Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the causal curves of Causal curves, chronological and causal reachability, and inextendibility, and let $S\subseteq M$.

(a) Spacelike hypersurfaces. A subset $\mathrm{\Sigma }\subseteq M$ is a spacelike embedded hypersurface when it is an embedded three-dimensional submanifold of $M$ (Embedded submanifolds and slice charts) such that the restriction of $g_p$ to $T_p\mathrm{\Sigma }\times T_p\mathrm{\Sigma }$ is positive definite for every $p\in \mathrm{\Sigma }$; equivalently, every nonzero vector tangent to $\mathrm{\Sigma }$ is spacelike (Timelike, null, spacelike and causal vectors of a Lorentz vector space). A spacelike embedded hypersurface is automatically a nondegenerate hypersurface, so its $g$-orthogonal complement in $T_{p}M$ is one-dimensional and nondegenerate, and it carries a unique future-directed unit timelike normal; that normal is constructed where it is used, not here.

(b) Achronality and acausality. $S$ is achronal when no two distinct points $p,q\in S$ satisfy $p\ll q$ or $q\ll p$; $S$ is acausal when no two distinct points $p,q\in S$ satisfy $p\le q$ or $q\le p$. Every acausal set is achronal, since $p\ll q$ implies $p\le q$.

(c) Cauchy sets. $S$ is a Cauchy set when for every inextendible causal curve $\gamma :I\to M$ the parameter set ${\gamma }^{-1}[S]:=\{t\in I:\gamma (t)\in S\}$ has exactly one element. Because a strictly increasing continuous surjection of intervals induces a bijection of parameter sets, this condition is invariant under reparametrization (Chart independence, concatenation, reparametrization and transitivity for causal curves clause (e)); it is stated in terms of parameters rather than of the point set $\gamma (I)\cap S$, so that a closed causal curve counts two visits to the same point as two parameter values. A Cauchy hypersurface is a Cauchy set that is also a spacelike embedded hypersurface in the sense of clause (a).

(d) Domains of dependence. For any $S\subseteq M$ define $D^{+}(S):=\{p\in M:\text{every past-inextendible causal curve through }p\text{ meets }S\},$ and dually $D^{-}(S):=\{p\in M:\text{every future-inextendible causal curve through }p\text{ meets }S\}$, where "meets $S$" means that some parameter value is mapped into $S$, and $D(S):=D^{+}(S)\cup D^{-}(S)$ is the domain of dependence of $S$.

(e) Cauchy horizon. For a closed set $S\subseteq M$ the future Cauchy horizon is $H^{+}(S):=D^{+}(S)\setminus I^{-}(D^{+}(S)),$ the set of points of $D^{+}(S)$ that are not chronologically preceded by any point of $D^{+}(S)$; dually $H^{-}(S):=D^{-}(S)\setminus I^{+}(D^{-}(S))$, and the Cauchy horizon is $H(S):=H^{+}(S)\cup H^{-}(S)$.

(f) Scope. These are mathematical definitions. Neither existence of Cauchy sets nor any relation between the objects defined here is asserted.