The Abraham-Lorentz force is the force a classical charged particle particle exerts on itself due to its own electromagnetic field. It has a rather simple formula that reads $$ \vec{F}_\mathrm{AL} = \frac{2 q^2}{3 c^2} \vec{\dddot{x}} \,. $$ My question is the following: Is there any realizable context where a classical charged body can be observed to experience the Abraham-Lorentz force?

It is unlikely that one could directly observe the immediate acceleration due to $\vec{F}_\mathrm{AL}$, but an indirect observation through long-term energy losses of the body might be observable.

  • 1
    $\begingroup$ In particle accelerators, whenever the dynamics has to take into account energy loss by radiation, then the radiation reaction is being taken into account, and consequently the Abraham-Lorentz force or one of its alternative formulations (they differ in detail but only at higher order in the size of the charged bodies). $\endgroup$ – Andrew Steane Jan 2 '19 at 18:47
  • $\begingroup$ @AndrewSteane I see, so one uses the classical approximation to compute the radiation losses for electrons/ions? Do you believe there are contexts where a classical computation using the Abraham-Lorentz force would provide an accurate prediction for the shape of a cathode ray inspiraling on an anode? $\endgroup$ – Void Jan 2 '19 at 20:08

Ok, I would like to thank Andrew Steane and Vladimir Kalitvianski for their input. I have done some digging myself and I believe I have gathered enough material to compile an answer to the question from the following sources:

  1. The 2017/2018 notes of Kirk T. McDonald On the History of the Radiation Reaction,
  2. the 2016 paper by Di Piazza et al. Investigation of classical radiation reaction with aligned crystals, and
  3. the 2017 paper by Wistisen et al. Experimental Evidence of Quantum Radiation Reaction in Aligned Crystals.

According to McDonald:

We noted earlier that while the radiation reaction for oscillating currents has clear manifestation in the so-called radiation resistance of antennas, there is no experimental evidence for the classical radiation reaction of an individual electric charge.

Now let me cite the abstract of Di Piazza et al.:

The self-consistent underlying classical equation of motion including radiation-reaction effects, the Landau-Lifshitz equation, has never been tested experimentally, in spite of the first theoretical treatments of radiation reaction having been developed more than a century ago. Here we show that classical radiation reaction effects, in particular those due to the near electromagnetic field, as predicted by the Landau-Lifshitz equation, can be measured in principle using presently available facilities, in the energy emission spectrum of 30-GeV electrons crossing a 0.55-mm thick diamond crystal in the axial channeling regime

By the Landau-Lifschitz equation they mean the approximation where the term $\dddot{x}$ is replaced by the jerk felt by a hypothetical "test particle" accelerated in the very same external field but without any radiation reaction. (I.e., simply an approximation of the Abraham-Lorentz force.)

The proposal of Di Piazza et al. was then experimentally realized by Wistisen et al. but they found that the classical approximation of the radiation reaction is not a valid model and that:

The measured photon emission spectra show features which can only be explained theoretically by including both 1) quantum effects related to the recoil undergone by the positrons in the emission of photons and the stochasticity of photon emission, and 2) radiation-reaction effects stemming from the emission of multiple photons.

However, none of their theoretical models really fit the spectra, so maybe one should wait for more experimental and theoretical studies for the final verdict.

But, as far as concerns the original question, we can state that there exist:

  1. Experimental realizations where ensembles of particles feel radiation reaction that can be modeled by the Abraham-Lorentz (or Landau-Lifschitz) formula. This includes radiation losses by beams of particles in particle accelerators as well as the "radiation resistance" of currents in antennas.
  2. Experimental realizations of quantum radiation-reaction that cannot be reasonably modeled by the Abraham-Lorentz (or Landau-Lifschitz) formula.

However, there currently seems to be no realizable experimental setup where effects of the Abraham-Lorentz force on a single classical body can be measured or observed.

  • $\begingroup$ The radiation reaction was observed in cyclotrons with relativistic electrons and it was described in some Russian textbook. The orbits without radiation reaction force remain, say, circular with a certain radius $R(E)$, but in reality, due to emitting hard photons, the electron orbit has a noticeable "width" determined with QED. Hard photons take a lot of energy and it influences the resulting radius, and since the emission happens spontaneously, it creates a distribution of different radii. Now, if you decrease the electron energy $E$ in the cyclotron, you will get a simply smaller width. $\endgroup$ – Vladimir Kalitvianski Jan 4 '19 at 16:32
  • $\begingroup$ Anyway, the original radius will be perturbed, this is a fact. You must ask accelerator physicists for their experience. $\endgroup$ – Vladimir Kalitvianski Jan 4 '19 at 16:33
  • $\begingroup$ Finally, electron-ion recombination may start from high orbits from which the electron descends practically quasi-classically and due to radiative losses by emitting soft photons. You may not deny this phenomenon. $\endgroup$ – Vladimir Kalitvianski Jan 4 '19 at 18:33
  • $\begingroup$ The LL equation uses the cause of the jerk -- the field, whereas the LAD equation uses the effect -- the acceleration of the charge. In the comoving frame these are $\dot E$ and $\dot a$ respectively. $\endgroup$ – Larry Harson Jan 5 '19 at 0:52
  • $\begingroup$ @LarryHarson: It is too superficial judgement. Roughly speaking, $\dot{a}$ is un unknown variable whereas $\dot{E}$ is some sort of an external field. They lead to different qualitatively solutions. $\endgroup$ – Vladimir Kalitvianski Jan 5 '19 at 7:40

The Abraham-Lorentz force is wrong since it leads to non physical solutions. It is not used in calculations. In practical calculations it is replaced with another force, namely with $\tau_0\dot{\vec{F}}_{\text{ext}}(x,t)$, see Landau-Lifshitz textbook and [1]. This force does not conserve the energy-momentum, strictly speaking, but it is much better option than the pure Abraham-Lorentz force since it behaves more physically: it indeed takes (approximately) into account radiative losses, which are obviously present in reality (in betatrons an other accelerators, for example, as well as in collisions).

[1] Fritz Rohrlich, The dynamics of a charged particle, (2008) http://arxiv.org/abs/0804.4614

  • $\begingroup$ Is LL really used today? I'm under the impression that Swinger's paper gave the full theory for accelerators using the LAD equation and there is no need to support the LL equation. $\endgroup$ – Larry Harson Jan 5 '19 at 0:59
  • $\begingroup$ @LarryHarson: I did not read the Schwinger's paper, so I cannot judge it, but LAD itself does not work. One needs a replacement. Often it is called an "approximation", but it is not of the third order in time derivatives, so it is another guess, like LL or so. (G. N. Plass tries to convince the reader that the third derivative is OK, forgetting an extreme instability of the solutions: journals.aps.org/rmp/abstract/10.1103/RevModPhys.33.37.) $\endgroup$ – Vladimir Kalitvianski Jan 5 '19 at 7:22
  • $\begingroup$ The classically exact equation is the retarded force over the classical radius; this is then approximated as a Taylor series over present time. I think the problem is that the effects of these higher order terms of acceleration can be zeroed for only special cases such as periodic and circular motion. $\endgroup$ – John McAndrew Jan 13 '19 at 23:46
  • $\begingroup$ @JohnMcVirgo:The Lotentz equation supplied with the retarded "self-force" contains physically understandable but a wrong result - self-induction of electron, i.e., a new mass addenda $\delta m_{\text{e}}\to\infty$, which is discarded anyway (mass renormalization). It is not a small radiative friction force. $\endgroup$ – Vladimir Kalitvianski Jan 14 '19 at 5:05
  • $\begingroup$ $r_e$ can be kept finite so maintaining a finite positive electromagnetic mass with no compensating negative mass prone to pre-acceleration. $\endgroup$ – John McAndrew Jan 14 '19 at 22:26

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