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Minkowski metric tensor computation

  1. Sep 13, 2009 #1
    Hi, I'm having problem with understanding tensors and the Einsteins summation convention, so I decided to start doing explicit calculations, and I'm doing it in the wrong way. Hope someone could help me to clarify the concepts.

    In flat spacetime we have [tex]\eta[/tex] with the signature (-+++). Under some coordinate change, say [tex]x_{\mu} \rightarrow x_{\overline{\mu}}[/tex], then the metric changes as [tex]g_{ \overline{\mu} \overline{\nu}}=\frac{ \partial x^{\rho}}{ \partial x_{\overline{\mu}}} \frac{ \partial x^{\sigma}}{ \partial x_{\overline{\nu}}}g_{ \overline{\rho} \overline{\sigma}}[/tex]. So, If I change the coordinate system from Cartesian [tex](t,x,y,z)[/tex] to spherical [tex](t,r, \theta, \varphi)[/tex] with the following equations

    [tex] x = r \cos(\varphi) \cos (\theta)[/tex], [tex] y = r \cos(\varphi) \sin (\theta)[/tex], [tex] z = r \sin(\varphi) [/tex], [tex] t = t [/tex]. The four non-zero componentes of the metric [tex]\eta[/tex] in spherical coordinates should be:

    [tex]g_{11} = (\frac{ \partial t }{ \partial t })^2 g_{1'1'}=-1[/tex]


    [tex]g_{22} =(\frac{ \partial x }{ \partial r })^2 g_{2'2'}=\cos^2(\varphi) \cos^2 (\theta)[/tex]


    [tex]g_{33} =(\frac{ \partial y }{ \partial \theta })^2 g_{3'3'}=r^2 \cos^2(\varphi) \cos^2 (\theta)[/tex]


    [tex]g_{44} =(\frac{ \partial z }{ \partial \varphi })^2 g_{4'4'}=r^2 \cos^2(\varphi)[/tex]

    And finally, the line element [tex]ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}=dt^2+\cos^2(\varphi) \cos^2 dr^2+ r^2 \cos^2(\varphi) \cos^2 d^2 \theta + r^2 \cos^2(\varphi) d^2 \varphi[/tex] wich is incorrect.

    Thanks for your time, any help will be appreciated.
     
  2. jcsd
  3. Sep 13, 2009 #2

    atyy

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    Did you sum over the repeated indices in the change of coordinates formula (summation convention)?
     
  4. Sep 13, 2009 #3
    I think I did, in fact it explains the square of the derivatives.
     
  5. Sep 13, 2009 #4

    atyy

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    I'm not sure about this, but I think in grr you should have terms like (dt/dr)^2.gtt+(dx/dr)^2.gxx+(dy/dr)^2.gyy+(dz/dr)^2+(dt/dr)(dx/dr)gtr + ....

    where in your formula I've taken u=r,v=r and rho and sigma must be summed over all combinations of rho=t,x,y,x and sigma=t,x,y,z
     
  6. Sep 13, 2009 #5
    Thanks for your posts.

    I think now I understand your point. Do you mean that, in the equation of the line element should be 4*4=16 summands instead of only four, by varying the ro and sigma indices between all their range?
     
  7. Sep 13, 2009 #6
    Ok, just answered too fast!

    Then this was my error, thank you so much!
     
  8. Sep 13, 2009 #7

    atyy

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    Wow, you sure compute fast! Good to know - I wasn't sure about this.
     
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