  By foliating the four-dimensional C-metric black hole spacetime, we consider
a kind of initial-value-like formulation of the vacuum Einstein's equation, the
holographic initial data is a double consisting of the induced metric and the
Brown-York energy momentum tensor on an arbitrary initial hypersurface. Then by
perturbing the initial data that generates the background spacetime, it is
shown that, in an appropriate limit, the fluctuation modes are governed by the
continuity equation and the compressible Navier-Stokes equation which describe
the momentum transport in non-relativistic viscous fluid on a flat Newtonian
space. It turns out that the flat space fluid behaves as a pure vortex and the
viscosity to entropy ratio is subjected to the black hole acceleration.
