First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense.
Let $(M, g)$ be a smooth, connected Lorentzian manifold of dimension $n$. Let $f: mathbb{R}to M$ be a smooth curve such that the pullback of $g$ through $f$ is everywhere negative (where we've chosen an orientation on $mathbb{R}$); we say that $f$ is time-like. Say that we can "factor" $f$ out of $M$ if there exists a manifold $S$ of dimension $n-1$ and an isomorphism $Msimeq Stimes mathbb{R}$ so that the map $mathbb{R}to Msimeq Stimes mathbb{R}to S$ is constant and the map $mathbb{R}to Msimeq Stimes mathbb{R}to mathbb{R}$ is the identity. Intuitively, this factorization exhibits $f$ as "time" in some reference frame, and $S$ as space. My question is:
For which $(M, g)$ can every time-like path be factored out?
Minkowski space seems like an obvious example unless I'm missing something; it seems one can take a tangent vector to $f$ at any point and consider a perpendicular subspace to that vector as $S$. I'd accept as an answer a characterization of all such $(M, g)$ in dimension $4$, or some nice sufficient condition on $M$ for factorization to always work.
If the motivation isn't obvious already, this is supposed to codify the intuition that in my reference frame, I seem to be standing still -- and that the same is true for everyone else, even if they seem to me to be moving. My apologies if I've overloaded terms, or used them incorrectly.
Added: Note that this condition is much stronger than stable causality; indeed, it certainly implies stable causality, as choosing any timelike path $f$ and then considering the given projection to $mathbb{R}$ gives a global time function. However, I am asking for (1) a product structure on $M$ for each path $f$ and (2) in order to formalize the notion that I seem to be standing still (to myself), the projection of $f$ to $S$ must be constant.
Added: I don't think global hyperbolicity suffices either. The theorem of Geroch (it and other splitting theorems are discussed here, for example) does indeed give a decomposition of $M$ as $mathbb{R}times S$. But I don't think this is enough. In particular, I am asking for the following---for every timelike path $f: mathbb{R}to M$, there is a product structure $Msimeq mathbb{R}times S$ such that the projection to $mathbb{R}$ is a section of $f$, and that $f$ is constant upon projection to $S$. This is much stronger than Geroch's splitting theorem, as far as I can tell.
Added: As the accepted answerer rightly points out in the comments to his question, I was wrong to claim that my condition is stronger than global hyperbolicity. They are in fact equivalent.
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