A Understanding the Tetrad Formalism in General Relativity

[strangerep]

It's also worth thinking about why the table about "Coordinate Conversions" is partially wrong (or at least sloppy)!

Oh, did I miss something? What do you think is wrong?

 

[vanhees71]

What I meant is the sloppy formula for the polar angle in polar coordinates (and the other analogous cases too). First of all one must recall the domains of the coordinates. For polar coordinates a convenient choice is $r \in (0,\infty)$ and $\varphi \in (-\pi,pi]$.

Then the correct formula for the angle in terms of the Cartesian coordinates is
$$\varphi=\arctan_2(y,x)=\text{sign} \; y \arccos \left (\frac{x}{\sqrt{x^2+y^2}} \right).$$