Local classical Einstein–Maxwell–charged-scalar development

For the periodic metric jet, put $Q^i:=h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk})$. The continuous positive metric $h$ has range in a compact subset of the open cone of positive-definite matrices. Smooth composition with matrix inversion and the $H^s$ product estimate in [L8] give $h^{-1}\in H^{s+1}$ and, by the Christoffel formula, $\mathrm{\Gamma }(h{)}^i{}_{jk}\in H^s$. Since $\gamma$ is smooth, $Q\in H^s$, $\tau =h^{ij}{K}_{ij}\in H^s$, and $h_{ij}{Q}^j\in H^s$. Thus all prescribed normal metric derivatives lie in $H^s$. The spatial derivatives of $h$ lie in $H^s$ as well, so $p_{ab}={\bar{\mathrm{\nabla }}}_0{g}_{ab}$ and $q_{abi}={\bar{\mathrm{\nabla }}}_i{g}_{ab}$, with the fixed smooth product background, have the required $H^s$ regularity. On the initial slice $g^{00}=-1$, $g^{0i}=0$, $g^{ij}=h^{ij}$. The Christoffel formula gives $\mathrm{\Gamma }(g{)}^0{}_{00}=\tau$, $\mathrm{\Gamma }(g{)}^0{}_{ij}=K_{ij}$, $\mathrm{\Gamma }(g{)}^i{}_{00}=Q^i$, and $\mathrm{\Gamma }(g{)}^i{}_{jk}=\mathrm{\Gamma }(h{)}^i{}_{jk}$. The product background has ${\bar{\mathrm{\Gamma }}}^0{}_{ab}={\bar{\mathrm{\Gamma }}}^i{}_{00}={\bar{\mathrm{\Gamma }}}^i{}_{0j}=0$ and ${\bar{\mathrm{\Gamma }}}^i{}_{jk}=\mathrm{\Gamma }(\gamma {)}^i{}_{jk}$. Therefore the definition of $H$ in [L7] gives $H^0=-\mathrm{\Gamma }(g{)}^0{}_{00}+h^{ij}\mathrm{\Gamma }(g{)}^0{}_{ij}=-\tau +\tau =0$ and $H^i=-\mathrm{\Gamma }(g{)}^i{}_{00}+h^{jk}(\mathrm{\Gamma }(h{)}^i{}_{jk}-\mathrm{\Gamma }(\gamma {)}^i{}_{jk})=-Q^i+Q^i=0$. These equalities hold in $H^s$ and hence pointwise for the continuous representatives by [L8].

The prescribed normal/tangential two-form components determine the unique real two-form on $\mathbb{R}n\oplus T\mathrm{\Sigma }$: $F(n,{\mathfrak{e}}_i)=-e_i$ and $F({\mathfrak{e}}_i,{\mathfrak{e}}_j)={ϵ}_{ijk}{b}_k$. Exactness $F_{\mathrm{\Sigma }}{\mathrm{\mid }}_{T\mathrm{\Sigma }}=d_{\mathrm{\Sigma }}{A}_{\mathrm{\Sigma }}$ and ${\pi }_i={\mathcal{D}}_i{\phi }_0$, together with ${\pi }_0={\pi }_n$ in the initial temporal gauge, make all field and potential entries compatible. In the periodic coordinate trivialization, Gram–Schmidt applied to the global coordinate frame and the positive matrix $h$ gives an $h$-orthonormal frame whose coefficients are smooth functions of $h$; by [L8] its coefficients lie in $H^{s+1}$, so the frame components $e_i,b_i$ lie in $H^s$. Also ${\mathrm{\partial }}_i{\phi }_0\in H^s$ and $A_i{\phi }_0\in H^s$ by the $H^s$ algebra estimate, hence ${\pi }_i={\mathrm{\partial }}_i{\phi }_0-i\lambda A_i{\phi }_0\in H^s$, while ${\pi }_0={\pi }_n\in H^s$. The remaining matter components have the stated $H^s$ regularity. After the fixed invertible rescalings every state component is real and in the stipulated $H^s$ class; the compact range lies in $K_0⋐K_1⋐O$.

On the unscaled state, the metric block has the positive time matrix $\mathrm{diag}(I,-g^{00}I,(g^{ij})I)$ from [L7]; the total stress is algebraic in $(g,F,\phi ,\pi )$, so the scalar contribution changes only zero-order terms. The Maxwell block has time matrix ${\alpha }^{-1}{I}_6$ and symmetric spatial matrices by [L4]; the sourced current is algebraic in $(g,\phi ,\pi )$ and hence is lower order. The charged scalar block has time matrix $\mathrm{diag}(-g^{00},(g^{ij}))$ on each real and imaginary part by [L5]. The $\phi$ and $A_i$ transport equations add identity time blocks and zero spatial blocks. These blocks have no off-diagonal derivative couplings, so their direct sum has symmetric principal matrices and positive time matrix. Let $v$ denote that unscaled state, $u=Sv$ the stated dimensionless state, and $S$ the fixed positive diagonal rescaling. Substituting $v=S^{-1}u$ in $A^{\mu }(x^0,x,v){\mathrm{\partial }}_{\mu }v+B(x^0,x,v)=0$ and multiplying by $S^{-T}$ gives principal matrices ${\tilde{A}}^{\mu }(x^0,x,u)=S^{-T}{A}^{\mu }(x^0,x,S^{-1}u)S^{-1}$ and source $\tilde{B}(x^0,x,u)=S^{-T}B(x^0,x,S^{-1}u)$. Congruence preserves symmetry and positive definiteness of the time matrix, so the same coefficient properties hold for the actual dimensionless state. All coefficient maps are smooth jointly on $I\times {\mathbb{T}}_{2\pi }^3\times O$: the background and reference metric are smooth, and state dependence uses smooth inversion of $g$, smoothness of $V$, and the stipulated open range. These displayed coefficient identities are algebraic in the finite state and its first coordinate derivatives; their derivation does not require an evolved solution to be smooth. At a point the relevant finite jets of a classical field can equally be realized by smooth Taylor polynomials, so the same coordinate identities and explicit coefficient maps apply to the classical domain in [F1]. This transfers an identity, not a smooth solution or a smoothness claim.

Apply [L3] to this verified coefficient family, the state $u_0\in {\mathcal{S}}_s(K_0,R)$, and compact sets $K_0⋐K_1⋐O$. The solver gives the common interval, values in $K_1$, $L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$ state regularity, uniqueness and stability, and smooth persistence for smooth initial data. It asserts only a solution to the reduced first-order system at this step.

The reduced coefficient system writes $u_s=-(A^0{)}^{-1}(A^i{\mathrm{\partial }}_{i}u+B)$ on a compact admissible range. Since $u\in C{H}^{s-1}$ and the coefficient maps are smooth, the algebra and composition estimates give $u_s\in C{H}^{s-2}$; differentiating this expression once gives $u_{ss}\in C{H}^{s-3}$. The metric derivative residual $q_i-{\bar{\mathrm{\nabla }}}_{i}g$ satisfies the homogeneous background-commutator ODE and has zero initial value, independently of the matter constraints, so $q_i={\bar{\mathrm{\nabla }}}_{i}g$; also $p={\bar{\mathrm{\nabla }}}_{0}g$ by its evolution. Thus the same Fourier norm and time-derivative argument as for the electrovacuum block gives $g\in L^{\mathrm{\infty }}{H}^{s+1}\cap C{H}^s$ and $g\in C^3$ as a spacetime metric at the base order $s=5$: third spatial derivatives come from two spatial derivatives of $q$, mixed third derivatives from spatial derivatives of $p_s$, and third time derivatives from $p_{ss}$, all continuous since $H^2\hookrightarrow C^0$. The algebraic reconstruction of $F$ from $(g,e,b)$ and the scalar component equations likewise give $F,A,\phi ,\pi \in C^2$ where needed. In particular the smooth coordinate time levels have a $C^1$ transverse normal, the normal residual-transport arguments are legitimate, and the subsidiary wave energy has Lipschitz principal coefficients and bounded lower-order coefficients. This verifies the finite-regularity applicability of [L6] before using its conclusion.

The initial metric, scalar, potential, and Maxwell compatibilities and all Gauss and Einstein constraints are exactly those of [L6]. Applying that subsidiary result recovers $p={\bar{\mathrm{\nabla }}}_{0}g$, $q_i={\bar{\mathrm{\nabla }}}_{i}g$, $\pi =\mathcal{D}\phi$, $F=dA$, both sourced Maxwell equations, total-stress conservation before Einstein recovery, and $H=0$. Therefore the reduced solution is a full solution of the classical mathematical system in [F1]: step 3.1 supplies $g\in C^3$, $A,\phi \in C^2$, while the compatibility just recovered identifies its $F$ and $\pi$ with $dA$ and $\mathcal{D}\phi$. Smooth persistence gives the original smooth system for smooth data.

Since $q_i={\mathrm{\partial }}_{i}g$ up to the fixed smooth background connection, the periodic Fourier norm identity gives $\mathrm{\parallel }g{\mathrm{\parallel }}_{H^{s+1}}\asymp \mathrm{\parallel }g{\mathrm{\parallel }}_{H^s}+{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_{i}g{\mathrm{\parallel }}_{H^s}$; similarly $p={\bar{\mathrm{\nabla }}}_{0}g$ gives the stated time-derivative regularity. From ${\pi }_i={\mathcal{D}}_i\phi$, one has ${\mathrm{\partial }}_i\phi ={\pi }_i+i\lambda A_i\phi$. The $H^s$ algebra estimate in [L8] makes $A_i\phi \in L^{\mathrm{\infty }}{H}^s$, and the same estimate in $C{H}^{s-1}$ gives continuity of the spatial derivatives. The Fourier norm identity $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{r+1}}\asymp \mathrm{\parallel }f{\mathrm{\parallel }}_{H^r}+{\sum }_i\mathrm{\parallel }{\mathrm{\partial }}_{i}f{\mathrm{\parallel }}_{H^r}$ then gives $\phi \in L^{\mathrm{\infty }}{H}^{s+1}\cap C{H}^s$. The field-strength, potential, and $\pi$ regularities are the corresponding components of the reduced state. The embedding in [L8] supplies the classical derivatives needed by the equations at $s\ge 5$.

For general smooth $\mathrm{\Sigma }$, regard $\mathrm{\Psi }=(A,\mathrm{Re}\phi ,\mathrm{Im}\phi )$ as the fixed covariant matter fields; $F=dA$, $\pi =\mathcal{D}\phi$, $j$, and $T$ are the fixed equations and tensorial expressions of [F1]. On each relatively compact initial chart choose a flat comparison metric $\bar{g}=-(d{x}^0{)}^2+{\delta }_{ij}d{x}^{i}d{x}^j$, set $g_{ij}=h_{ij}$, $g_{00}=-1$, $g_{0i}=0$, ${\mathrm{\partial }}_0{g}_{ij}=2K_{ij}$, ${\mathrm{\partial }}_0{g}_{00}=-2\tau$, and ${\mathrm{\partial }}_0{g}_{0i}=h_{ij}{h}^{k\mathrm{\ell }}\mathrm{\Gamma }(h{)}^j{}_{k\mathrm{\ell }}$. The Christoffel formula gives $H^0=\frac{1}{2}{\mathrm{\partial }}_0{g}_{00}+\tau =0$ and $H^i=-h^{ij}{\mathrm{\partial }}_0{g}_{0j}+h^{k\mathrm{\ell }}\mathrm{\Gamma }(h{)}^i{}_{k\mathrm{\ell }}=0$ initially. Set $q_{abi}={\bar{\mathrm{\nabla }}}_i{g}_{ab}$, $p_{ab}={\bar{\mathrm{\nabla }}}_0{g}_{ab}$, reconstruct $F$ from $(e,b)$, and use ${\pi }_i={\mathcal{D}}_i{\phi }_0$, ${\pi }_0={\pi }_n$, and $A_0=0$. Thus the complete data and all initial auxiliary constraints are specified; $A(n)=0$ fixes the initial temporal gauge. The equations and stress are covariant under local diffeomorphisms because $d$, the metric connection, contractions, and the oriented volume form commute with pullback. Steps 1.3 and 4.1 give the exact smooth symmetric-hyperbolic reduction, propagation of every auxiliary constraint, recovery of the matter equations, and ${\mathrm{\nabla }}^a{T}_{ab}=0$ before Einstein recovery. The initial constraints are precisely the normal projections in [L9]. These are the explicit reduction, initialization, conservation, and constraint hypotheses of [L10], so the local construction in that theorem gives a development near each initial point; the internal-gauge comparisons and collar patching are verified in the final comparison paragraphs below.

(Fix both coordinate and internal gauges before comparison.) Given two smooth developments of the same complete data, use the wave-coordinate construction in Local geometric uniqueness for a fixed covariant Einstein-matter model with complete data on each relatively compact initial ball: solve ${\mathrm{\square }}_g{x}^{\mu }=0$ with $x^0=0$, $x^i=y^i$, $n(x^0)=1$, and $n(x^i)=0$ on the initial surface. Its proof constructs these scalar waves by the same positive symmetric wave reduction and local solver used here, and their initial differential is invertible. The resulting charts are harmonic relative to the same flat comparison metric and have the identical initialized metric jet in step 5.2. A pulled-back potential need not satisfy $A_0^{\mathrm{\prime }}=0$. On each product collar set $\chi (s,x)=-{\int }_0^s{A}_0^{\mathrm{\prime }}(\sigma ,x)d\sigma$ and transform $A^{\mathrm{\prime }}\mapsto A^{\mathrm{\prime }}+d\chi$, ${\phi }^{\mathrm{\prime }}\mapsto e^{i\lambda \chi }{\phi }^{\mathrm{\prime }}$. Then $A_0^{\mathrm{\prime }}+{\mathrm{\partial }}_s\chi =0$; direct differentiation gives $D_{\text{\hspace{0.17em}⁣}{A}^{\mathrm{\prime }}+d\chi }(e^{i\lambda \chi }{\phi }^{\mathrm{\prime }})=e^{i\lambda \chi }{D}_{\text{\hspace{0.17em}⁣}{A}^{\mathrm{\prime }}}{\phi }^{\mathrm{\prime }}$, while $F$, current and stress are unchanged. Since $\chi {\mathrm{\mid }}_{\mathrm{\Sigma }}=0$, tangential initial $A$, ${\phi }_0$ and the covariant normal derivative are unchanged. Only after this temporal-gauge restoration do both developments extract to the identical coefficient system of step 1.3 with identical complete initial values.

(Local and whole-surface comparison.) On a smaller compact cylinder the two smooth states lie in a common convex admissible neighborhood. Subtract their equations; the coefficient mean-value formula puts their difference $v$ in $A^{\mu }(u_1){\mathrm{\partial }}_{\mu }v+Cv=0$, with bounded $C$. If $A^0\ge aI$ and $b\ge {\sup }_{\mathrm{\mid }\xi \mathrm{\mid }=1}\mathrm{\parallel }{\xi }_i{A}^i\mathrm{\parallel }$, choose a shrinking ball with speed $\sigma >b\mathrm{/}a$. Symmetry and spatial integration by parts give $E^{\mathrm{\prime }}\le CE$ for $E=\int v^{\mathsf{T}}{A}^{0}v$, since the side-flux matrix $\sigma A^0+{\nu }_i{A}^i$ is nonnegative. The zero initial difference gives $v=0$ in both time directions by Gronwall. This is the local coupled energy comparison in [L10], now applied after the internal gauge has actually been fixed. Undoing the gauges gives a local metric isometry and the stated matter-gauge equivalence. Future data-fixing metric isometries have unique germs: their differentials fix tangents and future normals, hence they agree along the normal geodesics from the initial patch. The injective variable-width normal collars and germ-patching construction of Local geometric uniqueness for a fixed covariant Einstein-matter model with complete data therefore yield one metric isometry near the entire initial surface; no uniform width is imposed. For the existence/gluing argument of [L10], choose the common normal product collar and put each local potential into its temporal gauge there, with unchanged initial $A_{\mathrm{\Sigma }}$. On overlaps the local isometry comparisons just proved identify the metrics and $F$; two temporal-gauge potentials with the same initial spatial restriction and same $F$ agree, since their difference has zero time component, zero exterior derivative and zero initial pullback. On the common smooth normal-coordinate metric, their temporal-gauge scalar/potential blocks have the same coefficients and complete data, so the scalar-block version of the local energy comparison gives equality of the scalar representatives; no harmonic-coordinate equation for that common metric is needed for this matter-only comparison. Thus the local fields patch on the literal collar, as required for the smooth-development application in step 5.2.

(One global real gauge near the initial surface.) Pull the second development back by the metric isometry from step 7.1, and write $B=A_2-A_1$ on a common product collar. The local comparisons give $dB=F_2-F_1=0$, and the common initial data give $B{\mathrm{\mid }}_{T\mathrm{\Sigma }}=0$. Define the global smooth real function $\chi (s,x)=-{\int }_0^s{B}_0(\sigma ,x)d\sigma$. Then ${\mathrm{\partial }}_s(B_i+{\mathrm{\partial }}_i\chi )={\mathrm{\partial }}_s{B}_i-{\mathrm{\partial }}_i{B}_0=0$ and $B_i+{\mathrm{\partial }}_i\chi =0$ initially; also $B_0+{\mathrm{\partial }}_s\chi =0$. Hence $A_2+d\chi =A_1$ on the collar. The local scalar gauge comparisons have this same differential and zero initial value, so their functions agree with $\chi$ and $e^{i\lambda \chi }{\phi }_2={\phi }_1$. This gives the asserted global-potential gauge equivalence, with $\chi {\mathrm{\mid }}_{\mathrm{\Sigma }}=0$, without requiring a nontrivial bundle construction or assuming that a closed one-form on the entire spacetime is exact.