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Spherically symmetric spacetime

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  1. May 18, 2013 #1
    I know from classical physics that, for example, an electric field is spherically symmetric if its magnitude depends only on the distance [itex]r[/itex] to the origin (and not on the angles [itex]\phi[/itex], [itex]\theta[/itex]) and it's in radially inward or outward direction.

    But, what does it mean when spacetime is spherically symmetric? Does it mean that the metric depends only on [itex]r[/itex] and not on the angles?
     
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  3. May 18, 2013 #2

    WannabeNewton

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    It means that the isometry group ##G## of the space-time ##(M,g_{ab})## has a subgroup ##H\subseteq G## such that ##H\cong SO(3)## and such that the orbits of the group action associated with ##H## are topological 2-spheres. You should think of spherical symmetry in this way and not in the way you tried to characterize it because that is a coordinate dependent characterization (and is false by the way the metric doesn't only depend on ##r## in the coordinate basis - the Schwarzschild metric also depends on ##\theta## in the coordinate basis) whereas spherical symmetry of the space-time is a geometric property independent of coordinates. The isometries are related to one-parameter families of local diffeomorphisms that generate killing vector fields related to rotational symmetry so very loosely put, a spherically symmetric space-time is "invariant under rotations".

    EDIT: See this introduction: http://www.physto.se/~ingemar/sfar.pdf
     
    Last edited: May 18, 2013
  4. May 18, 2013 #3
    You're using a 3D object and asking what it's like in 4D. Since the spatial part of an object does not apply to time then I don't see how any meaning can be given to this situation.
     
  5. May 19, 2013 #4
    Here http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution it says:

    Why is that true and how it is related to what WannabeNewton wrote?
     
    Last edited: May 19, 2013
  6. May 19, 2013 #5

    WannabeNewton

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    Ugh, I wish people would stop reading that wiki article. It is sacrilegious to describe a geometric property of space-time using meaningless coordinates so please don't take wiki's "definition" as an actual definition. It is merely a consequence of the definition I wrote above. As for why, see here (starting with page 171 of the PDF): http://arxiv.org/pdf/gr-qc/9712019v1.pdf
     
  7. May 19, 2013 #6
    Thanks WannabeNewton, you helped a lot!
     
  8. May 19, 2013 #7

    WannabeNewton

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    Anytime mate! Feel free to ask any further questions you might have after reading the PDF. I just wanna stress again, geometry > coordinates :smile: Cheers!
     
  9. May 19, 2013 #8

    Bill_K

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    Well, if you feel that way, the logical thing to do would be to edit the Wikipedia page and replace their definition with yours.
     
  10. May 19, 2013 #9

    WannabeNewton

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    Apparently there is already a wiki article already that has the coordinate independent definition: http://en.wikipedia.org/wiki/Spherically_symmetric_spacetime but I have no idea how to link this to the other one or edit wiki articles n' stuff :frown: It seems safer to just learn the definitions from a textbook rather than from wiki (the above article cites Wald for example, who gives the exact same definition in his text when deriving the Schwarzschild metric).
     
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