I had previously asked this question. This is kind of a continuation of that.

I recently found this expression which seems to be called the "Fierz-Pauli action" which is apparently the quadratic action for gravitons about $AdS_{d+1}$,

$S = \frac{c }{2 l_p ^{d-1} }\int d^{d+1}x \sqrt{-g} \left ( \frac{1}{4} \nabla_\mu h_{\rho \lambda}( \nabla^{\mu} h^{\rho \lambda} - 2\nabla^\rho h^{ \mu \lambda}) + \frac{1}{2}\nabla_\mu h^{\mu \nu}\nabla_\nu h - \frac{1}{4} \nabla_\mu h \nabla^\mu h - \frac{d(d-1) }{2L^2} ( h^{\mu \nu}h_{\mu \nu} - \frac{1}{2} h^2 ) + O(h^3) \right )$

- Can someone help derive (or reference) this?

I am aware of general expressions for curvature invariants expanded to second order in metric fluctuation about some chosen background. But its not clear to me as to how does one covariantly impose that the background is some $AdS$...

This post imported from StackExchange Physics at 2014-06-15 18:05 (UCT), posted by SE-user user6818