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Typically, a time-orientation of a Lorentzian manifold $(M,g)$ is defined in one of two ways, namely:

  1. A globally-defined timelike vector field $X$ on $M$, or
  2. A smooth choice of time-orientations at each $p \in M$.

where by "time-orientation" we mean a connected component of the subset of all timelike vectors in $T_pM$. It can be shown that these two definitions are equivalent.

My question: what is the mathematical definition of this "smooth choice" used in Def 2 above?

I guess it would be some kind of map from $M$ to the space of all time-orientations of all points in $M$, somehow equipped with a smooth structure.. but how is this done?

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At any given point in the manifold $M$, the associated Lorentzian metric field $g$ defines which tangent vectors are timelike, lightlike, or spacelike. At the given point, the space of timelike tangent vectors has two disconnected components. We can choose one of these to be "future-pointing" and the other to be "past-pointing." Suppose that we make such a choice independently for every individual point in $M$. The question is, how do we determine whether or not this collection of choices is "smooth"?

We already have a way to define "smooth" for a vector field. (The usual coordinate-independent definition of "vector field" has smoothness built into it, and I'm taking that for granted here.) Given a point $p\in M$, if $p$ has a neighborhood in which a smooth vector field can be chosen so that it is future-pointing everywhere within that neighborhood (according to our previous choices), then we say that our choice of "future-pointing" is smooth at $p$. If it is smooth at all points of $M$, then we have defined a time-orientation.

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