Exact small-field coupled residual order

Let $U$ be a smooth Lorentzian four-dimensional chart domain with metric $g$ as in Oriented Lorentzian fields, observers and coordinate conventions, and let $\mathrm{\Lambda }\in \mathbb{R}$ be constant. Write ${\mathrm{\Omega }}^k(U)$ for the smooth real $k$-forms on $U$. Here $d:{\mathrm{\Omega }}^k(U)\to {\mathrm{\Omega }}^{k+1}(U)$, $0\le k\le 3$, is the exterior derivative with coordinate formula $(d\beta {)}_{a_0\dots a_k}=(k+1){\mathrm{\partial }}_{[a_0}{\beta }_{a_1\dots a_k]}$, and ${\star }_g:{\mathrm{\Omega }}^2(U)\to {\mathrm{\Omega }}^2(U)$ is the Lorentzian Hodge map characterized by $\alpha \wedge {\star }_g\beta =\frac{1}{2}{\alpha }_{ab}{\beta }^{ab}{\mathrm{vol}}_g$ for $\alpha ,\beta \in {\mathrm{\Omega }}^2(U)$, as in Lorentzian exterior calculus and Hodge star. Both operators are real-linear. Suppose $G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}=0$. Let $F_1\in {\mathrm{\Omega }}^2(U)$ satisfy $d{F}_1=0$ and $d({\star }_g{F}_1)=0$. For any real dimensionless $\epsilon$, put $F_{\epsilon }:=\epsilon F_1$ and define

$T_{ab}^{(1)}:={\mu }_0^{-1}{\mathcal{Q}}_{ab}(g,F_1),{\mathcal{R}}_{ab}(g,F;\mathrm{\Lambda }):=G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}-\frac{\kappa }{{\mu }_0}{\mathcal{Q}}_{ab}(g,F),$

using $\kappa ,{\mu }_0,\mathcal{Q}$ of SI and geometrized Einstein–Maxwell parameters and Exact geometric electrovac normalization. Then $F_{\epsilon }$ remains an exact source-free Maxwell field on the fixed metric and

${\mathcal{R}}_{ab}(g,F_{\epsilon };\mathrm{\Lambda })=-\kappa {\epsilon }^2{T}_{ab}^{(1)}\mathrm{.}$

For a smooth covariant two-tensor $S$ and compact $K\subset U$, define the chart component norm of its restriction by $\mathrm{\parallel }S{\mathrm{\parallel }}_K:={\max }_{0\le a,b\le 3}{\sup }_{p\in K}\mathrm{\mid }{S}_{ab}(p)\mathrm{\mid }$ when $K\mathrm{\ne }\mathrm{\varnothing }$. In this estimate and its applications, use ${\sup }_{p\in \mathrm{\varnothing }}f(p):=0$ for nonnegative component functions $f$; equivalently, the empty restriction has $\mathrm{\parallel }S{\mathrm{\parallel }}_{\mathrm{\varnothing }}:=0$. It is finite for smooth $S$ and satisfies

$\mathrm{\parallel }\mathcal{R}(g,F_{\epsilon };\mathrm{\Lambda }){\mathrm{\parallel }}_K=\kappa {\epsilon }^2\mathrm{\parallel }{T}^{(1)}{\mathrm{\parallel }}_K\mathrm{.}$

If $\epsilon \mathrm{\ne }0$ and $T^{(1)}$ is not identically zero on $K$, the fixed pair $(g,F_{\epsilon })$ is not an exact coupled Einstein–Maxwell solution on $K$. This is a residual estimate only; it gives no nearby metric, solution-distance bound, point-source solution or backreaction limit.