Vacuum development realizing the exterior and local boost germs

Apply F5 and F4 to meet the full GR/SBS hypotheses, and use F1 to obtain $C_0$. Its induced data are smooth through the compact boundary and $g$ is Ricci-flat on an actual neighborhood. F6 gives $R_h+({\mathrm{tr}}_h\mathcal{K}{)}^2-\mathrm{\mid }\mathcal{K}{\mathrm{\mid }}_h^2=0$ and ${\mathrm{div}}_h\mathcal{K}-d{\mathrm{tr}}_h\mathcal{K}=0$ on the positive section, including the boundary by continuity. This checks the constraint hypothesis before applying the data-double supplier. The normal and sign conventions are precisely those in F4 and F6.

On the unchanged stationary end, F7 gives $h-\delta =O_6(r^{-1})$ and $\mathcal{K}=O_5(r^{-3})$. The compact radial truncations of F1 are carried to compact truncations of ${\bar{C}}_0$ by the compactly supported isotopy, which is the identity sufficiently far out; they exhaust it and have boundaries $S$ and the large coordinate spheres. Completeness with boundary included is supplied by F1. The ambient $T$ is Killing on the whole positive section and commutes with the local boost. Consequently every hypothesis of all the reflection and complete-double constructions in F2 is met. Use their signed collar and the retained Choice convention to obtain smooth complete boundaryless two-ended vacuum data and the global stationary KID.

Apply F3 to this full doubled data set to obtain its vacuum MGHD $(\widehat{M},\widehat{g})$ and the complete Killing field $\widehat{T}$. The original $E$ is a smooth globally hyperbolic vacuum development with Cauchy surface $C_0$, which is an open submanifold of $\widehat{C}$. The open-data embedding in F3 therefore gives a time-orientation-preserving isometric embedding $i:E\to \widehat{M}$ fixing those data.

We record the uniqueness needed below. For a Killing field vanishing along an open spacelike hypersurface, its Gaussian components $(N,Y)$ and their spatial derivatives vanish there. Equations (2.7)–(2.8) in F8 then give zero normal derivatives as well. At any point its local flow therefore fixes the point with identity differential; rigidity in F9 makes that flow the identity on the connected common domain. Differentiating in the flow parameter makes the field zero there. The same rigidity continues equality across overlapping connected coordinate neighborhoods. In particular equal KIDs give equal Killing fields: their difference vanishes along the initial surface. Apply this to $i_{\ast }T$ and $\widehat{T}$ on the connected embedded exterior, which have the same KID. Uniqueness of integral curves and completeness of both stationary fields give $i\circ {\mathrm{\Phi }}_t^T={\mathrm{\Phi }}_t^{\widehat{T}}\circ i$ for every finite $t$.

Restrict the doubled data to the open signed collar $A=(-ϵ,ϵ)\times S$. On its positive side take the KIDs induced by the actual local Killing fields $K$ and, in rotation, $Z$. They commute with the boost; their all-jet parities are therefore exactly those used in F2, with each field substituted for $T$. The componentwise smooth-pasting and sign calculation of the stationary-KID supplier is local, so it gives smooth vacuum KIDs on $A$. Their linear relation with the restriction of the stationary KID is preserved by reflection and continuity. The imported F8 applies to these smooth vacuum KIDs on $A$, with no completeness or nonzero-lapse requirement. It develops them in the same local vacuum development. Proposition 2.4 identifies that development with $D_{\widehat{M}}(A)$ near $S$, since $A$ is an open subset of the doubled Cauchy surface and hence achronal and spacelike. They are not being treated as global KIDs of $\widehat{C}$.

The bracket formula (2.15) in F8 vanishes for the positive-side commuting fields. Under the negative-copy change $(N,Y)\mapsto (-N,R^{\ast }Y)$ its lapse component changes sign under reflection and its shift component is reflected: each shift term $ND\bar{N}-\bar{N}DN$ contains two lapse signs. Thus their bracket KID is also zero on the negative side, and by smoothness on $S$. By F8 it is the KID of the developed commutator. Step 4.1 makes that commutator zero near $S$. The same uniqueness applied to the reflected linear relation gives $\widehat{T}=K-\mathrm{\Omega }Z$ there, with $Z$ omitted when $\mathrm{\Omega }=0$.

Along $S$ the developed boost field vanishes. Its first derivative agrees with the original boost derivative: tangential derivatives to $A$ follow from its smooth KID, and its remaining Gaussian normal derivatives follow from (2.7)–(2.8) in F8. In the future/outward frame $(e_0,e_1)$ it is therefore the constant boost matrix with coefficient $\kappa >0$, annihilating $TS$. F9 gives its differential flow as that boost. Use the normal tube of F9 with coordinates ${\exp }_p(((V-U)e_0+(V+U)e_1)\mathrm{/}2)$. Isometries commute with this exponential map, so differentiating the boost action gives exactly $K=\kappa (V{\mathrm{\partial }}_V-U{\mathrm{\partial }}_U)$ on a smaller tube.

In rotation the reflected rotation KID has zero lapse and shift tangent to the signed $r$ levels, because the original $Z$ preserves $U,V$. Its restriction to $S$ is the given $2\pi$-periodic action. Its local isometric flow preserves $A$, its chosen side and its future unit normal; hence along $S$ it sends $e_0,e_1$ to the corresponding future/outward frame at the rotated point. Exponential equivariance from F9 consequently identifies the local $Z$ flow with $(U,V,p)\mapsto (U,V,{\varphi }_s^{Z}p)$. This formula itself defines the flow for all $s$ on a uniform smaller normal tube, since $S$ is compact and the normal coordinates are unchanged. Its maps are isometries: for sufficiently short parameters they are local Killing isometries, and any finite parameter is a finite sum of such short parameters. The period $2\pi$ follows from the action on $S$. This proves the asserted local circle action and exact coordinate relations, without continuing $Z$ to the entire exterior.

Finally check the boundary germ, rather than infer it from the open-data embedding alone. In both the original metric and the developed metric, smooth Gaussian normal maps based on the closed positive section are defined near $S$: their differentials at zero map the section tangent space and its future unit normal isomorphically onto the ambient tangent space, so the local inverse assertion in F9 applies. The two original null sheets are transverse to these timelike normal curves; on a smaller chart they are graphs in the normal-time coordinate and enclose its right exterior interval containing time zero for every positive section point. Thus the segment from time zero to a point of that interval stays in $E$. On the open positive section $i$ fixes the data and carries the future normal to the future normal. Exponential equivariance therefore expresses $i$ in these charts as the identity wherever the normal geodesic segment stays in $E$. This gives its smooth extension and full differential at $S$ from the positive side; the metric identity holds there by continuity. On the section itself this is the restriction of the smooth embedding of the entire doubled Cauchy hypersurface, so it extends across $S$ as claimed. Its canonical radial curves are the outward normal geodesics from $S$ in both metrics: on $r>0$ their intrinsic acceleration and normal acceleration are determined by the same $h,\mathcal{K}$, and vanish as in the supplied original double-normal chart; smoothness includes $r=0$. Hence the normal tube of step 6.1 has the same positive canonical section. Equivariance from step 4.1 and the supplied stationary collar then match the positive exterior collar germ as well. This establishes the claimed compatibility at $S$, together with all the vacuum, completeness and symmetry conclusions. No domain-of-dependence identification for $E$ in a larger nonmaximal ambient spacetime is required.