Fixed-gauge finite-slab data and solution topology

Fix an integer $s\ge 5$, a finite number $T>0$, the closed interval $J=[0,T]$, and the flat spatial torus ${\mathbb{T}}_{2\pi }^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$. Use the spaces $H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ and one fixed periodic symmetric-hyperbolic system as defined in Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions: its state domain $O\subseteq {\mathbb{R}}^m$ is open, its coefficient maps $A^{\mu }(t,x,z),B(t,x,z)$ are smooth on the stated time, torus and state domain, each $A^{\mu }$ is symmetric, and $A^0$ is positive definite. The fixed-gauge Einstein state is the finite vector $z=(g_{ab},p_{ab},q_{abi},w)$, where $p_{ab}={\mathrm{\partial }}_0{g}_{ab}$, $q_{abi}={\mathrm{\partial }}_i{g}_{ab}$, and $w$ is the complete matter and auxiliary state of one fixed reduction.

For the SI interpretation, the system's time parameter $t$ is the length coordinate $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ in metres, while $x^i$ are dimensionless angular coordinates on the torus. The metric components then have coordinate-dependent units: $g_{00}$ is dimensionless, $g_{0i}$ has units of metres, and $g_{ij}$ has units of square metres. Accordingly $p_{ab}={\mathrm{\partial }}_0{g}_{ab}$ has units $[g_{ab}]{\mathrm{m}}^{-1}$ and $q_{abi}={\mathrm{\partial }}_i{g}_{ab}$ has units $[g_{ab}]$. The model-specific components of $w$ have the fixed units declared by that reduction. Divide each component by its fixed declared unit scale before forming the Euclidean-valued state and its Sobolev norm; the resulting numerical state is dimensionless. The coefficient and source fields are those of this same fixed normalized reduction. Thus $R_z$ is a bound on that normalized state, and $R_t$ is a separate bound on its derivative with respect to the length coordinate $x^0$ (units ${\mathrm{m}}^{-1}$). Neither is a unit-independent physical observable. If physical seconds are used instead, then ${\mathrm{\partial }}_{x^0}=c^{-1}{\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$; consequently the term $A^0{\mathrm{\partial }}_{x^0}$ becomes $c^{-1}{A}^0{\mathrm{\partial }}_{t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}$ unless the entire equation is multiplied by $c$.

A fixed-gauge finite-slab comparison context consists of one such coefficient family on an open time interval containing $J$, one coordinate gauge and reconstruction rule, compact state sets $K_0\subset \mathrm{int}{K}_1\subset \mathrm{int}{K}_2\subset O$, separate finite nonnegative bounds $R_z,R_t$, and constants $0<a\le b<\mathrm{\infty }$ such that

$a\mathrm{\mid }\zeta {\mathrm{\mid }}^2\le {\zeta }^{\mathsf{T}}{A}^0(t,x,z)\zeta \le b\mathrm{\mid }\zeta {\mathrm{\mid }}^2$

for every $t\in J$, $x\in {\mathbb{T}}_{2\pi }^3$, $z\in K_2$, and $\zeta \in {\mathbb{R}}^m$. The reference state takes values in $K_0$. Compared classical states take values in $K_1$, belong to $L^{\mathrm{\infty }}(J;H^s)\cap C^0(J;H^{s-1})$, have ${\mathrm{\partial }}_{t}z\in L^{\mathrm{\infty }}(J;H^{s-1})$, and satisfy ${\mathrm{ess\hspace{0.17em}sup}}_{t\in J}\mathrm{\parallel }z(t){\mathrm{\parallel }}_{H^s}\le R_z,{\mathrm{ess\hspace{0.17em}sup}}_{t\in J}\mathrm{\parallel }{\mathrm{\partial }}_{t}z(t){\mathrm{\parallel }}_{H^{s-1}}\le R_t\mathrm{.}$ The outer set $K_2$ supplies the positive coefficient margin for local existence restarted from data in compact subsets of $K_1$. Two states use the same chart grid, coefficient maps, gauge source functions, reconstruction rule, and fixed matter branch. Forcing terms, when included in a comparison, are continuous $H^{s-1}$-valued functions of the slab parameter; their size is controlled through the separate stated solution and time-derivative bounds when the comparison estimate is applied.

The Sobolev norms compare coordinate fields in this fixed gauge. They do not identify two arbitrary coordinate representatives by a diffeomorphism. The Einstein interpretation uses the same first-order reduction, full initial constraints, complete matter data, and local restart hypotheses throughout the finite slab.

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