Common subdevelopments, maximal common developments and corresponding boundary points

Let $(M,g,N,j,\psi )$ and $(M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ be two developments of the same data in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality.

(a) Common subdevelopment. A common subdevelopment of $M$ and $M^{\mathrm{\prime }}$ is a pair $(U,\phi )$ in which

1. $U\subseteq M$ is an open subset with $j(\mathrm{\Sigma })\subseteq U$ such that the restriction $(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)$ is a development of the same data (with the induced field equations on the open submanifold $U$); and
2. $\phi :(U,g{\mathrm{\mid }}_U,N{\mathrm{\mid }}_U,j,\psi {\mathrm{\mid }}_U)\to (M^{\mathrm{\prime }},g^{\mathrm{\prime }},N^{\mathrm{\prime }},j^{\mathrm{\prime }},{\psi }^{\mathrm{\prime }})$ is a development embedding in the sense of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality clause (c).

Here "the induced field equations on $U$" means the equations of clause (a)4 of Smooth developments of constrained initial data, data-fixing isometries, development embeddings and maximality restricted to $U$; the induced Lorentzian manifold is $(U,g{\mathrm{\mid }}_U)$ with the time orientation $N{\mathrm{\mid }}_U$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

(b) Maximal common development. A common subdevelopment $(U^{\ast },{\phi }^{\ast })$ is a maximal common development (MCD) of $M$ and $M^{\mathrm{\prime }}$ when for every common subdevelopment $(U,\phi )$ one has $U\subseteq U^{\ast }$ and ${\phi }^{\ast }{\mathrm{\mid }}_U=\phi$. Existence is not asserted by this definition; it is proved in Existence and uniqueness of the maximal common development of two developments.

(c) Corresponding boundary points. Let $(U,\phi )$ be a common subdevelopment. Its corresponding boundary points are pairs $(p,p^{\mathrm{\prime }})\in {\mathrm{\partial }}^{M}U\times {\mathrm{\partial }}^{M^{\mathrm{\prime }}}\phi (U)$, where the boundaries are taken in $M$ and in $M^{\mathrm{\prime }}$ respectively, such that for every open neighbourhood $V$ of $p$ in $M$ and every open neighbourhood $V^{\mathrm{\prime }}$ of $p^{\mathrm{\prime }}$ in $M^{\mathrm{\prime }}$ there is $x\in V\cap U$ with $\phi (x)\in V^{\mathrm{\prime }}$.

(d) Side of the initial surface. Write ${\mathrm{\Sigma }}_M=j(\mathrm{\Sigma })$ and ${\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}=j^{\mathrm{\prime }}(\mathrm{\Sigma })$. For a corresponding pair $(p,p^{\mathrm{\prime }})$, openness of $U\supseteq {\mathrm{\Sigma }}_M$ and of $\phi (U)\supseteq {\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}$ implies $p\mathrm{\notin }{\mathrm{\Sigma }}_M$ and $p^{\mathrm{\prime }}\mathrm{\notin }{\mathrm{\Sigma }}_{M^{\mathrm{\prime }}}$. The Cauchy-set trichotomy of Cauchy sets are closed, split the manifold into past, set and future, and are locally Lipschitz graphs clause (a) puts each boundary point in exactly one chronological side. These sides are open because $I^{\pm }({\mathrm{\Sigma }}_M)={\bigcup }_{s\in {\mathrm{\Sigma }}_M}{I}^{\pm }(s)$, and each point future and past is open by Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets clause (a); the same argument applies in $M^{\mathrm{\prime }}$. For $x\in U$, its ambient and intrinsic sides agree: the trichotomy in $U$ gives an intrinsic timelike curve between $x$ and ${\mathrm{\Sigma }}_M$ in one direction, and that curve is also an ambient curve; the disjoint ambient trichotomy excludes the other direction. The development embedding $\phi$ preserves these curves and the initial surface, so corresponding interior points have matching ambient sides in $M$ and $M^{\mathrm{\prime }}$. Choose neighbourhoods of $p,p^{\mathrm{\prime }}$ contained in their respective sides. Condition (c) supplies an interior witness $x$ in both neighbourhoods under $\phi$; hence the two sides must match. Every witness for neighbourhoods contained in these sides then has this same sign. Thus the pair lies either to the future of both initial surfaces or to their past, with no cofinal witnesses of the opposite sign. The future case is used below; the past case is its time reverse.

(e) Scope. These are mathematical relations between two fixed developments and their boundary points. Neither the existence of boundary points, nor the existence of a common subdevelopment with nonempty boundary, nor uniqueness of a corresponding point, is asserted here.

Clause (a) realises the common development as a subset of $M$, breaking the symmetry between $M$ and $M^{\mathrm{\prime }}$; this is the convention of Sbierski's second variant, chosen because the construction below patches maps on subsets of a fixed ambient development. Clause (c) is the definition of a corresponding boundary pair; the equivalent characterisation by timelike curves is proved, not assumed, in Structure of corresponding boundary points: achronality of the future boundary, equality of pasts, and uniqueness.