Compressible forced viscous fluid from product Einstein manifolds

We consider the fluctuation modes around a hypersurface $\Sigma_c$ in a $(d+2)$-dimensional product Einstein manifold, with $\Sigma_c$ taken either near the horizon or at some finite cutoff from the horizon. By mapping the equations that governs the lowest nontrivial order of the fluctuation modes into a system of partial differential equations on a flat Newtonian spacetime, a system of compressible, forced viscous fluid is realized. This result generalizes the non bulk/boundary holographic duality constructed by us recently to the case of a different background geometry.