Hawking radiation was calculated by Hawking in an idealized collapsing star, really an evolving Schwarzschild solution where the star moves into its horizon. But it is true and perhaps easier to understand in an already collapsed one, or really more accurately a perennial stationary black hole. Either way, he calculates what happens to the virtual particles in the vacuum. One way of interpreting it is that statistically some will follow geodesics away from the horizon and others towards it. The one directed outwards may escape the gravitational field, the other one will get pulled in. Since the escaping one has to become a real particle (it escapes and we may physically detect it), it is as if the other one going into the horizon has negative energy, and can be interpreted as a positive energy particle leaking out of the black hole.
There is a more accurate way to do this and it is to calculate it as photons tunneling out from inside the horizon. Tunneling can be explained quantum mechanically. The two explanations are equivalent. Hawking had to do this actually using a quantum field calculation in the background black hole general relativistic gravitational field.
See the Wikipedia article on Hawking radiation to get an idea.
EDIT after OP comment right below:
So, more on Hawking's calculation. His calculation is rather complicated, and is done in a number of other reviews and treatments. One that has his calculations, but also has material for the Unruh effect and Rindler coordinates in flat spacetime and other details that make it easier to follow, is openly available at https://arxiv.org/pdf/gr-qc/0010055.pdf
But a similar calculation I found a little easier to follow is here: http://kiso.phys.se.tmu.ac.jp/thesis/m.h.kuwabara.pdf
Kuwabara's pdf is a review of the Hawking radiation, and explains the approach and calculations. He first explains QFT in curved spacetime, with the difference from Minkowski (and thus standard QFT) that the metric is the GR metric, not Minkowski. The metric (i.e., gravitational field) is a fixed classical GR background, and is not quantized. The field is quantized. He does it for a free scalar field in a Schwarzschild metric (also does it for a sandwich expanding universe, but Hawking and others have done it with the same thermodynamic results for more general charged and rotating stationary black holes, and other quantum fields). He does not treat the Unruh or Rindler coordinates, the other online link I gave above does.
In both links noted above it is noted that one can change coordinate systems, and actually needs to in order to be able to solve and interpret the equations in the curved spacetime, and eventually get to a wave equation with a potential. It can be solved as an expansion in terms of the standard mode operators $a_k$ and its adjoints. It is also noted that the vacuum in diferent reference frames can look different, and the mode operators need to be transformed using Bogoliubov transformations in order to interpret the out states in terms of the in states. This is true also in flat spacetimes in an accelerated reference frame, where one gets Unruh radiation. In Kuwabara he then goes on to calculate the particle number for the radiation, in terms of the transformed mode operators, and gets the expression for the blackbody radiation that Hawking found. He discusses and you have to treat carefully the in and out states.
That's not exactly what you are asking about inbound or outbound geodesics, but this represents the QFT approach taken by Hawking. Maybe this is what you meant. I am not sure, so hope it may help.
