THE STRUCTURE OF THE RETARDED FIELD OF AN ETERNALLY UNIFORMLY ACCELERATED CHARGE, AND ITS IMPLICATIONS FOR THE EXISTENCE OF RADIATION AND THE PRINCIPLE OF EQUIVALENCE The accompanying PDF file is (currently) at https://www.math.tamu.edu/~fulling/ar/fulling1.pdf . Unlike George Matsas, I shall assume for the sake of argument that uniform acceleration for all time makes sense. The retarded electromagnetic field from such a motion, derivable from the Lienard-Wiechert formula, has been known since 1909, but its implications have been controversial from the start and still are. Understanding the unusual features of this solution is essential (though apparently not sufficient) for resolving the conundrums and persistent misunderstandings about the principle of equivalence, and even the very existence of radiation, in this system. The formulas become simpler and more transparent when the EM field is replaced by a massless scalar field, but the interpretational issues are, for the most part, unchanged. To require less effort on the part of both myself and readers, I'll discuss here mainly the scalar case, following Ren and Weinberg, Phys. Rev. D 49 (1994) 6526-6533. Let's start with a brief historical review of the EM solution. Breezing past such 19th-century worthies as Larmor and Lienard & Wiechert, I see these as the major steps: (Full references are in the PDF document.) Born (1909) derived the basic formulas for the E and B fields. He was rather nonchalant about the domains of the functions. Schott (1912-15) pointed out that the field was not, in fact, retarded if the formulas were applied in all space-time. The field had to be set to 0 by hand in the regions that causally cannot be influenced by the charge (the regions P and L, in the modern notation for discussing Rindler coordinates and the Unruh effect). Actually (according to Fulton and Rohrlich), Schott did not cite Born; he solved the problem independently. He also did pioneering work on resolving the apparent contradiction between the Larmor radiation law and the Abraham-Lorentz-Dirac radiation reaction law. Bondi and Gold (1955) observed that Schott's function was not, in fact, a solution of Maxwell's equations on the past horizon (the null plane separating P and L from the nontrivial regions, R and F). They obtained a valid solution by adding a delta-function term on the past horizon. Rohrlich (1960-63), in three papers (the first in collaboration with T. Fulton), made a thorough study of the situation. In the process of this work, Rohrlich of course learned a lot, and hence the papers sometimes contradict each other as he changed his mind. He came down squarely on the side of those who say that the uniformly accelerated charge does radiate, according to the Larmor formula (notwithstanding that the Abraham-Lorentz-Dirac radiation reaction force vanishes when the jerk does). On the other hand, he emphasized that "radiation" is well-defined only at a large distance from the charge, and radiation is in some sense a nonlocal phenomenon. In particular, a detector accelerating along with the charge would not detect radiation. This fact was used to argue that there is no contradiction between acceleration radiation and the principle of equivalence, since RELATIVE acceleration between charge and detector is, at least qualitatively, the key consideration. Unfortunately, Rohrlich does not seem to have appreciated the significance of the future horizon separating regions R and F -- note that he was writing before the appearance of Rindler's famous paper about horizons in Minkowski space -- and hence it is unclear in the papers to what extent the crucial property of the coaccelerated detector is its acceleration or its confinement to R. Boulware (1980) finally gave a modern treatment recognizing the division of the causally related region into R and F. His paper is the closest we have to a definitive account of the problem. Consider now a scalar field satisfying the massless Klein-Gordon equation with a scalar source in place of the electromagnetic charge-current density. (See the PDF file and the Ren-Weinberg paper for precise formulas.) Let the source q (also loosely called "charge") move along the standard hyperbolic trajectory with acceleration a, t = sinh(as)/a, z = cosh(as)/a, x = 0 = y. Let r^2 = x*2 + y^2. Then the retarded solution is \phi(t,x,y,z) = q/(4\pi R) \theta(t + z), where R = (a/2)\sqrt{(z^2 - t^2 + r^2 - a^{-2})^2 +4a^{-2}r^2}. (Sorry, we are stuck with two meanings of "R" throughout this discussion.) It looks like a Coulomb field somehow distorted by the acceleration. (See PDF for explication of the geometrical meaning of R.) Let us introduce Rindler coordinates in the region R (where z > |t|) by t = \sigma sinh(a\tau), z = \sigma cosh(a\tau). Then z^2 - t^2 = \sigma^2, and R depends on t and z only through the combination \sigma; it is independent of the Rindler time. Similarly, in F (where t > |z|), let t = \sigma cosh(a\tau), z = \sigma sinh(a\tau). Again R depends only on \sigma (and r), but now \sigma is a time coordinate. We can now catalog some properties of the function \phi. 1. It indeed satisfies the wave equation (with a source on the hyperbola), even on the horizon where the theta function sits. Normally, "chopping" a solution by a theta function would introduce into the partial differential equation terms proportional to the delta function and its derivative, thereby violating the PDE. In this case it can be shown (see PDF) that those terms vanish. However, nontrivial delta-function terms do appear in the FIRST derivatives of \phi. The true electromagnetic counterpart of \phi is not the E and B fields, but the vector potential. Conversely, the scalar counterpart of E and B is the gradient of \phi, whose components appear bilinearly in the components of the stress-energy-momentum tensor. So there is no conflict here with Bondi and Gold, and no real loss of smoothness in passing from scalar to EM. 2. It is indeed retarded, because of the theta function. (A technical objection might be raised to this claim, but I postpone it to the PDF.) 3. In region R, \phi is static, in the Rindler sense -- i.e., independent of the time coordinate \tau. It is also invariant under time reversal. It is hard to discern any radiation in this region, especially from the point of view of an observer moving along another hyperbolic orbit of the same Rindler Killing vector. This confirms Rohrlich's dictum that a coaccelerating observer sees no radiation. (However, I must say that all authors of the Rohrlich era were quite sloppy about what they meant by "coaccelerating". Their diagrams often show hyperbolas that are not members of the same Rindler congruence.) 4. In region F, \phi is dynamical -- a function of time but not of the spatial coordinate (which is now \tau, by my definition). So there may be "radiation" in F, but we need to be careful about what this means. For me the easy way out would be to cite the paper of Landulfo et al. (2019), which breaks up \phi into Unruh normal modes and shows that the amplitudes coincide with the results of an S-matrix analysis in the quantized theory. Traditionally the classical theory has been analyzed in terms of energy fluxes (the Poynting vector in the EM case). The most up-to-date treatments I know of are Sec. IV of Boulware's paper for the EM case and Sec. II of Ren & Weinberg's paper for the scalar case. In a nutshell, R&W (following Bourlware) do two calculations. First, they integrate T^{tj} over a sphere on the future light cone of the charge and get the analog of Larmor's formula, q^2a^2/(12\pi). [Sharp-eyed readers may object that this conclusion is inconsistent with my points 3 and 5 if the sphere is so small that it fits inside R. That is a tangled issue that I don't want to discuss right now.] Second, they do a calculation of the total energy flow through a certain space-time region entirely within R, and traversed by the charge's worldline, and get zero. This demonstrates that there is no contradiction between inertial and accelerated calculations inside R, but it leaves a murky issue of where the incoming energy comes from. Again, today is not the time to get into a critical analysis. I urge everyone to read the two papers. 5. We could have constructed an advanced solution, which is nonzero in R and P only and has its theta function on the future horizon, \theta(z - t). Then \phi[retarded] = \phi[advanced] in the interior of R! The time-symmetric function (\phi[ret] + \phi[adv])/2 is also a solution of the same PDE; it is zero only in L and has step singularities on both horizons. A different construction is to start with our unique \phi in the interior of R, continue it smoothly across the future horizon into F as \phi[ret], but also continue it smoothly across the past horizon into P as \phi[adv]. This is a smooth solution throughout the three regions; note that in F and P it differs from the symmetrized solution by a factor 2. Continuing this solution further, across the distant half-horizons into L, leads finally to disaster: it cannot satisfy the free wave equation in L; rather it has a source (with a negative sign) on the hyperbola that is a mirror image of the trajectory of the original source. (Some authors call this a charge "moving backwards in time", but, please, let's not.) In other words, as Boulware says, "The field in region [F] may be regarded as EITHER the field due to a uniformly accelerated charge, [q], in region [R] OR as the field due to a uniformly accelerated charge -[q] in region [L]; no measurements restricted to [F] cam ever distinguish the two situations." [His similar comment about the fact \phi[ret] = \phi[adv] in R is, "The field at any point in [R] may be interpreted EITHER as the Coulomb field plus OUTGOING radiation field at the retarded time, OR as the Coulomb field plus INCOMING radiation at the advanced time." This refers of course to the fact that there are a unique backward null line and a unique forward null line from the field point to the hyperbolic trajectory (Boulware, Fig. 4).] Let me finish this overlong essay with an introduction to the controversy over the principle of equivalence. That issue will surely merit much deeper discussion in the future. The point is illustrated very well by p. 1507 of the 1999 paper of Pauri and Vallisneri, which I've placed at https://www.math.tamu.edu/~fulling/ar/PVfig.pdf . The radiated circular waves there indicate what I'll call the Rohrlich orthodoxy: No radiation is detected if the source and detector are both at rest (of course) or if they are coaccelerating (as per my point 3). These are cases 4 and 1, respectively, in P&V's terminology. If the detector is in free fall and the charge is accelerated (case 3), radiation is detected. (Cf. point 4, where the same conclusion is reached at least when the detector is far away from the source.) Finally, if the charge is in free fall and the detector accelerated (case 2), by symmetry one would expect radiation to be detected. However, Rohrlich and P&V notwithstanding, this conclusion is not universally accepted, to put it mildly. Many physicists consider it blatantly absurd, and respond with either "This proves that the principle of equivalence simply does not apply to electromagnetic phenomena!" or "This shows that the other conclusion must also be false: Surely a charge at rest on a table in my lab is not radiating, even if I'm falling while I'm looking at it." In favor of the Rohrlich doctrine for the controversial case 2, one can cite parallel evidence from other scenarios. An accelerated Unruh detector is analogous to case 2, if one regards the detector as the "observer". On the other hand, if the detector is regarded as an emitter seen by an inertial observer (Unruh-Wald 1984), the analogue is case 3. For this system both radiation phenomena are now generally accepted, with, if anything, case 3 being slightly more controversial than case 2. Similarly, Scully et al. (2018) find that an atom freely falling into a black hole radiates as seen by a supported observer, provided that the quantized field is in a Boulware-like state; Svidzinsky et al. (2018) find the analogous phenomenon for a static atom in flat space if a Rindler-like field state is created (by a uniformly accelerated mirror). Finally, Fulling and Wilson (2019) carried the analogy down to a two-dimensional conformal scalar field interacting with a static mirror, which radiates if the field starts in a Rindler-like vacuum. In all these systems it seems that a qualitative version of the equivalence principle is satisfied and real radiation detection occurs. However, their relation to classical bremsstrahlung is muddied by the need to put the quantum field into some generalized "Rindler vacuum" when the emitter is in free fall. The distinction between Rindler (or Boulware) vacuum and Minkowski (or Unruh or Hartle-Hawking) vacuum has no counterpart in classical theory. On the other hand, upon rereading I find Rohrlich's argument for case 3 unconvincing. We have seen that the crucial point about case 2 (and its consistency with the null result for case 4) is that experiment 4 takes place entirely within Rindler space, R, whereas the radiation in experiment 2 is observed in region F. Clearly, the relation between Rindler and Minkowski space is not reciprocal! In case 3 there is no analog of F. Furthermore, from the point of view of a Rindler observer the Coulomb field of a stationary charge is never completely behind a Rindler horizon, even if the charge itself is. Therefore, that field is not truly "retarded" in the way that the Ren-Weinberg field is. This seems to cause problems for Rohrlich's argument (see his 1963 paper, p. 182 and the long footnote extending onto p. 183). A detailed critique unfortunately is beyond the scope of this essay.