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In Hawking's paper "Particle Creation by Black Holes" I'm not really able to pick apart what vacuum state Hawking is assuming the field $\phi$ to be in.

The paper "Hawking radiation as perceived by different observers" by Barbado, Barcelo and Garay seems to suggest that the field should be assumed to be in the Unruh state. But I don't see this assumption in Hawking's paper.

It makes sense that it is the Unruh state, although the Hartle-Hawking state would seem to make sense too. The Boulware vacuum does not seem right since it has a divergent stress-energy tensor at the horizon and so is unphysical in some sense.

In Hawking's result, what state is the field assumed to be in?

EDIT: Since Hawking deals with a Bogoliubov transform, does he not need to assume a precise vacuum?

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First, we need to distinguish between two different cases:

  • Case C (for Collapse): A black hole that formed by collapse.

  • Case E (for Eternal): An eternal black hole.

Case C is more interesting and realistic, but also more difficult. Hawking's original analysis used case C.

Case E is used as a short-cut. The states named in the question (Unruh, Boulware, and Hartle-Hawking) are various options that we can use in case E. Depending on what state we choose in case E, it can be made to immitate some features of the realistic case C.

Here's the key: We can't just say "let's use the same state in case E that we use in case C," because that's meaningless. Cases C and E are different spacetime backgrounds, so we need to be more specific about exactly what we want "same state" to mean.

The short-cut case E

In the short-cut case E, the black hole doesn't form, it was just always there. This is unrealistic, so we need to decide what state we want to use based on some indirect criterion, like deciding which feature of case C we want to immitate.

Which choice is best? "Best" depends on what we're trying to accomplish, but for immitating the most interesting (in my opinion) features of case C, the Unruh state is the standard choice. The Hartle-Hawking state represents a black hole in equilibrium with its surroundings (so it never evaporates), and the Boulware state doesn't look anything at all like empty space for observers falling through the horizon, as noted in the question. The abstract of Unruh's original paper says it this way:

A technique for replacing the collapse by boundary conditions on the past horizon is developed which retains the essential features of the collapse while eliminating some of the difficulties.

In this paper, Unruh introduces the idea of studying case E as a short-cut. But the question doesn't ask about case E. The question asks what state was used in Hawking's paper, which is case C.

The realistic case C

In the more interesting and realistic case C, we should (and Hawking did) use the state that matches the Minkowski vacuum state in the distant past, long before the black hole begins to form. After the black hole forms, spacetime is no longer flat, except very far away from the black hole, but the state has already been chosen. It was chosen based on the conditions long before the black hole began to form, so we just need to propopagate it forward in time. When we do that, we find that even far away from the black hole where spacetime is still flat, the state at late times is no longer the Minkowski vacuum state. Instead, it is a state with outgoing radiation. One way to see this is to observe that it's related to the Minkowski vacuum by a the Bogoliubov transform.

In Hawking's result, what state is the field assumed to be in?

Answer: In the distant past, long before the black hole begins to form, the state is the Minkowski vacuum state. The state at later times is determined by time-evolution in the usual way.

Which state in case E corresponds to the state Hawking used in case C? Again, that's ambiguous. Cases C and E are different spacetime backgrounds, so if we want to compare their states, then we need to be more specific about which special features we want to compare.

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    $\begingroup$ @Chiral_Anomaly Thanks for the nice answer. I have a question: is it really true that in Case C the spacetime is really Minkowski space before the collapse? It seems that the initial field solution $\phi = \sum_i ( f_i \mathbf{a}_i + \bar{f}_i \mathbf{a}_i^{\dagger} )$ in Hawking's eq.(2.3), is solved on past null infinity ($\mathscr{I}^{-}$ in the Penrose diagram). This seems to be part of Schwarzschild space (although at the very far "edge" of it). Wouldn't there have to still be a star of mass $M$ in the spacetime at early times (before it collapsed)? $\endgroup$ Mar 19, 2021 at 14:08
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    $\begingroup$ @QuantumEyedea You're right. I should have been more careful. The mass is already present in the distant past. It's much more spread out, but still, the spacetime isn't exactly flat. The important thing is that in case C, the past asymptotic region where the spacetime does approach Minkowski spacetime (past null infinity) can be used as a kind of Cauchy surface: if we specify the state there, then we've uniquely determined its whole future. Case E doesn't have that nice property, and Unruh worked around this by considering initial conditions on the past horizon, too. $\endgroup$ Mar 19, 2021 at 19:29
  • $\begingroup$ @Chiral_Anomaly Great, that makes sense! One more follow up question: what is different about the spacetime in the distant future then? Standing far from the black hole the spacetime is going to look the same right? (with a mass $M$ sitting at the middle) If there is a mass in the past and future, then in the future at $\mathscr{J}^{+}$ (far from the black hole), what has really changed? Is it that the event horizon is now "exposed" in the future? (not being the case in the past?) In Hawking's (2.4) he supplies Cauchy data on $\mathscr{J}^{+}$ and the horizon - is this the difference? $\endgroup$ Mar 20, 2021 at 0:15
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    $\begingroup$ @QuantumEyedea The wording is Hawking's paper is a little confusing, but he's really only specifying the state on past null infinity (equation (2.9)). Above equation (2.4), he says the fields "are completely determined by their data on [past null infinity]." The other text around equation (2.4) is really only defining the distinction between the $b$-operators and the $c$-operators, and then equations (2.5)-(2.6) show how the coefficients of these operators are determined by the state that was specified on past null infinity. So he's actually deriving, not supplying, the future Cauchy data. $\endgroup$ Mar 20, 2021 at 3:12

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