Equivariant global Harrison primitive under weak original data

The vacuum identities give $dF=d{\star }_{g}F=0$, $df=2{\iota }_{T}F$ and $d\psi =2{\iota }_T{\star }_{g}F$. Thus ${\iota }_T{\mathrm{\Theta }}_0=4((f-1){\iota }_{T}F+\psi {\iota }_T{\star }_{g}F)=d((1-f{)}^2+{\psi }^2)=d{D}_0$. In an oriented orthonormal coframe of signature $(-,+,+,+)$, direct expansion of a real two-form gives ${\star }_{g}F\wedge {\star }_{g}F=-F\wedge F$. Contracting this identity with $T$ gives $({\iota }_{T}F)\wedge F+({\iota }_T{\star }_{g}F)\wedge {\star }_{g}F=0$, hence $d{\mathrm{\Theta }}_0=0$. Alternation gives $T\sigma =2(F+i{\star }_{g}F)(T,T)=0$, so $D_0$ and ${\mathrm{\Theta }}_0$ are stationary; the Killing flow preserves the metric, its connection and its oriented Hodge star.

In the original stationary end, $F=\frac{1}{2}d{T}^{\mathrm{♭}}$ gives $F_{0i}=O_5(r^{-2})$ and $F_{ij}=O_5(r^{-3})$. The inverse metric has $g^{0i}=O_6(r^{-2})$, bounded diagonal blocks and bounded differentiated volume factors. In the component formula for ${\star }_{g}F$, the spatial components use $F_{0i}$ and are $O_5(r^{-2})$, while $({\star }_{g}F{)}_{0i}$ uses raised spatial components and is $O_5(r^{-3})$; the terms involving $F_{0i}$ in those raised spatial components have the additional factor $g^{0j}=O_6(r^{-2})$. Therefore $d\psi =O_5(r^{-3})$. The end normalization $\psi \to 0$ and integration along each radial ray give $\psi =O_0(r^{-2})$; the differential equation supplies its derivatives of orders one through six with the required gains, so $\psi =O_6(r^{-2})$. Since $f-1=-2\mathrm{/}r+O_6(r^{-2})$, the product rules give $D_0=O_6(r^{-2})$ and $({\mathrm{\Theta }}_0{)}_{ij}=O_5(r^{-4})$.

The mass in the original weak end is $1>0$, the metric is smooth, stationary and Ricci-flat, and a far tail has future timelike $T$. All end-normalization hypotheses in [F2] hold. Apply that supplier to obtain $(t^{\mathrm{\prime }},y)$ for the same $T$, with $R=\mathrm{\mid }y\mathrm{\mid }$, $g_{00}^{\mathrm{\prime }}=-1+2\mathrm{/}R+O_{\mathrm{\infty }}(R^{-2})$, $g_{0a}^{\mathrm{\prime }}=O_{\mathrm{\infty }}(R^{-2})$, $g_{ab}^{\mathrm{\prime }}={\delta }_{ab}+O_{\mathrm{\infty }}(R^{-1})$ and smooth transition $y=q(x)$, $t^{\mathrm{\prime }}=t+F_{\ast }(x)$. The supplied normalized potential has the same limit along this end diffeomorphism. Repeating the component calculation of step 2.1 at each finite order gives $F_{0a}^{\mathrm{\prime }}=O_{\mathrm{\infty }}(R^{-2})$, $F_{ab}^{\mathrm{\prime }}=O_{\mathrm{\infty }}(R^{-3})$, $\psi =O_{\mathrm{\infty }}(R^{-2})$ and $({\mathrm{\Theta }}_0{)}_{ab}^{\mathrm{\prime }}=O_{\mathrm{\infty }}(R^{-4})$. Together with step 1.1 and original $D_0=O_6(r^{-2})$, these are all the additional hypotheses of the smooth weak-end primitive in [F5]. It supplies a smooth stationary $V_{\mathrm{\infty }}$ on a far tail with contraction constant $c_0=D_0(p)$ and original spatial coefficients $O_5(r^{-3})$. The auxiliary chart is used only to construct this primitive.

On the original section, the sphere integral of ${\mathrm{\Theta }}_0$ is $O(r^{-2})$ by step 2.1, since Cartesian sphere tangent vectors contribute the area factor $r^2$. Use the supplied physical truncation $B_R$, oriented by the future normal and spacetime orientation, with outward end-sphere orientation on ${\mathrm{\Gamma }}_R$ and the orientation toward the exterior on $S$. Its boundary is ${\mathrm{\Gamma }}_R\bigsqcup (-S)$. The restricted form is smooth through $S$ and has compact support on this compact region. Stokes in [F3] gives ${\int }_{{\mathrm{\Gamma }}_R}{\mathrm{\Theta }}_0={\int }_S{\mathrm{\Theta }}_0$, so passage to infinity gives ${\int }_S{\mathrm{\Theta }}_0=0$. Independently, the topological product gives a compact smooth region $\{1\le \rho \le a_{\ast }\}$ for any $a_{\ast }>1$. Stokes on it gives zero integral over its sphere $\{\rho =a_{\ast }\}$. The stationary product and [F3] therefore supply a smooth primitive $A$ of ${\mathrm{\Theta }}_0$ on $E$. Its pullback $a$ to $\mathrm{\Sigma }$ satisfies $da=b$, where $b$ is the descended spatial form of ${\mathrm{\Theta }}_0-dt\wedge d{D}_0$. To obtain a surface primitive without extending $A$ to the boundary, set $\gamma ={\mathrm{\Theta }}_0{\mathrm{\mid }}_{TS}$ and pull it back to $\mathbb{R}\times (1,\mathrm{\infty })\times S$. This is a closed two-form of zero sphere period, so [F3] makes it exact on that product. Restricting its smooth primitive to one slice yields a smooth $\beta$ on $S$ with $d\beta =\gamma$. No physical sphere is identified with a topological coordinate sphere.

The fields $K,Z$ preserve the metric and commute with $T$, hence preserve $d\sigma =2{\iota }_T(F+i{\star }_{g}F)$. Their derivatives of $\sigma$ are constant on the connected neighborhood. Since $K{\mathrm{\mid }}_S=0$, $K\sigma =0$; integrating the constant $Z\sigma$ around one full circle gives $2\pi Z\sigma =0$. Thus $K,Z$ preserve $\sigma$ and ${\mathrm{\Theta }}_0$. Start with the normal frame supplied by the smooth spacelike boundary and [F4], average its future normal over the circle and normalize. Its unique orthogonal unit spacelike companion in the specified outward component is smooth and invariant as well; the connected circle preserves that component. Thus the resulting frame is orthonormal with the original time and side orientations. Pointwise, two such frames are related by $\widehat{n}=An+B\nu$, $\widehat{\nu }=Bn+A\nu$, where $A>0$ and $A^2-B^2=1$, by their orthonormality and the two orientations. If $an+b\nu =\widehat{a}\widehat{n}+\widehat{b}\widehat{\nu }$, then $b+a=(A+B)(\widehat{b}+\widehat{a})$ and $b-a=(A-B)(\widehat{b}-\widehat{a})$. Since $A\pm B>0$, the condition $b>\mathrm{\mid }a\mathrm{\mid }$ is equivalent to $\widehat{b}>\mathrm{\mid }\widehat{a}\mathrm{\mid }$. Consequently the prescribed outward wedge is unchanged in the invariant frame. Every isometry preserves the metric-selected connection by its uniqueness, and therefore preserves geodesics by the initial-value uniqueness in [F4]. Consequently its differential action on the normal bundle intertwines the exponential map. Fibre dilations commute with that linear action. Shrink the tube uniformly over compact $S$, retaining the full circle action and dilation stability; the $K$ equivariance is on every defined local flow domain. Average $\beta$ and apply relative homotopy and equivariant normalization with this tube. The resulting smooth $V_H$ satisfies $d{V}_H={\mathrm{\Theta }}_0$, local $K,Z,T$ invariance and $V_H(T)+D_0=D_0(p)$. At $p$, $T_p=0$ gives $f(p)=0$, so $D_0(p)=1+\psi (p{)}^2>0$.

In the invariant tube frame put $u=b+a$, $v=b-a$. Exponential equivariance and the specified normal boost make the local $T=K-{\mathrm{\Omega }}_{H}Z$ flow exactly $(s,u,v)\mapsto ({\varphi }_{-{\mathrm{\Omega }}_{H}t}^Z(s),e^{\kappa t}u,e^{-\kappa t}v)$ while it stays in the tube. Its outward wedge is $u,v>0$. The section $\bar{\mathrm{\Sigma }}$ is locally a graph $a=h(s,b)$ with $h(s,0)=0$: at $S$, projection to $(s,b)$ is nonsingular on its tangent space because its normal-plane tangent direction is spacelike. The smooth inverse function theorem, local extension across the boundary and compactness give a uniform graph neighborhood. Spacelikeness gives $\mathrm{\mid }{\mathrm{\partial }}_{b}h(s,0)\mathrm{\mid }<1$, and after shrinking there is one $c<1$ with $\mathrm{\mid }h(s,b)\mathrm{\mid }\le cb$ for $b\ge 0$. For a local wedge orbit set $j=uv>0$ and center its parameter so that $u=v=\sqrt{j}$ at time zero. Then $b(t)=\sqrt{j}\cosh (\kappa t)$ and $a(t)=\sqrt{j}\sinh (\kappa t)$. Choose $L>0$ with $\tanh (\mathrm{\mid }\kappa \mathrm{\mid }L)>c$. At the two times with $\kappa t=\pm \mathrm{\mid }\kappa \mathrm{\mid }L$, the continuous function $a(t)-h(s(t),b(t))$ has opposite signs. For sufficiently small $j$ the whole bounded-time segment is in the graph neighborhood, so [F6] gives an intersection with the section. It has $b\le C\sqrt{j}$ for a fixed $C$. The global stationary product makes this intersection unique on the full orbit.

Choose a section collar $C_H=\{1<\rho <1+{ϵ}_H\}$ small enough to lie in the tube and set $U_H=\mathrm{\Phi }(\mathbb{R}\times C_H)$. Transport $V_H$ from this collar by the global stationary product. Uniqueness of the product's section point defines one smooth stationary form ${\tilde{V}}_H$ on $U_H$, with the same exterior derivative and contraction constant, since these identities hold near each section point and are preserved by the stationary flow. For every point of a sufficiently small outward tube, step 5.1 places its unique section point in $C_H$. The segment joining them stays inside the larger tube: $u$ and $v$ vary monotonically, so each is bounded by its endpoint values, and the entire sphere factor is allowed. Local invariance of $V_H$ along this segment shows that the transport equals $V_H$ there. The section point tends uniformly to $S$ as $u,v\to 0$, by the bound $b\le C\sqrt{uv}$. Hence there is a smaller whole neighborhood $O$ of $S$ with $O\cap E\subset \mathrm{\Phi }(\mathbb{R}\times \{1<\rho <1+{ϵ}_H\mathrm{/}3\})$, and ${\tilde{V}}_H=V_H$ on $O\cap E$. The stated wedge attachment ensures that every point of $O\cap E$ is in this argument. No injectivity of a time product at $S$, where $T$ may vanish, is used.

Apply the stationary lift to $a$ from step 3.2, using $c_0=D_0(p)$, to obtain a smooth stationary $V=(c_0-D_0)dt+{\pi }^{\ast }a$ on $E$ with $dV={\mathrm{\Theta }}_0$ and $V(T)+D_0=c_0$. Choose $R_{\ast }>0$ sufficiently large that the physical tail $\{r>R_{\ast }\}$ lies in the end-primitive domain of step 3.1. Its closed tail is disjoint from $S$ by the supplied end-attachment hypothesis. Shrink the topological section collar so that its closure lies in the relative neighborhood $\bar{\mathrm{\Sigma }}\setminus \{r\ge R_{\ast }\}$ of compact $S$, and shrink $O$ again by step 6.1 for that smaller collar. The stationary product then makes $U_H$ disjoint from $U_{\mathrm{\infty }}=\mathrm{\Phi }(\mathbb{R}\times \{r>R_{\ast }\})$. Both $U_H$ and $U_{\mathrm{\infty }}$ are stationary products of $\mathbb{R}$ with a spherical annulus, hence have exact closed one-forms by [F3]. In particular the differences ${\tilde{V}}_H-V$ and $V_{\mathrm{\infty }}-V$ are exact there. Their contractions with $T$ vanish, so every scalar primitive $h$ of a difference satisfies $Th=dh(T)=0$. Use the smooth step function of [F6] in $\rho$ to choose ${\eta }_H=1$ for $1<\rho <1+{ϵ}_H\mathrm{/}3$ and ${\eta }_H=0$ for $\rho \ge 1+2{ϵ}_H\mathrm{/}3$; choose $R_{\ast }<R_1<R_2$ and define ${\eta }_{\mathrm{\infty }}$ using the same step function in the physical radius $r$, zero for $r\le R_1$ and one for $r\ge R_2$, extended by zero off the physical end. Both cutoffs are transported by the stationary product. The end cutoff is smooth across its extension seam because it vanishes on an end neighborhood of that seam; its support in $E$ lies in the closed stationary lift of $\{r\ge R_1\}\subset \{r>R_{\ast }\}$. The supplied closed-tail condition and the product diffeomorphism justify that closure assertion. The horizon cutoff has support inside $U_H$ because its nonzero set is confined to $1<\rho <1+2{ϵ}_H\mathrm{/}3$, with a positive gap from the outer edge of $C_H$. Thus these are smooth stationary functions with supports in the respective disjoint regions, using $\rho$ only at the horizon and $r$ only at the end. All hypotheses of stationary exact gluing now hold, including the whole-neighborhood intersection in step 6.1. Its output $W_0$ equals $V_H$ on a neighborhood of $S$, equals $V_{\mathrm{\infty }}$ on a far tail, and globally satisfies $d{W}_0={\mathrm{\Theta }}_0$, stationarity and $W_0(T)=D_0(p)-D_0$. It inherits the local $K,Z$ invariance and original $O_5(r^{-3})$ spatial decay. The original section and any of its supplied maximality remain fixed.