Selected invariant matter-action contract

For the action-based Einstein–Maxwell branch, choose one actual smooth matter configuration space $\mathcal{C}$ on the spacetime of Einstein–Maxwell primitive quantities and SI conventions, a smooth right principal $U(1)$ bundle $\pi :P\to M$, and a real differentiable functional $S_m[g,\omega ,\mathrm{\Psi }]$ on its declared domain, where $\mathrm{\Psi }\in \mathcal{C}$ and $\omega$ is a smooth principal connection on that fixed bundle. Use a supplied cover and fixed transition data with normalized real local connection representatives $a_i$; on phase-lift neighborhoods their convention is $a_j=a_i+d{f}_{ij}$ when $g_{ij}=e^{i{f}_{ij}}$. With the fixed normalization ${\kappa }_0>0$ of Einstein–Maxwell primitive quantities and SI conventions, the local expression for $\omega$ in a circle coordinate $z\in U(1)$ is $-i{z}^{-1}dz+a_i$. Define the physical local potentials $A_i:={\kappa }_0{a}_i$. They obey $A_j=A_i+{\kappa }_{0}d{f}_{ij}$, and their local exterior derivatives patch to the global physical curvature $F$, with $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. A global real potential is not assumed. The field class, transition data, gauge action, matter equations, and compact variation class must be specified for the chosen matter model.

More precisely, use the matter-field and differentiability definitions in Gauge/diffeomorphism matter identity and field exchange: supply a smooth finite-dimensional fiber bundle $E\to M$, take $\mathcal{C}={\mathrm{\Gamma }}^{\mathrm{\infty }}(E)$, and specify a real functional domain $\mathcal{D}$ of metric, fixed-sector connection and section triples. Compact variations are jointly smooth curves in $\mathcal{D}$ fixed off a common compact set. Require the freezing of any two components to remain admissible, admit every compact metric and connection direction, and require the derivative to depend real-linearly only on the three tangent directions and to split into the three partial variations. For each compact real gauge path and each horizontal-lifted compact base flow, supply the jointly smooth matter-bundle diffeomorphisms, their inverses on a common interval, and the admissible transported sections specified in that lemma. These are part of the selected model data; an arbitrary section bundle has no such transport by default.

If the selected matter model is given by a global physical potential $A\in {\mathrm{\Omega }}^1(M)$, use the trivial principal bundle $P=M\times U(1)$ with its global trivialization and normalized representative $a=A\mathrm{/}{\kappa }_0$; its principal connection is locally $-i{z}^{-1}dz+a$. This is the trivial-bundle global-potential branch, and the model may admit any real coupling when its gauge transformations are specified by compactly supported real lifts. A nontrivial bundle sector instead uses the stated local transition data and the model's compatible matter-field transition rules.

The functional is invariant under compactly supported gauge transformations of the fixed bundle sector and under orientation-preserving diffeomorphisms, with the connection and matter fields transported together. In particular, for every smooth compactly supported real gauge parameter $\chi$ with units Wb, its group map is $g_{\chi }=e^{i\chi \mathrm{/}{\kappa }_0}$; it acts by $a_i\mapsto a_i+{\kappa }_0^{-1}d\chi$ and hence by $A_i\mapsto A_i+d\chi$, together with the model's specified matter-field transformation. For a compactly supported vector field $\xi$, transport by the induced horizontal lift gives the physical connection variation ${\alpha }_a={\xi }^b{F}_{ba}$ used by Gauge/diffeomorphism matter identity and field exchange.

Adopt matter-field stationarity as part of this selected model: ${\delta }_{\mathrm{\Psi }}{S}_m=0$ for every admissible compactly supported matter variation. For compactly supported inverse-metric variations $q^{ab}=\delta g^{ab}$ and fixed-sector compact physical connection variations $\alpha \in {\mathrm{\Omega }}^1(M)$, vary the local potentials by $A_i(s)=A_i+s\alpha {\mathrm{\mid }}_{U_i}$, equivalently the normalized representatives by $a_i(s)=a_i+s{\kappa }_0^{-1}\alpha {\mathrm{\mid }}_{U_i}$. The transition data remain fixed, this is a variation of the same bundle connection sector, and $\delta F=d\alpha$. Require the remaining first variation to have the form

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

where $T^m$ is a smooth symmetric covariant two-tensor, the Hilbert stress-energy tensor in the convention of The Hilbert stress-energy tensor of a metric-dependent action, and $j$ is a smooth vector current. Here $\alpha$ is the physical-potential variation, not the normalized representative variation; the factor ${\kappa }_0^{-1}$ above makes the two conventions explicit. These normalization, connection-variation, and invariance hypotheses are exactly the formulation interface used by Gauge/diffeomorphism matter identity and field exchange. That conditional mathematical result does not assert that such an action exists for arbitrary proposed matter.

This postulate selects a differentiable action branch and its variation conventions. It does not determine a unique action, assert a local or global solution of its Euler–Lagrange equations, extend the model to singular sources, or establish empirical validity. The separate equation-based model does not depend on this action assumption.