Quoting from this Wikipedia article, if $(M,g)$ is a Lorentzian manifold then the tangent vectors at each point in the manifold can be classified into three different types. Using a $(+,-,-,-)$ metric signature, a tangent vector $X$ is
- timelike if $g(X,X)>0$
- null if $g(X,X)=0$
- spacelike if $g(X,X)<0$
The article then states that if $X$ and $Y$ are two timelike tangent vectors at a point $P$ of $M$, then we say that $X$ and $Y$ are equivalent (written $X\sim Y$) if $g(X,Y)>0$. It turns out (this is related to my question) that for each point there are two equivalence classes, which between them contain all timelike tangent vectors at that point. We then call one of these equivalence classes "future-directed" and the other "past-directed".
Question: How do we know that there are exactly two equivalence classes at each point? This could be stated mathematically as
For any three timelike tangent vectors $X$, $Y$ and $Z$ to a point $P$ of a Lorentzian manifold $(M,g)$, if we have $X\nsim Y$ and $X\nsim Z$, is it necessarily true that $Y\sim Z$? (using the equivalence relation defined earlier)
Disclaimer: I suspect this is a rather trivial result of the topology and geometry of Lorentzian manifolds, but I know very little of the mathematics of these two fields, so if your answer uses any terms other than those defined here then I'd greatly appreciate it if you could provide a definition of those terms in your answer.