Doppler shift of light in uniform circular motion of source

Question

Suppose, a monochromatic light source is undergoing uniform circular motion, and the observer is at the center of the circle. When the velocity of the source is perpendicular to the line joining the source and the observer, the observed frequency is $n_0\sqrt{1-\frac{u^2}{c^2}}$, where $u$ (constant) is the speed of the source with respect to the observer, and $n_0$ is the rest frequency of the source.

However, in this case, the source is accelerating, so rules of special relativity are not applicable. Is the observed frequency same as the above formula? If so, why?

If not, how to derive the correct formula?

Answer 1

It's commonly said that special relativity cannot deal with acceleration but this isn't true. I go into the details of how to calculate time dilation for circular motion in my answer to Can a ultracentrifuge be used to test general relativity? and also with more details in Is gravitational time dilation different from other forms of time dilation? It turns out that the time dilation of the circling body is the same as if it was moving in a straight line i.e.

$$ \frac{\tau}{t} = \sqrt{1 - \frac{v^2}{c^2}} $$

That's why the Doppler shift has the form you describe.

Answer 2

SR (special relativity) states the coordinates transformation laws between inertial reference frames. Nevertheless an inertial observer can describe an accelerated reference frame just comparing to a continuous set of inertial reference frames instantaneously at rest with the accelerated frame.

That means that, because of the configuration of the experiment, for each position of the light source along the circle you may apply the so-called transverse relativistic Doppler effect: the emitted frequency is reduced by the Lorentz factor when measured by the receiver.
$f_{receiver} = \gamma^{-1} f_{emitter}$
where:
$\gamma = 1 / \sqrt{1 - v^2/c^2}$ Lorentz factor

Note: The emitted frequency is red shifted as measured by the receiver, even if the distance receiver-emitter does not change. That is due to the time dilation measured by the stationary frame.