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 $M\simeq S\times \mathbb{R}$ so that the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to S$ is constant and the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to \mathbb{R}$ 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 $M\simeq \mathbb{R}\times S$ 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.

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 $M\simeq S\times \mathbb{R}$ so that the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to S$ is constant and the map $\mathbb{R}\to M\simeq S\times \mathbb{R}\to \mathbb{R}$ 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. As far as

Added: I can telldon'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 not implied by stable causality; global hyperbolicity seems to imply that there exists 1 enough. In particular, I am asking for the following---for every timelike path $f$ that can be factored outf: \mathbb{R}\to M$, but not that any there is a product structure$M\simeq \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$. This is much stronger than Geroch's splitting theorem, as far as I can be factored outtell. 2 Responded to answers. First of all, my knowledge of both GR and differential geometry is quite weak, so forgive me if the physics here doesn't make much sense. Let$(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$and an isomorphism$M\simeq S\times \mathbb{R}$so that the map$\mathbb{R}\to M\simeq S\times \mathbb{R}\to S$is constant and the map$\mathbb{R}\to M\simeq S\times \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. As far as I can tell, this is not implied by stable causality; global hyperbolicity seems to imply that there exists 1 timelike$f$that can be factored out, but not that any such$f\$ can be factored out.

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