Question

Theory predicts that uniform acceleration leads to experiencing thermal radiation (so called Fulling Davies Unruh radiation), associated with the appearance of an event horizon. For non uniform but unidirectional acceleration the shape of the experienced radiation changes from thermal to other spectral densities, but also is predicted to exist. But suppose the acceleration is periodic and oscillatory, i.e. no permanent horizon persists? In particular, what about the case of harmonic motion, for a full cycle, half a cycle, etc.?

Here is an even simpler related problem that makes the apparent paradox easier to see. Suppose at proper time t=0, I accelerate at constant acceleration k in the x direction for t0 seconds, presumably experiencing Unruh radiation. Then I accelerate with acceleration -k, (in the -x direction,) for 2*t0 seconds, seeing more Unruh radiation coming from the opposite direction, and then I finish with with acceleration +k for the final t0 seconds. At the end of the 4*t0 proper seconds, I'm back where I started, at rest, without any event horizon. Was the Unruh radiation I felt when reversing acceleration secretly correlated or entangled with the radiation I initially and finally saw? Otherwise, from a more macro scale, I didn't actually necessarily move much, and the acceleration event horizon was instantaneous, evanescent and fleeting, so whence arose the Unruh radiation?

Answer 1

I will attempt a qualitative answer to the question.

We assume that we have an observer that is performing an oscillatory motion in which we have a repetition of two constant acceleration phases. The trajectory would be the sum of parts of hyperbola, where each part would correspond to a constant acceleration phase (k or ?k) with the acceleration directed towards the x=0. In this motion there will be no event horizon, but there will be several apparent horizons made out of the corresponding Rindler horizons of each part of the orbit that has constant acceleration.

There is a lot of work that supports the fact that Unruh/Hawking-like radiation can be observed from apparent horizons (see here, here and here for example). So my guess would be that if the essential features that Visser describes exist, then you would see radiation. Thus in this particular case, for each accelerated branch the observer would se thermal radiation corresponding to acceleration k as long as the acceleration phase is long enough to satisfy the condition for slow evolution of the apparent horizon.

I am guessing that you can apply the same reasoning for the case of harmonic motion, as long as the period of oscillation is long enough. The spectrum of the radiation would evolve in time and would probably be something like an integrated Planck spectrum over a range of temperatures which would probably be something like a power law.