Date: Tue, 4 Aug 2020 21:34:50 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: Follow-up to a question by Don Page ============================================================== Date: Tue, 4 Aug 2020 11:01:46 -0600 From: Don Page 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 All, I took some time recently looking up online the Lienard-Wiechert potential and field in higher dimensions and found answers to my questions under (2) of my previous email below in these papers: Metin Gürses, and Özgür Sarıoğlu, "Liénard–Wiechert potentials in even dimensions," Journal of Mathematical Physics 44, 4672 (2003); https://doi.org/10.1063/1.1613040 [urldefense.com], and [LINK] [urldefense.com]B. P. Kosyakov, "ELECTROMAGNETIC RADIATION IN EVEN-DIMENSIONAL SPACE–TIMES," International Journal of Modern Physics A [urldefense.com]Vol. 23, No. 29, pp. 4695-4708 (2008). [urldefense.com] The second paper showed, if I interpreted it correctly, that in both D = 4n and in D = 4n-2, the retarded field 2-form F from a single point charge trajectory in D-dimensional Minkowski spacetime is the sum of n wedge products of 2 one-forms each, so that at each point of the spacetime, there are n independent algebraic invariants of the form tr(F^m), rather than the D/2 = 2n or 2n-1 of a generic EM field at each point in even spacetime dimension D. It is also interesting that the one-forms that appear in the n wedge products are all the one-form retarded potentials in other even dimensions. Thus the retarded EM field of a single charge worldline is a simple two-form (a single wedge product of 2 one-forms) only in D = 2 and D = 4.  However, for D = 4n, the rank of F_a^b is half the spacetime dimension, so half of the eigenvalues are zero (and the other half occur in pairs of equal magnitude, as usual for a two-form with one index raised). One conceivable physical difference between D = 4 and D = 6 is that in D = 4, the retarded field depends only on the null vector N from the charge to the retarded field location, the 4-velocity V of the charge, and the 4-acceleration a of the charge, so it is finite for all finite a.  Therefore, if one imagined that a charge were suddenly accelerated by a slab EM wave that had its EM field suddenly jump from zero to a nonzero value, even without any radiation reaction the retarded field from the charge that suddenly starts accelerating would be finite and carry away finite energy.  However, in D = 6, the retarded field depends on the proper time derivative (say b) of the acceleration a as well as on N, V, and a, so if a charge were suddenly accelerated by such a slab EM wave, at least to the extent that one could ignore the effect of radiation reaction on the charge, it would have b be a delta-function of the charge worldline proper time and hence give a delta-function retarded field that would carry off infinite energy.  So it naively seems that radiation reaction would be much more important in D = 6 than in D = 4, but I should leave it to you experts to comment on that speculation. Best wishes, Don ------------------------------------------------ [original message] On Thu, Jul 23, 2020 at 12:45 PM Don Page wrote: 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 , A Penin , Aksel Hallin , 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