Question

It's a well known fact that an observer that accelerates at a constant rate from ?c at past infinity to +c at future infinity sees a horizon in flat Minkowski spacetime. This is easy to see from a spacetime diagram once you realize that the union of past light cones on such a trajectory has a boundary that divides the spacetime into two regions - one inaccessible by the accelerating observer.

This leads to the classic result of Unruh radiation when one looks at the quantum field theory for such an observer. The horizon plays a crucial role here.

How does one go about determining whether a horizon is seen by a general class of worldines? In particular, is there any reason to believe that a horizon would exist for an observer that is stationary for all time except for a finite period of acceleration and deceleration?

Is there any other class of worldlines other than an indefinitely accelerating observer for which a horizon is known to exist in flat spacetime?

Answer 1

Since you're really interested in the horizon as part of an approach to understanding the Unruh effect, I will address that instead of the initial Question. What experimental resources are we to consider an accelerating observer A to have? If A can make only local measurements in the laboratory La in the diagram, then (in the Wightman or Haag-Kastler axiomatic backgrounds) A can make exactly the same deductions about the state from their measurements as can an observer who can use the laboratory L0.An inertial laboratory L0 and an accelerated laboratory La Because of the Reeh-Schlieder theorem, what is at space-like separation from the two laboratories is equally accessible to both (in the sense that both can get as close as they like to knowing what the statistics of measurement results would be everywhere in Minkowski space by making precise enough measurements in their local laboratory, on the assumption that the Wightman axioms hold or that the Haag-Kastler axioms hold). [It would have been better not to have drawn the two laboratories the same shape, because the shapes are not relevant to the Reeh-Schlieder argument, but it's uploaded now.]

A local laboratory, however, cannot tell whether the state of the universe is in the folium of the vacuum state, in the folium of a thermal state, or in the folium of some other less symmetrical state; the folium of a highly symmetric state is the set of states that are in the Hilbert space that is constructed by the GNS construction from the vacuum state (resp. from a thermal or some other state) over the net of algebras of observables associated with regions of Minkowski space.

I take your discussion about the Unruh effect that is prompted by the earlier Answers to boil down to the fact that it's only if our laboratory is not confined to a finite region of space-time that we can make any global claims about the state of the universe, despite the Reeh-Schlieder theorem. We could only possibly determine whether the state is in the folium of a thermal state instead of in the folium of the vacuum state if our laboratory is infinite in extent. Having available as our experimental resource a particle that interacts with the vacuum state in an ad-hoc way for the whole of an accelerating trajectory, as is proposed in Unruh and Wald's Phys. Rev. A 29,1047(1984), even though it's infinite in extent, seems unlikely to be enough to make such a fine-grained determination.

I don't plan on putting the terse comments that I made earlier on the Question into this Answer, however they do inform the perspective a little if you choose to think about the Unruh effect in something like the way I have done here. The above is essentially tentative, something I've been mulling for several years, so I'd be glad to have suggestions for how anyone here would fill the gaps or of what I haven't taken into account that will shoot it down.

Finally, thanks @dbrane for the Question. I held off giving it +1 because although it seemed definitely useful it didn't seem quite clear enough, but when I realized how much it was stimulating, I decided that it was better for me than had it been clear.

PS. Hans Halvorson's paper on "Algebraic Quantum Field Theory" (with Michael Mueger), http://arxiv.org/abs/math-ph/0602036, is a good on-line resource, although in many ways it takes you one step higher than Haag. His treatment of the concept of a folium is more sophisticated than Haag introduces in his Local Quantum Physics, in terms of quasi-equivalence of states. Hans is one of the few Philosophers of Physics who can really come up to scratch with Mathematicians on AQFT.