Finite-time weak/slow charged probe correspondence

Use the charged-probe action sector of the fixed-background model of Separately adopted fixed-background Maxwell and charged probes and let $\alpha =q\mathrm{/}m\mathrm{\ne }0$, with constant $m>0$, constant charge $q$, length scale $L>0$, and speed constant $c>0$. When the Lorentz-force form is obtained from the adopted action, the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis of that exact action-equivalence theorem is assumed; no stronger choice axiom is used. In SI, $[\alpha ]=\mathrm{C}\mathrm{k}{\mathrm{g}}^{-1}$, $[L]=\mathrm{m}$, and $[c]=\mathrm{m}{\mathrm{s}}^{-1}$; $ϵ$, $X$, $\varphi$, $\mathbf{e}$, and $\mathbf{b}$ are dimensionless. Choose a cutoff $0<{ϵ}_{\ast }\le 1$ sufficiently small in terms of the displayed uniform bounds below, as required by Controlled weak/slow charged test trajectory comparison, and fix $0<ϵ\le {ϵ}_{\ast }$. On an open $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$, let $\mathcal{K}⋐\mathrm{\Omega }$ be compact and convex, use length coordinates $x^0=ct$, $x^i=L{X}^i$, and suppose the prescribed metric is smooth, stationary and Lorentzian on a neighborhood of ${\mathbb{R}}_{x^0}\times L\mathcal{K}$, with ${\mathrm{\partial }}_0$ future timelike. For a smooth dimensionless $\varphi$ and some fixed finite $C_g$, assume for every spatial multi-index $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 2$ that

$\mathrm{\mid }{\mathrm{\partial }}_X^{\beta }{r}_{00}^{(ϵ)}\mathrm{\mid }+{\max }_{i,j}\mathrm{\mid }{\mathrm{\partial }}_X^{\beta }{r}_{ij}^{(ϵ)}\mathrm{\mid }\le C_g{ϵ}^2,{\max }_i\mathrm{\mid }{\mathrm{\partial }}_X^{\beta }{r}_{0i}^{(ϵ)}\mathrm{\mid }\le C_g{ϵ}^{3\mathrm{/}2},$

where $g_{00}^{(ϵ)}=-(1+2ϵ\varphi )+r_{00}^{(ϵ)}$, $g_{ij}^{(ϵ)}=(1-2ϵ\varphi ){\delta }_{ij}+r_{ij}^{(ϵ)}$, and $g_{0i}^{(ϵ)}=r_{0i}^{(ϵ)}$. Prescribe smooth stationary dimensionless fields $\mathbf{e},\mathbf{b}$ with values and first derivatives bounded on $\mathcal{K}$, and a smooth real closed two-form $F^{(ϵ)}$ on the same background neighborhood, with $d{F}^{(ϵ)}=0$ and SI components $F_{0i}^{(ϵ)}=-E_i^{(ϵ)}\mathrm{/}c$, $F_{ij}^{(ϵ)}={ϵ}_{ijk}{B}_k^{(ϵ)}$ and

$\alpha {\mathbf{E}}^{(ϵ)}=\frac{ϵc^2}{L}\mathbf{e},\alpha {\mathbf{B}}^{(ϵ)}=\frac{\sqrt{ϵ}c}{L}\mathbf{b}\mathrm{.}$

Closure $d{F}^{(ϵ)}=0$ ensures local potentials, not a global potential: in each sufficiently small open coordinate ball $D$ about a curve point, use the smooth radial primitive of Radial primitives for closed one- and two-forms as the supplied $A_D$ with $d{A}_D=F^{(ϵ)}{\mathrm{\mid }}_D$. Adopt fixed-endpoint action stationarity only on sufficiently short finite curve segments contained in such a ball, as in Separately adopted fixed-background Maxwell and charged probes and Stationary charged action and Lorentz force. There is no assumption that the whole trajectory lies in a single potential domain, that $F^{(ϵ)}$ is globally exact, or that the local potentials are stationary. No inhomogeneous Maxwell or Einstein equation is imposed on this prescribed background. The dimensionless comparison trajectory satisfies ${\mathbf{X}}_N^{\mathrm{\prime }}={\mathbf{W}}_N$, ${\mathbf{W}}_N^{\mathrm{\prime }}=-{\mathrm{\nabla }}_X\varphi ({\mathbf{X}}_N)+\mathbf{e}({\mathbf{X}}_N)+{\mathbf{W}}_N\times \mathbf{b}({\mathbf{X}}_N)$ on $[0,S_{\ast }]$, where primes denote $S$ derivatives, $S_{\ast }>0$, and $S=\sqrt{ϵ}ct\mathrm{/}L$. Write $\mathrm{\parallel }(\mathbf{X},\mathbf{W}){\mathrm{\parallel }}_1$ for the sum of the absolute values of the six components and $\mathrm{\parallel }Df{\mathrm{\parallel }}_{1\to 1}$ for its induced operator norm, as in Controlled weak/slow charged test trajectory comparison. Let the six-component comparison vector field $f$ have derivative norm at most $L_f<\mathrm{\infty }$ on a compact convex state set $\mathcal{Q}\subseteq \mathcal{K}\times {\mathbb{R}}^3$ containing both scaled trajectories throughout $[0,S_{\ast }]$.

Suppose a $C^2$ future-directed proper-time worldline exists through this finite interval, obeys this local action model (equivalently the external Lorentz law on each such segment), has coordinate speed $\mathrm{\mid }\mathbf{w}\mathrm{\mid }\le C_v\sqrt{ϵ}c$, has its scaled state remain in $\mathcal{Q}$, and has the same initial scaled data as the comparison trajectory. Set ${\mathbf{X}}_{ϵ}=\mathbf{x}\mathrm{/}L$ and ${\mathbf{W}}_{ϵ}=\mathbf{w}\mathrm{/}(\sqrt{ϵ}c)$. Then for some finite $C_r$ depending only on the displayed uniform bounds, the exact scaled motion obeys

${\mathbf{X}}_{ϵ}^{\mathrm{\prime }}={\mathbf{W}}_{ϵ},{\mathbf{W}}_{ϵ}^{\mathrm{\prime }}=-{\mathrm{\nabla }}_X\varphi ({\mathbf{X}}_{ϵ})+\mathbf{e}({\mathbf{X}}_{ϵ})+{\mathbf{W}}_{ϵ}\times \mathbf{b}({\mathbf{X}}_{ϵ})+{\mathbf{r}}_{ϵ}(S),\mathrm{\parallel }{\mathbf{r}}_{ϵ}(S){\mathrm{\parallel }}_1\le C_rϵ,$

and consequently

${\sup }_{0\le S\le S_{\ast }}(\mathrm{\parallel }{\mathbf{X}}_{ϵ}-{\mathbf{X}}_N{\mathrm{\parallel }}_1+\mathrm{\parallel }{\mathbf{W}}_{ϵ}-{\mathbf{W}}_N{\mathrm{\parallel }}_1)\le C_rϵS_{\ast }{e}^{L_f{S}_{\ast }}\mathrm{.}$

This is a finite-duration conditional model prediction. It does not construct the trajectories, prove common compact containment, solve Maxwell's equations for the prescribed field, or establish a coupled Einstein–Maxwell family. The magnetic scaling may violate a coupled stress hierarchy unless additional matter scales are supplied.