Understanding Wald's Book: Appendix E and the Einstein Equation

[atrahasis]

Hi,

I have a question concerning the Wald's book: General Relativity.
In the appendix E, he derived the Einstein equation by considering the surface term (GHY).
I do not understand what he said after the equation (E.1.38).

Actually he considers that $h^{bc}\nabla_c(\delta g_{ab})=0$, because we fix $\delta g_{ab}=0$ on the surface, but therefore why the other term in (E.1.38) is not null, the term $h^{bc}\nabla_a(\delta g_{bc})$.

They look the same for me, and after some algebra, where we replace the covariant derivative by the one compatible with the metric on the surface we should have a total derivative term on the surface that we can integrate away.

Thanks in advance

 

[atrahasis]

Ok finally I got it.