A Former Atheist

[Hatshepsut]

...use riemannian geometry even  if it's just SR. Otherwise I wouldn't know how one would correctly take the new coordinates into account.

I dare not clog the forum. Callaghan considers a special case. The non-rotating frame is presumed inertial. The two observers coincide in space and have no linear motion relative to one another. The rotation is uniform about the x-axis. The only world line tracked belongs to a point that remains stationary in the rotating frame. In this case, Callaghan just applied the classical rotation matrix to change the two space coordinates and a separate equation, T(t) = t * sqrt(1 - w^2*r^2/c^2) to find the proper time T on the world line, which is a helix in the non-rotating frame. It was simple because the point's radial distance r, common to both observers, and the  rotating frame's angular rate w are constant. I can see this won't work if the point begins moving about. But I admit I'm in pretty heavy seas with this kind of stuff.  

Nonetheless, though the point is stationary in the rotating frame, it's clock is slower relative to both observers the greater r is. Callagan also noted that the rotating frame has a boundary: it only covers events inside the cylinder of radius c/w centered on the x-axis. I agree it's a non-interesting situation: all this to describe a stationary dot as seen by two people, one of whom is dizzy!