Compact Einstein–Hilbert variation and SI coupling

Let $(M,g,N,o)$ be as in Oriented Lorentzian fields, observers and coordinate conventions. Write $R:=S=g^{ab}{\mathrm{Ric}}_{ab}:M\to \mathbb{R}$ for the smooth scalar curvature of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, with ${\mathrm{Ric}}_{bd}=R^a{}_{bad}$ and $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$ in that supplier's curvature convention. For a compactly supported metric variation, write $q^{ab}=\delta g^{ab}$, $h_{ab}=\delta g_{ab}=-g_{ac}{g}_{bd}{q}^{cd}$, $w=\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}$, and $V^a=g^{bd}\delta {\mathrm{\Gamma }}^a{}_{db}-g^{ab}\delta {\mathrm{\Gamma }}^d{}_{db}$, where $\delta {\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ad}({\mathrm{\nabla }}_b{h}_{cd}+{\mathrm{\nabla }}_c{h}_{bd}-{\mathrm{\nabla }}_d{h}_{bc})$; curvature and Einstein tensors are as in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. The local density identity is

$\delta ((R-2\mathrm{\Lambda })w)=w(G_{ab}+\mathrm{\Lambda }{g}_{ab})q^{ab}+{\mathrm{\partial }}_a(w{V}^a)\mathrm{.}$

Fix constants $c>0$, $G_N>0$, ${\mu }_0>0$, and $\mathrm{\Lambda }\in \mathbb{R}$. Regard these as formal coefficients; their physical interpretation and units are not assumptions of this mathematical lemma. For smooth metric variations with compact support in the interior, at fixed bundle connection and fixed matter fields, consider the action difference

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{c^3}{16\pi G_N}{\int }_M(R-2\mathrm{\Lambda }){\mathrm{v}\mathrm{o}\mathrm{l}}_g-\frac{1}{4{\mu }_{0}c}{\int }_M{F}_{ab}{F}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g+S_m,$

where noncompact action values are used only through compactly supported differences, $F$ is the curvature of a fixed smooth bundle connection, and $S_m$ has Hilbert tensor $T^m$ in the convention of The Hilbert stress-energy tensor of a metric-dependent action. Let $T^{\mathrm{E}\mathrm{M}}$ be the electromagnetic Hilbert tensor supplied by the compact Maxwell variation of Compact potential and metric variation of the Maxwell action, and set ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4$. Then for every compactly supported inverse-metric variation $q^{ab}=\delta g^{ab}$,

$\delta S_{\mathrm{t}\mathrm{o}\mathrm{t}}=\frac{c^3}{16\pi G_N}{\int }_M(G_{ab}+\mathrm{\Lambda }{g}_{ab}-{\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m))q^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

Consequently, stationarity under all such metric variations is equivalent to

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

Any tensor fields satisfying this equation necessarily have ${\mathrm{\nabla }}^a(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)=0$ because $\mathrm{\Lambda }$ is constant. This is a compatibility condition, not a claim that the coupled field equations have a solution or that matter and Maxwell equations follow from metric stationarity.

For dimensional bookkeeping in length coordinates one may assign $[R]=[\mathrm{\Lambda }]={\mathrm{m}}^{-2}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[G_N]={\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $[{\mu }_0]=\mathrm{H}{\mathrm{m}}^{-1}$, $[F_{ab}]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$, and $[T^m]=[T^{\mathrm{E}\mathrm{M}}]=\mathrm{J}{\mathrm{m}}^{-3}$. Then every action term has units $\mathrm{J}\mathrm{s}$ and ${\kappa }_{E}T$ has units ${\mathrm{m}}^{-2}$. These are formal assignments only; the lemma assumes no measured parameter value or physical field law.

• The theorem proves only the metric Euler–Lagrange equation and its necessary Bianchi compatibility. The matter and Maxwell Euler–Lagrange equations are separate variations.
• The M03 Maxwell-action variation is present as a draft and is used only at its inspected statement; the conditional proof here does not treat that draft as accepted. The former research token GR-G3 is reconciled at its actual use to the published contracted Bianchi theorem [F5]; no alias for its broader research content is asserted. The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02: Finite-real supremum and infimum domain, Real metric and pseudometric convention, Metric nonnegativity from the three axioms, Explicit rational and real power regimes, Sequence-limit criterion with explicit choice use, Boundaryless domain for geometric completeness, and Set-sized objects and definable-class schemas.