Zero stationary field gives the local bifurcate Killing-horizon and boundary data

The local flow ${\mathrm{\Phi }}_t$ of $T$ is defined on a common neighborhood for sufficiently small $t$. The flow law and the definition of Lie derivative give $\frac{d}{dt}{\mathrm{\Phi }}_t^{\ast }g={\mathrm{\Phi }}_t^{\ast }({\mathcal{L}}_{T}g)=0$, so each such time map is a local isometry. Pulling the Levi–Civita connection back by this map gives another torsion-free metric-compatible connection; uniqueness makes it the same connection. Thus ${\mathrm{\Phi }}_t$ takes affinely parametrized geodesics to affinely parametrized geodesics, with their pushed-forward initial velocities. Geodesic uniqueness yields ${\mathrm{\Phi }}_t({\exp }_{p}v)={\exp }_{{\mathrm{\Phi }}_t(p)}(d{\mathrm{\Phi }}_t{\mathrm{\mid }}_{p}v)$ whenever the geodesic segment and its image lie in the common local domain. These are local statements, requiring no complete flow.

Fix either null sheet $H$ near a point of $S$. Choose the local defining function $u$ from [F6] and set $L=(du{)}^{\mathrm{♯}}$. Along $H$, this is a smooth nonzero field spanning $(TH{)}^{\perp }$, which equals the radical line in $TH$ because $H$ is null. Torsion-freeness makes the Hessian of $u$ symmetric: its antisymmetric part on $U,V$ is $U(Vu)-V(Uu)-[U,V]u=0$. Metric compatibility therefore gives, for every $Y\in TH$, $g({\mathrm{\nabla }}_{L}L,Y)=\mathrm{Hess}u(L,Y)=\mathrm{Hess}u(Y,L)=\frac{1}{2}Y(g(L,L))=0$ on $H$. The last equality holds because $g(L,L)$ vanishes identically there. Hence ${\mathrm{\nabla }}_{L}L=fL$ on $H$ for a smooth scalar $f$; smoothness follows locally by dividing any nonzero component of $L$. This proves the pregeodesic property rather than assuming it from the horizon terminology.

Write $A_p(v)={\mathrm{\nabla }}_{v}T$ and consider $Z=\{p:T_p=0,A_p=0\}$. It is closed by smoothness. If $p\in Z$, flow uniqueness fixes $p$, and in coordinates the variational equation for $J_t=d{\mathrm{\Phi }}_t{\mathrm{\mid }}_p$ is $J_t^{\mathrm{\prime }}=(dT{)}_p{J}_t=0$, $J_0=\mathrm{i}\mathrm{d}$; here $(dT{)}_p=A_p$ because $T_p=0$. Hence $d{\mathrm{\Phi }}_t{\mathrm{\mid }}_p=\mathrm{i}\mathrm{d}$. The exponential identity of step 1.1 and the local diffeomorphism in [F3] make ${\mathrm{\Phi }}_t$ the identity on a neighborhood of $p$ for all sufficiently small $t$. One may choose the neighborhood and time interval together by shrinking inside the open local-flow domain and a small exponential-coordinate ball. Differentiating at $t=0$ makes $T$ zero there, hence also $\mathrm{\nabla }T=0$ there. Thus $Z$ is open. If nonempty it is all of connected $M$, contradicting the nontriviality of $T$. Therefore $Z$ is empty.

The radical line is transverse to $S$ inside $H$: a nonzero null vector cannot lie in the positive-definite plane $T_{q}S$. Let ${\mathrm{\Psi }}_r$ be the local flow of $L{\mathrm{\mid }}_H$. Applied within the three-manifold $H$ to a small embedded patch of its codimension-one submanifold $S$, [F5] makes $(r,q)\mapsto {\mathrm{\Psi }}_r(q)$ a diffeomorphism onto an open sheet neighborhood. Along each curve define $c_q(r)=\exp ({\int }_0^{r}f({\mathrm{\Psi }}_v(q))dv)$ and $s_q(r)={\int }_0^r{c}_q(v)dv$. Then $c_q>0$, $s_q^{\mathrm{\prime }}=c_q$ and $c_q^{\mathrm{\prime }}=f{c}_q$. Reparametrizing by $s=s_q(r)$ gives velocity $L\mathrm{/}{c}_q$ and acceleration $c_q^{-2}({\mathrm{\nabla }}_{L}L-(c_q^{\mathrm{\prime }}\mathrm{/}{c}_q)L)=0$. Its initial velocity is $L_q$, so [F3] identifies it with ${\gamma }_q(s)={\exp }_q(s{L}_q)$. The strictly increasing parameter change preserves this local coverage and has $s=0$ exactly on $S$. Thus these affine null-normal geodesics issuing from $S$ cover the sheet locally, with nonzero tangent $L\mathrm{/}{c}_q$.

At $q\in S$, differentiating the zero restriction $T{\mathrm{\mid }}_S$ gives $A_q{\mathrm{\mid }}_{T_{q}S}=0$. Metric compatibility, torsion-freeness and ${\mathcal{L}}_{T}g=0$ give $g({\mathrm{\nabla }}_{U}T,V)+g(U,{\mathrm{\nabla }}_{V}T)=0$, so $A_q$ is $g$-skew. Consequently its image lies in $(T_{q}S{)}^{\perp }$, and this Lorentzian normal two-plane is invariant. In a null basis ${\mathrm{\ell }}_{+},{\mathrm{\ell }}_{-}$ with $g({\mathrm{\ell }}_{+},{\mathrm{\ell }}_{-})=-1$, skewness forces $A_q{\mathrm{\ell }}_{+}={\lambda }_q{\mathrm{\ell }}_{+}$ and $A_q{\mathrm{\ell }}_{-}=-{\lambda }_q{\mathrm{\ell }}_{-}$. If ${\lambda }_q=0$ then $A_q=0$ on both the normal plane and $T_{q}S$, putting $q$ in $Z$, impossible by step 2.1. Thus ${\lambda }_q\mathrm{\ne }0$.

At each $q\in S$ the two transverse sheets have distinct radical lines in $(T_{q}S{)}^{\perp }$: coincident radicals would give equal tangent hyperplanes, each the orthogonal complement of that same line. Label their local nonzero initial fields ${\mathrm{\ell }}_{\pm }$ accordingly. Step 3.1 gives the respective eigenvalues $\pm {\lambda }_q$. Since the Killing flow fixes $q$, its differential satisfies the constant-coefficient variational equation with coefficient $A_q$ and hence sends ${\mathrm{\ell }}_{\pm }$ to $e^{\pm t{\lambda }_q}{\mathrm{\ell }}_{\pm }$. By step 1.1 and geodesic scaling, ${\mathrm{\Phi }}_t({\exp }_q(s{\mathrm{\ell }}_{\pm }))={\exp }_q(e^{\pm t{\lambda }_q}s{\mathrm{\ell }}_{\pm })$ for small parameters. Differentiating at $t=0$ gives $T=\pm {\lambda }_{q}s{\mathrm{\partial }}_s$ on the sheet. Step 2.2 ensures that these coordinates cover a neighborhood in each sheet and that ${\mathrm{\partial }}_s$ spans its radical. Thus $T$ is tangent, null-normal and nonzero away from $S$. Taking the union of sufficiently small such neighborhoods along $S$ gives the claimed neighborhood sense of a bifurcate Killing horizon. No uniform collar width for noncompact $S$, all-time flow, global causal completion or compact-core section is inferred.

Finally $N=-g(T,n)=0$ at $S$ and $X=T-Nn=0$ there by the orthogonal decomposition. Conversely if $T$ is tangent to both sheets and lies in the null-generator line of either at their spacelike intersection, it is a vector of $TS$ which is also null, and is therefore zero. These pointwise boundary consequences require no maximal slice.