Reply to "Comment on 'Absence of a consistent classical equation of motion for a mass-renormalized point charge'" (arXiv:2511.02865v1, 3 Nov 2025)

TL;DR

The paper analyzes the classical motion of a charged sphere approaching the point-charge limit to address whether mass renormalization leads to delta-function radiation. It reviews the derivation of a causal, Lorentz-covariant equation of motion based on Maxwell's equations, relativistic dynamics, and $E=mc^2$, emphasizing transition intervals of length $\Delta t_a \approx 2a/c$ where conventional self-force expansions fail. It shows that, for finite $a$, radiation during transitions is finite and that, as $a\to0$ with finite $m$, jumps in velocity are handled by a transition force within the modified LAD framework, with radiated energy obtained from transition-interval integrals rather than naive point-charge formulas. It also argues that without renormalization Maxwell's equations govern all times and jump effects vanish in the limit, while renormalization alters the direct Maxwellian calculation of transition radiation, thereby resolving apparent paradoxes and reinforcing energy-momentum conservation in both extended and (renormalized) point-charge regimes.

Abstract

By means of a brief review of the derivation of the causal modified Lorentz-Abraham-Dirac classical equation of motion from the renormalization of the mass in the modified equation of motion of an extended charged sphere as its radius approaches zero, it is shown that Zin and Pylak's objection that the jumps in velocity allowed across transition intervals near nonanalytic points in time of the externally applied force produce delta functions in the radiated fields is incorrect.