New Equation of Motion for Classical Charged Particles

Abstract

With the intuitive new ideas that (1) in classical electrodynamics, radiation reaction should be expressible by the external field and the charge's kinematics, (2) a charge experiences, in addition to the Lorentz forces, another "small" external force ${e}_{1}{F}^{\ensuremath{\mu}\ensuremath{\lambda}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{u}}_{\ensuremath{\lambda}}$ proportional to its acceleration, and (3) inertia plus radiation is balanced by these two external forces, we propose the new equation of motion, $m{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{u}}^{\ensuremath{\mu}}\ensuremath{-}(\frac{2{e}^{3}}{3m}){F}_{\mathrm{ext}}^{\ensuremath{\lambda}\ensuremath{\alpha}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{u}}_{\ensuremath{\lambda}}{u}_{\ensuremath{\alpha}}{u}^{\ensuremath{\mu}}=e{F}_{\mathrm{ext}}^{\ensuremath{\mu}\ensuremath{\lambda}}{u}_{\ensuremath{\lambda}}+{e}_{1}{F}_{\mathrm{ext}}^{\ensuremath{\mu}\ensuremath{\lambda}}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{u}}_{\ensuremath{\lambda}}$ , where mass conservation requires ${e}_{1}=\frac{2{e}^{3}}{3m}$. (The particle's spin is not considered in this work.) This equation for a classical charge is free from all the well-known difficulties of the Lorentz-Dirac equation. It conserves energy and momentum in a modified form in which the energy-momentum tensor contains a part ${t}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$ made of a new field-charge interaction ${\ensuremath{\varphi}}^{\ensuremath{\mu}}(x)$, in addition to the conventional "local" part made of ${F}_{\mathrm{ret}}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$ and ${F}_{\mathrm{ext}}^{\ensuremath{\mu}\ensuremath{\nu}}(x)$ only, and therefore it no longer satisfies the conventional "local" conservation laws. It predicts correct radiation damping, as demonstrated here by applying it to various cases of basic physical importance. Also, it implies that a massless particle follows a null geodesic and cannot interact with the electromagnetic field whether it be charged or not; this implication may add a new degree of freedom to the charge-conservation law.