What is the effect of time dilation due to rotational motion? Consider a lab frame and the frame of an object rotating within it. Set up up a light clock of two mirrors and a beam of light that rises and falls vertically having a height z0. While in the lab frame, the light beam goes up and down vertically, in the rotating frame, the beam travels along the surface of a cylinder, traversing a longer distance also at the speed of light. The surface of a cylinder has no intrinsic curvature and vertical motion is perpendicular to the direction of motion. Does this imply time dilation $t=\frac{t_0}{\sqrt{1-\frac{(\omega r)^2}{c^2}}}$?
Here $t$ is time to the observer at the center of the rotational frame , $t_0$ is time in the lab frame, $\omega$ is the rotational velocity, and $r$ is the radial distance between the axis of the rotating frame and the light lock.
I'm thinking something is off because this seems to imply $\omega r$ must be less than $c$, but that corresponding product for the rotation of the earth and the distance to Alpha Centauri is higher than that. It also seems to imply things go slower the further they are away from the axis.
I'm not sure where my reasoning fails though.
 A: If you put a clock in a centrifuge, the clock will indeed slow down (relative to the lab frame) by a factor of $\gamma$, just as you derived. Experiments in particle acclerators confirm this. And yes, $v = \omega r$ must be less than $c$: you can not physically spin an object faster than $c$ (your centrifuge or similar apparatus would fail long before it reaches that speed).
Note that this is all calculated in the lab frame, which is to a good approximation inertial. Calculating things in a rotating frame is a whole different matter. Such a frame is definitely non-inertial, and so the usual rules of mechanics do not apply. In particular, the coordinate speed of light may be greater than $c$ in a non-inertial frame, and the usual Lorentz transformations (which go between inertial frames) do not work. You can use some of the machinery of general relativity to do calculations in such a frame, but they are much more complicated.
Basically, there is no inertial frame in which Alpha Centauri revolves around the Earth. There are non-inertial frames (such as one rotating with the Earth) but the usual laws of kinematics do not have the same form in such frames, and the time dilation derivation you made would not work in rotating coordinates. The Earth's rotation does not cause time to slow down on Alpha Centauri. On the contrary, it would actually appear to speed up time (since the Earth's rotation does create a tiny time dilation for objects on the Earth, relative to distant stars).
A: Yes there is time dilation compared to the lab frame for a rotating body around a point in the lab space at distance $r$ with the Lorentz equation you are given and not only that but also for a  hypothetical observer on the surface of an object for ease say, a sphere sinning around its diameter axis and of $r$ radius.
Using the parallel axis theorem the observer on a point on the surface of the fast spinning sphere with a tangential velocity of $υ=ωr$ will experience time dilation relative to the lab's inertial frame of reference the same as he would if translating in a straight path through space with the same speed $υ$ thus both cases are equivalent in SR since SR time dilation is depending only at the speed thus the scalar value of the velocity vector.
However, there is an important condition for the above time dilation to be true. The stationary observer on the lab frame must be outside the volume area of rotation:

By P. Fraundorf - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=51215249
So, as shown in this animation the stationary observer red dot at the center and the bouncing red dot rotating observer are actually both in the same rotating absolute frame of reference. In this case no time dilation is observed. In order time dilation to be experienced the observer at the center must move out of the rotating frame inside the lab inertial frame.
