Smooth curved vacuum Maxwell constitutive adoption

Adopt the following physical field law for this Einstein–Maxwell model. Let $M$ be a smooth, oriented, time-oriented Lorentzian four-manifold with metric $g$ of signature $(-,+,+,+)$ as in Einstein–Maxwell primitive quantities and SI conventions, let $U\subseteq M$ be open, let $F\in {\mathrm{\Omega }}^2(U)$ be a smooth real electromagnetic two-form, and let $j$ be a smooth four-current vector field on $U$. Using the Levi-Civita connection ${\mathrm{\nabla }}^g$ of $g$, the SI constants ${\mu }_0>0$ and $c>0$, and the field/current conventions of Einstein–Maxwell primitive quantities and SI conventions, postulate

$dF=0,{\mathrm{\nabla }}_a^g{F}^{ab}=-{\mu }_0{j}^b\mathrm{.}$

This is the selected smooth vacuum constitutive law. The first equation is the homogeneous field equation; the second is the sourced field equation with the sign fixed by $F_{0i}=-E_i\mathrm{/}c$ and $j=(c\rho ,J)$ in a positively oriented inertial tetrad. No global potential is assumed: although $F=dA$ implies $dF=0$ when a smooth potential exists, the postulate applies to the two-form itself and does not assert that such an $A$ exists globally.

The postulate adopts one curved-spacetime continuation as a physical model assumption. The flat Minkowski and special-relativity suppliers Adopted Minkowski physical model of the electromagnetic framework, Adopted inertial covariance of the model equations, The microscopic vacuum Maxwell equations: smooth and weak branches, and Tensor Maxwell equations equal the four laboratory Maxwell equations fix the inherited SI, inertial, and flat-limit conventions; they do not impose a globally flat metric on $M$ or deduce this curved field law. General-relativity sources describe covariant Maxwell formulations in their own unit and sign conventions; those treatments motivate the chosen formulation but neither mathematical structure nor source authority proves that this is nature's law. No empirical result is adopted here.

The model has no magnetic-charge term and no material excitation pair such as $D,H$. It applies only to smooth $F$ and $j$ on the stated domain; point and sheet sources, distribution-valued fields, and products of distributions require separate definitions and are outside this postulate.