Gauge/diffeomorphism matter identity and field exchange

Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ with metric-selected connection $\mathrm{\nabla }$, as in Oriented Lorentzian fields, observers and coordinate conventions. Let $\pi :P\to M$ be a smooth principal $U(1)$ bundle with a smooth connection and global curvature two-form $F$, using the local-potential and fixed-sector conventions of Local potentials and smooth circle-bundle connection data and Circle gluing, curvature and connection variations. Specify the matter fields by a smooth fiber bundle $p_E:E\to M$: its local trivializations are smooth diffeomorphisms $p_E^{-1}(U)\cong U\times Q$ commuting with projection, for a fixed finite-dimensional smooth manifold $Q$. A matter configuration is a smooth section $\mathrm{\Psi }:M\to E$, so $p_E\circ \mathrm{\Psi }={\mathrm{i}\mathrm{d}}_M$; write $\mathcal{C}={\mathrm{\Gamma }}^{\mathrm{\infty }}(E)$. No linear structure on $Q$ is required. A compact matter variation is a jointly smooth family ${\mathrm{\Psi }}_s\in \mathcal{C}$, $\mathrm{\mid }s\mathrm{\mid }<\epsilon$, with ${\mathrm{\Psi }}_0=\mathrm{\Psi }$ and ${\mathrm{\Psi }}_s=\mathrm{\Psi }$ off a common compact subset of $M$. Its derivative $\eta (x)={\mathrm{\partial }}_s{\mathrm{\Psi }}_s(x){\mathrm{\mid }}_0$ lies in $V_{\mathrm{\Psi }(x)}E:=\ker (d{p}_E{)}_{\mathrm{\Psi }(x)}$, the vertical tangent space. Thus $\eta$ is a smooth compactly supported section of the vector bundle ${\mathrm{\Psi }}^{\ast }VE\to M$.

Let $S_m:\mathcal{D}\to \mathbb{R}$, where $\mathcal{D}$ is a declared subset of triples $(\tilde{g},\tilde{A},\tilde{\mathrm{\Psi }})$ consisting of smooth Lorentzian metrics of the fixed signature, smooth connections on $P$, and sections in $\mathcal{C}$; $A$ denotes the connection, not necessarily a global one-form. Admissible variations are jointly smooth curves in $\mathcal{D}$ equal to the fixed triple off a common compact set. Require that freezing any two components of an admissible curve still gives an admissible curve near zero, and that every compact metric and fixed-sector connection direction is admitted. Here differentiable means that along each admissible curve the derivative at zero exists, depends real-linearly only on $(q,\alpha ,\eta )$, and is the sum ${\delta }_g{S}_m[q]+{\delta }_A{S}_m[\alpha ]+{\delta }_{\mathrm{\Psi }}{S}_m[\eta ]$ of its three partial derivatives. The connection direction $\alpha$ is the physical one: $\delta A_i=\alpha {\mathrm{\mid }}_{U_i}$, equivalently $\delta \omega ={\kappa }_0^{-1}{\pi }^{\ast }\alpha$ for the normalized connection $\omega$ of Circle gluing, curvature and connection variations.

Matter transport is additional supplied data, not a canonical construction for an arbitrary $E$. For every compact real gauge parameter $\chi$, use the principal-bundle path $H_s(p)=p\cdot e^{is\chi (\pi (p))\mathrm{/}{\kappa }_0}$ covering $f_s={\mathrm{i}\mathrm{d}}_M$. For each compact smooth vector field $\xi$ use its flow $f_s$ and the flow $H_s$ of its horizontal lift ${\xi }^H$, uniquely specified by $d\pi ({\xi }^H)=\xi \circ \pi$ and $\omega ({\xi }^H)=0$. For each of these paths supply a jointly smooth family of bundle diffeomorphisms $H_s^E:E\to E$ covering $f_s$, with $H_0^E={\mathrm{i}\mathrm{d}}_E$, equal to the identity over the complement of the compact support of the generating parameter or vector field. The maps and their inverses are required on a common interval about zero. Define the transported section by ${\mathrm{\Psi }}_s=(H_s^E{)}^{-1}\circ \mathrm{\Psi }\circ f_s$, and require $(f_s^{\ast }g,H_s^{\ast }\omega ,{\mathrm{\Psi }}_s)$ to be admissible. This pathwise specification also allows models whose gauge action is defined using real phase paths; it asserts no action of every finite circle transformation on $E$.

Fix a configuration at which the matter Euler–Lagrange variation vanishes, meaning ${\delta }_{\mathrm{\Psi }}{S}_m=0$ for every smooth compactly supported matter-field variation. Suppose that for all compactly supported metric variations $q^{ab}=\delta g^{ab}$ and all compactly supported connection variations $\alpha \in {\mathrm{\Omega }}^1(M)$ in the fixed bundle sector, the remaining first variation is

$\delta S_m=-\frac{1}{2c}{\int }_M{T}_{ab}^m{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g+\frac{1}{c}{\int }_M{j}^a{\alpha }_a{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$

where $c>0$, $T^m$ is a smooth symmetric covariant two-tensor, and $j$ is a smooth vector field. Assume $S_m(f_s^{\ast }g,H_s^{\ast }\omega ,{\mathrm{\Psi }}_s)=S_m(g,\omega ,\mathrm{\Psi })$ for both types of supplied transport above. Only derivatives at $s=0$ are used; no global flow for all times is required. Then

${\mathrm{\nabla }}_a{j}^a=0,{\mathrm{\nabla }}_a{T}_m^{ab}=F^{bc}{j}_c,$

where $j_c:=g_{ca}{j}^a$. These are conditional variational identities; no Maxwell or Einstein field equation, existence theorem, or empirical premise is assumed.

For SI bookkeeping in length coordinates, take $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[A_a]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, and $[F_{ab}]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. If $S_m$ is assigned action units $\mathrm{J}\mathrm{s}$, the displayed variation convention assigns $[T^m]=\mathrm{J}{\mathrm{m}}^{-3}$ and $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$. These are formal dimensional assignments only; the lemma adopts no matter model or empirical law.

• The formula is global even when no global potential $A$ exists: the horizontal connection variation is ${\iota }_{\xi }F$, and $F$ is global.
• Matter stress is not separately divergence-free in the charged subsystem: its divergence is the Lorentz-force density, with the sign derived from the stated first-variation and flow conventions. Under the matching sourced-Maxwell convention, Curved electromagnetic stress divergence gives the opposite electromagnetic stress divergence, so the two cancel where both hypotheses hold.
• Sourced Maxwell sign and current conservation is a formulation prerequisite: it fixes the downstream source convention ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ (equivalently $d({\star }_{g}F)={\mu }_0{\iota }_j{\mathrm{v}\mathrm{o}\mathrm{l}}_g$ for ${\mu }_0>0$), specifying the sign with which this lemma's variational current is matched to the Maxwell source in a coupled formulation. This lemma's Statement and proof assume neither Maxwell equation; $j^a$ is fixed by the displayed matter first variation, and the stress-exchange sign is derived from the stated diffeomorphism and gauge conventions.
• The M02 and M03 supplier carriers are present as drafts and are used only at their inspected statements; the Maxwell-current dependency preserves the planned notation but no Maxwell equation is assumed; their proofs are not treated as accepted. The seven typed foundational replacements are now authored on emg-mg02; no independent acceptance is asserted: 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. The inherited geometry chain states ${\mathrm{A}\mathrm{C}}_{\omega }$; after the smooth data and local flows are given, the displayed calculation makes no further choice. The direct Countable Choice metadata dependency is inherited from that geometry chain.

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.

Let $(M,g,N,o)$ be an oriented time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$ as in Oriented Lorentzian fields, observers and coordinate conventions. Set $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with multiplication. A smooth right principal $U(1)$ bundle is a smooth manifold $P$ with a smooth surjective submersion $\pi :P\to M$, a smooth right $U(1)$ action, and smooth local equivariant product charts ${\pi }^{-1}(U_i)\cong U_i\times U(1)$; the action is free and transitive on each fibre. This is the smooth version of the right-action convention in Principal g bundle and associated fiber bundle. A principal connection on $P$ is a smooth distribution $H\subseteq TP$ with $T_{p}P=\ker (d{\pi }_p)\oplus H_p$ and $d{R}_z(H_p)=H_{pz}$. Equivalently, it is a smooth real one-form $\omega$ on $P$ that is right-invariant and satisfies $\omega ({\zeta }_P)=1$, where ${\zeta }_P(p):={\frac{d}{dt}\mid }_{0}p\cdot e^{it}$; then $H=\ker \omega$. Choose a fixed smooth right principal $U(1)$ bundle with a principal connection $\omega$. In a smooth trivialization ${\tau }_i:P{\mathrm{\mid }}_{U_i}\to U_i\times U(1)$, set ${\sigma }_i(x):={\tau }_i^{-1}(x,1)$ and $a_i:={\sigma }_i^{\ast }\omega$. Choose the normalized local data $(U_i,g_{ij},a_i;{\kappa }_0)$ as in Local potentials and smooth circle-bundle connection data; the potentials used below are their rescalings $A_i:={\kappa }_0{a}_i$. On any overlap neighborhood admitting a smooth real phase lift $f_{ij}$, write $g_{ij}=e^{i{f}_{ij}}$. Use the dual representation ${\rho }_k(z):=z^{-k}$ and define charged scalar representatives by the global transition rule ${\phi }_j=g_{ij}^k{\phi }_i$, equivalently $e^{ik{f}_{ij}}{\phi }_i$ on each phase-lift neighborhood; the connection rule there is

$A_j=A_i+{\kappa }_{0}d{f}_{ij}\mathrm{.}$

Here ${\kappa }_0>0$ is the chosen circle normalization with $[{\kappa }_0]=\mathrm{W}\mathrm{b}$, $k\in \mathbb{Z}$ is the representation weight, and $\lambda :=k\mathrm{/}{\kappa }_0$ has units ${\mathrm{W}\mathrm{b}}^{-1}$. Because $g_{ij}{g}_{jk}=g_{ik}$ and $k$ is an integer, the maps $g_{ij}^k$ satisfy the scalar transition cocycle. A charged scalar is a family of smooth local functions ${\phi }_i$ satisfying this rule; all phase-lift calculations below are local, and no global phase lift is assumed. The constant $c>0$ is the length-to-time conversion parameter of Oriented Lorentzian fields, observers and coordinate conventions, with units $\mathrm{m}{\mathrm{s}}^{-1}$. Define $F{\mathrm{\mid }}_{U_i}:=d{A}_i$; the transition law gives $d{A}_j=d{A}_i$, so $F$ is a global curvature two-form. In a global real-potential model on a trivial bundle, one may instead take a global $A$ and any real $\lambda$; no charge quantization is imposed there.

In a local trivialization define the gauge-covariant derivative and its lower-index one-form by

$P_a:=(D\phi {)}_a:={\mathrm{\partial }}_a{\phi }_i-i\lambda A_{i,a}{\phi }_i\mathrm{.}$

The transition laws give $P_{j,a}=e^{ik{f}_{ij}}{P}_{i,a}$, since $\lambda {\kappa }_0=k$. For any charged covector $Q_b$ with the same phase transition, define the gauge- and metric-covariant derivative by $D_a{Q}_b:={\mathrm{\nabla }}_a{Q}_b-i\lambda A_a{Q}_b$ in a local trivialization; for a charged scalar $\eta$, set $D_a\eta :={\mathrm{\partial }}_a\eta -i\lambda A_a\eta$. Conjugate fields use the opposite gauge sign. Thus $D_a{P}_b$ is a well-defined charged covariant two-tensor, and $D_a{P}^a:=g^{ab}{D}_a{P}_b$. Consequently $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ and $g^{ab}\overline{P_a}{P}_b$ are global scalar fields. Let $V:\mathcal{U}\to \mathbb{R}$ be smooth on an open set $\mathcal{U}\subseteq \mathbb{R}$ containing the range of $\mathrm{\mid }\phi {\mathrm{\mid }}^2$, write $V^{\mathrm{\prime }}$ for its ordinary real derivative, and choose a positive normalization constant $K_{\phi }>0$.

The kinetic scalar $g^{ab}\overline{P_a}{P}_b$ is real because its complex conjugate equals itself by symmetry of $g^{ab}$. Put

$L_{\phi }:=g^{ab}\overline{P_a}{P}_b+V(\mathrm{\mid }\phi {\mathrm{\mid }}^2),S_{\phi }:=-\frac{K_{\phi }}{2c}{\int }_M{L}_{\phi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

For SI dimensional bookkeeping in length coordinates, take $[\phi ]=1$, $[A_a]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, $[P_a]={\mathrm{m}}^{-1}$, $[V]={\mathrm{m}}^{-2}$, and $[K_{\phi }]=\mathrm{J}{\mathrm{m}}^{-1}$; then $S_{\phi }$ has units $\mathrm{J}\mathrm{s}$. These are formal dimensional assignments, not measured values or physical postulates. On noncompact $M$, the integral is used when finite; otherwise only differences between configurations agreeing outside a compact set are defined, so every variation below is a compactly supported integral.

Allowed first variations are compactly supported smooth complex-section variations $\delta \phi$, compactly supported real global one-form variations $\alpha =\delta A$ added to every local $A_i$ (so the same bundle transition functions are retained), and compactly supported symmetric inverse-metric variations $q^{ab}=\delta g^{ab}$. The formula defines a classical mathematical functional; it asserts no quantum interpretation, value of $\mathrm{\hslash }$, or material-response model.

• The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: 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.
• The published Lorentzian supplier carries the stated ${\mathrm{A}\mathrm{C}}_{\omega }$ convention through its tangent-bundle setup; the local patch calculation here makes no further choice. No full axiom of choice is used. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))
• In a nontrivial circle sector the integer weight condition $\lambda {\kappa }_0\in \mathbb{Z}$ is the representation condition selected for this bundle. In a trivial global real-potential sector $\lambda$ may be any real number; the two cases are not identified.

Use the scalar functional of Classical complex-scalar connection derivative and action on a fixed smooth bundle sector, with $K_{\phi }>0$, real $\lambda$, real smooth $V$, and smooth $\phi ,A,g$. Write $P_a:=D_a\phi$ and define the gauge- and metric-covariant derivative of its one-form by

$D_a{P}_b:={\mathrm{\nabla }}_a{P}_b-i\lambda A_a{P}_b,$

with the conjugate derivative using $+i\lambda A_a$. For compactly supported variations, define

$E_{\phi }:=D_a{P}^a-V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi ,j^a:=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a),$

$T_{ab}^{\phi }:=K_{\phi }\mathrm{Re}(\overline{P_a}{P}_b)-\frac{K_{\phi }}{2}{g}_{ab}{L}_{\phi }\mathrm{.}$

Then the scalar-field variation is ${\delta }_{\phi }{S}_{\phi }=\frac{K_{\phi }}{c}\mathrm{Re}{\int }_M\overline{\delta \phi }{E}_{\phi }{\mathrm{v}\mathrm{o}\mathrm{l}}_g$; stationarity in every compactly supported complex scalar direction is equivalent to $E_{\phi }=0$. The connection and inverse-metric variations are

${\delta }_A{S}_{\phi }=\frac{1}{c}{\int }_M{j}^a{\alpha }_a{\mathrm{v}\mathrm{o}\mathrm{l}}_g,{\delta }_g{S}_{\phi }=-\frac{1}{2c}{\int }_M{T}_{ab}^{\phi }{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

The following off-shell identities also hold:

${\mathrm{\nabla }}_a{j}^a=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{E}_{\phi }),{\mathrm{\nabla }}_a(T^{\phi }{)}^a{}_b=K_{\phi }\mathrm{Re}(\overline{E_{\phi }}{P}_b)+F_{ba}{j}^a\mathrm{.}$

In particular, the matter equation $E_{\phi }=0$ implies current conservation and the exact Lorentz-force exchange identity ${\mathrm{\nabla }}_a(T^{\phi }{)}^{ab}=F^{bc}{j}_c$. No coupled Einstein–Maxwell solution or empirical matter model is asserted.

Under the formal SI assignments of Classical complex-scalar connection derivative and action, $j^a$ has units $\mathrm{A}{\mathrm{m}}^{-2}$ and $T^{\phi }$ has units $\mathrm{J}{\mathrm{m}}^{-3}$; this dimensional check is not an empirical claim.

• The proof is local in bundle trivializations, while $j$, $T^{\phi }$, and the final identities are global because their defining expressions are gauge invariant.
• The current and stress formulas are classical variational identities; they make no assertion about a quantum field, charge quantization beyond the selected circle representation, or an observed material system.
• The M02/M03 supplier carriers are present as drafts and are used only at their inspected statement interfaces; their proofs are not treated as accepted. The seven typed foundational replacement carriers declared in deps are now authored on emg-mg02; no independent acceptance is asserted: 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.

Let $U$ be an oriented open subset of a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with metric-selected connection $\mathrm{\nabla }$. Let $F\in {\mathrm{\Omega }}^2(U)$ and $j\in \mathrm{\Gamma }(TU)$ be smooth, and let ${\mu }_0>0$. Assume

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

with the sign convention of Sourced Maxwell sign and current conservation. Define the Maxwell Hilbert tensor by the compact metric variation of Compact potential and metric variation of the Maxwell action,

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

Then

${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^{bc}{j}_c\mathrm{.}$

All fields and derivatives are smooth on $U$. In length-coordinate SI bookkeeping, assign $[F_{ab}]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$, $[{\mu }_0]=\mathrm{H}{\mathrm{m}}^{-1}$, $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, and $[T_{\mathrm{E}\mathrm{M}}]=\mathrm{J}{\mathrm{m}}^{-3}$; these are formal units and not observed values. No quadratic stress is assigned to singular point self-fields or distributional sources.