Existence of a universal development: the quotient of the set-sized family of canonical developments along pairwise maximal common developments

(Clause (a).) Each member is determined by four subsets of fixed sets as in [L1], so separation makes $\mathcal{X}$ a set; it is nonempty whenever some development of the data exists, by transporting that development through the flow representation [L6] (this transport is carried out in the proof of clause (e) below and uses no new hypothesis).

(Canonical representation of an arbitrary development.) Let $(M,g,N,j,\psi )$ be a development. By [L4] it is globally hyperbolic with Cauchy surface $j(\mathrm{\Sigma })$. Let $X,\mathrm{\Phi },T$ be as in [L6]. Each integral curve is defined on $\mathbb{R}$ and satisfies $T(\mathrm{\Phi }(t,y))=T(y)+t$, so it has no endpoint in $M$ and meets $j(\mathrm{\Sigma })$ exactly once. The map $\chi :\mathrm{\Sigma }\times \mathbb{R}\to M$, $\chi (x,t)=\mathrm{\Phi }(t,j(x))$, is therefore bijective and smooth. At $(x,t)$ its differential sends tangent vectors from $\mathrm{\Sigma }$ to $d{\mathrm{\Phi }}_t(dj(T_x\mathrm{\Sigma }))$ and the time direction to $X_{\mathrm{\Phi }(t,j(x))}=d{\mathrm{\Phi }}_t(X_{j(x)})$. Since $j(\mathrm{\Sigma })$ is spacelike, $X_{j(x)}$ is not tangent to it; since ${\mathrm{\Phi }}_t$ is a diffeomorphism with inverse ${\mathrm{\Phi }}_{-t}$, these images span $T_{\chi (x,t)}M$. Thus $\chi$ is a bijective local diffeomorphism, hence a diffeomorphism. Pulling back the metric, orientation representative and matter fields gives a tuple on $Z_0=\mathrm{\Sigma }\times \mathbb{R}$ in $\mathcal{X}$, with $\chi \circ j_0=j$. This proves nonemptiness from the assumed development. The associated quotient embedding will be concluded after the quotient structure is established.

(Clause (b): transitivity of the pairwise identifications.) The identity common development of a member with itself gives $U_{\alpha \alpha }=U_{\alpha }$ and ${\phi }_{\alpha \alpha }=\mathrm{id}$ by maximality [L2]; reversing a common-development embedding gives the same identifications in the opposite order, by the same maximality. Let $p\in U_{\alpha }$, $q\in U_{\beta }$, $r\in U_{\gamma }$ with $p\sim q$ via ${\phi }_{\alpha \beta }$ and $q\sim r$ via ${\phi }_{\beta \gamma }$. Then $p\in U_{\alpha \beta }$ and $q={\phi }_{\alpha \beta }(p)\in U_{\beta \gamma }$, so $p\in V={\phi }_{\alpha \beta }^{-1}(U_{\beta \gamma })$, and by [L3] the pair $(V,\chi )$ is a common subdevelopment of $\alpha$ and $\gamma$ mapping $p$ to $\chi (p)={\phi }_{\beta \gamma }({\phi }_{\alpha \beta }(p))=r$. By the maximality property of the maximal common development of $\alpha$ and $\gamma$, $V\subseteq U_{\alpha \gamma }$ and ${\phi }_{\alpha \gamma }(p)=\chi (p)=r$. Hence the identification of $p$ with $r$ holds at chain length two, and by induction at every finite chain length; the converse inclusion is the definition of the generating relations. This proves (b).

(Clause (c): the member maps and Hausdorffness.) By [L5] the map $\pi {\mathrm{\mid }}_{U_{\alpha }}$ is continuous; it is injective because step 2.1 reduces any identification of points in $U_{\alpha }$ to the maximal common development with itself, which is $(U_{\alpha },\mathrm{id})$ by the identity common development and maximality [L2]. Openness: the saturation of an open $A\subseteq U_{\alpha }$ is the union over $\gamma \in \mathcal{X}$ of $\{x\in U_{\gamma }:x\sim a\text{ for some }a\in A\}$, which by step 2.1 equals ${\bigcup }_{\gamma \in \mathcal{X}}{\phi }_{\alpha \gamma }(A\cap U_{\alpha \gamma }),$ a union of open sets, because each ${\phi }_{\alpha \gamma }$ is a diffeomorphism onto the open set ${\phi }_{\alpha \gamma }(U_{\alpha \gamma })$ and $A\cap U_{\alpha \gamma }$ is open. Hence each $\pi {\mathrm{\mid }}_{U_{\alpha }}$ is a homeomorphism onto the open set $Q_{\alpha }$, and the $Q_{\alpha }$ cover $Q$ because every class has a representative. For Hausdorffness let $[p]\mathrm{\ne }[q]$ with $p\in U_{\alpha }$, $q\in U_{\beta }$. Consider the pair quotient $Q_{\alpha \beta }$ of [L2] with its canonical map $\kappa :Q_{\alpha \beta }\to Q$; by step 2.1 the classes of $p$ and $q$ are distinct in $Q_{\alpha \beta }$ (otherwise they would be identified by a chain, hence by the single pairwise identification), and $Q_{\alpha \beta }$ is Hausdorff, so there are disjoint open $O_p\ni [p]$, $O_q\ni [q]$ in $Q_{\alpha \beta }$. The map $\kappa$ is injective (step 2.1), continuous and open (the saturation argument above applies to the two-member family), hence a homeomorphism onto its open image; thus $\kappa (O_p),\kappa (O_q)$ are disjoint open neighbourhoods of $[p],[q]$ in $Q$. This proves (c).

(The local smooth structure, fields and future cones.) Compose the charts of each member with $(\pi {\mathrm{\mid }}_{U_{\alpha }}{)}^{-1}$. On $Q_{\alpha }\cap Q_{\beta }=\pi (U_{\alpha \beta })$, the transitions use the smooth data-fixing diffeomorphism ${\phi }_{\alpha \beta }$, so these charts generate a unique maximal compatible smooth atlas. The metrics and matter fields agree on overlaps and descend to $\bar{g},\bar{\psi }$; the field equations hold locally. The pushforwards of $N_{\alpha }$ need not agree as vectors, but they select the same future timelike component on every overlap because ${\phi }_{\alpha \beta }$ preserves future cones. Thus there is a consistent locally smooth choice of future cones, before any global representative field is asserted. The maps $\pi {\mathrm{\mid }}_{U_{\alpha }}\circ j_0$ agree and define an embedded spacelike $S:=\bar{j}(\mathrm{\Sigma })$ with the prescribed induced data, since every member contains this entire surface and each member map is a diffeomorphism onto an open subset.

(A countable dense set without assuming second countability of $Q$.) Choose a countable dense $D_0\subseteq S$. At every newly occurring point choose one tangent basis and retain it at all later stages; let $R_x$ be its rational span. Recursively set $D_{n+1}=D_n\cup \{{\exp }_{x}v:x\in D_n,v\in R_x,{\bar{g}}_x(v,v)<0,{\exp }_{x}v\text{ defined}\}$ and $D={\bigcup }_n{D}_n$. These sets are countable by [L9], using the declared choice axioms for the recursion. To show density, take $p\in Q_{\alpha }$ and its member flow line from $z_0\in S$ to $p$ given by [L6]. If its flow parameter is zero, density of $D_0$ in $S$ suffices. Otherwise partition its compact segment into finitely many nonconstant timelike subsegments, each contained in a convex normal neighborhood within this member. Write their endpoints $z_0,\dots ,z_k=p$. For negative flow parameter the chords are past timelike; for positive parameter they are future timelike. We show inductively $z_i\in \overline{D\cap Q_{\alpha }}$: if $z_{i-1}$ has this property, choose $d\in D\cap Q_{\alpha }$ sufficiently near $z_{i-1}$. In the local two-point inverse the vector $v(d,z_i)$ is timelike by continuity and openness of the timelike set. Approximate it by a timelike $u\in R_d$ so closely that the member geodesic remains defined to time one and ${\exp }_{d}u$ lies in any prescribed neighborhood of $z_i$ within $Q_{\alpha }$. Such $u$ exists by rational density and openness of the local exponential domain. If $d\in D_m$, then ${\exp }_{d}u\in D_{m+1}\subseteq D$, with no claim that $m=i-1$. The exact inverse satisfies ${\exp }_d(v(d,z_i))=z_i$; no approximation error is added for this equality. This proves the induction and density of $D$ in $Q$.

(An explicit countable basis.) In each fixed tangent basis at $d\in D$, let $B_d(a,r)$ denote a Euclidean ball with rational center $a\in {\mathbb{Q}}^4$ and rational radius $r>0$. Form the family of all ${\exp }_d(B_d(a,r))$ for which the ball lies in the exponential domain and ${\exp }_d$ restricts to a diffeomorphism onto an open subset of $Q$. This family is countable by [L9] and consists of open sets by its definition. Given $p\in O$ with $O$ open, choose a member containing $p$ and use its joint exponential inverse near $(p,p)$. For all $d$ in a sufficiently small open neighborhood of $p$ in that member, there is a velocity $v(d,p)$ inside an open fiber domain $H_d$ on which ${\exp }_d$ is a diffeomorphism onto an open set containing $p$. Choose such $d\in D$ by step 5.1. The open set $H_d\cap {\exp }_d^{-1}(O)$ contains $v(d,p)$, so a rational translated ball in the fixed basis at $d$ contains $v(d,p)$ and lies in this open set. Its image is a family member containing $p$ and contained in $O$. This proves the basis criterion and hence second countability of $Q$.

(A smooth global orientation representative.) The quotient is now a smooth manifold in the full sense of [L10]. Choose a subordinate locally finite smooth partition of unity $({\rho }_{\alpha })$ for the member cover and put $\bar{N}={\sum }_{\alpha }{\rho }_{\alpha }(\pi {\mathrm{\mid }}_{U_{\alpha }}{)}_{\ast }{N}_{\alpha }$, extending each summand by zero outside $Q_{\alpha }$. Each summand is smooth because its support lies in $Q_{\alpha }$; the sum is locally finite. At every point the nonzero summands are positive multiples of vectors in the same future timelike component by step 4.1 and at least one coefficient is positive since the coefficients sum to one. Cone convexity makes $\bar{N}$ smooth, nowhere zero and future timelike. Every member map therefore preserves time orientation in the sense of preserving future cones. The metric and matter fields are the descended fields of step 4.1; the particular representative $\bar{N}$ is not claimed unique.

(Clause (d): the Cauchy property, at least one meeting.) Let $\gamma :\bar{I}\to Q$ be a causal curve inextendible in $Q$ and pick $t_0\in \bar{I}$ with $\gamma (t_0)\in Q_{\alpha }$. Let $J$ be the connected component of ${\gamma }^{-1}[Q_{\alpha }]$ containing $t_0$. By [L4] the restriction $\gamma {\mathrm{\mid }}_J$ is inextendible in the open set $Q_{\alpha }$, so its pullback under the diffeomorphism $\pi {\mathrm{\mid }}_{U_{\alpha }}$ is a causal curve in $U_{\alpha }$ inextendible in $U_{\alpha }$; by [L4] it meets $j_0(\mathrm{\Sigma })$ in exactly one parameter value, so $\gamma$ meets $\bar{j}(\mathrm{\Sigma })$.

(Compatible side decomposition.) In each member write $U_{\alpha }=I_{\alpha }^{-}(j_0\mathrm{\Sigma })\bigsqcup j_0\mathrm{\Sigma }\bigsqcup I_{\alpha }^{+}(j_0\mathrm{\Sigma })$ as in [L4]. For $x\in U_{\alpha \beta }$ the side computed in $U_{\alpha \beta }$ is the same as in $U_{\alpha }$: a timelike curve inside the subdevelopment gives membership in the corresponding ambient side, and these sides are disjoint. The isometry ${\phi }_{\alpha \beta }$ identifies these subdevelopment sides with those in its image in $U_{\beta }$, so the side in $U_{\beta }$ is identical. It follows that $Q^{-}:={\bigcup }_{\alpha }\pi (I_{\alpha }^{-}(j_0\mathrm{\Sigma }))$ and $Q^{+}:={\bigcup }_{\alpha }\pi (I_{\alpha }^{+}(j_0\mathrm{\Sigma }))$ are disjoint open sets and $Q=Q^{-}\bigsqcup S\bigsqcup Q^{+}$. In particular $S$ is closed. A future causal segment in a member that starts on its Cauchy surface lies, immediately afterwards, in its future side, and one ending on the surface lies immediately beforehand in its past side: extend the segment to an inextendible causal curve in that member; the unique Cauchy meeting and the disjoint side decomposition give the stated sides. More explicitly, any future-side point before the meeting would yield a timelike curve from the surface followed by a causal segment back to it, contrary to acausality; the past-side case after the meeting is its time reverse, using the push-up rule underlying the member side decomposition.

(At most one Cauchy meeting.) Suppose a future causal curve $\gamma$ has two parameter values on $S$. In a member neighborhood of any meeting it can meet $S$ at most once, since $S$ is the member's acausal Cauchy surface. Thus its meeting parameters form a relatively closed locally discrete subset; on the compact interval between the proposed two meetings it is finite, and we may choose consecutive meetings $t_1<t_2$. The connected image $\gamma ((t_1,t_2))$ misses $S$, so by step 8.2 it lies wholly in either $Q^{-}$ or $Q^{+}$. Near $t_1$ it lies in a member containing $\gamma (t_1)$, and the member side observation of step 8.2 puts it in $Q^{+}$. Near $t_2$ the same observation puts it in $Q^{-}$. This contradicts the disjointness of these sides. A past causal curve is treated by reversing its parameter. Together with step 8.1, every inextendible causal curve meets $S$ exactly once; all development conditions, including second countability, hold.

(Universality.) For an arbitrary development $M$, use its canonical representative $D:=\mathrm{\Sigma }\times \mathbb{R}\in \mathcal{X}$ and diffeomorphism $\chi :D\to M$ from step 1.2. The map $(\pi {\mathrm{\mid }}_D)\circ {\chi }^{-1}:M\to Q$ is a diffeomorphism onto an open subset, an isometry preserving future cones and matter fields, and takes $j$ to $\bar{j}$. Hence it is a data-fixing development embedding, proving (e).

(Conclusion.) Steps 1.1 and 1.2 establish (a), step 2.1 establishes (b), step 3.1 establishes (c), steps 4.1-9.1 establish (d), and step 10.1 establishes (e).