4
$\begingroup$

Given a generic 2-dimensional metric $ ds^2=E(x,y)dx^2+2F(x,y)dxdy+G(x,y)dy^2 $ what is the change of coordinates that move it into the conformal form $ ds^2=e^{\phi(\xi,\zeta)}(d\xi^2+d\zeta^2) $ being $\xi=\xi(x,y)$ and $\zeta=\zeta(x,y)$? Is it generally known? Also a good reference will fit the bill.

Thanks beforehand.

  • 0
    @WillieWong: Thanks! It sounds familiar... I probably learned it once and forgot it.2012-07-19

2 Answers 2

2

This problem is equivalent to solving the Beltrami equation $f_{\bar z}=\mu f_{z}$ where the coefficient $\mu$ comes from the given metric, as explained on the Wikipedia page linked by @WillieWong. A solution can be sometimes semi-guessed when the coefficient is really simple. You should at least try it. But in general the solution comes as an infinite series involving singular integral operators. This is carefully written out in the book by Astala, Iwaniec, Martin.

  • 0
    Thanks Leonid. I will check the approach devised in Wikipedia. The reference you gave seems really interesting.2012-07-20
1

Please check Chandrashekhar's Mathematical Theory of Black holes, section 11 of chapter 2.

  • 0
    Yes, the content of this answer is not worth downvote being very near to my aims and of course, flagging it as spam is blatantly wrong. I understand that Chandra was a physicist and the cited book is about physics, but this guy should be treated somewhat better.2012-07-22