Recently a physicist I know said he was interested in the following question: "would one observer following Kruskal-Szekeres coordinates in the presence of a Schwarzschild black hole (i.e., whose comoving coordinates are K-S) experience Hawking radiation?".

Now, if I interpreted this correctly, what he wants to know is: given a Schwarzschild spacetime $(M,g)$, we consider Kruskal-Szekeres coordinates, defined in terms of the usual $(t,r,\theta,\phi)$ coordinates by

$$T=\left(\dfrac{r}{2GM}-1\right)^{1/2}e^{r/4GM}\sinh\dfrac{t}{4GM},\quad X=\left(\dfrac{r}{2GM}-1\right)^{1/2}e^{r/4GM}\cosh\dfrac{t}{4GM}$$

on the exterior region $(r > 2GM)$ and $$T=\left(1-\dfrac{r}{2GM}\right)^{1/2}e^{r/4GM}\cosh\dfrac{t}{4GM},\quad X=\left(1-\dfrac{r}{2GM}\right)^{1/2}e^{r/4GM}\sinh\dfrac{t}{4GM}$$

for the interior region $(r<2GM)$.

Then we consider one observer, whose worldline is one of the coordinate lines of the coordinate function $T$. Namely, we fix $X,\theta,\phi$ and consider the curve $\gamma : \mathbb{R}\to M$ in coordinates given by

$$T\circ\gamma(\tau)=\tau,\quad X\circ\gamma(\tau)=X_0,\quad \theta\circ \gamma(\tau)=\theta_0,\quad \phi\circ\gamma(\tau)=\phi_0.$$

We thus consider one detector following these coordinates and we ask ourselves if it detects Hawking radiation. I believe this is the precise statement of the problem he has in mind. I believe the whole point to ask this question is that in these coordinates there is no apparent singularity corresponding to the event horizon.

My question here is: has this been discussed on the literature? Is there any reference on which this is discussed? If there is, where I can find it?