In 1943 CJ Eliezer published a paper claiming that the self-force prevents a zero angular momentum particle from ever reaching the center of an attractive Coulomb potential (and what's more that it can collide with a repulsive potential). As stated in the paper this result is somewhat counterintuitive, but the reasoning seems like a relatively straightforward differential equation argument.

In thinking about this the understanding of the self-force that I came to is that while you can derive it purely from energy and momentum conservation (and thus it must be valid in any theory of classical charged point particles), the resulting differential equation is better thought of as a consistency condition than equations of motion (i.e. only solutions of the 3rd order self-force equation correspond to point particle sources that have solutions to Maxwell's equations that lose or gain the correct amounts of energy and momentum at the point particle corresponding to its motion). And while one would like to be able to define a dynamical system of a point particle coupled to the electromagnetic field with physically plausible boundary conditions, even eliminating runaway solutions doesn't prevent you from being forced to include waves coming in from past infinity (i.e. in pre-acceleration solutions, which do not have to have runaway behavior, the particle will move before an external force is applied, meaning that it must be gaining energy from radiation from past infinity).

Assuming that Eliezer's result is correct it seems like every trajectory of a particle in a stationary Coulomb potential similary requires radiation adding energy to the particle (otherwise you can prove with a simple energy argument that it must fall in). So the question is, is my interpretation of the dynamics of the self-force correct and is there a physical or intuitive explanation for this extremely pathological behavior in the presence of a Coulomb potential?


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    $\begingroup$ There is no such thing as a "self-force" in physics. $\endgroup$ – CuriousOne May 6 '15 at 12:14
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    $\begingroup$ I thought it was another name for the Abraham-Lorentz-Dirac Force. $\endgroup$ – James Hanson May 6 '15 at 12:38
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    $\begingroup$ That's a misnomer, an accelerated charged particle doesn't interact with itself but with the physical vacuum. The deeper problem, of course, is that any theory with point particles necessarily breaks down in the limit for r->0, which leads to inconsistent classical results. Even quantum field theory suffers from consistency problems in that limit because it still hasn't done away with infinitely small quantities. A better theory will, of course, discover some inherent limit which will remove any need for such forces. $\endgroup$ – CuriousOne May 6 '15 at 12:45
  • $\begingroup$ Mostly I just needed a shorter term for the force. $\endgroup$ – James Hanson May 6 '15 at 13:11
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    $\begingroup$ I meant for reality. If I had cared much about titles, I would have become a book editor, rather than a physicist. $\endgroup$ – CuriousOne May 6 '15 at 13:46

is my interpretation of the dynamics of the self-force correct and is there a physical or intuitive explanation for this extremely pathological behavior in the presence of a Coulomb potential?

Eliezer makes his argument based on the equation with the Lorentz-Abraham-Dirac term.

This term was originally (Lorentz) devised as an approximate way to account for the action of charged sphere on itself (one charged part acts on another charged part and as a result, there is a net force). His derivation shows the LAD term is only approximate way to account for the interaction of the parts. Similarly in antenna theory it is possible to show that third derivative is only approximate way to model internal interactions.

Moreover, there are well-known cases where the model based on the LAD term fails completely (run-aways, preaccelerations).

All this holds for particles with finite charge density (the particle has non-zero dimensions).

If the particle is truly a point, there is no valid reason to even try to apply the LAD term to it. Its derivation is not valid for point particles (Dirac's paper has a "derivation" that is based on wrong premise - Poynting expressions for point particles). People have tried anyway and they consistently failed - there are always some fishy excuses made to make the edifice work apparently.

Consistent theories of charged point particles were described many times long time ago, e.g. by Frenkel:

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. http://dx.doi.org/10.1007/BF01331692

In English, this article also explains it concisely:

R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let- ters, 8, 3, (1964), p. 185-187. http://dx.doi.org/10.1016/S0031-9163(64)91989-4

  • $\begingroup$ Questions: Do you have a source on the antenna theory statement? Does this mean that Eliezer's result would still hold for sufficiently small but finite size charged objects? When you talk about Poynting expressions for point particles are you talking about equation (14) in Dirac's paper? That's the only thing I can find that sounds relevant. In there any difference between the different treatments of point particles in quantum field theories? $\endgroup$ – James Hanson May 7 '15 at 0:42
  • $\begingroup$ Landau, L. D., Lifshitz E. M., Classical theory of fields, §75. By Poynting expressions I mean expressions for EM energy density in vacuum $\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2$ and for momentum density in vacuum $\epsilon_0\mathbf E\times \mathbf B$. $\endgroup$ – Ján Lalinský May 7 '15 at 19:40
  • $\begingroup$ Thank you for the reference. What I don't understand is that my reading of Dirac's paper leaves me with the impression that all he's doing is calculating the four-momentum flux leaving the particle, which seems to be fixed by the Liénard–Wiechert potential and Poynting's theorem infinitesimally close to the point particle, and then figuring out what force is necessary to make the particle lose that much four-momentum. Are you saying that's not what his argument is or is there some hidden assumption in the argument that allows for other radiation reaction force equations? $\endgroup$ – James Hanson May 8 '15 at 2:21
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    $\begingroup$ Dirac says in his paper he assumes that the Poynting expressions from the macroscopic theory (and the corresponding tensor of energy-momentum, function of total electric and magnetic field) are valid even in the theory of point particle and proceeds from there. However, the Poynting theorem only holds for regions where $\mathbf E\cdot\mathbf j$ is integrable, which excludes the points of space where the point particles are located. The theorem thus cannot be interpreted as work-energy theorem relating work done on the particles to other quantities. $\endgroup$ – Ján Lalinský May 8 '15 at 4:22
  • $\begingroup$ I think I can see the problem now. Thank you for the explanation. $\endgroup$ – James Hanson May 11 '15 at 19:56

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