# D'Inverno - another query

1. Apr 16, 2010

### TerryW

The attached file contains part of page 151 from D'Inverno's Introducing Einstein's Relativity. Beneath the shaded equation (11.38) he continues with 'Integrating the first two terms in (11.37) by parts, we get....' ....equation (11.39). This doesn't look right to me because the original equation has one term on the left hand side and three terms on the right hand side but in the resultant equation, two of the terms (the LG on the LHS and the LG barred on the RHS) have not been integrated. How can you get away with integrating only part of an equation??

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2. Apr 22, 2010

### TerryW

I've had another look at this and worked out what is going on.
Basically, integration by parts is not being used at all. He is using the differentiation of a product.
(gab$$\Gamma$$abc),c gives two terms and (gab$$\Gamma$$cbc),a gives another two terms. A bit of index manipulation on two of the six terms in the two equations produces Qa,a and a bit more manipulation produces the LG and the L$$\bar{}$$G leaving two terms -gab,c$$\Gamma$$cab + gab,b $$\Gamma$$cac. These can then be organised into (11.39) as given.

Sorry about the Latex rendition. A couple of terms don't appear as they should but I can't see where the Latex is wrong!

3. Apr 22, 2010

### Altabeh

You got it just right! I think D'inverno thought of "integration by parts" simply as separating terms into two parts. Indeed, he applies the Leibniz rule to the two first terms and then rearrange terms in such a way that (11.39) gets produced!

AB

4. Apr 22, 2010