Interior stationary Einstein–Maxwell–matter action

Use the smooth oriented, time-oriented Lorentzian four-manifold, curvature convention and SI constants of Einstein–Maxwell primitive quantities and SI conventions and Oriented Lorentzian fields, observers and coordinate conventions. Choose the actual matter functional $S_m[g,\omega ,\mathrm{\Psi }]$ and its fixed smooth right principal $U(1)$ bundle $\pi :P\to M$ from Selected invariant matter-action contract. In its fixed normalized connection sector, the local representatives are $a_i$, the physical local potentials are $A_i={\kappa }_0{a}_i$, and their globally defined physical curvature satisfies $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. A global real potential is not required. Write $R:=g^{ab}{\mathrm{Ric}}_{ab}$ for the scalar curvature of the metric-selected connection, with the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. For configurations where the action is finite, or for compactly supported action differences on a noncompact manifold, adopt

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

Adopt that a configuration is dynamically admissible in this branch exactly when the first variation of this total action vanishes independently for every smooth compactly supported symmetric inverse-metric variation, every smooth compactly supported fixed-sector physical connection variation $\alpha \in {\mathrm{\Omega }}^1(M)$, and every admissible compactly supported matter-field variation. The connection variation changes $A_i$ by $\alpha {\mathrm{\mid }}_{U_i}$, equivalently changes $a_i$ by ${\kappa }_0^{-1}\alpha {\mathrm{\mid }}_{U_i}$, while leaving the bundle and transition data fixed; thus $\delta F=d\alpha$. If $M$ has a physical boundary, only variations vanishing on a neighbourhood of that boundary are included. This makes no claim about a Gibbons–Hawking–York or other boundary action.

The earlier postulate Adopt stationary Maxwell action as alternative physical formulation concerns its stated flat electromagnetic and prescribed-potential particle sectors. It supplies no unified matter–gravity action or coupled-solution theorem; the functional above is an additional, explicitly adopted model. An equation-based Einstein–Maxwell formulation is a separate alternative, not a premise of this action branch.

All displayed coefficients are model parameters with the units and interpretations in Einstein–Maxwell primitive quantities and SI conventions. Compact support is essential: the Einstein–Hilbert and Maxwell integrations by parts have boundary terms in general. The postulate asserts neither that its stationary configurations exist nor that the action describes nature.