Not quite, the laws of physics don't apply in a rotating reference frame. For example the speed of light stops being a constant.
So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.
So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.
That's what the Christoffel symbols already do - it's part of the formalism, there's nothing to add!
Consider a reference frame rotating at say 1 RPM. At 1 light year from the center a 'stationary' particle in a non rotation reference frame is now moving in a circle at 2 * π light years per minute. At 2 light years from the center it's moving 4 * π light years each minute and that continues the further from the center you get.
GR is fine with most reference frames as long as it's translation. Rotation is however not ok.
GR is fine with that - what will happen is that there'll be a sign change in the metric (the 'time' coordinate will become spacelike, whereas the 'angular' coordinate will become timelike), making it impossible to sit at a fixed value of the rotating frame's angular coordinate (which would correspond to faster-than-light motion in the non-rotating, inertial frame).
That's just a notation for doing exactly what I am describing.
GR deals with curved spacetime, but the only way to work in rotating spacetime is by changing how you calculate what's going on. Which means your calculations must be mapped and don't generalize.
EX: Try and do an actual calculation for say two electrons hitting each other at say 0.5c. In rotating, non rotating inertial, and non rotating non-inertial reference frames of your choice.
So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.