*Title:* In what sense uniformly accelerated charges radiate for inertial
observers but do not for Rindler ones? -- a simplified discussion.


*Summary: *We revisit the long-standing question of whether or not
uniformly accelerated sources radiate for coaccelerating observers; we keep
our discussion as down-to-Earth as possible to avoid nonphysical issues
that mystify the problem. We end up commenting that observing Larmor
radiation reflects the existence of the Unruh thermal bath.


Below we present the main statements. For more details, please download the
corresponding PDF file.


In 1992 it was shown (Higuchi, GM, Sudarsky) that the emission of a usual
photon from a uniformly accelerated charge in the Minkowski vacuum
corresponds to either the absorption from or the emission to the Unruh
thermal bath of a zero-energy Rindler photon (HMS92).


We assume that we all agree that accelerated charges with *physical
classical trajectories* do radiate according to the Larmor formula for
inertial observers. In particular, uniformly accelerated charges for long
enough have an emission rate (approximately) proportional to their proper
acceleration. Indeed, this is used as a benchmark in the discussion below.
We stress that we will keep out issues related to eternally accelerated
charges here.


Perhaps, because the concept of zero-energy Rindler photons may be
unfamiliar or because of the regularization procedure used in (HMS92), or
because of something else, it seems that the 1992 paper was not fully
appreciated. Our primary aim here is to render the original argument given
in (HMS92) in terms of Unruh-DeWitt detectors, which are more familiar.
(Hopefully this helps? I do not know.) Our secondary aim is to call
attention to some recent spin-offs.


The detector is minimally coupled to a massless scalar quantum field *phi*.
For the sake of physical reality, the detector is kept switched on for a
finite *(but very long)* amount of proper time *T*. We assume that the
field quantum state is the Minkowski vacuum.


The rate of particles radiated by the detector as defined by inertial
observers is


*P/T  \propto  \int_0^\infty d \omega_R \delta(\omega_R - E),*


where *E* is the detector energy gap, *\omega_R* is Rindler frequency and
the integral emphasizes that every detector emission of a Minkowski
particle will correspond to either the detector emission or absorption of a
Rindler particle with frequency *\omega_R=E.*


Taking the limit *E -> 0* in *P/T* above, *in which case our detector with
collapsed inner structure can only absorb and emit zero-energy particles*,
we obtain (see PDF file for details)


                                *P/T = (c_0^2 a)/ (4 \pi^2),*


where *c_0* is the coupling constant between detector and field. This is
the emission rate of Minkowski particles that inertial observers would see
being radiated from our collapsed (*E->0*) detector.

EDITOR'S NOTE:  
The PDF file is currently (late May 2020) available at
www.math.tamu.edu/~fulling/ar/matsas1.pdf .
:END EDITOR'S NOTE

Indeed, a straightforward inertial frame calculation for a uniformly
accelerated pointlike (structureless) classical scalar source *j(x)* leads
precisely to the *P/T* above.


*As a result, the emission of a usual (finite-energy) Minowski particle
from a uniformly accelerated source as defined by inertial observers
corresponds to either the absorption or emission of a zero-energy Rindler
particle as defined by Rindler observers.*


A way to understand why inertial observers do see radiation from a
uniformly accelerated source while Rindler observers do not (they only see
a deformed Coulomb field) is by recalling that zero-energy Rindler modes
concentrate on the horizon, where physical Rindler observers do not have
access to.


*Recently (2019), the relationship between zero-energy Rindler particles
and finite-energy Minkowski particles was strengthened even more (Landulfo,
Fulling, GM) by showing explicitly how the usual classical Larmor radiation
can be entirely built from zero-energy Rindler modes -- this will be
discussed further.*


Because the Unruh thermal bath was crucially used to build Larmor
radiation, we claim that under proper conditions the observation of Larmor
radiation is circumstantial evidence for the existence of the Unruh thermal
bath. The proper conditions we refer above was discussed in 2017-2018
(Cozzella, Landulfo, GM, Vanzella) and are mostly related to the fact that
the charge must have enough time to thermalize in the Unruh thermal bath.
Although the experiment is feasible under present technology, it was not
realized yet to the best of our knowledge.



GEORGE MATSAS

Professor of Physics
Institute for Theoretical Physics/ São Paulo State University
São Paulo State Academy of Science 
https://professores.ift.unesp.br/george.matsas/