Date: Wed, 29 Jul 2020 10:23:24 -0500 (CDT)
From: "Stephen A. Fulling" <fulling@math.tamu.edu>
To: Acceleration Radiation Community: ;
Subject: Comments July 29 (quick Unruh reply)

============================================================

Date: Tue, 28 Jul 2020 09:29:14 -0700 (PDT)
From: Bill Unruh <unruh@physics.ubc.ca>
Subject: Re: Four comments by three authors

>> ...
>> >
>> > P.S.  There was some mention in a comment I received today of a possible
>> > divergence if the acceleration of the charge worldline changed suddenly.
>> > However, the 4D form F = (e/r^3) N ^ [(1 - a.N) V + r a] does not show
>> any
>> > divergence in F, but just a discontinuity between different finite
>> values of
>> > F if a jumps suddenly.
>> 
>> I was basing my comment on the belief that The vector potential had a
>> discontinuity in that case which would give a delta function in the EM
>> field.
>> For example the field at the charge goes as the time derivatative of a
>> which
>> is a delta function if the acceleration is a step function.
>> 
>
> But the Lienard-Wiechert 4-vector potential (in units with c = 4 pi
> epsilon_0 = 1) is A = e V/(V.N) = eV/r, where r = V.N with V the 4-velocity
> of the charge worldline at the emission event and N the null vector from
> the emission event to the event where the vector potential is evaluated and
> the EM field is to be observed.  So long as V is continuous along the
> charge worldline, A is continuous, though if the 4-acceleration a of the
> charge worldline jumps discontinuously, the retarded EM Faraday 2-form F =
> (e/r^3) N ^ [(1 - a.N) V + r a] that one gets from the Lienard-Wiechert
> potential A can also jump discontinuously.  However, so long as a remains
> finite, so does F, and hence so does the stress-energy tensor of the EM
> field.

As I said I based my comment on the fact that the self field of the point
charge at the location of the charge has a delta function singularity if the
acceleration has a step function singularity at some time. Since the self
field can be determined by taking Kirkoff type integrals over the EM field
over the light cone far from the location of the charge (I wrote a paper on
this in Proc Roy Soc in the early 70's) , this means that the
average field over those light cones from that point must also have a delta function
singularity if the acceleration has a step function behaviour. 
It is hard to see how one could get a step function in the average field, if
the field itself is continuous.