Date: Sat, 25 Jul 2020 09:52:20 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: More comments, and a new query by Don Page ============================================================== 1. From: Daniel Sudarsky Subject: Re: Comments on comments on ... Date: Thu, 23 Jul 2020 13:00:36 -0500 [DANIEL] I agree that taking the acceleration to start and end suddenly is problematic but I think one might be to separate its effect on radiation , once the notion is fully clarified) by separating the contribution that scales with the proper time the charge is taken to undergo acceleration from the part that does not scale in that way. EDITOR'S NOTE: Daniel's comment is in reply to https://www.math.tamu.edu/~fulling/ar/fulling1_comments723.txt . Fragments thereof with additional Sudarsky comments follow: > On Jul 23, 2020, at 11:28 AM, Stephen A. Fulling = wrote: > 1. Date: Wed, 22 Jul 2020 16:45:41 -0700 (PDT) > From: Bill Unruh > Subject: Re: More comments by Daniel Sudarsky [LARGE AMOUNTS OF NOW IRRELEVANT MATERIAL DELETED] ... > [BILL:] > My proposal is as above. boosted static before t0, and retarded thereafter. [DANIEL] Probably I am misunderstanding something in the above proposal: In the boosted frame where the charge is static util t_0 you want $ E= ( q/r^2 ) \hat r $ , $ B=0$ on all spacetime to the past and including the hypresurface t_0, and then to the future of that hypersurface, you want to consider the world-line of the charge staring at t_0 as the source of the retarded solution. Is that correct? If that is the proposal then it seems that in the region to the future of the t_0 hypersurface , but outside the future light come starting at the location of the charge in the t_0 hypersurface we would have E=B=0. That does not seem right! ... > [BILL:] > Physics does not depend on coordinates. [DANIEL] It seems to me that in varios parts of the discussion one is dealing with two notions of energy fluxes: one associated to inertial observers and a second one associated to Rindler observers. And those do not necessarily coincide. In particular if one wants to compare results must recall that the region covered by Rindler observes is not spherically symmetric. ... > 2. From: Paul Davies > Subject: Re: More comments by Daniel Sudarsky > Date: Thu, 23 Jul 2020 00:01:41 +0000 ... > 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=3D 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=3D d retarded solutions to achieve this. But to harmonize = with standard Maxwell electrodynamics it needs a perfect absorber on the = future light cone. Tha=3D 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=3D 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. [DANIEL] I think at this point one is considering back reaction of the radiation on the charge, but I understood that the problem under consideration was a simpler one namely one in which the particle's world line is taken as given. > 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=3D 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 =3D 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=3D the agency couples to the = internal stress tensor, that has to be done cons istently. I would not = recommend going down that path. [DANIEL ] Again, are not these issues that would arise if one wants to consider the dynamics of the particle, but that will not show up if one considers the problem in which the particle's world line ( and that of its internal parts) is taken as given? ========================================================== 2. From: Don Page Date: Thu, 23 Jul 2020 12:45:09 -0600 Subject: Question for the EM experts in this discussion about whether the EM field from a single charge worldline is a simple 2-form in other dimensions Dear EM experts, Now that Stephen has organized this discussion, I wanted to ask a couple of old questions I had: (1) Has anyone made any use of the congruence of observers in flat 4D Minkowski spacetime for which the EM field of one particular but arbitrary smooth charged particle worldline has the Coulomb form with E = e/R^2 and B = 0 with R = U.N, the dot product of the 4-velocity U of the observer with the null vector N from the emission event on the charge worldline to the observation event on the observer worldline? (2) In flat 4D spacetime, 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 = e V/(V.N) = eV/r, where now r = V.N with V the 4-velocity of the charge worldline at the emission event and where a is the 4-acceleration of the charge worldline at the same emission event, is a simple 2--form, the outer or wedge product of two 1-forms, so F ^ F = 0 or E.B = 0. Is the retarded EM 2-form for a single charge worldline in flat spacetime in other spacetime dimensions D also a simple 2-form? What is the analogue of F = (e/r^3) N ^ [(1 - a.N) V + r a] in other spacetime dimensions D? Note that R is the spatial distance in the frame of the observer at the observation event of where the charge was at the retarded time, and r is the spatial distance in the frame of the charge at the emission event of where the observer would be when it received the field travelling at the speed of light (along the null vector N from the charge emission event to the observation event). Below is a more detailed email I sent to John David Jackson and to some colleagues of mine at the University of Alberta, but I never got answers to what I asked there, so now I thought I would ask you. Best wishes, Don 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 might _guess_ that in D > 4, there might be higher derivatives of the charged particle worldline, so that there would be a divergence for F in higher dimensions if the 4-acceleration changed suddenly, but I don't see one in D = 4. Are there indeed higher derivatives in higher dimensions? ---------- Forwarded message --------- From: Don Page Date: Fri, Nov 2, 2012 at 10:37 AM Subject: Coulomb's law works better than I had realized for an accelerated charge To: John David Jackson , Andrzej Czarnecki < andrzej.czarnecki@ualberta.ca>, A Penin , Aksel Hallin < aksel.hallin@ualberta.ca>, J Pinfold , Frank Marsiglio , Valeri Frolov , Dmitri Pogosyan , Sharon Morsink , Craig Heinke , Wojciech Rozmus , Andrei Zelnikov Dear Professor John David Jackson and U of A Physics Colleagues, I have been greatly enjoying using your Classical Electrodynamics (third edition) textbook for my graduate electrodynamics class, PHYS 524, at the University of Alberta, which I am teaching for the first time this year. After differentiating the Lienard-Wiechert 4-vector potential in Minkowski spacetime, A = e V/(V.N) = eV/r (using gaussian units and also c = 1, where A is the 4-vector potential, e is the particle charge, V is the particle 4-velocity, and N is the null vector from the retarded charge location event to the later field location event in spacetime, with the 4-dimensional dot product [using the Jackson metric signature + - - - so that 4-velocities are normalized to give V.V = +1 once I set c = 1] being V.N, which I shall henceforth call r, since it is the distance from the retarded position of the charge to the later field point event in the frame of the charge, so that the 4-vector potential on the light cone in the frame of the charge has precisely the Coulomb form), to get the electromagnetic field-strength tensor (or 2-form) F = (e/r^3) N ^ [(1 - a.N) V + r a], where a is the 4-acceleration of the charge and where the ^ is used for the wedge product, I noticed that in a frame at the field point event in which the electric and magnetic fields are parallel, the magnetic field B is zero (which I knew long ago, a simple consequence of the fact that F is a timelike simple 2-form, the wedge product of two 1-forms rather than being a sum of more than one independent such 1-forms, and hence giving the Lorentz invariant E.B = 0), and the electric field strength has precisely the Coulomb form (which I did not realize before, though it is a simple consequence of the fact that E.B = 0 and the fact that the other Lorentz invariant is E^2 - B^2 = e^2/r^4), E = e/r^2, with the electric field in that frame pointing in the spatial direction of the null vector N (since N is one of the vectors or 1-forms in the wedge product for the field strength) and hence directly away (for e > 0) from the direction of where the charge was at the retarded time in the observer's frame, and with, as above, r = V.N, the spatial distance of the field point event in the frame of the charge at the retarded source point event. There is a one-parameter set of observer 4-velocities U for which the magnetic field vanishes (corresponding to boosting along the direction of the electric field, which leaves its magnitude the same at the same spacetime event), and one can choose a unique particular preferred observer 4-velocity U so that the distance to the retarded position of the charge in the observer's frame, U.N, is the same r, the same value as the distance V.N in the frame of the charge that had 4-velocity V at the retarded time event that is the source of the electromagnetic field at the later field event. That is, for this preferred 4-velocity, E = e/r^2, where r is the distance to the retarded position of the charge in the frame of the observer (as well as in the frame of the charge at its retarded event). Of course, this preferred observer 4-velocity U in which the electromagnetic field has precisely the Coulomb form is not the same as the retarded charge 4-velocity V, unless the 4-acceleration of the charge, a, is zero at the retarded time. Assuming that I did the algebra correctly to find it (which I am less confident of than of the fact that such a preferred 4-velocity exists at each field point event in the electromagnetic field from a single charged particle with arbitrary motion in Minkowski spacetime and no other incoming electromagnetic waves), the preferred observer 4-velocity is U = (1 - a.N) V + r a + [2a.N - (a.N)^2 - r^2(a.a)] N/(2r). But it is interesting that there exists this preferred observer 4-velocity U in which the magnetic field from the accelerating charge is zero and the electric field has the value e/r^2 where r is the spatial distance along the null vector connecting the retarded charge event and the observation event in both the frame of the charge and in the frame of the observer. Do any of you know whether this simple observation has been noted before in the literature? Best wishes, Don ============================================================== Stephen A. Fulling Professor of Mathematics and Physics Texas A&M University College Station, TX 77843-3368 USA