Pre-acceleration from Landau-Lifshitz Series

TL;DR

This paper investigates the Landau-Lifshitz equation as an approximation to the Abraham-Lorentz-Dirac equation, revealing that the series expansion underlying the approximation diverges and resummation introduces pre-acceleration and runaway solutions.

Contribution

It demonstrates that the Landau-Lifshitz series diverges and that resummation of this series leads to pre-acceleration and runaway solutions, clarifying the differences between the two equations.

Findings

Landau-Lifshitz series diverges in general.

Resummation of the series introduces pre-acceleration.

Analysis primarily focuses on non-relativistic case.

Abstract

The Landau-Lifshitz equation is considered as an approximation of the Abraham-Lorentz-Dirac equation. It is derived from the Abraham-Lorentz-Dirac equation by treating radiation reaction terms as a perturbation. However, while the Abraham-Lorentz-Dirac equation has pathological solutions of pre-acceleration and runaway, the Landau-Lifshitz equation and its finite higher order extensions are free of these problems. So it seems mysterious that the property of solutions of these two equations is so different. In this paper we show that the problems of pre-acceleration and runaway appear when one consider a series of all-order perturbation which we call it the Landau-Lifshitz series. We show that the Landau-Lifshitz series diverges in general. Hence a resummation is necessary to obtain a well-defined solution from the Landau-Lifshitz series. This resummation leads the pre-accelerating and the runaway solutions. The analysis is focusing on the non-relativistic case, but we can extend the results obtained here to relativistic case at least in one dimension.