Canonical stationary end expansions give the precise source slice decay

The matrix $g_{ij}$ tends to the identity, hence is positive definite for large $r$. It is the induced metric $h$. Its inverse is bounded with the corresponding differentiated decay. In the decomposition $T=Nn+\beta$, the shift covector is ${\beta }_i=g_{0i}$, the vector is ${\beta }^i=h^{ij}{\beta }_j$, and $N^2=-g_{00}+h^{ij}{g}_{0i}{g}_{0j}$. Thus $N\to 1$, $\beta =o_3(r^{-1})$ and $h=(1+2m\mathrm{/}r)\delta +o_3(r^{-1})$.

Let $D$ be the unique Levi–Civita connection of the induced Riemannian metric $h$, so $Dh=0$ and $D$ is torsion-free. Its symbols are ${\mathrm{\Gamma }}^a{}_{ij}=\frac{1}{2}{h}^{ab}({\mathrm{\partial }}_i{h}_{jb}+{\mathrm{\partial }}_j{h}_{ib}-{\mathrm{\partial }}_b{h}_{ij})$, and its dual action on the shift covector is $D_i{\beta }_j={\mathrm{\partial }}_i{\beta }_j-{\mathrm{\Gamma }}^a{}_{ij}{\beta }_a$. Stationarity gives ${\mathrm{\partial }}_{t}h=0$. Differentiating the induced metric along $T=Nn+\beta$ yields $2Nk=\pm {\mathcal{L}}_{\beta }h$, with the sign determined by the second-form convention and irrelevant for the estimate. Explicitly its components are, up to that sign, $N^{-1}(D_i{\beta }_j+D_j{\beta }_i)\mathrm{/}2$. The first derivatives of $\beta$ and their derivatives through order two are $o(r^{-2})$ with the corresponding extra inverse powers. The $h$-Christoffel symbols are $O_2(r^{-2})$ by their coordinate formula, so their products with $\beta$ decay faster. Hence $k=o_2(r^{-2})$, which implies the weaker required $o_2(r^{-1})$, and the claim for $h$ follows from step 1.1.

The estimates for $h$ and $k$ in steps 1.1 and 2.1 have precisely the two differentiated decay forms in [F2], with the source coefficient $M$ replaced by $m$. The derivative convention in the Statement is the same as that of H1. This establishes the asserted comparison of decay requirements alone; neither an exterior-slice topology nor an ADM-mass identification follows from this calculation.