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(wald) method for calculating curvature

  1. Nov 21, 2012 #1
    R[itex]_{a}[/itex][itex]_{b}[/itex][itex]_{c}[/itex][itex]^{d}[/itex]ω[itex]_{d}[/itex]=((-2)[itex]\partial[/itex][itex]_{[a}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{b] }[/itex][itex]_{c}[/itex]+2[itex]\Gamma[/itex][itex]^{e}[/itex][itex]_{[a]}[/itex][itex]_{c}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{}[/itex][itex]_{e}[/itex])ω[itex]_{d}[/itex]

    good, me question is about of:

    1.- as appear the coefficient (-2) und the (2)?

    2.- it is assumed that:
    [itex]\partial[/itex][itex]_{[a}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{b]}[/itex][itex]_{c}[/itex]=[itex]\partial[/itex][itex]_{a}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{b}[/itex][itex]_{c}[/itex]+[itex]\partial[/itex][itex]_{b}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{a}[/itex][itex]_{c}[/itex]

    also the general form is: (maybe my problem is with the notation)

    R[itex]_{a}[/itex][itex]_{b}[/itex][itex]_{c}[/itex][itex]^{d}[/itex]ω[itex]_{d}[/itex]=([itex]\partial[/itex][itex]_{a}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{b}[/itex][itex]_{c}[/itex]-[itex]\partial[/itex][itex]_{b}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{a}[/itex][itex]_{c}[/itex]+[itex]\Gamma[/itex][itex]^{e}[/itex][itex]_{a}[/itex][itex]_{c}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{b}[/itex][itex]_{e}[/itex]-[itex]\Gamma[/itex][itex]^{e}[/itex][itex]_{b}[/itex][itex]_{c}[/itex][itex]\Gamma[/itex][itex]^{d}[/itex][itex]_{a}[/itex][itex]_{e}[/itex])ω[itex]_{d}[/itex]

    thank very much!
     
  2. jcsd
  3. Nov 21, 2012 #2

    Bill_K

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    Brackets around a pair of indices means antisymmetrize. So

    [aΓdb]c = ½(∂aΓdbc - ∂bΓdac)
     
  4. Nov 21, 2012 #3

    pervect

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    Trying to do the latex is giving me fits. But since we have three variables inside the brackets, shouldn't we write

    [tex] f([a,d,b],c) = \frac{1}{6} \left[ f(a,d,b,c) + f(d,b,a,c) + f(b,a,d,c) - f(a,b,d,c) - f(b,d,a,c) - f(d,a,b,c) \right] [/tex]

    i.e [itex]\frac{1}{n!} [/itex] (even permutations - odd permutations), where n=3?
     
  5. Nov 22, 2012 #4
    pervect, we have two variables inside the brackets, a and b.

    ok... but see the curvature tensor:

    R[itex]_{a}[/itex][itex]_{[b}[/itex][itex]_{c}[/itex][itex]_{d]}[/itex]=0

    it is definition equal of the tensor antisymmetric in the brackets?

    (where it origines ∂[aΓdb]c = ½(∂aΓdbc - ∂bΓdac) ? )


    thank very much!
     
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