Chronological and causal future

Definition

Let $(M,g)$ be a time-oriented Lorentzian manifold. The chronological future and causal future of $p\in M$ are

I^+(p)=\{q\in M:\text{a future-directed timelike curve runs from \(p\) to \(q\)}\},

J^+(p)=\{p\}\cup \{q\in M:\text{a future-directed [[differential-geometry/causal-curve|causal curve]] runs from \(p\) to \(q\)}\}.

The chronological and causal pasts $I^-(p)$ and $J^-(p)$ are defined with past-directed curves.

For $A\subseteq M$ , set

I^\pm(A)=\bigcup_{p\in A}I^\pm(p), \qquad J^\pm(A)=\bigcup_{p\in A}J^\pm(p).

One always has $I^\pm(p)\subseteq J^\pm(p)$ . The chronological future is open, whereas the causal future need not be closed without additional causal hypotheses.

The intersection $J^+(p)\cap J^-(q)$ is the causal diamond between $p$ and $q$ .

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983. Publisher record. Relevant: Chapter 14.
Ettore Minguzzi, “Lorentzian causality theory,” Living Reviews in Relativity 22 (2019), article 3. Journal record.