Finite WKB Maxwell residual, null rays and polarization quotient

Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ with the metric-selected connection $\mathrm{\nabla }$ of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Use the curvature operator, Ricci contraction, and sign convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Let $A$ be a real one-form whose coefficients in every smooth chart are $C^3$. On such coefficients, use the covariant-derivative component formula ${\mathrm{\nabla }}_a{A}_b:={\mathrm{\partial }}_a{A}_b-{\mathrm{\Gamma }}^c{}_{ab}{A}_c$ and define $dA$ as the $C^2$ two-form with chart components $(dA{)}_{ab}:={\mathrm{\partial }}_a{A}_b-{\mathrm{\partial }}_b{A}_a$; for smooth $A$ this agrees with the exterior derivative of Lorentzian exterior calculus and Hodge star. The transformation law for these antisymmetrized components follows by the chain rule, since the second-derivative chart-transition terms cancel by equality of mixed partials. Define $(\mathrm{\square }A{)}_b:=g^{ac}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_c{A}_b$, $(\mathrm{div}A):={\mathrm{\nabla }}^a{A}_a$, and $(PA{)}_b:=\mathrm{\square }{A}_b-{\mathrm{Ric}}_b{}^c{A}_c$. Then

${\mathrm{\nabla }}^a(dA{)}_{ab}=(PA{)}_b-{\mathrm{\nabla }}_b(\mathrm{div}A)\mathrm{.}$

The same identity holds for a complex one-form after complex-linear extension. The identity is local and uses no field equation, gauge condition, boundary condition, or physical assumption.

Use the smooth oriented, time-oriented Lorentzian four-manifold, metric, connection, curvature sign, length-coordinate SI convention and constants of Smooth formal matter, EM tensors and observer quantities. Let $U\subseteq M$ be open. Let $\theta \in C^4(U,\mathbb{R})$ be dimensionless, set $k:=d\theta$, and assume that $k$ is nowhere zero and $g^{-1}(k,k)=0$. Define the metric dual vector field $k^{\mathrm{♯}}$ by $g(k^{\mathrm{♯}},X)=k(X)$ for every tangent vector $X$; in coordinates, $(k^{\mathrm{♯}}{)}^a=g^{ab}{k}_b$. Let $a_0,a_1\in C^3(U;T^{\ast }M{\otimes }_{\mathbb{R}}\mathbb{C})$; this means that the real and imaginary parts of their coefficients are $C^3$ in every smooth chart. Complex tensor operations and $\mathrm{\nabla }$ are extended componentwise and complex-linearly. Define

$(\mathrm{\square }a{)}_b:=g^{ac}{\mathrm{\nabla }}_a{\mathrm{\nabla }}_c{a}_b,(Pa{)}_b:=(\mathrm{\square }a{)}_b-{\mathrm{Ric}}_b{}^c{a}_c,d_1:={\mathrm{\nabla }}^a(a_1{)}_a,$

and define the first-order operator

$(La{)}_b:=2k^c{\mathrm{\nabla }}_c{a}_b+({\mathrm{\nabla }}_c{k}^c)a_b\mathrm{.}$

Assume pointwise on $U$ that

$g^{-1}(k,a_0)=0,L{a}_0=0,iL{a}_1+P{a}_0=0,i{k}^a(a_1{)}_a+{\mathrm{\nabla }}^a(a_0{)}_a=0.$

These are finite-order ansatz constraints; they assert neither initial data nor a global solution of the transport equations or convergence of a longer series. For $0<ϵ\le 1$, set

$A_{ϵ}:=ϵe^{i\theta \mathrm{/}ϵ}(a_0+ϵa_1),F_{ϵ}:=d{A}_{ϵ}\mathrm{.}$

Here $d{A}_{ϵ}$ uses the complex-linear extension of the $C^3$ coordinate definition in Curved Lorentzian potential wave identity, so $F_{ϵ}$ has $C^2$ components. On a $C^2$ two-form $F$, define $dF$ by the $C^1$ three-form components $(dF{)}_{abc}:={\mathrm{\partial }}_a{F}_{bc}+{\mathrm{\partial }}_b{F}_{ca}+{\mathrm{\partial }}_c{F}_{ab}$; the antisymmetrized chain rule and equality of mixed second partials make this coordinate definition chart-independent (Clairaut--Schwarz theorem for continuous second partial derivatives).

Then $d{F}_{ϵ}=0$ and the exact Maxwell divergence residual is

${\mathrm{\nabla }}^a(F_{ϵ}{)}_{ab}=e^{i\theta \mathrm{/}ϵ}[-iϵk_b{d}_1+{ϵ}^2((P{a}_1{)}_b-{\mathrm{\nabla }}_b{d}_1)]\mathrm{.}$

The field-strength expansion is also exact:

$F_{ϵ}=e^{i\theta \mathrm{/}ϵ}[ik\wedge a_0+ϵ(d{a}_0+ik\wedge a_1)+{ϵ}^{2}d{a}_1]\mathrm{.}$

The acronym WKB means Wentzel–Kramers–Brillouin; here “finite WKB ansatz” means exactly the phase-amplitude family $A_{ϵ}$ just defined, subject to the four pointwise constraints above, and asserts no convergence of a longer series.

For units, take $[k]={\mathrm{m}}^{-1}$, $[a_0]=[a_1]=\mathrm{T}\mathrm{m}$, $[d_1]=\mathrm{T}$, $[P{a}_1]=[\mathrm{\nabla }{d}_1]=\mathrm{T}{\mathrm{m}}^{-1}$, and $[ϵ]=[\theta ]=1$. Fix a smooth chart $(V,y)$ and a compact set $K\subseteq y(V)$ in its coordinate image. Define

$\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{\infty },K}:=\{\begin{array}{ll}{\max }_I{\max }_{x\in K}\mathrm{\mid }{S}_I(x)\mathrm{\mid },& K\mathrm{\ne }\mathrm{\varnothing },\\ 0,& K=\mathrm{\varnothing },\end{array}$

where $I$ ranges over the finitely many components of the real or complex tensor in that chart. If $K\mathrm{\ne }\mathrm{\varnothing }$ and finite constants $K_0,D_0,C_0,B_0,B_1\ge 0$ satisfy

$\mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty },K}\le K_0,\mathrm{\parallel }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le D_0,\mathrm{\parallel }P{a}_1-\mathrm{\nabla }{d}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le C_0,\mathrm{\parallel }d{a}_0+ik\wedge a_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_0,\mathrm{\parallel }d{a}_1{\mathrm{\parallel }}_{\mathrm{\infty },K}\le B_1,$

then

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

Taking real parts gives real potentials and fields with the same bounds. Here a local integral curve of $k^{\mathrm{♯}}$ means a $C^2$ curve $\gamma :J\to U$, with $J\subseteq \mathbb{R}$ an open interval, satisfying $\dot{\gamma }(\lambda )=k_{\gamma (\lambda )}^{\mathrm{♯}}$. In length coordinates the parameter $\lambda$ has unit ${\mathrm{m}}^2$. An affinely parametrized null geodesic here means a $C^2$ curve with $\dot{\gamma }(\lambda )\mathrm{\ne }0$, $g(\dot{\gamma },\dot{\gamma })=0$, and ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=0$; every such local integral curve is one and lies in a level set of $\theta$. Along each such curve, choose any ${\lambda }_0\in J$ and any nonzero dimensionless complex initial value ${\mathcal{A}}_0\in \mathbb{C}\setminus \{0\}$, and define the scalar transport amplitude

$\mathcal{A}(\lambda )={\mathcal{A}}_0\exp (-\frac{1}{2}{\int }_{{\lambda }_0}^{\lambda }({\mathrm{\nabla }}_c{k}^c)(\gamma (s))ds),\lambda \in J\mathrm{.}$

The integral is well-defined on each compact segment between ${\lambda }_0$ and $\lambda$, and $\mathcal{A}$ is nowhere zero because ${\mathcal{A}}_0\mathrm{\ne }0$ and the exponential is nonzero. The one-form $e:=a_0\mathrm{/}\mathcal{A}$ is therefore well-defined and satisfies ${\mathrm{\nabla }}_{\dot{\gamma }}e=0$ and $g^{-1}(k,e)=0$ along the curve; these equations express parallel transport and transversality, respectively. At each $p\in U$, define $k_p^{\perp }:=\{a\in T_p^{\ast }M{\otimes }_{\mathbb{R}}\mathbb{C}:g_p^{-1}(k_p,a)=0\}$. Define the leading polarization at $p$ to be the class $[a_0(p)]$ in $k_p^{\perp }\mathrm{/}{\mathrm{span}}_{\mathbb{C}}\{k_p\}$. This quotient is a two-dimensional complex vector space with the positive Hermitian form induced by $g_p^{-1}$, and for every $\beta \in \mathbb{C}$ replacing $a_0(p)$ by $a_0(p)+\beta k_p$ leaves $k_p\wedge a_0(p)$ unchanged. If $a_0(p)$ is proportional to $k_p$, the leading field vanishes at $p$. No convergence of a formal series extending the finite ansatz, existence of amplitudes satisfying the constraints, exact-solution error estimate, or extension of the local rays or transport amplitude beyond their stated domains is asserted.

Use the field, stress, matter, unit and curvature conventions of Smooth formal matter, EM tensors and observer quantities. Let $U\subseteq M$ be open, let $\mathrm{\Lambda }\in \mathbb{R}$ be constant, and define the Einstein residual for the source-free field model by

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

Suppose $G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}=0$ on $U$, and let $F_0$ be a smooth real two-form with $d{F}_0=0$ and ${\mathrm{\nabla }}_a(F_0{)}^{ab}=0$, the source-free form of the Maxwell equations in Smooth formal matter, EM tensors and observer quantities. For every real dimensionless $\delta$, the field $F_{\delta }:=\delta F_0$ obeys those same source-free equations and

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

This is an exact residual identity on the fixed metric. If $\delta \mathrm{\ne }0$ and $F_0$ is nonzero at a point, its observer energy there is positive and the residual is nonzero there; this does not estimate the distance to a different coupled solution.

Separately, let $\rho ,\sigma \in C^1(\mathrm{\Omega })$ and a $C^2$ future timelike field $U$ with $g(U,U)=-c^2$ define charged dust on an open $\mathrm{\Omega }\subseteq U$, with $\rho >0$, $T_m^{ab}=\rho U^a{U}^b$, $j^a=\sigma U^a$, and the formal matter-exchange equation of Smooth formal matter, EM tensors and observer quantities. For any dimensionless $\eta >0$, replacing $(\rho ,\sigma )$ by $(\eta \rho ,\eta \sigma )$ multiplies $T_m$ and $j$ by $\eta$, leaves $\alpha =\sigma \mathrm{/}\rho$ unchanged, and leaves the dust acceleration equation $\rho U^a{\mathrm{\nabla }}_a{U}^b=\sigma F^b{}_c{U}^c$ unchanged when the prescribed $F$ and $U$ are held fixed. The sourced Maxwell equation must still match the changed current. If both $F$ and $j$ are scaled by $\eta$, Maxwell's equations remain linear but the electromagnetic stress scales by ${\eta }^2$, whereas the dust stress scales by $\eta$; no general coupled-solution family follows.

For a flat background with $\mathrm{\Lambda }=0$ and $T_m=0$, a nonzero source-free Maxwell field cannot solve the exact Einstein equation with the unchanged Minkowski metric, because its observer energy density is positive somewhere.

Finally, take $M={\mathbb{R}}^4$ with length coordinates $x^0=ct$ and $\mathbf{x}=(x^1,x^2,x^3)$, Minkowski metric $\eta =\mathrm{diag}(-1,1,1,1)$, standard orientation, and future direction ${\mathrm{\partial }}_0$. Let $\mathrm{\Omega }={\mathbb{R}}^3$, $I=\mathbb{R}$, and use the Euclidean norm $\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}_2=({\sum }_{i=1}^3(x^i{)}^2{)}^{1\mathrm{/}2}$ and Lebesgue volume $d^{3}x$ in these Cartesian coordinates. Consider only the prescribed Coulomb profile on the finite annulus ${\mathcal{A}}_{\mathrm{\ell }}:=\{\mathbf{x}\in {\mathbb{R}}^3:\mathrm{\ell }<\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}_2<2\mathrm{\ell }\}$, where $\mathrm{\ell }>0$ is a length and $Q\mathrm{\ne }0$ is a charge parameter with units $\mathrm{C}$. Put $r:=\mathrm{\mid }\mathbf{x}{\mathrm{\mid }}_2$, ${\mathbf{e}}_r:=\mathbf{x}\mathrm{/}r$, $\mathbf{E}(\mathbf{x})=Q(4\pi {\epsilon }_0{r}^2{)}^{-1}{\mathbf{e}}_r$, and $\mathbf{B}=0$. Its Maxwell energy on that annulus is

${\mathcal{E}}_{\mathrm{\ell }}={\int }_{{\mathcal{A}}_{\mathrm{\ell }}}\frac{{\epsilon }_0}{2}\mathrm{\mid }\mathbf{E}{\mathrm{\mid }}^2{d}^{3}x=\frac{Q^2}{16\pi {\epsilon }_0\mathrm{\ell }}\mathrm{.}$

Thus this fixed-charge Coulomb cutoff family has diverging shell energy as $\mathrm{\ell }\downarrow 0$. This finite-annulus calculation asserts no point-source construction, self-field evaluation, or obstruction theorem for arbitrary regularizations.

Use the mathematical Lorentzian and SI conventions of Smooth formal matter, EM tensors and observer quantities. Fix a length scale $L>0$, a speed $c>0$, a charge parameter $q\in \mathbb{R}$ with units $\mathrm{C}$, and a mass parameter $m>0$ with units $\mathrm{k}\mathrm{g}$; set $\alpha :=q\mathrm{/}m\mathrm{\ne }0$, with units $\mathrm{C}\mathrm{k}{\mathrm{g}}^{-1}$. Choose $0<ϵ\le {ϵ}_{\ast }\le 1$. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be open and let $\mathcal{K}⋐\mathrm{\Omega }$ be a compact convex set. Use the Cartesian coordinate domain ${\mathbb{R}}_{x^0}\times L\mathrm{\Omega }$ with standard orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and use $x^0=ct$, $x^i=L{X}^i$ with dimensionless $X\in \mathrm{\Omega }$. Assume a smooth stationary (independent of $x^0$) Lorentzian metric $g^{(ϵ)}$ of signature $(-,+,+,+)$ on an open neighborhood of ${\mathbb{R}}_{x^0}\times L\mathcal{K}$ in this domain, with ${\mathrm{\partial }}_0$ chosen future timelike and $t$ increasing along the considered curves, whose components have the form

$g_{00}^{(ϵ)}(X)=-(1+2ϵ\varphi (X))+r_{00}^{(ϵ)}(X),g_{ij}^{(ϵ)}(X)=(1-2ϵ\varphi (X)){\delta }_{ij}+r_{ij}^{(ϵ)}(X),g_{0i}^{(ϵ)}(X)=r_{0i}^{(ϵ)}(X)\mathrm{.}$

Here $\varphi$ is dimensionless and smooth. A spatial multi-index is $\beta =({\beta }_1,{\beta }_2,{\beta }_3)\in {\mathbb{N}}_0^3$, with $\mathrm{\mid }\beta \mathrm{\mid }={\beta }_1+{\beta }_2+{\beta }_3$ and ${\mathrm{\partial }}_X^{\beta }:={\mathrm{\partial }}_{X^1}^{{\beta }_1}{\mathrm{\partial }}_{X^2}^{{\beta }_2}{\mathrm{\partial }}_{X^3}^{{\beta }_3}$. For each such $\beta$ with $\mathrm{\mid }\beta \mathrm{\mid }\le 2$, assume the uniform bounds

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

on $\mathcal{K}$, with a fixed finite nonnegative $C_g$. Let $\mathbf{e},\mathbf{b}:\mathrm{\Omega }\to {\mathbb{R}}^3$ be smooth dimensionless fields whose values and first derivatives are bounded on $\mathcal{K}$. Since $\mathcal{K}$ is compact, the fixed smooth function $\varphi$ and its derivatives used below are also bounded there. Prescribe a smooth stationary (independent of $x^0$) real two-form $F^{(ϵ)}$ by its Cartesian components

$F_{0i}^{(ϵ)}=-E_i^{(ϵ)}\mathrm{/}c,F_{ij}^{(ϵ)}={ϵ}_{ijk}{B}_k^{(ϵ)},\alpha {\mathbf{E}}^{(ϵ)}(X)=\frac{ϵc^2}{L}\mathbf{e}(X),\alpha {\mathbf{B}}^{(ϵ)}(X)=\frac{\sqrt{ϵ}c}{L}\mathbf{b}(X)\mathrm{.}$

No Maxwell or Einstein equation for this prescribed background field or metric is assumed. Define the dimensionless potential $\varphi$ comparison system on $S\in [0,S_{\ast }]$ by

$\frac{d{\mathbf{X}}_N}{dS}={\mathbf{W}}_N,\frac{d{\mathbf{W}}_N}{dS}=-{\mathrm{\nabla }}_X\varphi ({\mathbf{X}}_N)+\mathbf{e}({\mathbf{X}}_N)+{\mathbf{W}}_N\times \mathbf{b}({\mathbf{X}}_N),$

where $(\mathbf{u}\times \mathbf{v}{)}_i:={\sum }_{j,k}{ϵ}_{ijk}{u}_j{v}_k$. Let $f(\mathbf{X},\mathbf{W})$ denote the six-component right-hand side of this system. Assume that on a compact convex state set $\mathcal{Q}\subseteq \mathcal{K}\times {\mathbb{R}}^3$, containing both trajectories, its derivative satisfies $\mathrm{\parallel }Df{\mathrm{\parallel }}_{1\to 1}\le L_f$ for a finite nonnegative $L_f$. Here $\mathrm{\parallel }(\mathbf{X},\mathbf{W}){\mathrm{\parallel }}_1$ is the sum of the absolute values of its six real components, and $\mathrm{\parallel }Df{\mathrm{\parallel }}_{1\to 1}$ is the operator norm induced by this ${\mathrm{\ell }}^1$ norm.

For each $ϵ$, let $z_{ϵ}$ be a $C^2$ future timelike curve, parameterized by proper time $\tau$ as in Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold, that solves the prescribed external Lorentz equation

${\mathrm{\nabla }}_{U}U=\alpha F^{(ϵ)\mathrm{♯}}(U),g^{(ϵ)}(U,U)=-c^2,U=\frac{d{z}_{ϵ}}{d\tau }\mathrm{.}$

Write its coordinate velocity as $\mathbf{w}=d\mathbf{x}\mathrm{/}dt$ and assume $\mathrm{\mid }\mathbf{w}\mathrm{\mid }\le C_v\sqrt{ϵ}c$ on $[0,S_{\ast }]$ for a fixed finite $C_v$. Suppose the relativistic scaled trajectory $({\mathbf{X}}_{ϵ},{\mathbf{W}}_{ϵ})$ and the comparison trajectory $({\mathbf{X}}_N,{\mathbf{W}}_N)$ exist throughout $[0,S_{\ast }]$, remain in $\mathcal{Q}$, and have the same initial data, where

$\mathbf{X}:=\mathbf{x}\mathrm{/}L,\mathbf{W}:=\mathbf{w}\mathrm{/}(\sqrt{ϵ}c),S:=\sqrt{ϵ}ct\mathrm{/}L\mathrm{.}$

Then, after choosing ${ϵ}_{\ast }>0$ sufficiently small in terms of the stated uniform constants, the exact relativistic trajectory satisfies

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

where $C_r$ depends only on the displayed uniform bounds and the fixed coefficient bounds on $\mathcal{K}$, and is independent of $ϵ$. Consequently,

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

The estimate is conditional on existence and common compact containment through the specified finite scaled interval. It does not construct such a long-time family, solve Maxwell's equations for the prescribed fields, or produce a coupled Einstein–Maxwell family. In particular the $\sqrt{ϵ}$ magnetic scaling can fail a coupled Einstein stress hierarchy unless additional matter scales are supplied.