Electromagnetic and matter stress exchange

Let $(M,g)$ be a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, and let $U\subseteq M$ be open. Restrict to $U$ the fixed smooth principal $U(1)$ bundle and connection selected in Selected invariant matter-action contract. In its normalized local representatives $a_i$, the physical local potentials are $A_i={\kappa }_0{a}_i$; their global physical curvature satisfies $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. The $F$ and $j$ below are the curvature and current of that same connection and selected action. Let $F$ and $j$ be smooth, and assume the adopted Maxwell equations $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ of Smooth curved vacuum Maxwell constitutive adoption. Let $S_m$ be the selected smooth invariant matter functional of Selected invariant matter-action contract, satisfying the section-bundle, real functional-domain, compact-variation splitting, and supplied smooth pathwise-transport hypotheses of Gauge/diffeomorphism matter identity and field exchange for this same connection. Suppose its matter fields are on shell. Define $T^{\mathrm{E}\mathrm{M}}$ by the local Maxwell variation and $T^m$ by the Hilbert variation of $S_m$. Then

${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^{bc}{j}_c,{\mathrm{\nabla }}_a{T}_m^{ab}=F^{bc}{j}_c,{\mathrm{\nabla }}_a(T_{\mathrm{E}\mathrm{M}}^{ab}+T_m^{ab})=0.$

Use the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions and define the scalar curvature by $R:=S=g^{ab}{\mathrm{Ric}}_{ab}$ and the Einstein tensor by $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$. Set ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4>0$ using the SI constants of Einstein–Maxwell primitive quantities and SI conventions. If, in addition, the pointwise Einstein equation $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$ holds with constant $\mathrm{\Lambda }$, the contracted Bianchi identity shows that the total conservation law is its necessary source-compatibility condition. This local identity is not a universal gravitational energy statement and does not imply existence of a solution.