Smooth event-boundary branches are achronal null geodesic sheets

Apply [F1] to $I^{+}(E)$ and its time-reversed past counterpart $I^{-}(E)$. Their ambient boundaries are achronal. Restricting to the specified open set $O$ cannot introduce a timelike curve between branch points, so each branch is achronal in $O$.

Work locally on either branch $H$ with a nonzero smooth radical field $L$; such local sections exist because the radical is a smooth line subbundle. For every smooth tangent field $Y$, metric compatibility and torsion freedom give $g({\mathrm{\nabla }}_{L}L,Y)=-g(L,{\mathrm{\nabla }}_{L}Y)=-g(L,{\mathrm{\nabla }}_{Y}L+[L,Y])=-\frac{1}{2}Yg(L,L)=0$, because $[L,Y]$ is tangent to $H$. Thus ${\mathrm{\nabla }}_{L}L$ lies in the normal line, which equals the radical line, so ${\mathrm{\nabla }}_{L}L=aL$. The coefficient $a$ is smooth: locally divide a component of ${\mathrm{\nabla }}_{L}L$ by a nonvanishing component of $L$.

By [F3], $L$ has local integral curves. On any such curve $\gamma :I\to H$, fix $t_0\in I$ and put $b(t)=\exp (-{\int }_{t_0}^{t}a(\gamma (u))du)$ and $s(t)={\int }_{t_0}^{t}b(u{)}^{-1}du$. Differentiation gives $b^{\mathrm{\prime }}=-a(\gamma )b$ and $s^{\mathrm{\prime }}=b^{-1}>0$, so $s$ is a smooth strictly increasing reparametrization with smooth inverse onto its interval image by [F4]. The rescaled velocity is $d\gamma \mathrm{/}ds=bL$, and $D_s(d\gamma \mathrm{/}ds)=b(b^{\mathrm{\prime }}+a(\gamma )b)L=0$. Its norm is zero since $L$ is null, proving the geodesic assertion. This is a rescaling along the curve, not an assertion of a single affine scale on the entire branch.

At $p\in S$, the null radical is orthogonal to $T_{p}S$. It cannot belong to $T_{p}S$ because that space has positive definite induced metric. Thus it is transverse to $S$ within either three-dimensional branch $H$. Choose a local smooth nonzero radical field $L$ near $p$ in $H$. By [F3], after shrinking, its local flow defines a smooth map $F:(-\epsilon ,\epsilon )\times W\to H$, $F(t,q)=\mathrm{\Phi }(t,q)$, where $W$ is an open neighborhood of $p$ in $S$. Its differential at $(0,p)$ sends $(c,v)\in \mathbb{R}\oplus T_{p}S$ to $c{L}_p+v$, an isomorphism onto $T_{p}H$ by transversality and dimension. By [F4], shrinking to a smaller product neighborhood makes $F$ a diffeomorphism onto an open branch neighborhood. This supplies the local sheets from $S$; their curves are the null geodesics up to reparametrization from step 2.1. Orthogonality at $S$ and step 1.1 give the remaining claims. Nothing in this construction identifies the branches with the entire local causal boundary of $S$.