Date: Thu, 20 Aug 2020 12:03:40 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: Three comments by two authors =========================================================== From: Ashok Kumar Singal Date: Sun, 16 Aug 2020 16:44:19 +0530 Subject: Re: Comment Aug 9 (Singal) I have a comment that concerns the definition of radiation in Fulton and Rohrlich 1960 paper: Classical Radiation from a Uniformly Accelerated Charge, section 3 'DOES A UNIFORMLY ACCELERATED CHARGE RADIATE?' where to examine the above question, the radiation is defined by the total rate of radiation energy emitted by the charge at time t', and is found by integrating over the surface of the light sphere in the limit of infinite R = c(t - t') for a fixed emission time t', where both R and t must approach infinity. Fulton and Rohrlich further write: "Pauli's argument (magnetic field, H = 0 everywhere at t = 0, implying no formation of a wave zone nor any corresponding radiation from a uniformly accelerated charge) cannot be valid, since in this statement a limit to large distances (R -> infinity) is implied at a fixed time (t = 0). This limit is not in accordance with the (Fulton and Rohrlich's) definition of radiation, where the limit R -> infinity is to be taken for a fixed source point (t' fixed), implying a limit t -> infinity as well. It can easily be seen that the two limiting procedures do not give the same result. From this, one conclude that the fact, that H = 0 at t = 0, is unusual for accelerated motion and of some interest, but it has nothing to do with the presence or absence of radiation." Now, I want to point out here a subtle point about Green's retarded solution, where the scalar potential phi at a field point X, at time t, is determined from a volume integral of rho/R, where rho is the charge density and R = |X- X'|, where X' is the location of charge density rho at the corresponding retarded time t' (Jackson 1999, eq. 6.48), and similar for vector potentials. Thus here X and t are first chosen and the volume integral of the charge density at the corresponding retarded time is then computed. Pauli's argument is consistent with that procedure. In fact Fulton and Rohrlich define radiation which, strictly speaking, is not in tune with Green's retarded time solution. I may add here that for a point charge e, first fixing the charge position (X') at the retarded-time t', to determine phi yields phi=e/R, while the more correct approach of first fixing the field point (X) gives phi=e/R(1-v cos theta/c), the correct expression for the potential (Feynman lectures, ii vol. eqs. 21.28 and 21.29). Ashok =============================================================== From: Ashok Kumar Singal Date: Tue, 18 Aug 2020 17:08:30 +0530 Subject: Re: Comment Aug 9 (Singal) In continuation of my comment about a cautious usage of Green's retarded solution, there is a similar cautious approach to used for applying Poynting's theorem of energy conservation in an electromagnetic system comprising charges and fields. According to the conventional wisdom, the radiative power loss is given correctly by Larmor's formula, (2e^2/3c^3)a^2, where a is the acceleration. In the text-book derivation of Larmor's formula (e.g., Jackson 1999, section 14.3), Poynting's theorem is applied to equate the radiated power at time t to the rate of loss of the mechanical energy of the charge at a retarded time t'=t-r/c. However, the consequential net rate of momentum loss to Larmor's radiation by such a charge turns out to be nil because the radiation pattern possesses a sin^2 theta angular symmetry, which contradicts the above power loss of the charge which will then be losing mechanical energy without losing momentum. Now, in Poynting's theorem, ALL QUANTITIES are supposed to be calculated for the SAME INSTANT of time (Jackson 1999, EQ. 6.107), and one cannot directly calculate the rate of loss of the mechanical energy of the charge at a retarded time t'=t-r/c, from the radiated power at time t. Therefore the Larmor's formula may not be an exact result and it is this oversight which could mostly be responsible for the confusion in this century-old problem. A correct application of the Poynting's theorem, gives instantaneous power loss of the charge P =-(2e^2/3c^3)\dot{a}.v, where \dot{a} is the instantaneous rate of change of acceleration vector and v is the velocity vector. (Ref. Eur. J. Phys., 37, 045210, 2016). An application of the electromagnetic momentum conservation, employing the Maxwell's stress tensor, where again ALL QUANTITIES are to be calculated for the SAME INSTANT of time, yields a rate of loss of mechanical momentum of the charge as F= -(2e^2/{3c^3)\dot{a} (Ref. Am. J. Phys., 85, 202, 2017). This is the radiation reaction formula derived in an independent manner. In the case of a periodic motion of period T, there is no difference in the radiated power integrated (or averaged) between t to t+T and t' to t'+T, therefore Larmor's formula, does yield a correct average power loss by the charge for a periodic case. It should be noted that Larmor's original paper was for an oscillating charged system (Larmor, J., Phil. Mag. 44, 503, 1897). Further, for a periodic motion, e.g., a harmonically oscillating charge in a radio antenna, it is easily verified that =-<(rate of change of a) . v>, therefore Larmor's formula yields the same time-averaged radiative power as from the formula derived from the famous Abraham-Lorentz radiation reaction formula. For instance, let x=x_0 cos(omega t+phi). Then v=\dot{x}=-omega x_0 sin(omega t+phi), a=\dot{v}=-omega^2 x_0 cos(omega t+phi)=-omega^2 x, \dot{a}=omega^3 x_0 sin(omega t+phi)=-omega^2 v. Then Larmor's formula yields radiative power propto \dot{v}^2=omega^4 x^2_0 cos^2(omega t+phi), while the power loss from the radiation reaction turns out propto -\ddot{v}.v= omega^4 x^2_0 sin^2(omega t+phi). Though the two expressions yield equal radiated energy when integrated or averaged over a complete cycle, the instantaneous rates are quite different. As any actual motion of the charge could be Fourier analyzed, the above statement would be true for individual Fourier components, and the power spectrum, which gives average power in the cycle for each frequency component, would be the same in the two formulations. Of course there will be cases where a Fourier analysis is not possible, for instance, in the case of a uniformly accelerated charge. In such cases the two formulas could yield conflicting answers. Once this fact is realized, much of the doubt or confusion in this long drawn out controversy disappears. Ashok ======================================================== Date: Thu, 20 Aug 2020 11:53:13 -0500 (CDT) From: "Stephen A. Fulling" Subject: On the definition of radiation, and our future activities Let me take the occasion of Dr. Singal's messages to reinforce the point that the precise definition of radiation is crucial to deciding the correct statements to make about it. As I indicated earlier, there are many points in Rohrlich's great trilogy that are inconsistent or dubious. Yet I still consider those papers to be foundational, and therefore a modern reconstruction of them is much to be desired. These two things (clarifying the meaning of radiation -- perhaps distinguishing among more than one essential concept -- and producing a modern summary of Rohrlich) are two of the tasks that I hoped this group would accomplish in its collective wisdom. Unfortunately, my teaching duties are so heavy this semester that I cannot take the initiative. So, please volunteer.... Please remember also that we have a dedicated future issue of Symmetry to receive contributions for publication. The deadline for submission is in March. The publisher's representative is beginning to wonder when papers are going to appear in their office. Stephen A. Fulling Professor of Mathematics and Physics Texas A&M University College Station, TX 77843-3368 USA =================================================================