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Vacuum solution with static, spherical symmetric spacetime

  1. Dec 8, 2012 #1
    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 [itex]d\theta [/itex] and [itex] d\phi [/itex]


    2. Relevant equations
    ds must be invariant under reflections
    [itex]\theta \rightarrow \theta'=\pi - \theta [/itex]
    and
    [itex]\phi \rightarrow \phi' = -\phi [/itex]

    3. The attempt at a solution
    Well I just put in this in the equation for the line element. assuming t=r=konstant.
    [itex]ds^2=Ad\theta^2 + Bd\phi^2 + Cd\theta d\phi [/itex]
    and the line element after reflection:
    [itex]ds^2=Ad\theta^2 + Bd\phi^2 + Cd\theta d\phi [/itex]
    Ah, and for a 2 sphere [itex]A=R^2[/itex] and [itex]B=R^2sin^2\theta[/itex]
    How can I show that C=0?
     
  2. jcsd
  3. Dec 8, 2012 #2

    TSny

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    Homework Helper
    Gold Member

    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.
     
  4. Dec 9, 2012 #3
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