Coupled reduced Einstein–Maxwell local evolution

Given: The fixed source-free Einstein–Maxwell equations, the positive symmetric Maxwell reduction, a smooth background connection, the admissible compact state sets, and the draft local solver interface.

Draft definition Electrovacuum tensor Cauchy system fixes the curvature/sign convention, $\kappa >0$, ${\mu }_0>0$, $\mathrm{\Lambda }\in \mathbb{R}$, the trace-free Maxwell stress $T_{ab}={\mu }_0^{-1}{Q}_{ab}(g,F)$, and the equations $G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}$, $dF=d{\star }_{g}F=0$. The item file is listed in batch 6; its status is draft and its verification precheck is pending; this use is conditional on the actual draft statement.

Draft periodic-state definition Periodic high-Sobolev state spaces uses ${\mathbb{T}}_{2\pi }^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ and the real componentwise Fourier norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}^2={\sum }_{k\in {\mathbb{Z}}^3}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2$, under ${\mathrm{A}\mathrm{C}}_{\omega }$. Its item file is listed in batch 6; its status is draft and its verification precheck is pending; this use remains conditional on that actual draft.

Draft result Exact periodic positive symmetric-hyperbolic supplier: for $s\ge 5$, smooth periodic symmetric matrices $A^{\mu }$ and smooth $B$ on an open finite-dimensional state set, a time matrix uniformly positive on compact admissible neighborhoods, and bounded $H^s$ initial data valued in $K⋐\mathrm{int}K\mathrm{\prime }$, there is a common $T>0$ and a solution on $(-T,T)$ with $L^{\mathrm{\infty }}{H}^s\cap C{H}^{s-1}$, ${\mathrm{\partial }}_{s}U\in L^{\mathrm{\infty }}{H}^{s-1}$ and values in $K\mathrm{\prime }$. Its proof applies the current GR existence, uniqueness/stability and smooth-persistence suppliers. The draft file is listed in batch-6 inventory and its precheck is pending; this conclusion remains conditional.

Positive symmetric Maxwell field-strength reduction gives the six-dimensional Maxwell principal block $S(\xi )$, which is symmetric, has norm $\mathrm{\mid }\xi \mathrm{\mid }$, and has positive time matrix ${\alpha }^{-1}{I}_6$.

Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction gives the reduced metric wave block with time matrix $\mathrm{diag}(I,-g^{00}I,g^{ij}I)$ and states that a smooth algebraic stress map of additional first-order matter variables introduces no metric principal-derivative coupling.

Draft initial-data interface Harmonic initial data relative to a fixed smooth background uses a fixed smooth positive reference $\gamma$ and sets $g_{ij}=h_{ij}$, $g_{00}=-1$, $g_{0i}=0$, ${\mathrm{\partial }}_0{g}_{ij}=2K_{ij}$, ${\mathrm{\partial }}_0{g}_{00}=-2{\mathrm{tr}}_{h}K$, and ${\mathrm{\partial }}_0{g}_{0i}=h_{ij}{h}^{k\mathrm{\ell }}(\mathrm{\Gamma }(h{)}^j{}_{k\mathrm{\ell }}-\mathrm{\Gamma }(\gamma {)}^j{}_{k\mathrm{\ell }})$. Its actual proof gives the required $H=0$ and $H^s$ regularity. The item file is listed in batch 6 and its precheck is pending; this use remains conditional.