Date: Thu, 16 Jul 2020 15:59:45 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; ============================================================ 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)?) 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. 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. 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.? ------------------------------------------------------- I [original message] 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.) ============================================================ 2. From: Don Page Date: Wed, 15 Jul 2020 14:53:24 -0600 The following example might possibly be interesting to work out: Take the static spherically symmetric metric given by a thin massive neutral shell to give flat spacetime inside but Schwarzschild outside. Let a massive (but much lighter than the shell) charged particle fall from infinity through the shell and back out to infinity, and take the retarded electromagnetic solution in this otherwise charge-free space. (Assume that the charge does not directly interact with the shell and so can pass through it with no impulse given to the charge.) I think everyone would agree that as the charge is falling toward the shell from the outside (or rising back up on the other side), it will be effectively accelerating relative to a static observer far away, so that observer will see radiation. I think that the static observer will also see radiation over the interval of the retarded time when the charge is crossing the inside of the shell, when it has constant velocity in this flat space region (ignoring radiation reaction; if that would alter the effects, just assume some other force on the charge to cancel the radiation reaction force and keep the charge worldline on a radial geodesic of the spacetime), since the part of the particle's electromagnetic field outside the shell will effectively be accelerating, So this would be a case of radiation even without any local or global acceleration of the charge itself. The situation is a bit subtle in that in the Schwarzschild metric outside the shell, the EM Green function does not obey Huygens' principle even in the even spacetime dimensions being assumed (D=4) and so does not vanish off the future light cone as it does in empty flat 4D spacetime (where I might express Huygens' principle by saying "the speed of dark equals the speed of light"). Instead there will be tails, so even if the charge did not radiate while crossing the flat spacetime inside the shell at constant velocity, outside one would still see tails from what the charge radiated while falling in from outside the shell (and there speeding up as seen by the exterior static observer). I haven't tried to check how the scaling of these tails with the mass per radius of the shell or other parameters compares with the scaling for the effect from the effective acceleration of the electromagnetic field attached to the charge but outside the shell while the charge is crossing the inside of the shell, which scaling (if different) conceivably might allow one to say unambiguously that there is radiation from the effective acceleration of the field outside the shell rather than from the tails of the Green function. (I'm not even sure that these really are two different effects, or whether they are possibly the same effect just in different words.) If anyone has seen this example analyzed and an attempt made to disentangle the radiation while the charge itself is moving with constant velocity across the flat spacetime inside the shell from the tails of the Green function in the Schwarzschild region outside, I would be interested in learning the results. ============================================================= Stephen A. Fulling Professor of Mathematics and Physics Texas A&M University College Station, TX 77843-3368 USA