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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 isenter image description here

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?

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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".

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  • $\begingroup$ 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 $X$? $\endgroup$ – User1 Jun 8 at 18:53
  • $\begingroup$ 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. $\endgroup$ – Ivo Terek Jun 8 at 19:03
  • $\begingroup$ Oh yes of course thanks. And if, as in the definition from Wikipedia, no concrete vector field is determined as the vector field from which we define future/past-directed vectors, does that mean this vector field is random or is there a concrete vector field which is meant if none is mentioned (my english is maybe not that good here, I hope ypu can understand me, I mean something like a default vector field which is meant ) $\endgroup$ – User1 Jun 8 at 19:14
  • $\begingroup$ If we say that a Lorentzian manifold is time-oriented without explicit mention to the distinguished timelike field, it means that one such field has been chosen and fixed a priori throughout the entire discussion. $\endgroup$ – Ivo Terek Jun 8 at 19:57
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    $\begingroup$ Ok, thank you very much! $\endgroup$ – User1 Jun 9 at 9:05

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