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Minkowski metric - to sperical coordinates transformation

  1. Jul 7, 2012 #1

    soi

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    I need to transform cartesian coordinates to spherical ones for Minkowski metric.
    Taking:
    (x0, x1, x2, x3) = (t, r, α, β)

    And than write down all Christoffel symbols for it.

    I really have no clue, but from other examples i've seen i should use chain rule in first and symmetry of Christoffel symbol Tab=Tba
     
  2. jcsd
  3. Jul 8, 2012 #2

    stevendaryl

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    Staff Emeritus
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    The spherical form of the Minkowsky metric is just

    [itex]ds^{2} = c^2 dt^2 - dr^2 - r^2 d\theta^2 - r^2 sin^2(\theta) d\phi^2[/itex]

    So the metric components are
    [itex]g_{tt} = c^2[/itex]
    [itex]g_{rr} = -1[/itex]
    [itex]g_{\theta\theta} = -r^2[/itex]
    [itex]g_{\phi\phi} = -r^2 sin^2(\theta)[/itex]

    The connection coefficients [itex]\Gamma_{uvw}[/itex] are computed in terms of the metric components via:

    [itex]\Gamma_{uvw} = \frac{1}{2} (\partial_{v} g_{uw} + \partial_{w} g_{vu} - \partial_{u} g_{vw})[/itex]
     
  4. Jul 11, 2012 #3

    soi

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    OK, great thanks for your help.

    To look if I understand it, i calculated it using formula
    http://upload.wikimedia.org/wikipedia/en/math/f/f/d/ffdb897152259f912ad9c4d5ab3d474d.png

    And i got what you got (not surprisingly) but with -1 everywhere:

    gtt=-1
    grr=1
    gθθ=r^2
    gββ=r^2(sinθ)^2

    And Christoffel symbols (nonzoro, numering metric matrix from 0 to 3):
    T221=1/r
    T122=-r
    T331=1/r
    T332=1/2(rsinθ)^2
    T133=-r (sinθ)^2
    T233=(sin2θ)/2

    Is it okay?
     
  5. Jul 11, 2012 #4

    soi

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