Conditional semiclassical Einstein–Maxwell source contract

Let $M$ be a smooth Hausdorff, second-countable four-manifold without boundary. Equip it with a smooth nondegenerate Lorentzian metric $g$ of signature $(-,+,+,+)$, a smooth timelike vector field $N$ selecting a time orientation, and an independent orientation $o$. Let $\mathrm{\nabla }$ be the unique torsion-free, metric-compatible connection selected by $g$, as in the oriented Lorentzian geometry convention recorded in the Definition references. For SI coordinate bookkeeping, use length coordinates with $x^0=ct$, where $c>0$ is the coordinate-conversion scale with unit $\mathrm{m}{\mathrm{s}}^{-1}$; this assigns no measured value to $c$. Fix ${\mu }_0>0$ in the inherited SI convention. A smooth real two-form $F$ assigns to each $p\in M$ an alternating bilinear map $F_p:T_{p}M\times T_{p}M\to \mathbb{R}$, with coefficients smooth in charts.

For a vector bundle $E\to M$ with projection ${\pi }_E$, write $\mathrm{\Gamma }(E)$ for its smooth sections, namely smooth maps $s:M\to E$ such that ${\pi }_E\circ s={\mathrm{id}}_M$. The bundle ${\mathrm{Sym}}^2{T}^{\ast }M$ is the subbundle of $T^{\ast }M\otimes T^{\ast }M$ whose fibers are symmetric bilinear forms. A unital complex $\ast$-algebra ${\mathfrak{A}}_{g,F}$ means an associative complex algebra with unit $1$ and an antilinear involution satisfying $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$ and $1^{\ast }=1$. A state on this algebra means a complex-linear functional $\omega$ such that $\omega (1)=1$ and $\omega (A^{\ast }A)\ge 0$ for every $A\in {\mathfrak{A}}_{g,F}$. Choose a specified state class ${\mathcal{S}}_{g,F}$ of such states, with any additional regularity conditions stated as part of that choice.

A renormalization prescription $\mathcal{R}$ is a fixed item of input recording the subtraction rule, scale, and finite local counterterms used for the source assignments. A candidate assignment consists of maps $\omega ⟼T_{\mathcal{R}}[g,F,\omega ]\in \mathrm{\Gamma }({\mathrm{Sym}}^2{T}^{\ast }M),\omega ⟼j_{\mathcal{R}}[g,F,\omega ]\in \mathrm{\Gamma }(TM),$ for $\omega \in {\mathcal{S}}_{g,F}$, with smooth spacetime fields. In the inherited SI bookkeeping, $F$ has units $\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$, $j_{\mathcal{R}}$ has units $\mathrm{A}{\mathrm{m}}^{-2}$, and $T_{\mathcal{R}}$ has units $\mathrm{J}{\mathrm{m}}^{-3}$; these are formal units, not observations. The maps may depend nonlocally on the state.

Use the inverse metric $g^{-1}$ to raise indices, with ${\mathrm{\nabla }}^a:=g^{ac}{\mathrm{\nabla }}_c$.

The assignment is contract-admissible when, for every $\omega \in {\mathcal{S}}_{g,F}$, ${\mathrm{\nabla }}_a{j}_{\mathcal{R}}^a=0,{\mathrm{\nabla }}^a{T}_{\mathcal{R},ab}=F_{bc}{j}_{\mathcal{R}}^c\mathrm{.}$ The current convention is the one in which the classical sourced Maxwell equation is ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ and the Maxwell stress divergence is ${\mathrm{\nabla }}^a{T}_{ab}^{\mathrm{E}\mathrm{M}}=-F_{bc}{j}^c$. The identities displayed in this definition are interface conditions; they do not construct ${\mathfrak{A}}_{g,F}$, ${\mathcal{S}}_{g,F}$, $\mathcal{R}$, or either source map, and they do not assert that any assignment is contract-admissible.

Oriented Lorentzian fields, observers and coordinate conventions

Curved electromagnetic stress divergence

The state class and renormalization prescription are part of the interface, not suppressed choices. A concrete QFT application must supply them and prove that its source maps have the stated regularity and identities. Nonlocal state dependence and higher-curvature counterterms can change the differential order of the coupled equations; order reduction is a separate approximation. Hawking spectra, KMS/Unruh states, generalized entropy, and black-hole thermodynamic laws are contextual applications, not prerequisites here.