Trivial-potential classical charged-scalar Einstein–Maxwell system

Let $(M,g,N,o)$ be an oriented, time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with its metric-selected Levi–Civita connection and independent orientation as in Oriented Lorentzian fields, observers and coordinate conventions. Use length coordinates $x^0=ct$ and the curvature and Hodge conventions of Electrovacuum tensor Cauchy system and Lorentzian exterior calculus and Hodge star. Fix $c>0$, $G_N>0$, ${\mu }_0>0$, $\mathrm{\Lambda }\in \mathbb{R}$, and put $\kappa =8\pi G_N\mathrm{/}{c}^4$.

The trivial-potential sector consists of a global smooth real one-form $A\in {\mathrm{\Omega }}^1(M)$ and a global smooth complex scalar field $\phi \in C^{\mathrm{\infty }}(M;\mathbb{C})$. It defines the exact real two-form $F:=dA$ and the charged derivative

${\pi }_a:=(D\phi {)}_a:={\mathrm{\nabla }}_a\phi -i\lambda A_a\phi ={\mathrm{\partial }}_a\phi -i\lambda A_a\phi ,$

where $\lambda \in \mathbb{R}$ is an arbitrary fixed coupling. Define $D_a{\pi }_b:={\mathrm{\nabla }}_a{\pi }_b-i\lambda A_a{\pi }_b$, with the conjugate derivative using $+i\lambda A_a$, and $D_a{\pi }^a:=g^{ab}{D}_a{\pi }_b$. Let $\mathcal{U}\subset \mathbb{R}$ be open with $\mathrm{\mid }\phi {\mathrm{\mid }}^2(M)\subset \mathcal{U}$, let $V\in C^{\mathrm{\infty }}(\mathcal{U};\mathbb{R})$, and fix $K_{\phi }>0$. Define

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

where on a noncompact $M$ the integral is used when finite and otherwise only compactly supported action differences are used. The scalar equation, current vector, and scalar stress tensor are

$D_a{\pi }^a-V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi =0,j^a:=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{\pi }^a),$

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

Put

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

The associated classical charged-scalar Einstein–Maxwell system is

$G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab},dF=0,{\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b,D_a{\pi }^a-V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi =0.$

Here $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$; all indices are raised with $g^{-1}$. This is a definition of a mathematical system, not an adopted physical postulate or a claim about nature. It is restricted to a trivial real-potential sector: $F$ is exact, no nontrivial $U(1)$ bundle is included, and $\lambda$ is not quantized. The scalar is classical, and neither $K_{\phi }$ nor $\lambda$ is tied to a Planck constant.

For formal SI bookkeeping in length coordinates, assign $[g_{ab}]=1$, $[x^a]=\mathrm{m}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[G_N]={\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $[{\mu }_0]=\mathrm{H}{\mathrm{m}}^{-1}$, $[\mathrm{\Lambda }]={\mathrm{m}}^{-2}$, $[A_a]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, $[F_{ab}]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$, $[\phi ]=1$, $[\lambda ]=\mathrm{W}{\mathrm{b}}^{-1}$, $[{\pi }_a]={\mathrm{m}}^{-1}$, $[V]={\mathrm{m}}^{-2}$, and $[K_{\phi }]=\mathrm{J}{\mathrm{m}}^{-1}$. These give $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, $[T_{ab}]=\mathrm{J}{\mathrm{m}}^{-3}$, $[\kappa T_{ab}]={\mathrm{m}}^{-2}$, and $[S_{\phi }]=\mathrm{J}\mathrm{s}$. They are dimensions, not measured values.