I'm reading the book "Modeling Black Hole Evaporation" by Alessandro Fabbri and Jose Navarro-Salas, and in section 3.3.2 they talk about wavepackets at $\mathscr{I}^+$.

It all starts like this: one considers solutions to the massless KG equation which vanish at the future horizon and have positive frequency $\omega$ at $\mathscr{I}^+$. This gives one orthonormal set $\{p_\omega\}$ with

$$p_\omega=\dfrac{1}{4\pi\sqrt{\omega}}\dfrac{e^{-i\omega u}}{r}.$$

This gives rise to the operators $a_{\text{out}}(\omega),a_{\text{out}}^\dagger(\omega)$ which create and annihilate quanta on the mode $p_\omega$, and to the associated number operators $N^{\text{out}}(\omega)=a_{\text{out}}^\dagger(\omega)a_{\text{out}}(\omega)$.

The authors them say:

Let us clarify the meaning of the quantity $\langle \text{in}|N^\text{out}(\omega)|\text{in}\rangle.$ It gives the mean particle number detected at $\mathscr{I}^+$ with a definite frequency $\omega$. The use of this type of states, with a definite frequency, implies absolute uncertainty in time. Therefore $\langle \text{in}|N^\text{out}(\omega)|\text{in}\rangle$ provides, indeed, the number of particles with frequency $\omega$ emitted at any time $u$. But we are mainly interested in evaluating the mean particle number produced at late retarded times $u\to +\infty$, when, in a realistic situation, the black hole has settled down to a stationary configuration. To properly evaluate the late time particle production one has to replace the plane wave type modes, which are completely delocalized, by wave packets.

One can introduce a complete orthonormal set of wave packet modes at $\mathscr{I}^+$, with discrete quantum numbers, as follows:

$$p_{jn}=\dfrac{1}{\sqrt{\epsilon}}\int_{j\epsilon}^{(j+1)\epsilon} d\omega e^{2\pi i\omega n/\epsilon}p_\omega$$

with integers $j\geq 0,n$. These wave packets are peaked about $u=2\pi n/\epsilon$ with width $2\pi/\epsilon$. Taking $\epsilon$ small ensures that the modes are narrowly centered around $\omega \approx j\epsilon$. Therefore, the computation of $\langle\text{in}|N^{\text{out}}_{jn}|\text{in}\rangle$, associated with the wavepacket, has a clear physical interpretation. It gives the counts of a particle detector sensitive only to frequencies within $\epsilon$ of $\omega_{j}$ which is turned on for a time interval $2\pi/\epsilon$ at time $u=2\pi n/\epsilon$.

Now, why $p_{jn}$ can be interpreted so that the associated number operator's $N_{jn}^{\text{out}}$ mean value gives the count of a particle detector sensitive only to frequencies within $\epsilon$ of $\omega_{j}$ which is turned on for a time $2\pi/\epsilon$ at $u = 2\pi n/\epsilon$?

I do understand that if $p_\omega$ is a state of definite frequency so that the frequency operator acts as $\Omega p_\omega = \omega p_\omega$, then if $\epsilon$ is small, $p_{jn}$ has mean frequency close to $j\omega$ and the probabilities are equaly distributed in the interval.

I don't understand the reasoning about time of detection which I wrote in bold. Why does that interpretation holds?

Notice that the same thing could be stated in flat spacetime replacing $p_\omega$ by the plane wave of positive frequency $\omega$ and replacing $u$ by the inertial time $t$.


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