such a seed solution is obtained by simply performing a coordinate transformation of the well known planar Schwarzschild-AdS black hole.

But this is precisely the set-up of [@Bhattacharyya:2008ji; @Bhattacharyya:2008mz], the only difference being the fact that in these works the Dirichlet data is imposed on the boundary at $r =\infty$. Let’s call this boundary metric $g_{\mu\nu}$, which is also by definition slowly varying etc.. The solution to the asymptotic Dirichlet problem is characterized by the boundary metric $g_{\mu\nu}$, a distinguished velocity field $u^\mu$ (which is unit normalized) and a scalar function $b$ (determining the temperature or equivalently the local energy density). Let us denote these variables collectively as ${\mathfrak X}$. The boundary Brown-York stress tensor (up to counter-terms) takes the fluid dynamical form and is built out of the data contained in ${\mathfrak X}$.

Now given the space of solutions to the asymptotic Dirichlet problem, we can reparametrize that space of solutions appropriately to obtain the solutions of the new bulk Dirichlet problem. The only condition we have to satisfy is that the induced metric on $\Sigma_D$ in the solutions obtained this way[^10] be equal to ${\hat g}_{\mu\nu}$. Furthermore, we can extract the stress tensor on $\Sigma_D$[^11]. We will argue 