Causality conditions

Definition

Let $(M, g)$ be a time-oriented Lorentzian manifold, with chronological and causal futures and pasts $I^\pm(p)$, $J^\pm(p)$ as in the causal structure. The main causality conditions are the following.

1. $(M, g)$ is chronological if it contains no closed timelike curve.
2. $(M, g)$ is causal if it contains no closed future-directed causal curve.
3. $(M, g)$ is distinguishing if for all $p, q \in M$, $I^+(p) = I^+(q)$ implies $p = q$, and $I^-(p) = I^-(q)$ implies $p = q$.
4. $(M, g)$ is strongly causal if every point has arbitrarily small open neighbourhoods $U$ that no causal curve leaves and then re-enters.
5. $(M, g)$ is stably causal if, for some continuous timelike vector field $X$, the metric $g - X^\flat \otimes X^\flat$ has no closed timelike curve. Here $X^\flat = g(X, \cdot)$; the new metric has wider light cones than $g$.
6. $(M, g)$ is causally simple if it is causal and the sets $J^+(p)$ and $J^-(p)$ are closed for every $p$.