What are the quadratic order fluctuation terms for curvature tensors about some background metric?

Question

Like one can see the linear order terms on page 18 here, http://www.ugr.es/~bjanssen/text/fierz-pauli.pdf

- I would like to know of a reference for the derivation of the quadratic order terms for the Riemann tensor, scalar curvature and Ricci tensor and for $\sqrt{-g}$?

- Further on such an expression how does one impose that the background is say $AdS$? (even for the linear case as linked above)

- Is there a way by which Mathematica can be made to derive these expressions?

Answer 1

- I would like to know of a reference for the derivation of the quadratic order terms for the Riemann tensor, scalar curvature and Ricci tensor and for $\sqrt{-g}$?

Many introductory GR books and lectures: Carroll, Weinberg, Feynman lectures on gravitation, Misner, etc. You should work it out on your own, though. You just need to split the metric in a background plus a fluctuation and treat the fluctuation as a tiny perturbation.

- Further on such an expression how does one impose that the background is say $AdS$? (even for the linear case as linked above)

The background metric is the AdS metric instead of Minkowski. What in a flat background are the components of Minkowski metric are the components of AdS.