I have a simple metric:
$$ ds^2=dr^2+r^2(1+kr)d\theta^2 $$
Near the origin $r=0$, I wanted to switch to (smooth) coordinates $x=r\cos\theta$, $y=r\sin\theta$, but it seems this is only possible when $k$ is non-positive. Assuming my conjecture is correct, what is the easiest way to confirm this and show the breakdown with $k$ positive? We are used to spherical/polar coordinates not behaving at the origin, but for $k$ positive with this metric, it is the Cartesian coordinates that do not behave. Also, if smoothness vs. mere continuity is important please explain. Also, if you can tie it to Lie brackets, geodesics, affine parameters, etc. that would be nice. (This question emerged from pondering Wald's General Relativity Chapter 11 Problem 3b.)
Thanks in advance!