at $r = r_D$. The second boundary condition in general can take the form of a regularity boundary condition in the interior of the spacetime. If we were working in a spacetime with a horizon this would demand that the mode functions of interest are purely ingoing at the future horizon.

The question we wish to pose is the following: usually in an asymptotically spacetime we know that the solution to the scalar wave equation above has two linearly independent modes with power-law fall-off characterized by the source $J_\phi$ and vev $\phi$ of the dual boundary operator ${\cal O}_\Phi$ (which we recall is a conformal primary). We wish to ask what is the characterization of the boundary data as a functional of the Dirichlet hypersurface data. In this simple linear problem it is easy to see that there is a one-one map between the two sets of data. Essentially we are asking how to tune $J_\phi$ and $\phi$ so that the value of the scalar field on the Dirichlet hypersurface at $r=r_D$ takes on its given value.

While it is possible to derive a formal answer to the above question, it is useful to first visit the simple setting of 