I apologize in advance if this ia too basic question.

Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-orientation, i.e. a global timelike vector field. In other words, $g_{p}(X_{p},X_{p})<0$ for all $p\in M$. Now, it is well-known that this allows me to split the space of timelike tangent vectors $v\in TM$ into two classes:

• future-directed (or future-pointing) if $g(v,X)<0$
• past-directed (or past-pointing) if $g(v,X)>0$

So far so good. Now, my confusion arises because in many references, people use the concept of "future-directed" and "past-directed" also for covectors $\xi\in T^{\ast}M$, without explaining the concept. I was trying to search the internet for quite some while but it was never explicitely explained. My problem is essentially that there are two possibilities:

1. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is future (resp. past) directed;
2. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is past (resp. future) directed;

Both definitions, which are exactly the opposite, seem meaningful to me. While Definition 1. seems a bit more "symmetric", the second one also kind of makes sense, since lowering/raising indices somehow involves a sign flip. Furthermore, in the second definition, $\xi(v)>0$ for $\xi,v$ future-directed and $\xi(v)<0$ for $\xi,v$ past-directed.

Example: To make it a bit more clear: Consider Minkowski spacetime $\eta=-dt\otimes dt+\sum_{i}dx^{i}\otimes dx^{i}$

Then, a global time-orientation is provided by the canonical choice $X:=\partial_{t}$. In particular, $X^{\flat}=-dt$ due to the sign convention. Now, Definition 1. would hence imply $dt$ to be future-directed, while Definition 2. implies that $dt$ is past-directed. So, which one is usually used?

I apologize in advance if this is a too basic question.

Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-orientation, i.e. a global timelike vector field. In other words, $g_{p}(X_{p},X_{p})<0$ for all $p\in M$. Now, it is well-known that this allows me to split the space of timelike tangent vectors $v\in TM$ into two classes:

• future-directed (or future-pointing) if $g(v,X)<0$
• past-directed (or past-pointing) if $g(v,X)>0$

So far so good. Now, my confusion arises because in many references, people use the concept of "future-directed" and "past-directed" also for covectors $\xi\in T^{\ast}M$, without explaining the concept. I was trying to search the internet for quite some while but it was never explicitely explained. My problem is essentially that there are two possibilities:

1. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is future (resp. past) directed;
2. You call $\xi$ future (resp. past) directed if $\xi^{\sharp}$ is past (resp. future) directed;

Both definitions, which are exactly the opposite, seem meaningful to me. While Definition 1. seems a bit more "symmetric", the second one also kind of makes sense, since lowering/raising indices somehow involves a sign flip. Furthermore, in the second definition, $\xi(v)>0$ for $\xi,v$ future-directed and $\xi(v)<0$ for $\xi,v$ past-directed.

Example: To make it a bit more clear: Consider Minkowski spacetime $\eta=-dt\otimes dt+\sum_{i}dx^{i}\otimes dx^{i}$

Then, a global time-orientation is provided by the canonical choice $X:=\partial_{t}$. In particular, $X^{\flat}=-dt$ due to the sign convention. Now, Definition 1. would hence imply $dt$ to be future-directed, while Definition 2. implies that $dt$ is past-directed. So, which one is usually used?