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Homework Help: Line element in spherical coordinates

  1. Dec 2, 2005 #1
    Hi,

    I was just reading up on some astrophysics and I saw the line element (general relativity stuff) written in spherical coordinates as:

    [tex]ds^2 = dr^2 + r^2(d\theta^2 + \sin\theta\d\phi)[/tex]​

    I don't get this. dr is the distance from origo to the given point, so why isn't ds^2 = dr^2 without the other stuff?
     
  2. jcsd
  3. Dec 2, 2005 #2

    Tom Mattson

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    Because you aren't after the distance between some point and the origin, you're after the distance between 2 arbitrary points in space. If you want to see how this expression comes about then start from the more intutive expression for the line element in Cartesian coordinates:

    [tex]ds^2=dx^2+dy^2+dz^2[/tex]

    Then use the following transformation equations:

    [tex]x=r\sin(\theta)\cos(\phi)[/tex]
    [tex]y=r\sin(\theta)\sin(\phi)[/tex]
    [tex]z=r\cos(\theta)[/tex]

    Take the differentials [itex]dx[/itex], [itex]dy[/itex], and [itex]dz[/itex] and verify that [itex]ds^2 \neq dr^2[/itex] in general.
     
    Last edited: Dec 2, 2005
  4. Dec 2, 2005 #3
    Thanks, Tom!
     
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