While the first order metric variation of Hilbert-Einstein action plus Gibbons-Hawking-York boundary term is well-known and takes the form:
$\delta S_{HE}+\delta S_{GHY}=-\frac{1}{16\pi G}\int d^3x \sqrt{-\gamma}~(K^{\mu\nu}-K\gamma^{\mu\nu})\delta \gamma_{\mu\nu}$
($K_{\mu\nu}$ and $\gamma_{\mu\nu}$ are the extrinsic curvature and the projection metric), does anybody know the second order variation of this combined action? Thanks to anyone who can help me!
A long time ago, when I was in graduate school, a friend and I went to Bryce DeWitt with a similar question. He thought about it for a while, and then said to us:
"Do you know what your problem is? Too much book learning."
He explained that the answer could almost certainly be found in a book or paper, if we looked hard enough, but we would learn much more by working out the answer ourselves. This may be the best advice I ever received as a graduate student.
Why am I telling you this? Because the answer to your question about the first variation of the action (and the notes I linked to) provides you with everything you need to solve this problem. Give it a try! If you get stuck on a specific step in the derivation, come back here and ask about it -- I'll be happy to work through the details with you. But this is a great chance for you to apply what you learned from your last question.
