# Vacuum solution with static, spherical symmetric spacetime

1. Dec 8, 2012

### zardiac

1. The problem statement, all variables and given/known data
I am trying to derive the line element for this geometry. But I am not sure how to show that ds cant contain any crossterms of $d\theta$ and $d\phi$

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

3. 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?

2. Dec 8, 2012

### 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.

3. Dec 9, 2012

### andrien

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