Flat Lorentz–Abraham–Dirac effective equation

Restrict Charged and spinning test-body quantities to its flat affine Minkowski branch, with signature $(-,+,+,+)$, proper time $\tau$ in seconds, future four-velocity $U=dx\mathrm{/}d\tau$ satisfying $\eta (U,U)=-c^2$, constant primitive $m>0$ in kilograms and nonzero charge $q\in \mathbb{R}$ in coulombs. Let the prescribed external electromagnetic two-form $F$ be $C^2$ on an open neighborhood of the worldline; use the component convention $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$ of The electromagnetic two-form, its observer decomposition and the four-current. Let $a:=dU\mathrm{/}d\tau$ and

$\alpha :=q\mathrm{/}m,{\tau }_e:=\frac{q^2}{6\pi {ϵ}_{0}m{c}^3}>0,f_0^a:=\alpha F^a{}_b{U}^b,$

where ${ϵ}_0=({\mu }_0{c}^2{)}^{-1}$ follows Framework-local SI units, the vacuum constants and the exact/measured status of their values. Adopt the flat LAD effective equation

$a^a=f_0^a+{\tau }_e(\frac{d{a}^a}{d\tau }-\frac{{\eta }_{bc}{a}^b{a}^c}{c^2}{U}^a)\mathrm{.}$

The derivative is the ordinary derivative in an inertial affine chart. Its dimensions are consistent: $a$ and $f_0$ are accelerations, the parenthesized term has acceleration per second, and ${\tau }_e$ is a time. This is an adopted effective model equation for sufficiently regular curves (at least $C^3$); it is not asserted to be an exact law of nature, a point-source Maxwell construction, or a globally well-posed dynamics. In particular, no existence, uniqueness, stability, or empirical claim is part of this postulate.

At each proper-time value $\tau$, define the particle's instantaneous rest frame to be the tuple $(x(\tau );e_0,e_1,e_2,e_3)$, where $e_0=U(\tau )\mathrm{/}c$ and $(e_1,e_2,e_3)$ is a positively oriented $\eta$-orthonormal basis of $U(\tau {)}^{\perp }$. This is a pointwise basis at the event $x(\tau )$, not one inertial chart along an accelerated curve; the rest-space norm is $\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}:=\sqrt{\eta (v,v)}$ for $v\in U(\tau {)}^{\perp }$. Use of the equation as a reduced physical approximation carries the source's qualitative validity qualification that radiation reaction be small relative to the applied Lorentz force in this instantaneous rest frame. This postulate neither asserts that condition for arbitrary $F$ nor supplies a numerical threshold; the qualification is not used to deduce the local ODE results or the force-free counterexample.

The equation is taken as a modeling premise. The source discussion motivates radiation-reaction modeling and order reduction but does not turn this postulate into an experimentally proved universal law. No self-field is evaluated at the worldline here.