Question on definition of time-orientability and future-directed curves I can't properly understand the definitions for a future directed tangent vector.
Now I know the following definitions:
A spacetime $(M,G)$ is called time-orientable, if there exists a vector field $X$ s.t. $g(X,X)<0$.
Now the definition for a future directed curve (from Wikipedia) is:
A timelike curve $c:I \rightarrow M$ (timelike means $g(c^{'},c^{'})<0$) is called future directed if $c^{'}$ is future directed.
And the definition for a future directed tangent vector is
I can't see why you get $\underline{2} $ equivalence classes here..? Is that obvious? And how can it be random which class you call future directed?
 A: In any Lorentzian manifold $(M,g)$, the index of the metric is $1$, so there's no way we can have two orthogonal timelike vectors. Otherwise they together would span a negative subspace of dimension $2$. Because of this, given $p \in M$ and $v,w \in T_pM$ timelike vectors, there are only two choices: $g_p(v,w)<0$ or $g_p(v,w) > 0$. We say that $v$ and $w$ have the same orientation if $g_p(v,w) < 0$. It is important to note that here we're not talking about vectors being future or past-directed yet. This says that in each tangent space $T_pM$, the subset consisting of all timelike vectors has two components, and $g_p(v,w)<0$ effectively says that $v$ and $w$ are in the same component. This is not enough, however, to make a smooth choice of components of the timecone along the entire manifold. This can be made if we have a distinguished timelike vector field $X$ defined globally -- this vector field will make the choice. Then we say that $v \in T_pM$ is future-directed if $g_p(v,X_p)<0$, and past-directed if $g_p(v,X_p)>0$. The choice is "random" in the sense that choosing $-X$ instead of $X$, future and past would be switched, but there's no reason to prefer one field over the other whatsoever.
Take as a particular example Lorentz-Minkowski space $\Bbb R^3_1 = (\Bbb R^3, {\rm d}x^2+{\rm d}y^2-{\rm d}z^2)$. The lightcone is described by the equation $z^2 = x^2+y^2$, and here you can see the timecones clearly. For example, if we choose $X$ to be the timelike field $X = \partial/\partial z$, future-directed timelike vectors point "up" and past-directed timelike vectors point "down".
