Adopted trivial-potential classical charged scalar model

Use a smooth Hausdorff, second-countable underlying oriented four-manifold with a time-oriented $C^3$ Lorentzian metric, signature $(-,+,+,+)$, Levi-Civita connection, curvature convention, and SI length-coordinate units of Einstein–Maxwell primitive quantities and SI conventions. Use the explicit classical domain of Classical finite-regularity charged-scalar system and relative action. Let $A$ be a $C^2$, globally defined real one-form with components in tesla-metres, and set $F:=dA$, so $F$ is a $C^1$ exact real two-form. The globally defined potential is the selected trivial-potential sector: it excludes bundles with no global potential and does not impose an integer charge condition.

Let $\phi :M\to \mathbb{C}$ be a $C^2$ dimensionless classical scalar field. Smooth configurations remain a subclass. Choose $\lambda \in \mathbb{R}$ with units $(\mathrm{T}{\mathrm{m}}^2{)}^{-1}$, a constant $K_{\phi }>0$ with units $\mathrm{J}{\mathrm{m}}^{-1}$, and an open set $\mathcal{U}\subseteq \mathbb{R}$ containing the range of $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ together with a smooth real function $V:\mathcal{U}\to \mathbb{R}$ whose values are in ${\mathrm{m}}^{-2}$. No sign, polynomial-degree, or smallness condition is imposed on $V$.

Define the gauge-covariant scalar derivative and charged covector by $P_a:=D_a\phi :={\mathrm{\partial }}_a\phi -i\lambda A_a\phi$. For a charged covector $P_b$, define $D_a{P}_b:={\mathrm{\nabla }}_a{P}_b-i\lambda A_a{P}_b$, where $\mathrm{\nabla }$ is the Levi-Civita derivative, and set $D^a{P}_a:=g^{ab}{D}_a{P}_b$. The adopted classical matter action is

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

With the stated SI dimensions, $S_{\phi }$ has unit joule-second. For a gauge function $\chi$ in $A\mapsto A+d\chi$, assign $[\chi ]=\mathrm{T}{\mathrm{m}}^2=\mathrm{W}\mathrm{b}$, so $d\chi$ has the same unit as the components of $A$ and $\lambda \chi$ is dimensionless.

On a noncompact domain the integral is used when finite; otherwise the formulation concerns compactly supported action differences. Variations are smooth and compactly supported in the fixed global-potential sector. Adopt this functional as a classical matter model. In particular, this item does not assert that the model describes nature, that its coupled equations have a solution, or that the scalar is a quantum field.

The scalar action is adopted in this global real-potential domain. The Einstein–Hilbert and free Maxwell coefficients below agree with the smooth action convention of Interior stationary Einstein–Maxwell–matter action, but the present model is declared directly: an arbitrary real $\lambda$ is not required to define an integer-weight representation of a principal circle bundle. For the declared classical class, separately adopt the same total functional on the domain and compact smooth directions of Classical finite-regularity charged-scalar system and relative action:

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

Every integral here is a relative compact action when a global integral is not finite. Adopt independent stationarity for all smooth compact symmetric inverse-metric, real one-form and complex scalar directions. Compact charged-scalar action variations at classical finite regularity proves all first variations and their pointwise equation equivalence at $g\in C^3$, $A,\phi \in C^2$. This declares the classical action model, including all its smooth configurations, without requiring a circle-bundle matter representation. The separate smooth circle-sector action formulation is not a prerequisite for this global-real-potential branch.

Its gauge action is $A\mapsto A+d\chi$, $\phi \mapsto e^{i\lambda \chi }\phi$ for a smooth real function $\chi$; direct substitution gives $D_a^{\mathrm{\prime }}{\phi }^{\mathrm{\prime }}=e^{i\lambda \chi }{D}_a\phi$, so the scalar action density is gauge invariant. It is also invariant under orientation-preserving diffeomorphisms because its integrand is a scalar multiplied by the metric volume form. These are properties of the selected functional, not evidence that it describes nature.

For clarity, the mathematical supplier Classical charged-scalar action, current and stress variation gives the compact first variations and defines

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

$T_{ab}^{\phi }:=K_{\phi }\mathrm{Re}(\overline{P_a}{P}_b)-\frac{K_{\phi }}{2}{g}_{ab}(g^{cd}\overline{P_c}{P}_d+V(\mathrm{\mid }\phi {\mathrm{\mid }}^2))\mathrm{.}$

Here $E_{\phi }=0$ is the scalar Euler–Lagrange equation, $j$ is the current from varying $A$, and $T^{\phi }$ is the Hilbert stress tensor from varying the inverse metric. Those formulas are mathematical consequences of the action under the cited supplier's hypotheses; they are not extra physical laws. For smooth fields those formulas use the cited smooth variation result; for the declared classical fields the identical formulas and compact stationarity equivalence are proved in Compact charged-scalar action variations at classical finite regularity. Both are mathematical deductions, separate from adoption of the physical model.