This involves looking at the fluid at a scaling limit and this can be formalized as taking the incompressible non-relativistic limit of the fluid on the hypersurface in a manner entirely analogous to the scaling limit described in for generic relativistic fluids in [@Bhattacharyya:2008kq; @Fouxon:2008tb].[^13]

Having understood the Dirichlet problem for generic $\Sigma_D$ away from the horizon ${\mathcal H}^+$, we then proceed to push this surface deeper into the spacetime and ask what happens as we approach the horizon. In this regime $\Sigma_D$ dynamics continues to be described by incompressible Navier-Stokes equations in the limit, though with some slight differences from the BMW limit mentioned above. Zooming in onto the region between $\Sigma_D$ and the horizon, we provide an embedding of the construction of [@Bredberg:2011jq; @Compere:2011dx] into the fluid/gravity correspondence [@Bhattacharyya:2008jc]. Further, we demonstrate that in this limit both the bulk metric in the region between $\Sigma_D$ and the boundary, and the boundary metric degenerate from metrics on a Lorentzian manifold to Newton-Cartan like structures. This raises interesting questions about the natural emergence of the Galilean structures in the AdS/CFT correspondence which we postpone for future work.

The plan of this paper is as follows: In we first address the 