Equation-based Einstein–Maxwell model

Let $M$ be the smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold of Einstein–Maxwell primitive quantities and SI conventions, with metric $g$ of signature $(-,+,+,+)$ and the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $U\subseteq M$ be open. Take a smooth real two-form $F$ on $U$. For each application, fix one explicitly named smooth classical matter-model instance $\mathcal{M}=({\mathcal{C}}_m,{\mathcal{E}}_m,{\mathsf{J}}_m,{\mathsf{T}}_m)\mathrm{.}$ For each admitted pair $(g,F)$, ${\mathcal{C}}_m[g,F]$ is a specified set of smooth matter-field configurations on $U$, including their field bundles and regularity; ${\mathcal{E}}_m[g,F;\mathrm{\Psi }]$ is the explicit system of equations and constraints that defines the admissible configurations; and ${\mathsf{J}}_m[g,F;\mathrm{\Psi }]$ and ${\mathsf{T}}_m[g,F;\mathrm{\Psi }]$ are specified maps from those configurations to, respectively, smooth vector fields and smooth symmetric covariant two-tensors on $U$. For an admissible configuration $\mathrm{\Psi }\in {\mathcal{C}}_m[g,F]$, set $j:={\mathsf{J}}_m[g,F;\mathrm{\Psi }]$ and $T^m:={\mathsf{T}}_m[g,F;\mathrm{\Psi }]$. A consuming item must name the selected model and state the exact field variables, equations, constraints, regularity, and additional domain or initial-data hypotheses it uses. Those matter equations and constraints are premises of that model instance; they are not supplied or proved by this postulate. In particular, arbitrary independent smooth $j$ and $T^m$ are not a matter-model instance merely because their types match.

There is also a named zero-matter instance ${\mathcal{M}}_0$: its configuration set is the singleton $\{\ast \}$, it has no matter-field variables or additional constraints, and ${\mathsf{J}}_0(\ast )=0$, ${\mathsf{T}}_0(\ast )=0$. Whenever a use sets $j=0$ and $T^m=0$ without additional matter fields, it is this source-free instance.

For an admissible $\mathrm{\Psi }$, adopt $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ under the adopted convention of Smooth curved vacuum Maxwell constitutive adoption. This equation-based postulate then adds the gravitational field law below, independently of any matter action.

Define the Einstein tensor by $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$, using the curvature contractions of Oriented Lorentzian fields, observers and coordinate conventions, and define the smooth-field electromagnetic stress tensor by

$T_{ab}^{\mathrm{E}\mathrm{M}}:=\frac{1}{{\mu }_0}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})\mathrm{.}$

For constant cosmological parameter $\mathrm{\Lambda }\in \mathbb{R}$ and ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4$, adopt the additional equation-based gravitational field law

$G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)\mathrm{.}$

Here $c>0$, $G_N>0$, and ${\mu }_0>0$ have the SI roles and units fixed in Einstein–Maxwell primitive quantities and SI conventions; in length coordinates, $G_{ab}$ and $\mathrm{\Lambda }{g}_{ab}$ have units ${\mathrm{m}}^{-2}$, $T^{\mathrm{E}\mathrm{M}}$ and $T^m$ have units $\mathrm{J}{\mathrm{m}}^{-3}$, and ${\kappa }_E$ has units $\mathrm{m}{\mathrm{J}}^{-1}$. The source-free and sourced Maxwell equations remain smooth-field postulates: point or sheet sources, distribution-valued products, material excitation laws, and a global potential are outside this scope.

This is an equation-based adoption. It does not assume or imply the existence of a matter action and is not derived from an action principle. The sources give formulations in their own conventions; they motivate comparison but do not establish a physical law or a solution-existence theorem.