The references below describe two interrelated relativistic issues that come up when you try to describe radiation from point charges:(adsbygoogle = window.adsbygoogle || []).push({});

1. We normally expect that an accelerating charge would radiate, but this would seem to violate the equivalence principle.

2. Classical E&M can't describe point particles, because the energy in the field of a point particle diverges, and by relativistic mass-energy equivalence, this would give it infinite inertia.

Issue #2 is normally blamed for the pathological behaviors of the solutions to the equation of motion for a point charge according to the Lorentz-Dirac equation for the radiation reaction force, ##F^c=(2/3)kq^2(\delta^c_b+v^cv_b)\dot{a}^b##, where ##a=dv/d\tau##, and the dot means ##d/d\tau##. These behaviors, described in Poisson, are runaway solutions and preacceleration (violations of causality in response to an external force).

Brown discusses the relationship between issues #1 and #2, but I'm getting confused by the details, and I'm posting to ask if anyone can help me to clear up my confusion. The basic idea is that if a charge is sitting on a tabletop, conservation of energy dictates that it shouldn't radiate. But the tabletop is actually accelerating compared to an inertial (free-falling) frame, so this implies that a charge undergoing constant acceleration shouldn't radiate. This is all consistent with the Lorentz-Dirac equation, which predicts zero reaction force when the proper acceleration is constant. After all, if there is no radiation, you shouldn't get a reaction force. So far, so good.

The first thing I'm confused about is the reasoning behind the runaway solutions. Poisson uses the low-velocity version of the equations of motion, under which the Lorentz-Dirac force becomes a Newtonian force three-vector proportional to ##\dot{a}##. The equations of motion are then of the form ##a=(\ldots)\dot{a}##, and this gives an exponential as a solution. But now suppose we try to do this without the low-velocity approximation. If you do this fully relativistically, then clearly the runaway motion has to look the same at any arbitrarily chosen moment in time, in the instantaneously comoving frame; therefore the proper acceleration has to be constant. But in that case the Lorentz-Dirac force is zero, so any runaway solution would have to violate the equations of motion. What's going on here?

My other confusion has to do with the Medina paper. In the abstract of the paper, he says that if the particle is not pointlike but has a certain minimum size, you don't get runaway solutions. (He states it as a condition on the mass, but it amounts to the same thing.) The runaway solutions exist because the particle is acted on by its own radiation reaction force, even in the absence of any external force. If there are no runaway solutions for big enough particles, then apparently the radiation reaction shows qualitatively different behavior for big particles than for pointlike ones. This seems strange, because I don't see how the resolution of the issues with the equivalence principle can be qualitatively different in these two cases. My understanding was that the resolution of this issue was that observers who are accelerated with respect to one another can disagree on whether a particular field pattern constitutes radiation or not. But when we judge whether a field constitutes radiation, we're talking about itslarge-distance behavior, and this doesn't seem to be compatible with the idea that there are two qualitatively differentsmall-range behaviors.

Does anyone have any insights?

Brown, http://www.mathpages.com/home/kmath528/kmath528.htm

Medina, Radiation reaction of a classical quasi-rigid extended particle, J.Phys. A39 (2006) 3801-3816, http://arxiv.org/abs/physics/0508031

Morette-DeWitt, "Falling Charges," Physics, 1,3-20 (1964), http://www.scribd.com/doc/100745033/Dewitt-1964

Poisson, http://arxiv.org/abs/gr-qc/9912045

Rohrlich: The dynamics of a charged sphere and the electron Am J Phys 65 (11) p. 1051 (1997), http://www.lepp.cornell.edu/~pt267/files/teaching/P121W2006/ChargedSphereElectron.pdf

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# Runaway solutions and the equivalence principle

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