Local Hawking and axial fields, boost and pole

Apply [F1] under [A1]. Its identification of $C$ supplies the spacelike exterior collar of $S$, while regularity supplies the two smooth transverse non-expanding chronological end-boundaries. The five geometry lemmas in [F2] therefore establish the full local AIK pair definition, including the local causal-boundary condition rather than adding it as an extra assumption. Apply [F3] to obtain $K_0$ and the periodic $Z_0=T+{\lambda }_0{K}_0$.

The local symmetry neighborhood meets the exterior collar. The stationary-product lemma in [F4], using the diffeomorphism in [F1], gives ${\lambda }_0\mathrm{\ne }0$. The common-period subgroup of the nontrivial local $Z_0$ action is closed and has no positive periods tending to zero: otherwise differentiation at a point where $Z_0\mathrm{\ne }0$ would force that vector to vanish. Its positive infimum is consequently a least period $P$; division by $P$ shows every period is an integer multiple. Set $r=P\mathrm{/}(2\pi )$ and $Z=r{Z}_0$. Then $Z$ has least positive common period $2\pi$. Put $K=-{\lambda }_0{K}_0=T-Z_0=T-r^{-1}Z$, so ${\mathrm{\Omega }}_H=-r^{-1}$; reversing the orientation of $Z$ reverses the sign of ${\mathrm{\Omega }}_H$ without changing $K$. Constant linearity and $[T,K_0]=0$ give every asserted commutator.

The circle-pole lemma in [F4] gives $p\in S$ with $Z(p)=T(p)=0$ and ${\mathrm{\nabla }}_{\nu }Z=0$. The axis lemma in [F2] identifies ${\mathcal{H}}^{+}$ with $a-b=0$ and ${\mathcal{H}}^{-}$ with $a+b=0$ in the normal tube ${\exp }_s(an+b\nu )$, so their future generator lines at $S$ are ${\mathbb{R}}_{+}(n+\nu )$ and ${\mathbb{R}}_{+}(n-\nu )$, respectively. With the source labels fixed in [F3], the imported affine profiles give, on $S$, ${\mathrm{\nabla }}_L{K}_0=L(\underset{‾}{u})L=L$ and ${\mathrm{\nabla }}_{\underset{‾}{L}}{K}_0=-\underset{‾}{L}(u)\underset{‾}{L}=-\underset{‾}{L}$; the terms containing the vanishing affine parameters are zero. Thus ${\mathrm{\nabla }}_{n+\nu }{K}_0=n+\nu$ and ${\mathrm{\nabla }}_{n-\nu }{K}_0=-(n-\nu )$ at every point of $S$. Subtracting these equations and dividing by two gives ${\mathrm{\nabla }}_{\nu }{K}_0=n$; adding gives ${\mathrm{\nabla }}_n{K}_0=\nu$. Since $K=-{\lambda }_0{K}_0=T-r^{-1}Z$, the normalized-boost lemma with [F1] now gives $-{\lambda }_0=\nu (N){\mathrm{\mid }}_p>0$. Thus $c=-{\lambda }_0$ is a positive constant with the required coefficient of $T$.

On ${\mathcal{H}}^{+}$ the source profile in [F3] gives $K=c\underset{‾}{u}L$, with $L(\underset{‾}{u})=1$ and ${\mathrm{\nabla }}_{L}L=0$, so ${\mathrm{\nabla }}_{K}K=c^2\underset{‾}{u}L=cK$. On ${\mathcal{H}}^{-}$ it gives $K=-cu\underset{‾}{L}$, with $\underset{‾}{L}(u)=1$ and ${\mathrm{\nabla }}_{\underset{‾}{L}}\underset{‾}{L}=0$, so ${\mathrm{\nabla }}_{K}K=c^{2}u\underset{‾}{L}=-cK$. Equivalently these are the signed affine-profile conclusions of the normalized-boost supplier in [F4], whose profile and normalization hypotheses were verified in step 3.1. Its Killing covector identity gives $d(g(K,K))=-2cg(K,\cdot )$ on ${\mathcal{H}}^{+}$ and the opposite sign on ${\mathcal{H}}^{-}$. The imported affine parameters have zero set exactly $S$, and the generators are nowhere zero, so $K$ is nonzero on either branch off $S$. The normal eigenvalue calculation in step 3.1 holds at every point of $S$ with this same constant $c$, giving the boost matrix with $\kappa =c>0$; the existing normal-boost supplier then gives the corresponding flow action. At $S$ the acceleration equations are merely their smooth zero extensions. This proves every original local assertion with its stationary and periodic normalizations.