A finite-time bound from a test-field residual

Let $U$ be a smooth four-dimensional Lorentzian chart domain with length coordinate $t$, let $\mathrm{\Lambda }\in \mathbb{R}$, and suppose $G_{ab}(g)+\mathrm{\Lambda }{g}_{ab}=0$ on $U$, using the Einstein tensor and curvature convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Let $F_1$ be a smooth real Maxwell two-form with $d{F}_1=0$ and $d({\star }_g{F}_1)=0$ on this fixed metric. For dimensionless $\epsilon \in \mathbb{R}$, put $F_{\epsilon }:=\epsilon F_1$. Use the common geometric normalization of the current mathematical suppliers:

${\mathcal{Q}}_{ab}(g,F):=F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd},{\mathcal{R}}_{ab}(g,F;\mathrm{\Lambda }):=G_{ab}+\mathrm{\Lambda }{g}_{ab}-\frac{\kappa }{{\mu }_0}{\mathcal{Q}}_{ab}(g,F)\mathrm{.}$

The tensor $\mathcal{Q}$ is the quadratic Maxwell tensor in that normalization. This residual uses the Maxwell tensor; the charged scalar model on the A page is a separate model.

For a smooth covariant two-tensor $S$ and nonempty compact chart set $K$, use the component norm $\mathrm{\parallel }S{\mathrm{\parallel }}_K:={\max }_{0\le a,b\le 3}{\sup }_{p\in K}\mathrm{\mid }{S}_{ab}(p)\mathrm{\mid }$, as in Exact small-field coupled residual order. Fix $\tau >0$ and a nonempty compact spatial chart set $K_0$ such that $K_{\tau }:=[0,\tau ]\times K_0$ is a compact subset of $U$. Let ${\mathcal{R}}_{\epsilon }(t)$ denote the restriction to $K_0$ of $\mathcal{R}(g,F_{\epsilon };\mathrm{\Lambda })$ at time $t$.

Let $(\mathcal{X},\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathcal{X}})$ be a specified normed space. Let $z_{\epsilon },{\tilde{z}}_{\epsilon }:[0,\tau ]\to \mathcal{X}$ be any two supplied state curves, interpreted in an intended application as a coupled state and its test-field approximation. No existence or regularity theorem for these curves is asserted. Assume the separate stability inequality

$\mathrm{\parallel }{z}_{\epsilon }(t)-{\tilde{z}}_{\epsilon }(t){\mathrm{\parallel }}_{\mathcal{X}}\le K(\tau ){\int }_0^t\mathrm{\parallel }{\mathcal{R}}_{\epsilon }(s){\mathrm{\parallel }}_{K_0}ds(0\le t\le \tau ),$

with a specified finite $K(\tau )\ge 0$. Set

$T_{ab}^{(1)}:={\mu }_0^{-1}{\mathcal{Q}}_{ab}(g,F_1),C:=\kappa \mathrm{\parallel }{T}^{(1)}{\mathrm{\parallel }}_{K_{\tau }}\mathrm{.}$

The exact test-field residual proposition gives $\mathrm{\parallel }\mathcal{R}(g,F_{\epsilon };\mathrm{\Lambda }){\mathrm{\parallel }}_{K_{\tau }}=C{\epsilon }^2$. Since $K_{\tau }=[0,\tau ]\times K_0$ and there are finitely many components, this equals ${\sup }_{0\le t\le \tau }\mathrm{\parallel }{\mathcal{R}}_{\epsilon }(t){\mathrm{\parallel }}_{K_0}$. Consequently,

${\sup }_{0\le t\le \tau }\mathrm{\parallel }{z}_{\epsilon }(t)-{\tilde{z}}_{\epsilon }(t){\mathrm{\parallel }}_{\mathcal{X}}\le K(\tau )C\tau {\epsilon }^2\mathrm{.}$

In the geometric normalization, $[\kappa T_{ab}^{(1)}]={\mathrm{m}}^{-2}$ and $[C]={\mathrm{m}}^{-2}$; $[\tau ]=\mathrm{m}$. If solution-error norm values have fixed unit ${\mathsf{u}}_{\mathcal{X}}$, then the assumed coefficient has unit $[K(\tau )]={\mathsf{u}}_{\mathcal{X}}\mathrm{m}$. The estimate is conditional on the supplied curves and stability inequality; it proves neither their existence nor the availability of that inequality.

Separately adopted fixed-background Maxwell and charged probes

Exact small-field coupled residual order

SI and geometrized Einstein–Maxwell parameters

Exact geometric electrovac normalization

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

The residual proposition concerns a scaled source-free Maxwell field, not the charged scalar postulate on this page. This thought experiment uses the Maxwell test-field postulate directly and keeps the two test models distinct. The factor $\tau K(\tau )$ may grow with the observation interval; no limit $\tau \to \mathrm{\infty }$ is justified.