Stationarity, timelikeness and uniform inverse-metric bounds for stationary weak-field data

(Smallness and timelikeness.) Over the spatial set $K$, [L1] gives $m(E)\le 2C\epsilon +C{\epsilon }^2+C{\epsilon }^{3\mathrm{/}2}\le 4C\epsilon$ for $\epsilon \le 1$, hence $m(H)\le 16C\epsilon \le 1\mathrm{/}8$ for $\epsilon \le (128C{)}^{-1}$; and $g(\epsilon {)}_{00}=-(1+2\epsilon \phi )+r_{00}\le -1+2C\epsilon +C{\epsilon }^2\le -1\mathrm{/}2$ for $\epsilon$ small as chosen. Thus ${\mathrm{\partial }}_0$ is timelike at every spacetime point over $K$.

(Stationarity.) For each $p\in \mathrm{\Omega }$, the local flow of ${\mathrm{\partial }}_0$ is the translation of [L3] for sufficiently small $s$ on a neighborhood of $p$. Its differential is the identity, and the metric components have equal values at source and target by [L1]. The local pullback therefore fixes $g(\epsilon )$, so differentiating at $s=0$ in the definition of [L3] gives ${\mathcal{L}}_{{\mathrm{\partial }}_0}g(\epsilon )=0$ throughout $\mathrm{\Omega }$. For each $p$, choose a timelike $v\in T_p\mathrm{\Omega }$ and extend its coordinate components constantly on a chart neighborhood; smoothness and the strict inequality $g_p(v,v)<0$ make the extension timelike after shrinking the neighborhood. This gives a local time orientation on which Killing fields of a Lorentzian metric applies. Its defining equation is independent of that local choice and holds on every such neighborhood, hence globally in the sense specified in (a). Thus ${\mathrm{\partial }}_0$ is Killing; neither a global time orientation nor a global translation flow on $\mathrm{\Omega }$ is required.

(Convergent matrix inverse.) Write $g(\epsilon )=\eta (I+H(\epsilon ))$. By step 1.1, $m(H)\le 1\mathrm{/}8$. From [L2], induction gives $m(H^k)\le 4^{k-1}m(H{)}^k\le (4m(H){)}^k$ for $k\ge 1$. Since $4m(H)\le 1\mathrm{/}2$, the scalar geometric series For $\mathrm{\mid }r\mathrm{\mid }<1$, ${\sum }_{k\ge 0}{r}^k=1\mathrm{/}(1-r)$, and for $\mathrm{\mid }r\mathrm{\mid }\ge 1$ the series diverges shows that every entry of $S_N:={\sum }_{k=0}^N(-H{)}^k$ converges as $N\to \mathrm{\infty }$; let $S$ be the limiting matrix. The finite geometric identity $(I+H)S_N=S_N(I+H)=I+(-1{)}^N{H}^{N+1}$ and the same bound $m(H^{N+1})\to 0$ imply $(I+H)S=S(I+H)=I$, so $S=(I+H{)}^{-1}$. Moreover, $m(S)\le 1+{\sum }_{k\ge 1}(4m(H){)}^k\le 2$.

(Exact inverse expansion.) The finite geometric identity also gives $S=I-H+H^{2}S$. Since $g^{-1}=S{\eta }^{-1}$, substituting $H={\eta }^{-1}E$ yields $g^{-1}={\eta }^{-1}-{\eta }^{-1}E{\eta }^{-1}+{\eta }^{-1}E{\eta }^{-1}ES{\eta }^{-1}\mathrm{.}$ Thus the first two terms are the zeroth- and first-order parts and the final term is exactly the remainder $R$ in (b), with the stated component convention.

(Derivative bounds.) Since $H={\eta }^{-1}E$ and $E$ is smooth with $m({\mathrm{\partial }}^{\alpha }E)\le 4C\epsilon \mathrm{/}{L}^{\mathrm{\mid }\alpha \mathrm{\mid }}$ for $1\le \mathrm{\mid }\alpha \mathrm{\mid }\le 2$ on the spatial compact (using ${\epsilon }^{3\mathrm{/}2}\le \epsilon$), [L2] gives $m({\mathrm{\partial }}^{\alpha }H)\le 4C\epsilon \mathrm{/}{L}^{\mathrm{\mid }\alpha \mathrm{\mid }}$. Differentiating $(I+H)S=I$ for $S=(I+H{)}^{-1}$ gives ${\mathrm{\partial }}_{c}S=-S({\mathrm{\partial }}_{c}H)S$; differentiating again gives ${\mathrm{\partial }}_d{\mathrm{\partial }}_{c}S=-S_d{H}_{c}S-S{H}_{cd}S-S{H}_c{S}_d$. Step 2.1 bounds $m(S)\le 2$, and repeated use of [L2] in these identities yields $m({\mathrm{\partial }}^{\alpha }S)\le C^{\mathrm{\prime }}\epsilon \mathrm{/}{L}^{\mathrm{\mid }\alpha \mathrm{\mid }}$ for $1\le \mathrm{\mid }\alpha \mathrm{\mid }\le 2$ (terms quadratic in $\epsilon$ are bounded by the same order since $\epsilon \le 1$). Differentiating the exact identity $g^{-1}={\eta }^{-1}-{\eta }^{-1}E{\eta }^{-1}+{\eta }^{-1}E{\eta }^{-1}ES{\eta }^{-1}$ from step 3.1, every derivative of the last term still contains two factors drawn from $E,\mathrm{\partial }E,{\mathrm{\partial }}^{2}E$, so $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{R}^{ab}\mathrm{\mid }\le C_1{\epsilon }^2\mathrm{/}{L}^{\mathrm{\mid }\alpha \mathrm{\mid }}$ for $0\le \mathrm{\mid }\alpha \mathrm{\mid }\le 2$. The first two terms give $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{g}^{ab}\mathrm{\mid }\le C_1\mathrm{/}{L}^{\mathrm{\mid }\alpha \mathrm{\mid }}$ for $1\le \mathrm{\mid }\alpha \mathrm{\mid }\le 2$, with $C_1=C_1(C,L)$ enlarged as needed. This proves (b).

(Conclusion.) Step 1.1 gives the three bounds and timelikeness of (a), step 1.2 gives the Killing property of (a), and steps 3.1 and 4.1 give the expansion and all bounds of (b); the constants depend only on $C$ and $L$ and on the fixed dimension $4$.