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

hey all,

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
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:

Bill_K
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.

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:
Bill_K