Date: Fri, 17 Jul 2020 21:04:04 -0500 (CDT) From: "Stephen A. Fulling" To: Acceleration Radiation Community: ; Subject: More comments, July 17 ============================================================ 1. Date: Thu, 16 Jul 2020 15:32:54 -0700 (PDT) From: Bill Unruh Subject: Re: Comments, July 16 > 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)?) But these are answered with the retarded solution. For the charge at rest (or constant velocity) the retarded solution is the same as the advanced in the areas of overlap, if one takes the data on sci- to be zero, and the extention into the spacelike region to the end of the constant velocity is also unique. Thus one can reasonably take that as the Initial data on some spacelike hypersurface which runs through the constant velocity part of the charges motion. Ie, taking 0 on sci- and retarded solution for the charge contribution seems like a very reasonable choice. (One could of course always add to this a solution of the source free equation which would be non-zero on sci-) > > 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. The above gives a well defined way of taking that 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. Why is that reasonable. The charge has an infinite EM energy. But anyway, the static charge in the past has a coulomb energy and the Homeogeneous field has some positive energy and two have a zero interference term (one is spherically symetric about the location of the charge and the other is not-- there is no monopole radiation) So the minimum energy is just the Coulomb energy of the charge with the homogeneous solution's energy being zero. . > > 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.? > > ============================================================ 2. From: Don Page Date: Fri, 17 Jul 2020 14:13:44 -0600 Subject: comment on No. 2 (mine) from Comments, July 16 I thought of a couple of examples in which there is almost certainly electromagnetic radiation even though the charge worldline is static in a natural coordinate system: a) Have a charge Q stay at the spatial origin of Minkowski spacetime (neglecting gravity) while two perfectly conducting spheres of radius R pass by with constant opposite velocities on opposite sides of the charge at impact parameter b > R (so that at each moment of the Minkowski time t whose hypersurfaces are flat and orthogonal to the static charge worldline, the charge is in the middle of the line segment between the centers of the two spheres). For example, the center of one of the conducting spheres could be at x = b, y = 0, and z = vt; the other could be at x = - b, y = 0, z = - vt. In this case the charge is static, and each of the two spheres has constant velocity, so there is no acceleration of any charge or object at all, but I am pretty sure there will be radiation that can be detected by static observers at large distances. I included two moving conducting spheres so that the back reaction would cancel and the charge would stay at x = y = z = 0 even with no external forces on it. There would be forces on the spheres, but one can make them sufficiently massive that their velocities stay very nearly constant even with no extra forces on them to cancel the electromagnetic forces on them. b) Have a charge Q in the middle of two symmetrically arranged identical black holes of mass M orbiting around nearly circular orbits at slowly decreasing distance r from the charge at the center of mass (orbiting because of the gravity of the other black hole; one can assume that the charge has much lower mass so that its gravitational effect is negligible). I feel very nearly certain that the time-dependent distortion of the Coulomb field of the charge by the gravitational field of the orbiting black holes would cause electromagnetic radiation to be emitted that can be detected by static observers very far away in the asymptotically flat spacetime. I got curious how the electromagnetic power P_E would compare with the gravitational wave power P ~ (M/r)^5 (with G = c = 4 pi epsilon_0 = 1 and dropping numerical coefficients of the general order of unity). My very crude _guess_ is the following: Since the black holes have solid angles ~ (M/r)^2 in the sky as seen by the charge, I would guess that this situation is crudely like having two charges of magnitude ~ Q (M/r)^2 orbiting around each other at distance ~ r with angular frequency omega ~ (M/r^3)^{1/2}. If this is right, it gives an effective electric quadrupole moment ~ Q M^2 and electromagnetic power P_E ~ (omega)^6 (Q M^2)^2 ~ Q^2 M^7 r^{-9} ~ Q^2 M^2 r^{-4} P (remember that r is the separation of each black hole from the charge, not the distance to the observer that I am assuming is in the far zone, at a distance >> 1/omega), so one would need |Q| > r^2/M > M for the electromatic power to exceed the gravitational wave power (modulo some numerical coefficient that I would not know how to calculate and suspect would need numerical integration), which (unless the coefficient for the gravitational wave power is sufficiently smaller than the coefficient for the electromagnetic quadrupole power) is impossible if the charge is a black hole, which with G = c = 4 pi epsilon_0 = 1 has |Q| < M, or, in the extreme charge case, |Q| = M. Have either of these two cases (as well as the case I gave yesterday, of a charge falling through the gravitational field of a thin mass shell) been calculated explicitly? Since it seems obvious that they will all give radiation, it is not obvious to me that an explicit calculation is needed, but if it would have pedagogical value and has not been done before, I might be interested in joining a collaboration with others who are much better in doing the numerical calculations than I. Or, if I could not contribute much other than my present suggestions of these situations, I am fine if someone else wants to do the calculations without me and perhaps just acknowledge that I posed the questions. Perhaps just let me know first, in case someone else wanted to work with me on it. Best wishes, Don [previous message:] > 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. > > ============================================================= 3. The next (probably) mailing will introduce a new topic (#3 on the original list): Does a uniformly accelerated charge radiate at all? Pauli, for one, said no. Stephen A. Fulling Professor of Mathematics and Physics Texas A&M University College Station, TX 77843-3368 USA