I have heard of the fact (see e.g. here) that there exists no everywhere-regular Hadamard (vacuum) state for quantum field in Kerr spacetime. My understanding is that the Hadamard condition provides "good" short-distance (UV) property which is to some extent universal, and related to the fact that locally spacetime is approximately flat (and this is reflected in the properties of the vacuum state).
Does this pose a big difficulty for QFT in curved spacetime paradigms? For one, most astrophysical black holes are rotating (and not very slowly), so even if one day we could measure things like Hawking effect directly, Schwarzschild black hole as a background metric would not provide a good model.
Another alternative would be simply to try to find two states that are regular e.g. on the horizon and also at (spatial) infinity, and then interpolate in between. Has there been any progress on this front for QFT on top of Kerr black holes?
Update: we should probably distinguish eternal Kerr black hole from those formed by collapse. I didn't consider the latter because it looks to me that there should be "no problem", since before the black hole forms the vacuum should have good Hadamard property. At the same time, it is not clear to me what is the final state after the black hole has formed, and how it relates to the eternal black hole vacua.