The intersection of two developments of the same data sharing the initial Cauchy surface is again a development

(Setup and side split.) Let $\gamma :I\to W$ be a future causal curve that is inextendible in $W$, and suppose for contradiction that $\gamma$ misses $\mathrm{\Sigma }$. Then $\gamma$ is a causal curve in each $U_i$ and in $M$. Since $I$ is an interval and $\gamma$ is continuous, $\gamma (I)$ is connected; by [L2] it is contained in exactly one of $I^{+}(\mathrm{\Sigma })$ or $I^{-}(\mathrm{\Sigma })$. We treat these two cases separately while keeping the given time orientation and matter fields fixed.

(The compact diamond in the future-side case.) Assume $\gamma (I)\subseteq I^{+}(\mathrm{\Sigma })$ and fix $t_0\in I$. By [L3] the set $K:=J^{+}(\mathrm{\Sigma })\cap J^{-}(\gamma (t_0))$ is compact. For every $t\le t_0$ in $I$, $\gamma (t)\le \gamma (t_0)$ and $\gamma (t)\in I^{+}(\mathrm{\Sigma })\subseteq J^{+}(\mathrm{\Sigma })$, so $\gamma (t)\in K$.

(The compact diamond in the past-side case.) Assume instead $\gamma (I)\subseteq I^{-}(\mathrm{\Sigma })$ and fix $t_0\in I$. By [L3] the time-dual set $K_{-}:=J^{+}(\gamma (t_0))\cap J^{-}(\mathrm{\Sigma })$ is compact. For every $t\ge t_0$ in $I$, $\gamma (t_0)\le \gamma (t)$ and $\gamma (t)\in I^{-}(\mathrm{\Sigma })\subseteq J^{-}(\mathrm{\Sigma })$, so $\gamma (t)\in K_{-}$.

(The future-side past tail converges.) For $s<t_0$ put $L(s)={\mathrm{\ell }}_h(\gamma {\mathrm{\mid }}_{[s,t_0]})$, bounded by $L(K)$ by [L3], and let $L_{\ast }={\sup }_{s<t_0}L(s)$. For $s<t<t_0$, additivity gives ${\mathrm{\ell }}_h(\gamma {\mathrm{\mid }}_{[s,t]})=L(s)-L(t)\le L_{\ast }-L(t)$, which tends to zero as $t$ approaches the past parameter end. A decreasing coinitial sequence and completeness of $d_h$ therefore give a limit $y$; the same tail bound gives convergence of the whole past tail. Since $K$ is closed, $y\in K\subseteq J^{+}(\mathrm{\Sigma })$. This works at finite and infinite parameter ends without reparametrizing $\gamma$.

(The past-side future tail converges.) For $s>t_0$ put $L(s)={\mathrm{\ell }}_h(\gamma {\mathrm{\mid }}_{[t_0,s]})$, bounded by $L(K_{-})$ by [L3], and let $L_{\ast }={\sup }_{s>t_0}L(s)$. For $t_0<s<t$ before the future parameter end, additivity gives ${\mathrm{\ell }}_h(\gamma {\mathrm{\mid }}_{[s,t]})=L(t)-L(s)\le L_{\ast }-L(s)$, which tends to zero as $s$ approaches that end. A cofinal increasing sequence and completeness of $d_h$ give a limit $y$; the same tail bound gives convergence of the whole future tail. Since $K_{-}$ is closed, $y\in K_{-}\subseteq J^{-}(\mathrm{\Sigma })$.

(The future-side limit lies in the intersection.) Fix $s<t_0$. For every earlier $t<s$, $\gamma (t)\le \gamma (s)$, so closedness of $\le$ and step 3.1 give $y\le \gamma (s)$. If $y\mathrm{\notin }{U}_1$, then $y\mathrm{\notin }\mathrm{\Sigma }$ and, since $y\in J^{+}(\mathrm{\Sigma })$, there is a causal curve from $\mathrm{\Sigma }$ to $y$. Concatenating it with a causal curve from $y$ to $\gamma (s)\in U_1\cap I^{+}(\mathrm{\Sigma })$ contradicts past convexity [L1]. The same argument gives $y\in U_2$, hence $y\in W$. Only closedness of the causal relation is used to obtain the segment to $\gamma (s)$; no endpoint extension preserving the original parameter is asserted.

(Closed causality and future convexity put $y$ in the intersection.) Fix $s>t_0$. For all later $t>s$, $\gamma (s)\le \gamma (t)$, so closedness of $\le$ and step 3.2 give $\gamma (s)\le y$. If $y\mathrm{\notin }{U}_1$, then $y\mathrm{\notin }\mathrm{\Sigma }$ and $y\in J^{-}(\mathrm{\Sigma })$ gives a causal curve from $y$ to some point of $\mathrm{\Sigma }$. Concatenate a causal curve from $\gamma (s)\in U_1\cap I^{-}(\mathrm{\Sigma })$ to $y$ with that curve to $\mathrm{\Sigma }$. Future convexity [L1] forces the concatenation, including $y$, to lie in $U_1$, a contradiction. The same argument gives $y\in U_2$, hence $y\in W$.

(The future-side contradiction.) By steps 3.1 and 4.1, $\gamma (t)\to y\in W$ as $t$ approaches its past parameter end. This contradicts inextendibility in $W$ under the no-limit convention [L5]. Thus the future-side case cannot occur.

(The past-side contradiction.) By steps 3.2 and 4.2, $\gamma (t)\to y\in W$ as $t$ approaches its future parameter end. This contradicts inextendibility in $W$ under the no-limit convention [L5]. Thus the past-side case also cannot occur, and the curve must meet $\mathrm{\Sigma }$.

(Uniqueness of the meeting and conclusion.) Suppose $t_1<t_2$ in $I$ with $\gamma (t_1),\gamma (t_2)\in \mathrm{\Sigma }$. By [L2] there is an inextendible causal curve $\widehat{\gamma }:J\to M$ and a strictly increasing affine $\rho$ with $\widehat{\gamma }\circ \rho =\gamma {\mathrm{\mid }}_{[t_1,t_2]}$; since $\widehat{\gamma }$ is a causal curve in $M$ with Cauchy surface $\mathrm{\Sigma }$, it meets $\mathrm{\Sigma }$ in at most one parameter value, but $\widehat{\gamma }(\rho (t_1)),\widehat{\gamma }(\rho (t_2))\in \mathrm{\Sigma }$ with $\rho (t_1)<\rho (t_2)$, a contradiction. Hence the meeting is unique. A past-directed parametrization is changed to a future-directed one by reversing its parameter; the image and number of visits to $\mathrm{\Sigma }$ are unchanged, so the same conclusion holds for past causal curves. Thus $\mathrm{\Sigma }$ is Cauchy in $W$. Moreover, $W$ is an open submanifold of $M$ with the restricted metric, time orientation and matter fields; the field equations are local and hold on $M$, hence on $W$; and $j:\mathrm{\Sigma }\to W$ has the original induced data. Therefore $W$ is a development of the same data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (a).