Can the S-Matrix formalism (or an extension of it) describe the transition of matter falling into a black hole into Hawking radiation?

What I mean is start with an initial state $| i> = | \psi_i(-\infty)>$ that describes the infalling matter far away from the black hole and assume a final state $| f> = | \psi_f(\infty)>$ where the black hole is evaporated and only Hawking radiation is left.

The transition probability would as usually be $w_{i\rightarrow f} = | M_{if}|^2$ where $M_{if} = \langle f |\hat{S} |i \rangle$ defines an element of the S-Matrix

\[\hat{S} = \hat{T}\exp\left({-\frac{i}{\hbar}} \int\limits _{-\infty}^{\infty} dt \hat{W}_{IP}(t)\right )\]

$\hat{T}$ is the time ordering operator and $\hat{W}_{IP}$ describes perturbation caused by the black hole in the interaction picture.

If the S-Matrix formalism can be applied to this situation, how would one in particular construct $\hat{W}_{IP}$?