Vacuum solution with static, spherical symmetric spacetime

[zardiac]

Homework Statement

I am trying to derive the line element for this geometry. But I am not sure how to show that ds can't contain any crossterms of $d\theta$ and $d\phi$

Homework Equations

ds must be invariant under reflections
$\theta \rightarrow \theta'=\pi - \theta$
and
$\phi \rightarrow \phi' = -\phi$

The Attempt at a Solution

Well I just put in this in the equation for the line element. assuming t=r=konstant.
$ds^2=Ad\theta^2 + Bd\phi^2 + Cd\theta d\phi$
and the line element after reflection:
$ds^2=Ad\theta^2 + Bd\phi^2 + Cd\theta d\phi$
Ah, and for a 2 sphere $A=R^2$ and $B=R^2sin^2\theta$
How can I show that C=0?

 

[TSny]

Maybe you need to consider more than just reflections. A cube is invariant under reflections about the center of the cube, but it is not spherically symmetric.

 

[andrien]

are you looking for schwarzschild metric,you can see it here
http://en.wikipedia.org/wiki/Schwarzschild_metric
c=0,because that is just off diagonal element of metric tensor.