Is D'Inverno's Equation (11.38) Correctly Integrated?

[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 L_G on the LHS and the L_G barred on the RHS) have not been integrated. How can you get away with integrating only part of an equation??

 

[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.
(g^ab\Gamma^a_bc),_c gives two terms and (g^ab\Gamma^c_bc),_a gives another two terms. A bit of index manipulation on two of the six terms in the two equations produces Q^a,_a and a bit more manipulation produces the L_G and the L\bar{}_G leaving two terms -g^ab,_c\Gamma^c_ab + g^ab,_b \Gamma^c_ac. 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!

 

[Altabeh]

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.
(g^ab\Gamma^a_bc),_c gives two terms and (g^ab\Gamma^c_bc),_a gives another two terms. A bit of index manipulation on two of the six terms in the two equations produces Q^a,_a and a bit more manipulation produces the L_G and the L\bar{}_G leaving two terms -g^ab,_c\Gamma^c_ab + g^ab,_b \Gamma^c_ac. 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!

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

 

[madness]

I worked from d'Inverno and came across the same thing. I think if you revisit the regular integration by parts you will see that it is just the dervitative of a product but done in reverse.

 

[TerryW]

Well, not quite! You have to take one term over to the other side of the equation to get the mantra "First times the integral of the second minus the integral of the differential of the first times the integral of the second" - I still remember that after 45 years!