scalar field $\Phi(r,x^\mu)$ of mass $m$. Generalizations to other linear wave equations such as the free Maxwell equation are straightforward.

We will let this scalar field propagate on a background asymptotically geometry with spatio-temporal translational symmetries so that the background metric can be brought to the form $$ds^2 =r^2\, g_{\mu\nu}(r) \, dx^\mu \, dx^\nu + \frac{g_{rr}(r)}{r^2}  \, dr^2 
\label{bggen}$$ The boundary of the spacetime is at $r \to \infty$ and we will assume that the boundary metric is the Minkowski metric on ${\mathbb R}^{d-1,1}$ for simplicity, so that asymptotically $g_{\mu\nu} \to \eta_{\mu\nu}$ and $g_{rr} \to 1$ (as $r\to \infty$).

The dynamical equation of motion for the scalar is the free Klein-Gordon equation which can be written as an ODE in the radial direction for the Fourier modes $\Phi_{k}(r)$ of $\Phi(r,x^\mu)$ $$\Phi(r,x^\mu)=\int \frac{d^dk}{(2\pi)^d} \,e^{i\, k\cdot x} \, \Phi_k(r)\ ,$$ and takes the form $$\frac{1}{\sqrt{-g\, g_{rr}}}\, \partial_r \, \left(\sqrt{-g\, g^{rr}} \,  \partial_r \Phi_{k}(r) \right) - \left(g_{\mu\nu} \,k^\mu\,k^\nu +m^2 \right) \Phi_{k} = 0$$ As a second order equation we need to specify two boundary conditions. We are going to restrict attention to the finite part of the geometry as illustrated in and impose Dirichlet boundary conditions for the field at some hypersurface $\Sigma_D$ 