AIK perturbative effective constant and its coordinate data

An AIK perturbative quantitative chart package consists of the preceding GR/SBS data with the choices and bounds of AIK §2.1. Write ${\mathrm{\Sigma }}_r={\mathrm{\Phi }}_0(E_r)$ and let $n$ be the future unit normal to ${\mathrm{\Sigma }}^0$. Choose $A_1,R_1\ge R_0$ with $R_1\ge A_1^4$, such that on ${\mathrm{\Sigma }}_{3\mathrm{/}4}$, $A_1^{-1}\mathrm{\mid }v{\mathrm{\mid }}^2\le g_{ij}{v}^i{v}^j\le A_1\mathrm{\mid }v{\mathrm{\mid }}^2,{\sum }_{i=1}^3\mathrm{\mid }g({\mathrm{\partial }}_i,T)\mathrm{\mid }+\mathrm{\mid }g(n,T)\mathrm{\mid }\le A_1\mathrm{.}$ On the end $r\ge R_1$, require the sum of weighted derivatives through order six of the three remainders to be bounded by $A_1$: their weights for $g_{ij}-{\delta }_{ij}$, $g_{00}+1-2M_0\mathrm{/}r$ and $g_{0i}+2{ϵ}_{ijk}{S}^j{x}^k{r}^{-3}$ are respectively $r^{m+1}$, $r^{m+2}$ and $r^{m+3}$ at derivative order $m$.

Choose a smooth cutoff $\eta :\mathbb{R}\to [0,1]$ with $\eta (u)=1$ for $u\le 1$ and $\mathrm{supp}\eta \subset (-\mathrm{\infty },2]$. Supply the smooth ambient field $T^{\mathrm{\prime }}$ of the AIK construction on a neighborhood of ${\mathrm{\Sigma }}_{3\mathrm{/}4}$ and on the distant stationary end, with $T^{\mathrm{\prime }}{\mathrm{\mid }}_{{\mathrm{\Sigma }}_{3\mathrm{/}4}}=\eta (r\mathrm{/}{R}_1)n+(1-\eta (r\mathrm{/}{R}_1))T,T^{\mathrm{\prime }}=T\text{ on }{\mathrm{\Phi }}_0(\mathbb{R}\times E_{2R_1})\mathrm{.}$

Here $r$ in the slice formula is the radius in ${\mathrm{\Phi }}_0$; the choice of smooth extension off the slice is part of the package. Let $F_s$ be its local flow: $F_0(p)=p$ and $\frac{d}{ds}{F}_s(p)=T^{\mathrm{\prime }}(F_s(p))$. Choose $0<{\epsilon }_0<1\mathrm{/}4$ sufficiently small that ${\mathrm{\Phi }}_1(s,x)=F_s({\mathrm{\Phi }}_0(x)):(-{\epsilon }_0,{\epsilon }_0)\times E_{1-{\epsilon }_0}⟶\tilde{M}$ is a diffeomorphism onto an open neighborhood of ${\mathrm{\Sigma }}^0\cap \bar{E}$, with ${\mathrm{\Phi }}_1(0,x)={\mathrm{\Phi }}_0(x)$, agreement with the stationary chart for $r>2R_1$, and ${\mathrm{\partial }}_0=T^{\mathrm{\prime }}$ in these coordinates. The restriction on ${\epsilon }_0$ puts its initial points in ${\mathrm{\Sigma }}_{3\mathrm{/}4}$. Writing $O$ for the supplied SBS neighborhood of $S_0$, require ${\mathrm{\Phi }}_1((-{\epsilon }_0,{\epsilon }_0)\times \{x:1-{\epsilon }_0<\mathrm{\mid }x\mathrm{\mid }<1+{\epsilon }_0\})\subset O\mathrm{.}$

Require ${\sum }_i\mathrm{\mid }{g}_{0i}\mathrm{\mid }+\mathrm{\mid }{g}_{00}+1\mathrm{\mid }\le A_1\mathrm{/}(R_1+r)$ throughout $\tilde{M}$. The availability of this source-specific field and flow chart is the imported construction recorded in External input. With $\mathrm{\mid }{\mathrm{\partial }}^m\mathrm{\mid }$ denoting the sum over all spacetime coordinate derivatives of total order $m$ and all displayed tensor indices, set $A_2={\sup }_{\tilde{M}}{\sum }_{m=0}^6({\sum }_{a,b=0}^3\mathrm{\mid }{\mathrm{\partial }}^m{g}_{ab}\mathrm{\mid }+{\sum }_{a=0}^3\mathrm{\mid }{\mathrm{\partial }}^m{T}^a\mathrm{\mid }),{\bar{A}}_{\mathrm{P}\mathrm{K}}=\max \{R_1,A_2,{\epsilon }_0^{-1},(M_0^2-J{)}^{-1}\}\mathrm{.}$ Require $A_2<\mathrm{\infty }$. These are the choices and bounds of AIK (2.1)–(2.5), not an unspecified coordinate-invariant norm of the metric.

AIK perturbative global regularity and bifurcation package

Differentiated asymptotic bounds and parameter uniformity

Spacelike sections, maximality and lapse–shift data

AIK, section 2.1, (2.1)–(2.5) and the flow construction between (2.2) and (2.3)

The field extension and its flow chart are supplied by the cited AIK construction, rather than inferred from the three general definitional suppliers alone. All coefficient arrays and coordinate derivatives are defined on the supplied charts. The prescribed spacelike metric makes the two-sided quadratic bound meaningful, and the normal exists on the supplied slice. Supremum over a specified set is an extended nonnegative number; the explicit finiteness condition gives a finite $A_2$. Since $M_0^2-J>0$ in the source GR package and ${\epsilon }_0>0$, the displayed maximum is a finite positive number. This definition records the actual chart package on which the source threshold depends; it proves neither existence of uniformly bounded packages in a family nor independence from their choices.

AIK, section 2.1, between (2.2) and (2.3) supplies the smooth interpolating field and its flow chart under the preceding source GR/SBS data and bounds (2.1)–(2.2). The cutoff, slice value, distant-end agreement, flow equation, chart domain and SBS collar restriction are transcribed in Definition. Equations (2.3)–(2.5) supply the subsequent chart bound and effective-constant conventions. The full imported statement, hypotheses and reading limits are in the source reference. This is an imported coordinate construction; the finite-constant justification above does not claim a local proof of its existence.