R for receiver, and the arrows are their velocities. The positions of the moving objects are their positions at the time of emission/reception.

Special relativity indeed predicts that there is a redshift in the case on the left, and a blueshift in the case on the right—even though they seem to be equivalent under a Lorentz transformation.

The resolution of the paradox is that they are not really equivalent. There is a third coordinate (time) not shown in the diagram, and the time of emission is earlier than the time of reception in both cases, breaking the symmetry.

If you Lorentz-transform both configurations into the rest frame of the receiver, you get something like this:

Redshift:                Blueshift:

E-->                   E-->

R                        R


The angle between the ER line segment and the velocity vector is different after the boost because of aberration. In the case on the right, E's velocity has a nonzero radial component relative to R, resulting in a blueshift that is larger than the transverse redshift.

• Why does the emission have to occur prior to the closest position to the receiver in your final blue shift illustration? Based on the receivers frame, shouldn't that light come out of the emitter without regard to the emitter's velocity, and then miss the receiver? Dec 12, 2021 at 19:38
• @HardlyCurious In any Doppler shift calculation, the light is assumed to reach the receiver. You can imagine the emitter is emitting light at all times in all directions (like the sun), and we are only considering the bit of light that reaches the receiver at a particular time. Dec 12, 2021 at 19:46
• Ok, and what about privileged frames? Doesn't this enable us to determine motion beyond simply relative motions? Dec 12, 2021 at 19:50
• @HardlyCurious You can distinguish the two cases because they really are different. It's just hard to visualize the difference. The relationship between E and R is not just their relative velocity, but also their relative position in spacetime. Dec 12, 2021 at 20:39
• Based on this article: en.wikipedia.org/wiki/… It seems you are comparing the orbital scenario to fig 2a, but the article says its equivalent is fig 2b. Dec 13, 2021 at 12:45

The center of mass is well within the sun because the sun is so much more massive than the Earth, so it's OK to say "orbiting the sun". We can likewise assume that if the sun were orbiting the Earth, then the Earth would be much more massive than the sun.

And that is the answer to your problem: the massive object has outgoing radiation redshifted and the incoming radiation blueshift much more than the less massive object has incoming radiation blue shifted and the outgoing radiation redshifted.

• If the question is about special relativity, the mass can be neglected. The reason why the orbiting object is objectively redshifted has to do with the constant acceleration Aug 18, 2019 at 23:23
• It sounds like you're talking about gravitational Doppler shift, which has nothing to do with this special-relativistic problem. The sign of the effect you're talking about is opposite the sign of the special-relativistic effect. Dec 12, 2021 at 19:01

In special relativity rotation and acceleration means that you are constantly switching your inertial frame. The object in the center is unaccelerated, so it stays in its inertial frame. The object in orbit is rotating and therefore constantly accelerating, so its light will be redshifted relatively to the unaccelerated object in the center, and the object in the center will be blueshifted relative to the orbiting object. That is because the accelerating object on a circular orbit is constantly switching into a frame in which the proper time of the unaccelerated object in the center runs faster than its own. For more information see https://en.wikipedia.org/wiki/Born_coordinates and the references therein.