tackle this problem directly we will take a slightly different tack in this paper, one which we believe clarifies some aspects of evolution in the radial direction and its possible connection to RG flows. One of our main conclusions will be that imposing rigid cut-offs in AdS is more naturally viewed in terms of perturbing the CFT by some non-local deformation or equivalently by introducing explicit state-dependent sources in the boundary theory.

A second motivation is the recent work [@Bredberg:2010ky; @Bredberg:2011jq] which derives an explicit map between solutions of vacuum Einstein equations (with no cosmological constant) and those of incompressible Navier-Stokes equations, thereby making direct contact with some of the ideas of the black hole membrane paradigm in asymptotically flat spacetime [@Damour:1978cg; @Thorne:1986iy]. This problem, which has been further generalized in [@Compere:2011dx], is the zero cosmological constant analog of the problem we consider (see also [@Eling:2009pb; @Eling:2009sj] for another approach and [@Cai:2011xv] for related work). The idea is to consider a fixed timelike hypersurface with Dirichlet data enforcing a flat metric on the slice. Given these boundary conditions one wants to solve vacuum Einstein equations so as to obtain a solution which has a regular future horizon.[^7] By explicit construction 