Date: Mon, 13 Jul 2020 11:15:16 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: Comments, mostly on Fulling's presentation The second presentation has proved to be less controversial than the first, but several comments have accumulated -- presented here not in chronological order. As always, the posts are archived at https://www.math.tamu.edu/~fulling/ar/ . Stephen A. Fulling Professor of Mathematics and Physics Texas A&M University College Station, TX 77843-3368 USA ============================================================== 1) From: Robert Wald Date: Sun, 5 Jul 2020 21:58:33 -0500 Subject: Re: Second discussion topic: The retarded solution Here are my comments on Steve's posting. Since the calculation of the classical retarded solution of an accelerating charge does not seem to be in question, the whole issue would appear to be what is meant by "radiation." I should say that I haven't tried to read any of the papers that Steve referred to---although I read the Boulware paper a long time ago and believe that everything he says there is correct. (I'd be willing to take a look at the other papers if anyone feels that I've overlooked some relevant point in what I say below.) Even without my having read the papers, it is clear from Steve's posting that the question of whether a charge "radiates" is being posed quite sloppily in the literature. The stress-energy tensor T_{\mu \nu} of the electromagnetic field is unambiguously well defined. However, to define "energy" (which should be the first step toward defining "radiation") one additionally must specify a time translation Killing field t^\mu. One then can define a corresponding conserved energy current J_\mu = T_{\mu \nu} t^\nu. The only notion of "radiation" that I could imagine defining would be the flux of this current through some timelike or null surface. If one chooses this surface to be future null infinity, this gives rise to the standard, well defined notion of the flux of energy radiated to infinity. For a uniformly accelerating charge, this flux is nonzero, so a uniformly accelerating charge "radiates" to infinity in this sense. It is very hard for me to imagine that there could possibly be any controversy about this. One also could choose a finite surface and use the flux of the energy current through it to define a local notion of "radiation." (I can't imagine what else one could do.) However, if one does so, then it is extremely important to say exactly what that surface is. It is also important to say what Killing field is being considered, particularly given that it would be natural, e.g., for an accelerating observer to define "time translations" as Lorentz boosts. In any case, it is clear that the local energy current flux will generically be non-zero for any charged particle motion and any timelike 3-surface unless there is some symmetry reason for it to vanish. One case where there is a symmetry reason for it to vanish is the case of a static charge surrounded by a static 3-surface. A second example is a uniformly accelerating charge with J_\mu defined using the boost Killing field associated with the charge orbit and the timelike surface chosen to be invariant under this Killing field. (This "second example" is really just another case of a static charge in a static spacetime.) From Steve's posting, I would guess that this fact underlies what Rohrlich seems to be groping towards, except the the "coaccelerating" has to mean following the orbits of the same boost Killing field as the accelerating charge AND energy has to be defined with respect to the boost Killing field. The idea that this lack of "radiation" has something to do with the relative acceleration of the charge and the observer (as opposed to having to do with symmetry) is completely off track. Aside from these cases where the flux of J_\mu vanishes by symmetry arguments, there always should be "radiation," i.e., there is no reason for the flux of J_\mu for a generic solution through a generic timelike surface to vanish. This statement also applies to radiation to infinity: Unless the spacetime is stationary and the charge is following an orbit of the Killing field, there should generically be radiation to infinity. Thus, a freely falling charge will radiate unless the geodesic it moves on also happens to be an orbit of a stationary Killing field. A stationary charge in a (globally) stationary spacetime, of course, will not radiate to infinity. =============================================================== 2) I (Fulling) asked Bob also to comment on the uniqueness (and existence) of the retarded solution for an eternally accelerated source. This is a nontrivial issue because the support of the source extends to infinity in a lightlike direction. Here is Bob's reply: Dear Steve, The retarded Green's function is uniquely defined, and it gives a unique (distributional) solution for any (distributional) source of compact support. Of course, the accelerating charge is not of compact support. Given the support properties, there is a unique retarded solution "above" the past horizon and, of course, the retarded solution vanishes "below" the past horizon. However, a priori, there is no reason why these solutions can be combined to yield a (distributional) solution including the past horizon. In the scalar case, one could start with the solution where the charge is created at a finite time (guaranteeing a distributional solution) and then take the limit as the creation time goes to -\infty. This presumably gives what people standardly call the retarded solution, but there is no reason why this limit need have existed. In fact, in the scalar case, for a null particle there is no distributional solution limit, as shown in my paper with Tolish, arXiv:1401.5831. (In the electromagnetic case, the limit of the field strengths for a null charge does exist as a distribution.) =========================================================== 3) I (Fulling) revised my presentation to correct a simple but important error: In the commentary on the figure by Pauri and Vallisneri, I had interchanged the interpretations of scenarios 1 and 4. I fell into a trap that illustrates how subtle the topic of the equivalence principle is: The pictures are drawn from the perspective of a terrestrial lab frame, so according to GR the bodies drawn in motion (#4) are "at rest" (i.e., in free fall), while the bodies drawn fixed (#1) are "coaccelerating". =========================================================== 4) Gerald Moore wrote to point out this review article on the dynamical Casimir effect: V. Dodonov, "Fifty Years of the Dynamical Casimir Effect", Physics (Basel, MDPI) 2 (2020) 67-104 DOI:10.3390/physics2010007 ===========================================================