Date: Wed, 22 Jul 2020 13:48:08 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: More comments by Daniel Sudarsky ============================================================== 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 clarify my motivation and concerns. When considering the question of radiation from a point like charge undergoing constant proper acceleration we are usually forced to deal with certain aspects that I think must be regarded as ultimately "unphysical" in at least two senses, which I think are ultimately related: 1) They involve consider taking certain limits of situations we might view as representing the physical situations before the limit is taken, but that become a bit less clear (to me at least) after the limiting procedure. 2) The analysis involves dealing with singular behaviors in the solution, which in my view obscure the process of extraction of fully trustable conclusions, at least as the physical problem before the limiting procedures 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 dealing with EM fields that diverge at the locus of the charge). 2) The consideration of charges that undergo constant proper acceleration for all times. I felt things would become more transparent if one could introduce physically reasonable modifications to regulate the problem and, then consider taking 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 past (t = -t*) and stops accelerating at some finite time in the future (t = t*), study the issue of radiation and then taking the limit when t* goes to infinity. Now let me add a proposal for dealing with 1): 1*) Consider a finite size charge (characterized by say a radius R*), study the problem of radiation and then consider taking the limit as R* goes to zero. It seems to me that if one manage to implement an analysis incorporating at the same time 1*) & 2*) one would have a problem in which divergences (or distributional aspects) would not come up, before the limits are taken and in my view that could provide a cleaner (or at least to my mind a more transparent) way of dealing with the problem. Of course implementing the proposals require us to consider some issues more carefully; 2*) Requires considering the statement of the problem corresponding physical problem in somewhat more precise terms regarding the initial data formulation. 1*) Requires us to consider the precise characterization of the charge distribution of "radius R* " and in particular its behavior during the phase of constant proper acceleration. One would be tempted to declare it to be a rigid sphere radius R with constant charge density. However, the requirement of rigidity might force us to deal with a notion that is known to be problematic in relativistic contexts. The requirement of constant charge density would be an issue as well (i.e. we would not be able to demand that to simultaneously hold for Rindler and Minkowski observers). One might require all the points of the charge to follow integrals of the Killing field, but this would not cover the regime where the charges passes from being inertial to that in which it is undergoing constant proper acceleration (and back).. Hopefully those issues would be of the type that go away in the 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 limiting procedures commute! Now regarding 2*) I offered some specific considerations in my previous message: One of them involved questioning the established wisdom (as far as addressing the question of radiation by the accelerating charge) by focussing just on the retarded solution. My point was that, if we were going to extract solid conclusion regarding the question in the manner I proposed, i.e by considering 2) in the finite t* regime before taking the limit and addressing at that stage the question of radiation by examining the form of the field at late times, it might well be rather relevant the manner in which we chose initial conditions, and that therefore focussing just on the retarded solution might not be the "right 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 below: > On Jul 17, 2020, at 9:04 PM, Stephen A. Fulling = wrote: > > 1. Date: Thu, 16 Jul 2020 15:32:54 -0700 (PDT) > From: Bill Unruh > Subject: Re: Comments, July 16 > >> 1. From: Daniel Sudarsky >> Date: Mon, 13 Jul 2020 12:28:07 -0500 >> >> I have a doubt related to the point 2) below, and Bob's reply. >> >> My concern is the following : It seems we take as a given that "the = physical solution" for the field corresponding to a certain spacetime = distribution of charges and currents, coincides with the one obtained = using the retarded Green function. >> >> However, it seems to me that what we want is the solution to the = field equations given initial data on a Cauchy hypersurface. It seems = to me that before taking limits, one should have complete control of the = " finite problem " (i.e. the problem before taking the limit should be = completely under control) so that one might have full control of the = process by which one takes the limit. >> >> I am thinking here of consideration of the situation in which a = charge has been accelerating always (i.e. it corresponds to one = complete orbit of the boost Killing field) taken as the limit of a = situation in which the charge is created (or better: starts = accelerating) at a finite point in the past. After solving this, then = one can consider the limit in which that time is taken to minus = infinity. [Creation of the charge being problematic in the EM case due = to strict charge conservation and its ties with Gauge Invariance.] >> >> 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 = fields for the finite problem at the initial hypersurface, (which = includes the point where the particle acceleration starts)?) > >[Bill:] > But these are answered with the retarded solution. For the charge at = rest (or constant velocity) the retarded solution is the same as the = advanced in the areas of overlap, if one takes the data on sci- to be = zero, and the extention into the spacelike region to the end of the = constant velocity is also unique. Thus one can reasonably take that as = the Initial data on some spacelike hypersurface which runs through the = constant velocity part of the charges motion. Ie, taking 0 on sci- and = retarded solution for the charge contribution seems like a very = reasonable choice. (One could of course always add to this a solution of = the source free equation which would be non-zero on sci-) [Daniel:] It is not evident to me that one can take data = zero (for say E and B) fields on sci- and be consistent with the constrains. I.e. what is the form the E-M constraints take on Sci-? (It seems we would need to include i- as well in the characterization of the initial situation, because we would have there all the finite charge with would again be described in distributional language) Furthermore how would things look once the limit of eternal accelerating charge is taken? The charge would hit sci- at some finite point, and it seems to me that the causal future of that point will have just a single ray on lying on sci-... so it is not clear to me how the corresponding "Gauss constraint" would be satisfied . > >> >> Given an answer to Q1, we might then consider the physical question: = what is the solution to the field equations given those data, and the = given charge current spacetime distribution (i.e. ignoring of course = back reaction). >> >> That solution involves using of the retarded solution + a part = involving the solution to the homogeneous equation so that the full = solution adjusts to the initial data. Energetic considerations can be = expected to be affected by the interference between the two terms. >> >> Then one might consider taking the limit. > > [Bill:] > The above gives a well defined way of taking that limit. > >> [Daniel:] >> If one has a time-like KF there seems to be a natural proposal for = answering Q1: The natural configuration of the fields on the initial = hypersurface would be THE (I am assuming it is unique) configuration = that minimizes (given the constraints) the Killing energy associated = with the field on that hypersurface. > > [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 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 = Homeogeneous field has some positive energy and two have a zero = interference term (one is spherically symetric about the location of the = charge and the other is not-- there is no monopole radiation) So the = minimum energy is just the Coulomb energy of the charge with the = homogeneous solution's energy being zero. . [Daniel:] Well I am not so sure about these considerations, specially if we are going to be interested on the energy radiated according to Rindler observers, as a Rindler wedge does not cover but a fraction of Minkowski spacetime and that fraction is not spherically symmetric. In the limiting case of eternal acceleration the charge world-line would intersect null infinity at two points breaking spherical symmetry, and it is not evident to me how one would disentangle there the various contributions to total energy fluxes. In any event if your argument could be made despite this objection (i.e. that there would be no interference), do you think that (when amended with 1*) my proposal would coincide with yours? >> >> Am I missing something? or is the logic above correct? Anyone knows = if the problem has been formulated in that way? And if so what is the = 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 = as "the physical solution"; everyone is free to consider any solution to = 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 ``radiation at future null infinity", we ought to make sure we did not put extra energy at the start. [Bob:] > The solution people are usually most interested in is the one with no = incoming radiation from null infinity. (If one has this solution, it is = easy to add a homogeneous solution to account for incoming radiation if = one wishes.) [Daniel:] As I wanted to rephrase everything in terms of initial data at a finite time, my concern was making sure that at the initial time there was not radiation present. That is, Instead of talking about radiation coming from past null infinity I wanted a scheme that would allow us to say "there is no more energy in the field beyond that strictly necessary in order to satisfy the EM = constraints (Div E = 4\pi \rho)." [Bob:] > I don't see any way of arguing why one person's choice of initial data = on a Cauchy surface would be "better" than someone else's choice = (unless, e.g., the charges are known to be static for all time in the = past, in which case the "no incoming radiation" condition would pick out = a "natural" choice of initial data). [Daniel:] That is the just the setting I was considering, and I think the proposal I talked about, would coincide with your "no incoming radiation" condition. Best wishes Daniel ===============================================================