- #1
leo.
- 96
- 5
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 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##.
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?
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 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?