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Error in Bel's 1958 article ? Lanczos formula incorrectly reported

  1. Sep 26, 2012 #1
    hey all,
    in this link :http://gallica.bnf.fr/ark:/12148/bpt6k7258/f122.image

    The equation 2b is not correct I believe. Please correct me if I'm wrong.
    Here we go :
    The equation mentioned reads : Rαβλμ Rαβλρ = 2 * 1/8 * Rαβλμ Rαβλμ gμρ
    Now multiply out by gμρ on both sides and elevate the μ index on the LHS, we would end up having :
    Rαβλεgμε Rαβλρgμρ =
    2 * 1/8 * Rαβλμ Rαβλμ gμρ gμρ
    This becomes :
    Rαβλεδερ Rαβλρ =
    2 * 1/8 * Rαβλμ Rαβλμ gμρ gμρ
    which leads to :
    RαβλρRαβλρ = 2 * 1/8 * Rαβλμ Rαβλμ gμρ gμρ
    On the RHS, do we agree that the indices on the g's are not to be summed over since they are fixed ones ? This means that the product of the g's is 1 (if they were summation indices, we would get 4 and not 1) , which results in :
    RαβλρRαβλρ = 2 * 1/8 * Rαβλμ Rαβλμ
    But THIS IS WRONG ! the μ on the LHS of the equation should not be the same μ as on the RHS since it is a fixed index I believe.
    thank you,
    Cheers,
    PS: I have seen the referenced Lanczos paper but there is no way you can tell how Bel ended up with that one !
    PS: Here :http://arxiv.org/pdf/1006.3168v4
    in the first paragraph, the author is referring to another variant of the Lanczos formula ! and this is correct I think
     
    Last edited: Sep 26, 2012
  2. jcsd
  3. Sep 26, 2012 #2

    Bill_K

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    For the RHS he says 2 A gμν where A = (1/8) Rαβλμ Rαβλμ. Well, before substituting A into the RHS, one must change the indices so that μ does not appear 3 times! So let A = (1/8) Rαβλσ Rαβλσ instead.
     
  4. Sep 26, 2012 #3
    I see . But do you agree then that the product of the g's is 4 in order for the equation to be correct? the indices on the g's are summation indices ? but if this is so, then why on the LHS we can see them unrepeated. I would have thought that they were fixed ones.
    sorry for my confusion with the index gymnastics...
     
    Last edited: Sep 26, 2012
  5. Sep 26, 2012 #4

    Bill_K

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    Yes, gμρgμρ is δμμ, the trace of the Kronecker delta, which is the dimensionality N, that is 4 in 4 dimensions.
     
  6. Sep 26, 2012 #5
    Indeed. I'm realizing now that when we multiplied the LHS by the gμρ , the indices become repeated and lose their 'fixation' so to speak...the g gives degrees of freedom to the indices somehow...
    Now I can sleep at night and so does professor Bel happilly in his tomb...
     
  7. Sep 26, 2012 #6
    The link that I had provided above interestingly provides an answer to the question 15.2 in chapter 15 of the Book gravitation by MTW which concerns the derivative of the Bel Tensor...
    It took me so many days for this challenging yet rich and illuminating Ex!
     
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