A Quantum amplitude for a particle falling into a black hole

[leo.]

Here we consider a black hole formed by gravitational collapse classically. We also consider a scalar massless Klein-Gordon field propagating on this background.

To quantize the field we expand it in appropriate modes. The three sets of modes required are:

• The incoming modes, appropriate for an observer at past null infinity. This is a complete set of solutions $\{f_i\}$ to the KG equation of positive frequency with respect to advanced time. It decomposes the field as $$\phi=\sum a_i f_i+a_i^\dagger f_i^\ast$$
• The outgoing modes, appropriate for an observer at future null infinity. This is a complete set of solutions $\{p_i\}$ to the KG equation of positive frequency with respect to retarded time and zero at the horizon.
• The horizon modes, which are required because future null infinity alone isn't a Cauchy surface. They are any complete set of solutions $\{q_i\}$ to the KG equation which are zero at future null infinity.

The horizon and outgoing modes together allow for the expansion of the field as $$\phi=\sum b_i p_i+b_i^\dagger p_i^\ast + c_i q_i + c_i^\dagger q_i^\ast$$
The incoming modes define the in vacuum $a_i|0\rangle_{\text{in}}=0$. The outgoing modes define the out vacuum $b_i|0\rangle_{\text{out}}=0$. Although the horizon modes do not have one unambiguous meaning, we still define a Fock space and a vacuum by $c_i|0\rangle_{\text{hor}}=0$.

Now, physically, it seems meaningful to ask the following question: we have one incoming particle $|\psi\rangle = a_i^\dagger |0\rangle_{\text{in}}$. The particle interacts with the gravitational background. What is the probability amplitude for it being absorbed by the black hole?​

If on the one hand the question seems physically meaningful, I am having a very hard time to work this out.

First, by the Hawking effect we know that the meaning of particles change when go all the way to future null infinity. Also, on the horizon particles seem meaningless. This is very weird, it seems in a sense the incoming particle "looses its identity". So if we can't track it, how can we even talk about an amplitude for it falling in the black hole?

More than that, I'm unaware of any interaction through which we can compute one $\cal{S}$-matrix for this

So how can we make sense of this? Can we mathematicaly make precise the situation on which a particle is thrown into a black hole and compute matrix-elements for, e.g., the absorption of the particle by the whole, or its escape to future null infinity?

 

[thierrykauf]

Are you reading Hawking's paper?

 

[leo.]

Are you reading Hawking's paper?

Yes, the "Particle Creation by Black Holes" as well as some other references like Parker's QFT in Curved Spacetimes book. But all of them seem to discuss a different matter: how one observer at $\mathcal{I}^+$ perceives the natural vacuum for an observer at $\mathcal{I}^-$. This is answered by the Bogolubov transformations, and if I got this right it envolves a classical wave scattering to be studied on the way. I didn't find up to this point in what I read how to set up a quantum scattering like that in such a spacetime, with for instance, the probability of a particle falling into the hole being computed with quantum field theory.

 

[thierrykauf]

If the particle falls into the black hole, is it still scattering? How do you define your S matrix? (I'm rusty on this)

 

[leo.]

If the particle falls into the black hole, is it still scattering? How do you define your S matrix? (I'm rusty on this)

That's exactly my problem. I don't know how to define the S matrix. It may really be the problem that this doesn't fit scattering theory at all, although I have one impression that it can be seen as scattering. In Hawking's papers for instance, one considers $\mathcal{I}^-$ to be one "initial Cauchy surface" and $\mathcal{H}^+\cup \mathcal{I}^+$ to be one "final Cauchy surface".

My issue with the definition of an S matrix is more or less along the following lines:

1. Given one initial incoming state $|\psi\rangle_\text{in}$ defined at $\mathcal{I}^-$ what are the states we would project it against in order to define the S matrix?
2. Also, what is the interaction generating the S matrix? In usual QFT we have one interaction potential, here I don't see it at all.

Sorry if anything I'm saying is totally silly, I'm trying to set up this properly, but I confess that I'm still very confused on how quantum scattering theory works out in the presence of black holes.