long-wavelength non-linear analog of the double trace deformation seen for the scalar toy model. Moreover, this solution allows us to see that the dynamics of the fluid on the Dirichlet surface, as given by the conservation equation on $\Sigma_D$, ‘emerges’ as collective dynamics of the boundary CFT. In particular, the boundary fluid lives on a ‘dynamical background’, and the effects of the background can be suitably subsumed into the fluid description. This suggests that the correct way to think about the hypersurface physics is in terms of a ‘dressed fluid’ living on an inert geometry. Thus, the effective description of a fluid on this dynamical background is geometrically encapsulated in terms of the Dirichlet hypersurface dynamics.

Examining the resulting dynamics on $\Sigma_D$ we find that the hypersurface or effective fluid suffers from a possible pathology for $r_D$ smaller than some critical value $r_{D,snd}$. At $r_{D,snd}$ the sound mode of the effective fluid starts to propagate outside the inert background $\Sigma_D$’s light-cone. We suggest that in the CFT, this effect is due to the extreme forcing of the fluid on the boundary by the ‘dynamical’ metric, and moreover propose that one can obtain sensible dynamics by projecting out the sound mode. 