Conservation of Momentum and Energy in the Lorenz-Abraham-Dirac Equation of Motion

TL;DR

The paper addresses the problem of momentum-energy conservation in the Lorentz-Abraham-Dirac framework for a relativistically rigid, extended charged sphere by incorporating transition forces at nonanalytic points in time when external forces turn on or off. It derives expressions for the radiated energy and momentum across each transition interval, $W_{\mathrm{TI},n}$ and $G_{\mathrm{TI},n}$, and establishes precise conditions, including Born rigidity $|\Delta u_n^0|/c \ll 1$ and small transition changes $\tau_e|\Delta F_{\mathrm{ext}}^n|/(m c) \ll 1$ (with $\tau_e = e^2/(6\pi \epsilon_0 m c^3)$), under which the modified equations remain causal and conserve momentum-energy with nonnegative radiated energy. A parallel-plate capacitor scenario provides explicit forms for these quantities and demonstrates the limitations when mass renormalization to a finite $m$ is employed, showing that both $W_{\mathrm{TI},n}$ and $G_{\mathrm{TI},n}$ cannot generally be made zero. The work highlights that while the transition-force approach yields a consistent classical description under certain constraints, a fully satisfactory, universal classical point-charge theory likely requires quantum effects and a unified treatment of inertial/gravitational and electrodynamic forces. It also notes that the Landau-Lifshitz approximation, though causal, can still yield unphysical energy behavior across transition ends in some regimes. Overall, the results delineate the regimes where causality and momentum-energy conservation hold and point to the need for deeper theoretical integration beyond classical electrodynamics.

Abstract

After a brief review of the modified causal Lorentz-Abraham classical equation of motion for an extended charged sphere and its limit to the mass-renormalized modified causal Lorentz-Abraham-Dirac equation of motion as the radius of the charged sphere approaches zero, a concise derivation is given for the conditions on the velocity and external force required for these modified equations of motion to satisfy conservation of momentum and energy. The solutions to the unmodified and modified LAD equations of motion as well as the Landau-Liftshitz approximate solution to the unmodified LAD equation of motion are obtained for a charge traveling through a parallel-plate capacitor.