Date: Thu, 23 Jul 2020 11:28:29 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: Comments on comments on ... ============================================================= 1. Date: Wed, 22 Jul 2020 16:45:41 -0700 (PDT) From: Bill Unruh Subject: Re: More comments by Daniel Sudarsky I agree completely with your proposal. Clearly acceleration into the infinite past is unphysical, and acceleration for a finite time is not. However, I think the transition would also have to be smooth, which of course makes the calculation somewhat more difficult (otherwise a step function is the acceleration would, I think, create delta function in E and in energy radiated. That of course makes things a bit more complicated. But taking the limit will, I believe, lead to delta functions (or worse) E and B fields along the past horizon and thus those are as physical as anything else. This of course produces a delta function going into the future horizon. The particle produces radiation which goes through the future horizon. Furthermore it produces fields within the whole reagion F and across the future horizon. Of course whether one calls this radiation is the whole point of these discussions. I think (but do not know) that the finite size of the charge is of lesser concern. I think but do not know, that this is a smaller problem. EDITOR'S NOTE: Bill intersperses more comments in Daniel's message (below), including the part addressed to Bob. I've tried to label them all by "[BILL:]" but I may have missed a few. William G. Unruh __| Canadian Institute for|____ Tel: +1(604)822-3273 Physics&Astronomy _|___ Advanced Research _|____ Fax: +1(604)822-5324 UBC, Vancouver,BC _|_ Program in Cosmology |____ unruh@physics.ubc.ca Canada V6T 1Z1 ____|____ and Gravity ______|_ https://urldefense.com/v3/__h= ttp://www.theory.physics.ubc.ca/__;!!KwNVnqRv!XTU7MkgP0iZbcySm1KGsMMHnxZec0= JS5hn2buW7Ny_0_yrLDZMHMBsfwLeeLs1yYrS8$=20 > From: Daniel Sudarsky > Subject: Re: More comments, July 17 > Date: Tue, 21 Jul 2020 15:07:21 -0500 > > Dear Colleagues > > Following up on my comment and the responses from Bill and Bob. Let me=20 > clarify my motivation and concerns. > > When considering the question of radiation from a point like charge=20 > undergoing constant proper acceleration we are usually forced to deal wit= h=20 > certain aspects that I think must be regarded as ultimately "unphysical" = in=20 > at least two senses, which I think are ultimately related: > > 1) They involve consider taking certain limits of situations we might vie= w as=20 > representing the physical situations before the limit is taken, but that= =20 > become a bit less clear (to me at least) after the limiting procedure. > > 2) The analysis involves dealing with singular behaviors in the solution,= =20 > which in my view obscure the process of extraction of fully trustable=20 > conclusions, at least as the physical problem before the limiting procedu= res=20 > are taken, > > > For the problem at hand there are fact at least two such aspects; > > 1) The consideration of a point like charge (a feature which forces us to= =20 > dealing with EM fields that diverge at the locus of the charge). > > > 2) The consideration of charges that undergo constant proper acceleration= for=20 > all times. > > > I felt things would become more transparent if one could introduce physic= ally=20 > reasonable modifications to regulate the problem and, then consider takin= g=20 > well defined limiting procedures. > > > In my first message I focussed mainly on problem 2) and proposed: > > 2*) Considering a charge that starts acceleration at a finite time in the= =20 > past (t =3D -t*) and stops accelerating at some finite time in the future= (t =3D=20 > t*), study the issue of radiation and then taking the limit when t* goes = to=20 > infinity. > > Now let me add a proposal for dealing with 1): > > 1*) Consider a finite size charge (characterized by say a radius R*), stu= dy=20 > the problem of radiation and then consider taking the limit as R* goes to= =20 > zero. > > > It seems to me that if one manage to implement an analysis incorporating = at=20 > the same time 1*) & 2*) one would have a problem in which divergences (or= =20 > distributional aspects) would not come up, before the limits are taken an= d in=20 > my view that could provide a cleaner (or at least to my mind a more=20 > transparent) way of dealing with the problem. > > Of course implementing the proposals require us to consider some issues m= ore=20 > carefully; > > > 2*) Requires considering the statement of the problem corresponding physi= cal=20 > problem in somewhat more precise terms regarding the initial data=20 > formulation. > > > 1*) Requires us to consider the precise characterization of the charge=20 > distribution of "radius R* " and in particular its behavior during the ph= ase=20 > of constant proper acceleration. One would be tempted to declare it to be= a=20 > rigid sphere radius R with constant charge density. However, the require= ment=20 > of rigidity might force us to deal with a notion that is known to be=20 > problematic in relativistic contexts. The requirement of constant charge= =20 > density would be an issue as well (i.e. we would not be able to demand t= hat=20 > to simultaneously hold for Rindler and Minkowski observers). One might=20 > require all the points of the charge to follow integrals of the Killing=20 > field, but this would not cover the regime where the charges passes from= =20 > being inertial to that in which it is undergoing constant proper accelera= tion=20 > (and back).. Hopefully those issues would be of the type that go away in = the=20 > process of taking the limit as R* goes to zero. > > At the end we would of course need to consider whether or not the two=20 > limiting procedures commute! > > Now regarding 2*) I offered some specific considerations in my previous=20 > message: > > One of them involved questioning the established wisdom (as far as addres= sing=20 > the question of radiation by the accelerating charge) by focussing just o= n=20 > the retarded solution. > > My point was that, if we were going to extract solid conclusion regarding= the=20 > question in the manner I proposed, i.e by considering 2) in the finite t*= =20 > regime before taking the limit and addressing at that stage the question = of=20 > radiation by examining the form of the field at late times, it might well= be=20 > rather relevant the manner in which we chose initial conditions, and that= =20 > therefore focussing just on the retarded solution might not be the "right= =20 > thing to do". In that case my Q 1 "becomes very relevant". > > Now Bill has offered a specific proposal in that regard, which I address= =20 > below: > > >> On Jul 17, 2020, at 9:04 PM, Stephen A. Fulling =3D > wrote: >>=20 >> 1. Date: Thu, 16 Jul 2020 15:32:54 -0700 (PDT) >> From: Bill Unruh >> Subject: Re: Comments, July 16 >>=20 >>> 1. From: Daniel Sudarsky >>> Date: Mon, 13 Jul 2020 12:28:07 -0500 >>>=20 >>> I have a doubt related to the point 2) below, and Bob's reply. >>>=20 >>> My concern is the following : It seems we take as a given that "the =3D > physical solution" for the field corresponding to a certain spacetime =3D > distribution of charges and currents, coincides with the one obtained =3D > using the retarded Green function. >>>=20 >>> However, it seems to me that what we want is the solution to the =3D > field equations given initial data on a Cauchy hypersurface. It seems =3D > to me that before taking limits, one should have complete control of the = =3D > " finite problem " (i.e. the problem before taking the limit should be =3D > completely under control) so that one might have full control of the =3D > process by which one takes the limit. >>>=20 >>> I am thinking here of consideration of the situation in which a =3D > charge has been accelerating always (i.e. it corresponds to one =3D > complete orbit of the boost Killing field) taken as the limit of a =3D > situation in which the charge is created (or better: starts =3D > accelerating) at a finite point in the past. After solving this, then =3D > one can consider the limit in which that time is taken to minus =3D > infinity. [Creation of the charge being problematic in the EM case due = =3D > to strict charge conservation and its ties with Gauge Invariance.] >>>=20 >>> Then it seems to me that the natural question is Q1: what do we >>> take those initial data to be? (what should be initial data for the =3D > fields for the finite problem at the initial hypersurface, (which =3D > includes the point where the particle acceleration starts)?) >>=20 >> [Bill:] >> But these are answered with the retarded solution. For the charge at =3D > rest (or constant velocity) the retarded solution is the same as the =3D > advanced in the areas of overlap, if one takes the data on sci- to be =3D > zero, and the extention into the spacelike region to the end of the =3D > constant velocity is also unique. Thus one can reasonably take that as =3D > the Initial data on some spacelike hypersurface which runs through the =3D > constant velocity part of the charges motion. Ie, taking 0 on sci- and =3D > retarded solution for the charge contribution seems like a very =3D > reasonable choice. (One could of course always add to this a solution of = =3D > the source free equation which would be non-zero on sci-) > > > [Daniel:] > It is not evident to me that one can take data =3D zero (for say E and B)= =20 > fields on sci- and be consistent with the constrains. I.e. what is the f= orm=20 > the E-M constraints take on Sci-? (It seems we would need to include i- a= s=20 > well in the characterization of the initial situation, because we would h= ave=20 > there all the finite charge with would again be described in distribution= al=20 > language) [BILL:] But we know that you want to take the the solution before some time t to be the charge travelling geodesically (no acceleration) and we certainly know what the solution is for the charge at rest. Just take that to be the initi= al data on t=3D constant suface before the charge starts to accelerate and tak= e the retarded solution thereafter. > > Furthermore how would things look once the limit of eternal accelerating= =20 > charge is taken? The charge would hit sci- at some finite point, and it= =20 > seems to me that the causal future of that point will have just a single = ray=20 > on lying on sci-... so it is not clear to me how the corresponding "Gaus= s=20 > constraint" would be satisfied . [BILL:] Take the limit and see. > >>=20 >>>=20 >>> Given an answer to Q1, we might then consider the physical question: =3D > what is the solution to the field equations given those data, and the =3D > given charge current spacetime distribution (i.e. ignoring of course =3D > back reaction). >>>=20 >>> That solution involves using of the retarded solution + a part =3D > involving the solution to the homogeneous equation so that the full =3D > solution adjusts to the initial data. Energetic considerations can be =3D > expected to be affected by the interference between the two terms. >>>=20 >>> Then one might consider taking the limit. >>=20 >> [Bill:] >> The above gives a well defined way of taking that limit. >>=20 >>> [Daniel:] >>> If one has a time-like KF there seems to be a natural proposal for =3D > answering Q1: The natural configuration of the fields on the initial =3D > hypersurface would be THE (I am assuming it is unique) configuration =3D > that minimizes (given the constraints) the Killing energy associated =3D > with the field on that hypersurface. [BILL:] My proposal is as above. boosted static before t0, and retarded thereafter. >>=20 >> [Bill:] >> Why is that reasonable. The charge has an infinite EM energy. > > [Daniel:] Yes, you are right . That is why I would amend it with 1*) (i.= e in=20 > order to avoid having to deal with any infinite quantity) > >> [Bill:] >> But anyway, the static charge in the past has a coulomb energy and the = =3D > Homeogeneous field has some positive energy and two have a zero =3D > interference term (one is spherically symetric about the location of the = =3D > charge and the other is not-- there is no monopole radiation) So the =3D > minimum energy is just the Coulomb energy of the charge with the =3D > homogeneous solution's energy being zero. . > > [Daniel:] > Well I am not so sure about these considerations, specially if we are goi= ng=20 > to be interested on the energy radiated according to Rindler observers, a= s a=20 > Rindler wedge does not cover but a fraction of Minkowski spacetime and th= at=20 > fraction is not spherically symmetric. In the limiting case of eternal [BILL:] Physics does not depend on coordinates. > acceleration the charge world-line would intersect null infinity at two=20 > points breaking spherical symmetry, and it is not evident to me how one w= ould=20 > disentangle there the various contributions to total energy fluxes. [BILL:] Spherical symmetry is on the spacelike hypesurface orthogonal to the consta= nt time slice. What happens to it in the limit, is taken care of by how you ta= ke that limit. > > In any event if your argument could be made despite this objection (i.e. = that=20 > there would be no interference), do you think that (when amended with 1*)= my=20 > proposal would coincide with yours? [BILL:] No idea. > >>>=20 >>> Am I missing something? or is the logic above correct? Anyone knows =3D > if the problem has been formulated in that way? And if so what is the =3D > answer taken to Q1.? > > ----------------------------------------------------------- > > Regarding Bob's remarks in response to mine: > > >> Daniel's remarks: I agree with Bill's comments. There is no such thing = =3D > as "the physical solution"; everyone is free to consider any solution to = =3D > Maxwell's equations that they want to consider. > > [Daniel:] > Yes of course. But if we want to address the issue at hand, in terms of= =20 > ``radiation at future null infinity", we ought to make sure we did not pu= t=20 > extra energy at the start. > > [Bob:] >> The solution people are usually most interested in is the one with no =3D > incoming radiation from null infinity. (If one has this solution, it is = =3D > easy to add a homogeneous solution to account for incoming radiation if = =3D > one wishes.) > > [Daniel:] > > As I wanted to rephrase everything in terms of initial data at a finite t= ime,=20 > my concern was making sure that at the initial time there was not radiati= on [BILL:] Since we have no idea what we mean by radiation, that is part of the proble= m. > present. That is, Instead of talking about radiation coming from past nu= ll=20 > infinity I wanted a scheme that would allow us to say "there is no more=20 > energy in the field beyond that strictly necessary in order to satisfy th= e EM=20 > =3D constraints (Div E =3D 4\pi \rho)." > > [Bob:] >> I don't see any way of arguing why one person's choice of initial data = =3D > on a Cauchy surface would be "better" than someone else's choice =3D > (unless, e.g., the charges are known to be static for all time in the =3D > past, in which case the "no incoming radiation" condition would pick out = =3D > a "natural" choice of initial data). > > [Daniel:] > > That is the just the setting I was considering, and I think the proposal = I=20 > talked about, would coincide with your "no incoming radiation" condition. > > Best wishes > Daniel > ========================================================= 2. From: Paul Davies Subject: Re: More comments by Daniel Sudarsky Date: Thu, 23 Jul 2020 00:01:41 +0000 Steve, I have some disorganized thoughts about this email thread, and I'm not sure (i) whether they are relevant, (ii) how to communicate them. (I have been distracted on other priorities of late.) But here they are: 1. Re the charge being created and Bill's comment on the problems of that. The Lyttleton-Bondi cosmological model had charged particle creation an= d I did some work on it in the early 1970s. If charges are created, 'longitudinal' modes of the electromagnetic field are excited, i.e. there is a third helicity state. I would think this would be a drastic and unwelcome change to the problem. 2. Wheeler-Feynman theory was developed precisely to avoid charge self-action and the associated problems of divergences. It uses equal advanced an= d retarded solutions to achieve this. But to harmonize with standard Maxwell electrodynamics it needs a perfect absorber on the future light cone. Tha= t would be a problem if acceleration is eternal, as the particle would hit the absorber at some stage, but for finite duration acceleration it might b= e made to work. 3. The problem of the sudden onset of acceleration is pre-acceleration, much discussed over the decades, and the need to add an additional boundary condition because of the third derivative in the equations of motion. There is an extensive ancient literature on it. 4. The problem of a finite-sized charged particle is that one needs to specify not only the charge distribution but the internal stresses that hold= the particle together, which have to transform in such a way that the particle retains its shape under Lorentz boosts. There is also the question of = the accelerating agency. You don't want the agency's electromagnetic fields to intermingle in a confusing way with those of the accelerated charge. If= the agency couples to the internal stress tensor, that has to be done cons istently. I would not recommend going down that path. Paul _______________________________________________________________________ Paul Davies Regents' Professor at Arizona State University Director of the Beyond Center for Fundamental Concepts in Science | beyond.= asu.edu Co-Director of the Arizona State University Cosmology Initiative | cosmolog= y.asu.edu Tel: +1 480 965 3240 Personal Website: cosmos.asu.edu ===================================================================