The Lorentz coupling: external force on a test charge and the continuum force and power densities

Adopt as physical postulates of this framework, on the spacetime and particle model of Adopted Minkowski physical model of the electromagnetic framework and Worldlines, four-velocity, chart readings and the frame/observer distinction and on the special-relativity particle model Adopted massive-particle momentum, observer measurement and isolated-collision conservation:

(i) [Test charge in a prescribed external field] Let $q\in \mathbb{R}$ be a constant charge parameter (Charge parameter, frame charge and current densities, volume charge and flux, and the four-current), $m>0$ an externally supplied rest mass, $x:J\to A$ a future timelike $C^2$ massive worldline with four-velocity $U$, and let $F$ be a prescribed field of The electromagnetic two-form, its observer decomposition and the four-current that is $C^1$ on a neighbourhood of the graph of $x$ and is generated externally to the particle. Adopt the equation of motion

$m\frac{dU}{d\tau }=q\mathrm{\Phi }(U),$

where, at each original parameter $s\in J$, $\mathrm{\Phi }={\mathrm{\Phi }}_{x(s)}$ is the raised-index operator of the field at $x(s)$, defined by $\eta ({\mathrm{\Phi }}_{x(s)}(u),w)=F(x(s))(w,u)$ (Antisymmetric contractions vanish and $m{U}^{\mathrm{\prime }}=q\mathrm{\Phi }(U)$ preserves $\eta (U,U)$); equivalently $md{U}^{\mu }\mathrm{/}d\tau =q{F}^{\mu \nu }{U}_{\nu }$. Here $dU\mathrm{/}d\tau$ is the derivative of $U$ with respect to its proper-time function $\tau (s)$: explicitly $dU\mathrm{/}d\tau =U^{\mathrm{\prime }}(s)\mathrm{/}{\tau }^{\mathrm{\prime }}(s)$, with ${\tau }^{\mathrm{\prime }}(s)=c^{-1}\sqrt{-\eta (x^{\mathrm{\prime }}(s),x^{\mathrm{\prime }}(s))}>0$. It is the four-acceleration of the worldline (Worldlines, four-velocity, chart readings and the frame/observer distinction, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition). The particle's own field is excluded from $F$: this is the external-field coupling, and it defines no self-force.

(ii) [Continuum matter coupling] For a smooth source $(\rho ,J)$ of Charge parameter, frame charge and current densities, volume charge and flux, and the four-current interacting with the field through an explicitly stated matter model to which this coupling applies, adopt the force density $\rho E+J\times B$ (unit N m${}^{-3}$) and the power density $J\cdot E$ (unit W m${}^{-3}$) as the electromagnetic source terms of that matter model, with the spatial cross product of The cross product in ${\mathbb{R}}^3$; they describe the local interaction of the field with matter and are not a property of a prescribed current alone.

(iii) [Scope] The rest mass and the charge are inputs; $m$ is not derived within electromagnetism. The coupling is adopted for constant $q$ and $m>0$; no magnetic-monopole coupling, no radiation reaction and no self-force are included. In particular the nonrelativistic coordinate statement $m{r}^{\mathrm{\prime }\mathrm{\prime }}=q(E+r^{\mathrm{\prime }}\times B)$ is not adopted as exact: the adopted statement is the covariant one of clause (i), and any low-speed statement requires its own approximation hypotheses and error bounds.