# 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?

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".
• Ok, thank you for the detailed answer. I have some question left: Do I understand it right that the two components of $T_pM$ are $A,B \subset T_pM$, $A \cup B=T_pM$ s.t. for all pairs $a_1,a_2$ elements in $A$ it holds $<a_1,a_2><0$, for all pairs $b_1,b_2$ elements in $B$ it holds $<b_1,b_2><0$ and for all $a\in A, b \in B$ it holds $<a,b>>0$? And then, to define future/past-directed vectors we choose a specific vector field $X$? Does that mean that the definition of future-directed depend on the choice of $X$? – User1 Jun 8 at 18:53
• The components $A$ and $B$ are not of the whole $T_pM$, but only of the collection of timelike vectors. Other than that, everything you said is correct. – Ivo Terek Jun 8 at 19:03