  The problem of gravitational fluctuations confined inside a finite cutoff at
radius $r=r_c$ outside the horizon in a general class of black hole geometries
is considered. Consistent boundary conditions at both the cutoff surface and
the horizon are found and the resulting modes analyzed. For general cutoff
$r_c$ the dispersion relation is shown at long wavelengths to be that of a
linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent
line-integral formula for the diffusion constant $D(r_c)$ is derived. The
dependence on $r_c$ is interpreted as renormalization group (RG) flow in the
fluid. Taking the cutoff to infinity in an asymptotically AdS context, the
formula for $D(\infty)$ reproduces as a special case well-known results derived
using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound
goes to infinity, the fluid becomes incompressible and the Navier-Stokes
dispersion relation becomes exact. The resulting universal formula for the
diffusion constant $D(horizon)$ reproduces old results from the membrane
paradigm. Hence the old membrane paradigm results and new AdS/CFT results are
related by RG flow. RG flow-invariance of the viscosity to entropy ratio $\eta
/s$ is shown to follow from the first law of thermodynamics together with
isentropy of radial evolution in classical gravity. The ratio is expected to
run when quantum gravitational corrections are included.
