I am trying to find out about the Unruh effect in the case of a finite period of low acceleration.
I understand the following:
- The calculation leading to a thermal spectrum at the Unruh temperature assumes the idealized case of permanent acceleration.
- For a finite period of acceleration, there is still some excitation of a detector, but it is no longer expected to be exactly thermal.
- I think (but please confirm) that in order that the thermal excitation at Unruh temperature be a good first approximation it is sufficient for the starting and finishing velocities to be of order $c$. However I can't remember where I read this.
- But what if the starting and finishing velocities are not of order $c$? That is my question. I am not looking for an exact treatment, merely an order of magnitude estimate. The issue is whether the effect is suppressed in some sort of exponential way as $\Delta v \rightarrow 0$ (where $\Delta v = a t \ll c$ for a duration $t$ of low acceleration $a$).
I have read Martinetti and Rovelli, "Diamond’s temperature: Unruh effect for bounded trajectories and thermal time hypothesis", Class. Quantum Grav. 20 (2003) 4919–4931. They seem to be interested in the same thing but I can't find a clear statement of the case I am interested in. They do obtain a non-zero "temperature" of some kind even for no acceleration (an inertial observer with a finite lifetime) but this seems to be just something to do with energy-time uncertainty.
Another paper I looked at is Akhmedov, Cheremushkinskaya and Singleton, "On the physical meaning of the Unruh effect", arXiv:0705.2525v3 (2007). They state in the introduction, "The real question is whether or not a detector which moves with linear, constant acceleration for a finite time will see particles" but when it comes to it in the body of the paper, they don't present such a calculation, instead stating: "Instead of performing a new calculation for a finite time, linearly, accelerating detector we turn our attention to circular motion." They also reference Sriramkumar and Padmanabhan, "Probes of the vacuum structure of quantum fields in classical backgrounds" (2001) but that paper does not answer my question I think. (But I am not able to follow all the methods employed).
Another way to pose the question is this. If we take the results that have been obtained for eternal circular motion, and apply them to a detector which just executes one revolution, or half a revolution, around a circle, then will be get a totally wrong estimate, or a reasonable one?