I'm reading the book of Wald "Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics", and I'm pondering on this problem:
In Minkowski spacetime, we usually quantize our fields with respect to the $t$ coordinate, and a Cauchy surface we use is a t-constant slice (say $t=0$) - corresponding to an inertial observer. We can also quantize our field with respect to an accelerated observer in the region $\mid x\mid > t$, now with the slice $t=0, x>0$, and taking for time coordinate the proper time of one uniformly accelerated observer. The link between the two quantizations is what gives the Unruh effect. Now, instead of using $t=0$, we use another slice $t>0$, $S$ such that this slice now penetrates the region III of the below diagram. For the inertial observer, this will also give the same quantization as before. But for the accelerated one, assuming that our time coordinate still is the parameter $t$ is not clear to me that the resulting quantization will be unitarily equivalent to the preceding one (in the sense of Haag's theorem).
Using the similarity of Minkowski and maximally extended Schwarzschild (ES) spacetimes (this time the region III corresponds to the interior of the black hole) and the equivalent of the surface $t=0$ to the ES yields again two quantizations with respectively associated vacua: Hartle-Hawking (HH) and Boulware (B) (this is well explained in Wald's book). However, what would happen to the quantizations based on the same Cauchy surface $S$ as above? Is it possible that the resulting quantizations are respectively unitarily equivalent to those giving the HH and B vacua?