The weighted AIK perturbation hypothesis PK

For the AIK GR/SBS and quantitative chart packages, use the smooth weighted tensor $\mathcal{W}$ on a neighborhood of ${\mathrm{\Sigma }}^0\cap \bar{E}$. Let $h=g{\mathrm{\mid }}_{T{\mathrm{\Sigma }}^0}$ and let $n$ be the future unit normal. Fix, as part of the PK data, one smooth adapted orthonormal frame $e=(e_0,e_1,e_2,e_3)$ along ${\mathrm{\Sigma }}^0$: $e_0=n$, $e_i\in T{\mathrm{\Sigma }}^0$ and $h(e_i,e_j)={\delta }_{ij}$ for $i,j\in \{1,2,3\}$. Such a frame exists; one available choice is the ordered Gram–Schmidt orthonormalization of the three tangent coordinate fields supplied by the global slice parametrization ${\mathrm{\Phi }}_0$. The frame is held fixed when testing the inequality. Local tests use restrictions of that selected frame, agreeing on overlaps; a different selection may require a tolerance adjustment.

The PK smallness condition at tolerance $\bar{\epsilon }>0$, relative to this selected frame, requires at every point of the displayed slice portion and for every displayed index triple: $\mathrm{\mid }\mathcal{W}(T,e_a,e_b,e_c)\mathrm{\mid }\le \bar{\epsilon }\text{on }{\mathrm{\Sigma }}^0\cap \bar{E},a,b,c\in \{0,1,2,3\}\mathrm{.}$ The bars denote modulus of a complex scalar, not a Lorentzian tensor norm. In the source notation this is $\mathrm{\mid }(1-\sigma )\mathcal{S}(T,e_a,e_b,e_c)\mathrm{\mid }\le \bar{\epsilon }$, using the smooth weighted extension at $\sigma =1$. For the rigidity theorem the tolerance must be sufficiently small depending only on the fixed ${\bar{A}}_{\mathrm{P}\mathrm{K}}$ from the preceding package. This is not generic smallness of $g-g_{\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{r}}$, not a bound only on $S_0$, and not merely small ADM angular momentum. This definition asserts no rigidity from the inequality without the separately imported source theorem.

AIK perturbative global regularity and bifurcation package

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AIK, PK (1.4), effective constant (2.5), and weighted tensor (2.18)–(2.19)

Given: The slice, parametrization, metric and normal of [F1], tensor and effective constant of [F2], frame terminology and finite Gram–Schmidt result of [F3], and source PK formulation of [F4].

1.1 Let $v_i=({\mathrm{\Phi }}_0{)}_{\ast }(\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^i)$, $i=1,2,3$, be the globally defined smooth independent tangent fields from the supplied slice parametrization. Apply [F3] in each positive-definite tangent space with $u_i=v_i-{\sum }_{1\le j<i}h(v_i,e_j)e_j$ and $e_i=u_i\mathrm{/}\sqrt{h(u_i,u_i)}$, starting with the empty sum. The cited recursion gives $u_i\mathrm{\ne }0$ and an $h$-orthonormal basis. The coefficients are smooth by finite sums, products and division by a positive smooth square root, inductively for the three fields. The same formula on the whole parametrized slice agrees on every overlap. Appending $e_0=n$ gives the required smooth adapted frame along the slice. No ambient extension, global tetrad on $M$, or additional choice of a frame at each point is needed. [F1, F3, construct]

2.1 Evaluation of the smooth tensor of [F2] on $T,e_a,e_b,e_c$ gives a smooth complex scalar along the slice, with ordinary nonnegative modulus. The polynomial value of $\mathcal{W}$ keeps it defined at $\sigma =1$. To compare any other selected adapted frame $\tilde{e}$ with $e$, orthonormality and the common $e_0=n$ give ${\tilde{e}}_0=e_0$ and ${\tilde{e}}_i={\sum }_{j=1}^3{R}_i{}^j{e}_j$ for a pointwise orthogonal matrix $R$. Every entry has modulus at most one. Expansion in the three varying tensor slots has at most $3^3=27$ terms, so a component bound $\bar{\epsilon }$ in one frame gives a component bound $27\bar{\epsilon }$ in the other, uniformly on the entire slice portion; the inverse comparison has the same bound. Thus the fixed-tolerance predicate is unambiguous, and sufficiently small tolerances transfer after a universal factor, preserving dependence only on the previously fixed ${\bar{A}}_{\mathrm{P}\mathrm{K}}$. This establishes well-definedness and the stated frame convention, without deducing rigidity from the definition. [F1, F2, F3, F4, step 1.1, algebra] ∎

AIK, section 1.1 PK (1.4), section 2.1 (2.5), and section 2.3 (2.18)–(2.19) supplies the source-specific weighted component inequality and its effective-constant dependence. The reference records its statement, hypotheses and reading limits. The frame construction and comparison in Justification are local arguments; the source rigidity theorem is not proved or imported by this definition.