Change of coordinates for functions

[tut_einstein]

I am trying to solve a problem with spacetime metrics and a change of coordinates for my research project.

I have an unspecified function R that depends on two coordinates t' and r' (which are the time and radial coordinates) and the only condition I'm imposing on R is that it be positive. Now, suppose I change to a new set of coordinates t and r and define R = r^2, is it possible to find an explicit relation between the old coordinate r' and r without knowing the exact dependence of R on t' and r'? I feel like it's not possible, but maybe there's a way using a Jacobian or something that I'm forgetting.

More importantly, if there's another function P that depends on t' and r', is it possible t o find P as a function of t' and r using the above information? Thanks!

 

[HallsofIvy]

Except in trivial situations, r will necessarily depend on both r' and $\theta'$, not just r.