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.