Separately adopted fixed-background Maxwell and charged probes

On an open region of the oriented, time-oriented Lorentzian spacetime and length-coordinate SI conventions of Smooth matter quantities, current and observer observables, adopt a prescribed-background Maxwell model. The metric $g$ is fixed input. Choose a real $C^2$ electromagnetic two-form $F$ and either adopt the source-free equations $dF=0$ and $d({\star }_{g}F)=0$, or supply a $C^1$ current $j$ and adopt $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$, with the exact signs and units of that definition. The source current is prescribed independently of the probe; this postulate does not impose an Einstein equation, matter evolution equation, or field equation for the sources on the unchanged metric.

For a massive charged probe, use the adopted local action and finite-curve conditions of External-field charged-particle action: $m>0$ and $q$ are constant, the path is a $C^2$ future-directed timelike segment parameterized by proper time, and its endpoints are fixed in variations supported in a local potential domain $V$ with a supplied $C^2$ one-form $A$ satisfying $dA=F$. The potential and metric are external data independent of the path. Under the exact hypotheses of Stationary charged action and Lorentz force, action stationarity is equivalent to $m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U)$; that theorem's ${\mathrm{A}\mathrm{C}}_{\omega }$ assumption is inherited only when this equivalent force-law formulation is used. No global potential or trajectory is included.

The probe's own singular field and self-gravity are excluded. No radiation reaction, variable mass or charge, finite-body model, or backreaction estimate is adopted. In particular, calling $g$ or $F$ a test background records this model's omission of their response; it does not claim that the unchanged pair solves the coupled Einstein–Maxwell equations or is close to a coupled solution.

When geometric optics is invoked, it is restricted to the exact $C^4$ real phase, $C^3$ complex amplitudes, four pointwise ansatz constraints, and fixed compact-chart coefficient bounds of Finite WKB Maxwell residual, null rays and polarization quotient. That result supplies finite residual and leading-field estimates, local null rays, and a polarization quotient; it supplies no convergent series, global transport through caustics, exact-solution error estimate, or backreaction theorem.

This is a separately adopted physical model with no empirical premise. The Maxwell and probe assumptions are not deductions from one another, from mathematical axioms, or from source authority.

Use the fixed-background Maxwell test model of Separately adopted fixed-background Maxwell and charged probes on a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$ and its adopted length-coordinate SI conventions. Let $U\subseteq M$ be open. Let $\theta \in C^4(U,\mathbb{R})$ be dimensionless, set $k=d\theta$, and assume $k$ is nowhere zero and $g^{-1}(k,k)=0$. Let $a_0,a_1\in C^3(U;T^{\ast }M{\otimes }_{\mathbb{R}}\mathbb{C})$, with covariant derivatives and contractions extended complex-linearly. Use $[k]={\mathrm{m}}^{-1}$, $[a_0]=[a_1]=\mathrm{T}\mathrm{m}$, and dimensionless $ϵ$, as in Finite WKB Maxwell residual, null rays and polarization quotient. Define

$(\mathrm{\square }a{)}_b:=g^{ac}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_c{a}_b,(Pa{)}_b:=(\mathrm{\square }a{)}_b-{\mathrm{Ric}}_b{}^c{a}_c,(La{)}_b:=2k^c{\mathrm{\nabla }}_c{a}_b+({\mathrm{\nabla }}_c{k}^c)a_b\mathrm{.}$

Assume the four pointwise constraints $g^{-1}(k,a_0)=0$, $L{a}_0=0$, $iL{a}_1+P{a}_0=0$, and $i{k}^a(a_1{)}_a+{\mathrm{\nabla }}^a(a_0{)}_a=0$. For $0<ϵ\le 1$, form the complex ansatz $A_{ϵ}=ϵe^{i\theta \mathrm{/}ϵ}(a_0+ϵa_1)$ and its real potential ${\widehat{A}}_{ϵ}:=\mathrm{Re}{A}_{ϵ}$ with real field ${\widehat{F}}_{ϵ}:=d{\widehat{A}}_{ϵ}$. Use the fixed-chart component max norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty },K}$ of Finite WKB Maxwell residual, null rays and polarization quotient, which is zero on an empty compact set. On a nonempty compact coordinate set $K$, assume finite constants $K_0,D_0,C_0,B_0,B_1$ satisfy $\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty },K}\le K_0$, $\mathrm{\parallel }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le D_0$, $\mathrm{\parallel }P{a}_1-\mathrm{\nabla }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le C_0$, $\mathrm{\parallel }d{a}_0+ik\wedge a_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_0$, and $\mathrm{\parallel }d{a}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_1$. The real field obeys $d{\widehat{F}}_{ϵ}=0$ and

$\mathrm{\parallel }{\mathrm{\nabla }}^a({\widehat{F}}_{ϵ}{)}_{ab}{\mathrm{\parallel }}_{\mathrm{\infty },K}\le ϵK_0{D}_0+{ϵ}^2{C}_0,\mathrm{\parallel }{\widehat{F}}_{ϵ}-\mathrm{Re}(e^{i\theta \mathrm{/}ϵ}ik\wedge a_0){\mathrm{\parallel }}_{\mathrm{\infty },K}\le ϵB_0+{ϵ}^2{B}_1\mathrm{.}$

Each local integral curve of $k^{\mathrm{♯}}$ is an affinely parameterized null geodesic. Along it $a_0$ is transported according to the supplied scalar transport amplitude, and the leading physical polarization is the class in $k^{\perp }\mathrm{/}{\mathrm{span}}_{\mathbb{C}}\{k\}$, a two-dimensional complex space with positive induced Hermitian form. These are exact finite-order deductions under the displayed ansatz constraints. The residual is a component norm in the chosen chart, not an observer-independent tensor norm. The conclusion asserts no convergent WKB series, existence of amplitudes across caustics, exact Maxwell solution for a nonzero residual, solution-distance estimate, or coupled Einstein–Maxwell family.

Use the smooth oriented, time-oriented Lorentzian four-manifold, Levi–Civita connection $\mathrm{\nabla }$, and length-coordinate SI conventions of Smooth matter quantities, current and observer observables and the separately prescribed Maxwell test model of Separately adopted fixed-background Maxwell and charged probes. Here $U$ is open, $g$ is the fixed metric, a field strength $F$ is a smooth real two-form, and a prescribed current $j$ is a smooth real vector field. The formal stress tensor and constants are

$T_{\mathrm{E}\mathrm{M}}^{ab}(g,F)=\frac{1}{{\mu }_0}(F^{ac}{F}^b{}_c-\frac{1}{4}{g}^{ab}{F}_{cd}{F}^{cd}),T_{ab}^{\mathrm{E}\mathrm{M}}=g_{ac}{g}_{bd}{T}_{\mathrm{E}\mathrm{M}}^{cd},\kappa =\frac{8\pi G_N}{c^4}>0.$

The adopted Maxwell alternatives are $dF=0$, $d({\star }_{g}F)=0$ when source-free, or $dF=0$, ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ for a separately prescribed current. Let $g$ satisfy $G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}=0$ on $U$, where $\mathrm{\Lambda }$ is constant. Define the electromagnetic stress residual relative to this unchanged vacuum metric, with no matter stress included, by

${\mathcal{R}}_{ab}(g,F;\mathrm{\Lambda }):=G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}-\kappa T_{ab}^{\mathrm{E}\mathrm{M}}(g,F)\mathrm{.}$

First let $F_0$ be a smooth source-free Maxwell two-form on $U$. For real dimensionless $\delta$, $F_{\delta }:=\delta F_0$ remains source-free and

${\mathcal{R}}_{ab}(g,F_{\delta };\mathrm{\Lambda })=-\kappa {\delta }^2{T}_{ab}^{\mathrm{E}\mathrm{M}}(g,F_0)\mathrm{.}$

The same conclusion holds for an actual smooth sourced Maxwell pair: let $F_0$ be a smooth real two-form and $j_0$ a smooth real vector field on $U$ satisfying $d{F}_0=0$ and ${\mathrm{\nabla }}_a(F_0{)}^{ab}=-{\mu }_0{j}_0^b$. With $F_{\delta }:=\delta F_0$ and $j_{\delta }:=\delta j_0$, the pair still satisfies the same sourced Maxwell equations, and

${\mathcal{R}}_{ab}(g,F_{\delta };\mathrm{\Lambda })=-\kappa {\delta }^2{T}_{ab}^{\mathrm{E}\mathrm{M}}(g,F_0)\mathrm{.}$

For a compact set $K$ in a fixed smooth coordinate chart contained in $U$ and a smooth covariant two-tensor $S$, define $\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{\infty },K}:={\max }_{0\le a,b\le 3}{\sup }_{p\in K}\mathrm{\mid }{S}_{ab}(p)\mathrm{\mid }$ when $K\mathrm{\ne }\mathrm{\varnothing }$, and set $\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{\infty },\mathrm{\varnothing }}:=0$. In either Maxwell case,

$\mathrm{\parallel }\mathcal{R}(g,F_{\delta };\mathrm{\Lambda }){\mathrm{\parallel }}_{\mathrm{\infty },K}=\kappa {\delta }^2\mathrm{\parallel }{T}^{\mathrm{E}\mathrm{M}}(g,F_0){\mathrm{\parallel }}_{\mathrm{\infty },K}\mathrm{.}$

If $\delta \mathrm{\ne }0$ and $F_0$ is nonzero at a point, its observer energy is positive there, so this electromagnetic/vacuum-background residual is nonzero. In the sourced case this does not test a full Einstein–Maxwell–matter equation, because no source-matter stress is included in $\mathcal{R}$. The residual is a consistency calculation, not an estimate of distance to a different metric solution.

Separately, on an open region where smooth charged-dust quantities have $\rho >0$, $T_m^{ab}=\rho U^a{U}^b$, $j^a=\sigma U^a$, and the formal matter-exchange equation holds, replacing $(\rho ,\sigma )$ by $(\eta \rho ,\eta \sigma )$ for $\eta >0$ scales matter stress and current linearly, leaves $\sigma \mathrm{/}\rho$ unchanged, and preserves the dust force equation when the prescribed $F,U$ are held fixed. Maxwell's sourced equation must still match the scaled current. If instead both $F$ and $j$ are scaled by $\eta$, the sourced Maxwell system is linear but electromagnetic stress scales as ${\eta }^2$ while dust stress scales as $\eta$; these degree differences provide no general coupled-solution family.

Finally, for a fixed $Q\mathrm{\ne }0$ and $\mathrm{\ell }>0$, prescribe only on the finite Minkowski annulus ${\mathcal{A}}_{\mathrm{\ell }}=\{\mathbf{x}:\mathrm{\ell }<\mathrm{\mid }\mathbf{x}\mathrm{\mid }<2\mathrm{\ell }\}$ the radial electric field $\mathbf{E}=Q(4\pi {\epsilon }_0{r}^2{)}^{-1}{\mathbf{e}}_r$, $\mathbf{B}=0$. Its field energy on that annulus is $Q^2\mathrm{/}(16\pi {\epsilon }_0\mathrm{\ell })$ and diverges as $\mathrm{\ell }\downarrow 0$. This calculation defines no field or source at the origin and supplies no point-charge construction or regularization theorem.

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.