Causal structure

Definition

Let $(M, g)$ be a Lorentzian manifold of dimension $n \ge 2$ with signature $(-, +, \dots, +)$. A tangent vector $v \in T_pM$ is timelike if $g(v, v) < 0$, null if $g(v, v) = 0$ and $v \ne 0$, causal if it is timelike or null, and spacelike if $g(v, v) > 0$. A time orientation is a continuous timelike vector field $\tau$ on $M$; given one, a causal vector $v$ is future-directed if $g(v, \tau) < 0$. Assume $(M, g)$ is time-oriented. A piecewise smooth curve is a future-directed timelike (resp. causal) curve if all its tangent vectors are future-directed timelike (resp. causal). At break points this is required of both one-sided tangents. For $p, q \in M$:

1. $p \ll q$ if there is a future-directed timelike curve from $p$ to $q$;
2. $p \le q$ if $p = q$ or there is a future-directed causal curve from $p$ to $q$;
3. the chronological future of $p$ is $I^+(p) = \set{q : p \ll q}$ and the causal future is $J^+(p) = \set{q : p \le q}$; the pasts $I^-(p)$ and $J^-(p)$ are defined with the reversed relations.

The causal structure of $(M, g)$ is the pair of relations $\ll$ and $\le$ on $M$.