Ambiguity in deriving Larmor's formula from the LAD equation of motion

TL;DR

This paper investigates the difficulties in deriving Larmor's formula from the LAD equation, revealing that common methods fail to resolve the energy balance paradox across general cases.

Contribution

It demonstrates that the energy balance problem is inherent in the LAD equation and critiques the bound field approach and Schott term as insufficient solutions.

Findings

The energy balance paradox occurs beyond uniform acceleration.

The bound field technique cannot resolve the energy discrepancy.

The Schott term is not a valid four-momentum.

Abstract

The power radiated by a moving charge is given by Larmor's formula which can be derived by integrating the Li\'enard-Wiechert potential over the whole past history of the charge. However, extracting the same result from the Lorent-Abraham-Dirac (LAD) equation of motion is problematic. This is well known for uniform proper acceleration for which case the radiation force vanishes and hence the very origin of the radiated energy is not clear, leading to an energy balance paradox. The purpose of this letter is first to evince that this problem occurs in the general case and not only for uniform acceleration. Second we show that the widely accepted treatment based on the bound field technique cannot fix the energy balance discrepancy. Indeed the related Schott term is not a legitimate four-momentum for being indefinite and not conserved.