Local evolution of the adopted classical charged scalar model

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ and Dependent Choice (DC). No stronger choice principle is assumed. Use a smooth Hausdorff, second-countable, boundaryless, oriented underlying four-manifold with a time-oriented $C^3$ Lorentzian metric $(M,g,N,o)$, signature $(-,+,+,+)$, curvature convention, length coordinates $x^0=ct$, and SI assignments of Einstein–Maxwell primitive quantities and SI conventions. Fix $c,G_N,{\mu }_0,K_{\phi }>0$, $\lambda ,\mathrm{\Lambda }\in \mathbb{R}$, and a real $V\in C^{\mathrm{\infty }}(\mathcal{U})$ on an open interval $\mathcal{U}\subseteq \mathbb{R}$ containing the attained values of $\mathrm{\mid }\phi {\mathrm{\mid }}^2$. The classical domain is Classical finite-regularity charged-scalar system and relative action: the model uses a $C^2$ complex scalar $\phi$ and a globally defined real $C^2$ one-form $A$ with $F=dA$. The smooth configurations are retained as a subclass. In length coordinates $[A_a]=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-1}$, $[\lambda ]=\mathrm{W}{\mathrm{b}}^{-1}$, $[K_{\phi }]=\mathrm{J}{\mathrm{m}}^{-1}$, and $[V]={\mathrm{m}}^{-2}$.

Adopt the explicit classical compact-interior stationary total action of Adopted trivial-potential classical charged scalar model. The same declared classical action includes smooth configurations; no integer-weight circle representation is assumed for arbitrary real $\lambda$. Write ${\mathcal{D}}_a\phi :={\mathrm{\partial }}_a\phi -i\lambda A_a\phi$ and $P_a:={\mathcal{D}}_a\phi$. For a charged covector set ${\mathcal{D}}_a{P}_b:={\mathrm{\nabla }}_a{P}_b-i\lambda A_a{P}_b$, where $\mathrm{\nabla }$ is the spacetime Levi-Civita derivative; indices are raised with $g^{-1}$. The scalar equation, current and Hilbert tensor are

$E_{\phi }:={\mathcal{D}}_a{P}^a-V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi =0,j^a:=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a),$

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

Set $T_{ab}^{\mathrm{E}\mathrm{M}}:={\mu }_0^{-1}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd})$, $T^{\mathrm{t}\mathrm{o}\mathrm{t}}:=T^{\mathrm{E}\mathrm{M}}+T^{\phi }$, and ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4$. The equations of the adopted action model are

$dF=0,{\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b,G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E{T}_{ab}^{\mathrm{t}\mathrm{o}\mathrm{t}},E_{\phi }=0.$

Let $\mathrm{\Sigma }$ be a smooth spacelike initial hypersurface, $h$ its induced Riemannian metric, $n$ its future unit normal, and $\mathcal{K}(X,Y):=g({\mathrm{\nabla }}_{X}n,Y)$ its second fundamental form. Let ${\mathrm{\nabla }}^{\mathrm{\Sigma }}$ denote the Levi-Civita derivative of $h$ and $\tau :={\mathrm{tr}}_h\mathcal{K}$. Let $i_{\mathrm{\Sigma }}:\mathrm{\Sigma }\hookrightarrow M$ be the inclusion. The complete scalar data are ${\phi }_0:=\phi {\mathrm{\mid }}_{\mathrm{\Sigma }}$, ${\pi }_i:=({\mathcal{D}}_i\phi ){\mathrm{\mid }}_{\mathrm{\Sigma }}$, and the independently prescribed ${\pi }_n:=n^a{\mathcal{D}}_a\phi {\mathrm{\mid }}_{\mathrm{\Sigma }}$. In a spatial chart this means ${\pi }_i={\mathrm{\partial }}_i{\phi }_0-i\lambda A_{\mathrm{\Sigma },i}{\phi }_0$. The globally defined spatial potential is $A_{\mathrm{\Sigma }}:=i_{\mathrm{\Sigma }}^{\ast }A$, with exact-potential compatibility $F{\mathrm{\mid }}_{T\mathrm{\Sigma }}=d_{\mathrm{\Sigma }}{A}_{\mathrm{\Sigma }}$. Use the temporal potential gauge $A_0:=A({\mathrm{\partial }}_0)=0$ in local normal coordinates with ${\mathrm{\partial }}_0=n$ initially, so $A(n)=0$ on $\mathrm{\Sigma }$ and ${\pi }_0={\pi }_n$ initially.

For the periodic high-regularity branch take $\mathrm{\Sigma }={\mathbb{T}}_{2\pi }^3:={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ with dimensionless periodic coordinates $y^i$, and fix $L_0>0$ metres so that the length coordinates are $x^i=L_0{y}^i$. Sobolev norms use $y$; tensor components and derivative units use the length coordinates $x$. Let $s\ge 5$ be an integer. Prescribe a Riemannian metric $h\in H^{s+1}$ whose continuous representative satisfies $h\ge c_{0}I$ for some $c_0>0$, a symmetric $\mathcal{K}\in H^s$, spatial vector fields $e,b\in H^s$, a global real $A_{\mathrm{\Sigma }}\in H^s$, ${\phi }_0\in H^{s+1}$, and an independent complex ${\pi }_n\in H^s$. Here the Sobolev spaces are componentwise real, with complex components split into real and imaginary parts. For the periodic state, let $({\mathsf{e}}_A(h){)}_{A=1}^3$ be the positively oriented $h$-orthonormal frame obtained by Gram–Schmidt from the ordered global length-coordinate frame, as specified in Periodic high-Sobolev state spaces. Put ${\widehat{e}}_A:=h(e,{\mathsf{e}}_A(h))$ and ${\widehat{b}}_A:=h(b,{\mathsf{e}}_A(h))$. The initial two-form convention below then gives $F_{\mathrm{\Sigma }}({\mathsf{e}}_A(h),n)={\widehat{e}}_A$ and $F_{\mathrm{\Sigma }}({\mathsf{e}}_A(h),{\mathsf{e}}_B(h))={ϵ}_{ABC}{\widehat{b}}_C$, with ${ϵ}_{123}=+1$. Thus ${\widehat{e}}_A=E_A\mathrm{/}c$ and ${\widehat{b}}_A=B_A$ in the observer convention of Einstein–Maxwell primitive quantities and SI conventions, and both components have unit tesla. Sobolev regularity of the vector fields $e,b$ refers to their components in the fixed periodic coordinates; the orthonormal-frame state components are those specified by the named convention. The initial two-form $F_{\mathrm{\Sigma }}$ is fixed by $F_{\mathrm{\Sigma }}(X,n)=h(e,X)$ and $F_{\mathrm{\Sigma }}(X,Y)={\mathrm{v}\mathrm{o}\mathrm{l}}_h(b,X,Y)$ for spatial $X,Y$. Assume ${\pi }_i={\mathcal{D}}_i{\phi }_0$ and $F_{\mathrm{\Sigma }}{\mathrm{\mid }}_{T\mathrm{\Sigma }}=d_{\mathrm{\Sigma }}{A}_{\mathrm{\Sigma }}$. With $j,T^{\mathrm{t}\mathrm{o}\mathrm{t}}$ evaluated on the complete initial fields, impose the Maxwell constraints

${\mathrm{\nabla }}_i^{\mathrm{\Sigma }}{b}^i=0,{\mathrm{\nabla }}_i^{\mathrm{\Sigma }}{e}^i={\mu }_0{j}^0,j^0:=-n_a{j}^a,$

and the Einstein constraints in the stated sign convention,

$R(h)+{\tau }^2-\mathrm{\mid }\mathcal{K}{\mathrm{\mid }}_h^2=2\mathrm{\Lambda }+2{\kappa }_E{T}^{\mathrm{t}\mathrm{o}\mathrm{t}}(n,n),({\mathrm{div}}_h\mathcal{K})(X)-{\mathrm{\nabla }}_X^{\mathrm{\Sigma }}\tau ={\kappa }_E{T}^{\mathrm{t}\mathrm{o}\mathrm{t}}(n,X)(X\in T\mathrm{\Sigma })\mathrm{.}$

Here $({\mathrm{div}}_h\mathcal{K})(X):=h^{ij}({\mathrm{\nabla }}_{{\mathrm{\partial }}_i}^{\mathrm{\Sigma }}\mathcal{K})({\mathrm{\partial }}_j,X)$, so the momentum equation is an equality of covectors and does not mix coordinate and orthonormal components.

Fix a smooth positive reference metric $\gamma$ on $\mathrm{\Sigma }$ and the product comparison metric $\bar{g}:=-(d{x}^0{)}^2+\gamma$ near the initial slice. Define $H^a:=g^{bc}(\mathrm{\Gamma }(g{)}^a{}_{bc}-\mathrm{\Gamma }(\bar{g}{)}^a{}_{bc})$ and $Q^i:=h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk})$. Use the initial harmonic metric jet

$g_{ij}=h_{ij},g_{00}=-1,g_{0i}=0,{\mathrm{\partial }}_0{g}_{ij}=2{\mathcal{K}}_{ij},{\mathrm{\partial }}_0{g}_{00}=-2\tau ,{\mathrm{\partial }}_0{g}_{0i}=h_{ij}{Q}^j\mathrm{.}$

Let $\bar{\mathrm{\nabla }}$ be the Levi-Civita derivative of $\bar{g}$ and put $p_{ab}:={\bar{\mathrm{\nabla }}}_0{g}_{ab}$ and $q_{abi}:={\bar{\mathrm{\nabla }}}_i{g}_{ab}$ for the first-order metric derivative variables. For a fixed field scale $F_0>0$ tesla, form the real first-order state from $(g_{ab},p_{ab},q_{abi})$, the six orthonormal-frame Maxwell components $({\widehat{e}}_A,{\widehat{b}}_A)$, the three spatial potential components $A_i$, and the real and imaginary parts of $\phi ,{\pi }_0,{\pi }_i$. Rescale each first metric derivative and each ${\pi }_a$ by multiplying by $L_0$, each ${\widehat{e}}_A,{\widehat{b}}_A$ by dividing by $F_0$, and each $A_i$ by dividing by $L_0{F}_0$; $g_{ab}$ and $\phi$ are already dimensionless. These fixed invertible scales are bookkeeping choices, not extra equations. Denote the resulting componentwise-real periodic state by $u_0$. Suppose its continuous range is a compact $K_0$ contained in the interior of a compact $K_1$ inside an open admissible state set $O$, that the coefficient maps are smooth on $O$, the potential range lies in $\mathcal{U}$, and the combined time matrix is positive definite throughout $O$. Require $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^s}\le R$ for some finite $R\ge 0$. Thus $u_0\in {\mathcal{S}}_s(K_0,R)$ in the componentwise real periodic Fourier and weak-derivative convention used by the CP04 supplier.

Under the stated mathematical hypotheses of the current authored supplier Local classical Einstein–Maxwell–charged-scalar development, these periodic data have a local solution of the full Einstein, sourced Maxwell, and classical charged-scalar equations on $(-T_{\ast },T_{\ast })\times {\mathbb{T}}_{2\pi }^3$. Here $T_{\ast }>0$ is a width in the length coordinate $x^0=ct$ (so the corresponding time width in seconds is $T_{\ast }\mathrm{/}c$). In its coordinate gauge, the first-order state lies in $L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$ with time derivative in $L^{\mathrm{\infty }}{H}^{s-1}$; the metric has $g\in L^{\mathrm{\infty }}{H}^{s+1}\cap C{H}^s$ and ${\mathrm{\partial }}_{0}g\in L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$, while $F,\pi ,A_i\in L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$ and $\phi \in L^{\mathrm{\infty }}{H}^{s+1}\cap C{H}^s$. The Maxwell, scalar-potential, metric-derivative, and harmonic constraints propagate. More generally, for smooth data $(h,\mathcal{K},e,b,A_{\mathrm{\Sigma }},{\phi }_0,{\pi }_n)$ on any smooth oriented three-manifold without boundary, define $F_{\mathrm{\Sigma }}$ by $F_{\mathrm{\Sigma }}(X,n)=h(e,X)$ and $F_{\mathrm{\Sigma }}(X,Y)={\mathrm{v}\mathrm{o}\mathrm{l}}_h(b,X,Y)$ for spatial $X,Y$. The current authored supplier states local smooth development near each initial point, provided $h$ is positive definite, $A_{\mathrm{\Sigma }}$ is global, $F_{\mathrm{\Sigma }}{\mathrm{\mid }}_{T\mathrm{\Sigma }}=d_{\mathrm{\Sigma }}{A}_{\mathrm{\Sigma }}$, ${\pi }_i={\mathcal{D}}_i{\phi }_0$, $A(n)=0$, the sourced Maxwell and total-stress Einstein constraints hold, and $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ remains in $\mathcal{U}$. Its uniqueness conclusion is local near the initial surface, up to a data-fixing time-orientation-preserving isometry and electromagnetic gauge $A\mapsto A+d\chi$, $\phi \mapsto e^{i\lambda \chi }\phi$ with $\chi {\mathrm{\mid }}_{\mathrm{\Sigma }}=0$ and $[\chi ]=\mathrm{W}\mathrm{b}$. This is a local conditional deduction in the adopted action model. It asserts no nontrivial bundle sector, singular source, global-in-time existence, geodesic completeness, empirical validity, or quantum field.

• The global real potential forces $F$ to be exact; nontrivial $U(1)$ bundles are outside this result. The classical coupling $\lambda$ is arbitrary and is not tied to $\mathrm{\hslash }$ or a quantized charge.
• The action, field equations and development are adopted or deduced model statements. No measurement or observation is used, and the proof does not establish that the model describes nature.
• The local mathematical theorem is used with its complete charged data, integer Sobolev order, choice and admissible-state hypotheses. The conditional physical conclusion still depends on adoption of the stated action model.