Quantum Mechanical amplitude for particle falling into a black hole

Consider a black hole spacetime originated by gravitational collapse, like the following Vaidya geometry

$$ds^2=-\left(1-\frac{2M\theta(v)}{r}\right)dv^2+2dvdr+r^2(d\theta^2+\sin^2\theta d\phi^2).$$

Next, consider a massless Klein-Gordon field propagating in said background. Consider the asymptotic quantization on $$\mathcal{I}^-$$. We can introduce a set of solutions $$f_i$$ which are positive frequency with respect to advanced time and are orthonormal in the KG inner product. This decomposes the field as $$\phi=\sum_i a_i f_i+a_i^\dagger f_i^\ast.$$

This defines a vacuum $$a_i|0\rangle_{\text{in}}=0$$ and one associated (in) Fock space $$\mathscr{F}_{\text{in}}$$. One such state is a state on which we have a number of ingoing massless particles.

Next suppose the field is on some one-particle state $$a_i^\dagger|0\rangle=|1_i\rangle.$$ In other words: an observer in the asymptotic past Minkowskian region will see one particle in some mode.

Now, the particle enters spacetime and interact only gravitationaly (in other words, the field doesn't interact with anything else apart from gravity).

The question is: how can I mathematically describe quantum mechanically the amplitude for the particle fall through the horizon into the black hole?

I don't see how to do this because I'm used to getting amplitudes out of observables and eigenstates of such observables. So if $$A$$ is an observable with eigenstates $$|a_i\rangle$$ the probability amplitude for result $$i$$ is $$\langle a_i|\phi\rangle$$ on state $$|\phi\rangle$$. This doesn't seem to fit here.

Edit: this question of mine is related. Some authors like Parker and Hawking claim that a certain quantity is the "absorption cross section for a mode of frequency $$\omega$$". In particular, I believe that the right answer to this question should in the end connect to that and yield the absorption cross section these authors mention.

• Don't we need quantum gravity for this? – my2cts Dec 4 '18 at 22:12
• I personally think we don't. We are treating the gravitational field as classical and studying quantum matter propagating on it. I'm not even considering the backreaction of the quantum field on the metric. It is just a matter of how do we make sense of a particle falling into the black hole. The central problem seems to be that the idea of particle just seems well defined at $\mathcal{I}^\pm$ – user1620696 Dec 5 '18 at 1:55
• From a classical viewpoint there seems to be a problem of coordinates: to an outside observer a (classical) particle will never cross the event horizon of a black hole, so one must use coordinates that are relevant to the object that is falling, which I am not sure if it would be valid quantum-mechanically. – Quantumness Feb 26 at 0:56
• Thanks for an interesting question! Wouldn't we inevitably need to extend the definition of our quantum theory inside the horizon (i.e., to include particle-states localized inside the horizon) to compute this amplitude? Otherwise, even after assuming we have the propagator $U$, what would we bra the ket $U|1_i\rangle_{\mathcal{I^{-}}}$ with? – Dvij Mankad Feb 26 at 1:11
• @DvijMankad I think that if we could define particle-states localized inside the horizon, then this would help, but as far as I know this can't be done because inside the horizon the spacetime isn't static. Even if it could be done I see another issue. It is well-known that the definition of particle in curved spacetimes is a complicated one which is observer-dependent. So what the observer at $\mathcal{I}^-$ sees need not agree with this idea of particle inside the horizon. Actually, the question "what would we bra the ket $U|1_i\rangle_{\mathcal{I}^-}$" is my main doubt here. – user1620696 Feb 26 at 1:39