Reduced-order Abraham-Lorentz-Dirac equation and the consistency of classical electromagnetism

Abstract

It is widely believed that classical electromagnetism is either unphysical or inconsistent, owing to pathological behavior when self-force and radiation reaction are non-negligible. We argue that there is no inconsistency as long as it is recognized that certain types of charge distribution are simply impossible, such as, for example, a point particle with finite charge and finite inertia. This is owing to the fact that negative inertial mass is an unphysical concept in classical physics. It remains useful to obtain an equation of motion for small charged objects that describes their motion to good approximation without requiring knowledge of the charge distribution within the object. We give a simple method to achieve this, leading to a reduced-order form of the Abraham-LorentzDirac equation, essentially as proposed by Eliezer, Landau and Lifshitz, and derived by Ford and O'Connell.

Electromagnetic Self-Force of a Point Charge from the Rate of Change of the Momentum of Its Retarded Self-Field

“…There are good reasons, among them the existence of runaway, and pre-and post-acceleration solutions of the LAD equation, to regard the very concept of a point charge as unphysical in classical electrodynamics. The equations of motion that admit no unphysical solutions, like the Landau-Lifshitz, Eliezer and Ford-O'Connell equations, assume explicitly or implicitly the charge to have a finite spatial extension of the order of the classical radius corresponding to its mass [7]. Be it as it may, our result supports the commonly held notion of the LAD equation as the "exact" equation of motion of a point charge.…”

Towards the nonlinear regime in extensions to GR: assessing possible options

“…For instance, if the total physical time of interest is t 0 = 0 to T , one could choose to integrate the homogenous problem θ (0) = 0 over this full range (and iterate over it), or over a smaller interval ∆T and advance over this interval to cover the full interval of interest. The introduction of such an interval is natural, and is analogous to what is done, for instance in the integration of the Abraham-Lorentz-Dirac equation [28] (see also [39,40]) through a reduction of order approach. There it is found, as naturally expected, that more accurate solutions are obtained with smaller intervals.…”

Comments on initial conditions for the Abraham–Lorentz(–Dirac) equation

“…Some considerations of the origin of the ALD force in the microscopic structure of the so-called point-particle have even lead to problems such as the " 4 3 problem" [14][15][16][17], negative mass [18,19], and even suggestions that the fundamentals of EM of point particles be altogether replaced [10,22]. Most attempts for solving these problems have suggested modifying the equation itself [23][24][25][26][27][28][29][30][31][32][33][34][35], predominantly by reducing the equation for the particle's trajectory to a more familiar 2 nd order ODE, although this * ofek.birnholtz@mail.huji.ac.il 1 Relativistically, x becomes x µ , time derivatives are taken w.r.t…”