Future sets, past sets, achronal boundaries and their elementary properties

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with the chronological and causal relations $\ll$, $\le$ and the sets $I^{\pm }$, $J^{\pm }$ of Causal curves, chronological and causal reachability, and inextendibility.

(a) Future and past sets. A subset $F\subseteq M$ is a future set when $I^{+}(F)\subseteq F$, where $I^{+}(F)={\bigcup }_{p\in F}{I}^{+}(p)$; it is a past set when $I^{-}(F)\subseteq F$. A future set is open when it is open in the manifold topology; for an open future set one has $F=I^{+}(F)$.

(b) Achronal boundary. A set $A\subseteq M$ is an achronal boundary when $A=\mathrm{\partial }F$ for a future set $F$; this is equivalent to being the boundary of an open future set. Indeed, put $H:=I^{+}(F)$. Chronological openness makes $H$ open, and the future-set property gives $H\subseteq F$. Arbitrarily short future timelike segments from each $p\in F$ give $p\in \bar{H}$, so $\bar{H}=\bar{F}$. If $p\in \mathrm{Int}F$, a sufficiently short past timelike segment ending at $p$ starts inside $F$, hence $p\in H$; therefore $H=\mathrm{Int}F$. It follows that $\mathrm{\partial }H=\bar{H}\setminus H=\bar{F}\setminus \mathrm{Int}F=\mathrm{\partial }F$. The short segments and openness are supplied by The geodesic initial-value problem and the exponential map of the metric-selected connection and Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets ↗.

The derived elementary properties and boundary equivalence use Countable Choice and Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain) through the stated causal-calculus, chronology and push-up suppliers. The future/past-set predicates themselves are the set definitions above.

(c) Elementary properties. The following are immediate from the openness of $I^{\pm }$ and the transitivity of $\ll$ (Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets ↗ clause (a), Chart independence, concatenation, reparametrization and transitivity for causal curves clauses (d) and (f)):

• if $F$ is a future set then its complement is a past set, and conversely;
• $I^{\pm }(S)$, $J^{\pm }(S)$ and $\overline{J^{\pm }(S)}$ are future (respectively past) sets for every $S\subseteq M$;
• if $F$ is a future set then $F$ is open if and only if $I^{+}(F)=F$, and then $\mathrm{\partial }F$ is achronal. Indeed, if $p,q\in \mathrm{\partial }F$ and $p\ll q$, openness of the chronological relation gives a neighborhood $V$ of $p$ such that every $p^{\mathrm{\prime }}\in V$ satisfies $p^{\mathrm{\prime }}\ll q$. Since $p\in \bar{F}$, choose $p^{\mathrm{\prime }}\in V\cap F$. The future-set property gives $q\in I^{+}(F)\subseteq F$, contradicting $q\in \mathrm{\partial }F$;
• a set $S$ is achronal in the sense of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (b) if and only if $\bar{S}$ is achronal, because $I^{\pm }$ is open in its arguments.

(d) Scope. These are set-theoretic definitions inside the smooth Lorentzian model; they assert nothing about existence of achronal boundaries with additional regularity, and they carry no physical interpretation here.

• Well-definedness. No choice principle beyond the stated assumptions is used. Transitivity proves the future/past-set property for $I^{\pm }(S)$; the push-up rule supplies the corresponding property for $J^{\pm }(S)$, and openness of chronology passes it to their closures (Chart independence, concatenation, reparametrization and transitivity for causal curves, The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion ↗, Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets ↗). For the boundary claim, the needed openness is in the first endpoint: if $p\ll q$, then $I^{-}(q)$ is an open neighborhood of $p$, so it meets $F$ when $p\in \mathrm{\partial }F$. To show an open future set satisfies $F=I^{+}(F)$, choose for each $p\in F$ a sufficiently short past timelike geodesic segment ending at $p$ and contained in $F$ (The geodesic initial-value problem and the exponential map of the metric-selected connection). The statement that achronal boundaries are locally Lipschitz graphs is not part of this definition; no regularity of achronal boundaries beyond closedness and achronality is asserted here.