1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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]
    [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


    User Avatar
    Homework Helper
    Gold Member
    2017 Award

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook