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In the original computation of Hawking radiation one starts with a gravitational collapse spacetime (like the Vaidya geometry with $M(v)=M_0\theta(v-v_0)$). Then one introduces three complete sets of modes:

The ingoing modes defned to be positive frequency with respect to advanced time at $\mathscr{I}^-$. These define the "in" Fock space. The outgoing modes defined to be positive frequency with respect to retarded time at $\mathscr{I}^+$ and to vanish at the horizon. These define the "out" Fock space. The horizon modes. These are defined to vanish at $\mathscr{I}^+$. They are also chosen to be positive frequency with respect to some parameter on the horizon, although there is amibiguity in this choice. These define the "horizon" Fock space. With these definitions one can show that if a massless scalar quantum field is in the vacuum state of the "in" Fock space, namely $|0\rangle_{\text{in}}$ then it is

$$|0\rangle_{\text{in}}=\prod_{\omega}\dfrac{1}{\cosh r_\omega}\sum_{n=0}^\infty (\tanh r_\omega)^n |n_\omega\rangle_{\text{out}}|n_\omega\rangle_{\text{horizon}}, $$

where $\tanh r_\omega = e^{-4\pi M\omega}$. Then one traces the horizon part out and gets Hawking's thermal density matrix.

There is one issue with this though. The outgoing modes are what one observer on the far asymptotic future detects as particles. Now if the black hole evaporates, in the far asymptotic future there should be no horizon anymore, right?

How can we have a state in a tensor product $\mathscr{H}_{\text{out}}\otimes \mathscr{H}_{\text{horizon}}$, if it seems that for the observer in the asymptotic future there is no horizon anymore?

Or there is an interval on $\mathscr{I}^+$, say $u\in (-\infty,u_0)$ such that the horizon still exists and it is only after $u_0$ that the horizon has disappeared?

I believe the source of my confusion is that since $\mathscr{I}^+$ is the asymptotic future, it seems that somehow all events comprising the horizon "lie to the past of it", so that the splitting wouldn't make sense.

What am I missing? How that splitting make sense?

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The original Hawking's calculation does not include the effects of backreaction. So, while outgoing radiation carries away energy it does not (in this calculation) modify the background metric and does not change the position of the horizon. Therefore black hole does not really evaporates.

This means that the calculation becomes inapplicable in situations where quanta of radiation carry away significant portion of the black hole mass. So, the UV limit of radiation spectrum and final moments of black hole evaporation would require different approach. However, if we restrict ourselves to situations where only tiny portion of black hole mass evaporates during characteristic time of interaction, and if we specify our boundary conditions at a finite distances large enough that particles detected there could be identified with particles at infinity, yet small enough that black hole does not change considerably during the round-trip of radiation, this approximation should work quite well.

An example calculation that does include the backreaction of the radiation could be found here.

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