3
$\begingroup$

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me.

As far as my understanding goes, given a Riemannian metric $ds^2=E\,dx^2+2F\,dx\,dy+G\,dy^2$ we somehow have to find other variables $u,v$ so that $ds^2=du^2+G(u,v)\,dv^2$.

  1. How can I convert $ds^2={dr^2+r^2\,d\theta^2\over f(r)^2}$ where $f(r)$ is some function of $r$ into that form?
  2. Is there an "algorithm" to achieve this change of form?
  3. Is there some significant benefit in doing so?

Thank you.

1 Answers 1