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For istance, d'Inverno's textbook instructs as about how to solve the geodesic equations in comoving coordinate systems even in flat spacetime but does not mention anything about their transformations. The same holds for Carroll's notes, where the fact that SR can handle accelerations is mentioned ( as once @DaleSpam pointed out in another thread :) ), but there are no explicit maths given! The list goes on...

Moreover it is crucial to emphasize that coordinate transformations do depend on the relativity of the frames. That means, if every coordinate system is equivalent, one must distinguish only between inertial and non-inertial frames to calculate the generalized transformations. To me, the transformation from an accelerated to an accelerated frame would be different from the transformation from an accelerated to an inertial and so on and so forth.