Normal deformation families and an outgoing null gauge

Openness of the exponential domain and compactness of $S$ give one interval on which $t{X}_p$ lies in that domain for all $p$. The smooth exponential map gives a smooth family. By [F4], ${\mathrm{\Phi }}_t(p)={\gamma }_{i(p),X_p}(t)$ on that interval, so ${\mathrm{\Phi }}_0(p)=i(p)$ and ${{\mathrm{\partial }}_t{\mathrm{\Phi }}_t(p)\mid }_{t=0}=X_p$ by the initial data of this geodesic. Choose finitely many coordinate neighborhoods $V_j$ covering $S$ whose closures $K_j=\overline{V_j}$ are compact and contained in larger coordinate neighborhoods $W_j$. To obtain these, take around each point a coordinate ball with closed ball contained in the chart image, pull back a smaller concentric ball, and take a finite subcover by compactness. In the coordinates on $W_j$, put $q_{t,AB}=g({\mathrm{\partial }}_A{\mathrm{\Phi }}_t,{\mathrm{\partial }}_B{\mathrm{\Phi }}_t)$. On each $K_j$, both $q_{0,11}$ and $\det (q_{0,AB})$ are positive continuous functions and hence have positive minima by [F6]. Continuity on these compact sets, and finiteness of the cover, give a common smaller parameter interval on which $q_{t,11}>0$ and $\det (q_{t,AB})>0$ throughout every $K_j$. These two principal-minor inequalities make the symmetric two-by-two Gram matrix positive definite, so the differential is injective and each deformed surface is spacelike.

To check global injectivity, suppose there are $t_j\to 0$ and distinct $p_j,q_j$ with ${\mathrm{\Phi }}_{t_j}(p_j)={\mathrm{\Phi }}_{t_j}(q_j)$. Compactness gives convergent subsequences, and injectivity of $i$ makes their limits a common point $p$. Choose a coordinate projection of an ambient chart to two tangent coordinates whose differential on $di(T_{p}S)$ is invertible. The map $(t,y)\mapsto (t,\pi {\mathrm{\Phi }}_t(y))$ has invertible differential at $(0,p)$; its local inverse contradicts equality for these pairs once they lie in that neighborhood. Thus ${\mathrm{\Phi }}_t$ is injective for small $t$. By [F6], the continuous bijection $S\to {\mathrm{\Phi }}_t(S)$ is a homeomorphism, since $S$ is compact and ${\mathrm{\Phi }}_t(S)$ is a Hausdorff subspace of $M$; together with the rank condition, this makes ${\mathrm{\Phi }}_t$ an embedding by [F5].

On each deformed surface, subtract from the ambient time-orientation vector $N$ its tangential projection: $U=N-q_t^{AB}g(N,E_A)E_B$, where $E_A={\mathrm{\partial }}_A{\mathrm{\Phi }}_t$. The tangent term is spacelike and orthogonal to $U$, so $g(U,U)=g(N,N)-\mathrm{\mid }{N}^{\mathrm{\top }}{\mathrm{\mid }}^2<0$, and $g(U,N)=g(U,U)<0$. Therefore $n_t=U\mathrm{/}\sqrt{-g(U,U)}$ is a smooth future unit normal. The normal plane is Lorentzian. Its orientation from the orientations of $M$ and $S$, together with $n_t$, selects a smooth unit spacelike normal ${\nu }_t$; its sign is chosen to match the outgoing null ray at $t=0$. This can be computed by ordinary two-vector orthogonalization in each normal-plane chart; uniqueness of the oriented unit vector glues these formulas. Put ${\widehat{\mathrm{\ell }}}_t=n_t+{\nu }_t$, ${\widehat{k}}_t=(n_t-{\nu }_t)\mathrm{/}2$.

The initial outgoing ray agrees with that of $\mathrm{\ell }$, so $\mathrm{\ell }=a{\widehat{\mathrm{\ell }}}_0$ for a positive smooth function $a$ on $S$. The normalization then gives $k=a^{-1}{\widehat{k}}_0$. Setting ${\mathrm{\ell }}_t=a{\widehat{\mathrm{\ell }}}_t$ and $k_t=a^{-1}{\widehat{k}}_t$, with $a$ constant in the variation parameter, supplies the required pair. All denominators are nonzero, and compactness permits the uniform shrinkings already used. Countable Choice is inherited from the exponential-map interface; no stronger choice is used in this construction.