Error in Bel's 1958 Article: Correcting Lanczos Formula

[zn5252]

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 : R^αβλ_μ R_αβλρ = 2 * 1/8 * R^αβλμ R_αβλμ g_μρ
Now multiply out by g^μρ on both sides and elevate the μ index on the LHS, we would end up having :
R^αβλεg_με R_αβλρg^μρ =
2 * 1/8 * R^αβλμ R_αβλμ g_μρ g^μρ
This becomes :
R^αβλεδ_ε^ρ R_αβλρ =
2 * 1/8 * R^αβλμ R_αβλμ g_μρ g^μρ
which leads to :
R^αβλρR_αβλρ = 2 * 1/8 * R^αβλμ R_αβλμ g_μρ g^μρ
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 :
R^αβλρR_αβλρ = 2 * 1/8 * R^αβλμ R_αβλμ
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

 

[Bill_K]

For the RHS he says 2 A g_μν where A = (1/8) R^αβλμ R_αβλμ. 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) R^αβλσ R_αβλσ instead.

 

[zn5252]

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...

 

[Bill_K]

Yes, g_μρg^μρ is δ_μ^μ, the trace of the Kronecker delta, which is the dimensionality N, that is 4 in 4 dimensions.

 

[zn5252]

Indeed. I'm realizing now that when we multiplied the LHS by the g^μρ , 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...

 

[zn5252]

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!