I'm trying to understand the relativistically spinning disk within the framework of SR (if that is even possible). I thought to first simplify the problem by considering a spinning ring/annulus, but I don't know if my analysis is correct. I imagined a spinning ring of radius R, spinning at an angular velocity of ω, and I wrote down an expression for its 4-velocity. I then used u⋅u=-c^2 to normalize it. This gave me an expression for the Lorentz factor due to rotation, which I called γ_ω. Can I then premultiply this 4-velocity by the Lorentz matrix (for linear motion in the x direction) to find the 4 velocity of an annulus spinning at angular velocity ω and moving at some speed v w.r.t some inertial observer? Or are there issues due to the fact that a rotating annulus is noninertial? The center of the annulus is inertial, right? So as long as I have the basic parametrization for a circle in polar coordinates, which parametrizes the circle with respect to the center, I should be fine, right? I'm asking because I tried this, and then tried to get the 3-velocity from the 4-velocity by u_x/u_t=dx/dt, and in this expression both the γ_v and γ_ω terms cancel out, γ_v being the Lorentz factor coming from the linear motion w.r.t. to an inertial observer, i.e. the typical (1-v^2/c^2)^(-1/2). This seems wrong, because I expect there to be length contraction due to the spinning, but then further length contraction due to its linear motion w.r.t. an inertial observer. The dy/dt term contains the γ_v term only, and I can't figure out if that makes sense or not. I'm working on this problem because I was trying to suss out how the geometry of the annulus would change, because if it is both spinning and moving linearly, then one side of the annulus will be moving forward more quickly than the opposite side w.r.t. to the inertial observer, so what would happen there? How would the Einstein velocity addition mess with the geometry of the annulus? Or, being a rigid body (I'm aware that the definition of rigidity needs to be amended in SR) would the annulus slow down to match the speed of its slowest part? Can anyone share any insight on this?