Can spacetime with a compact dimension be foliated into spatial hypersurfaces?

[shoehorn]

I'm familiar with how one foliates, say, a four-dimensional spacetime $(\mathcal{M},g)$ so that one can identify it as a sequence of spatial hypersurfaces $\mathcal{M}\simeq\Sigma\times I$ where $\Sigma$ is some spatial three-manifold and $I\subseteq\mathbb{R}$. However, suppose that we're interested in a spacetime which has at least one compact dimension. For clarity, let's look at

$$\mathcal{N} \simeq \mathcal{M}\times S^1,$$

i.e., something like a Kaluza-Klein spacetime with one compact periodic dimension per spacetime point. Does anybody know if any work has been done on foliating a spacetime like this into spatial hypersurfaces? Is this even possible?

 

[Chris Hillman]

Periodic directions?

Hi, Shoehorn, since you mentioned "spatial hyperslices", I take it that you have in mind a periodic timelike coordinate vector field. If so, consider Minkowski spacetime with $$t=0$$ and $$t=1$$ identified. Is that the kind of thing you have in mind?

Chris Hillman

 

[shoehorn]

Hi, Shoehorn, since you mentioned "spatial hyperslices", I take it that you have in mind a periodic timelike coordinate vector field. If so, consider Minkowski spacetime with $$t=0$$ and $$t=1$$ identified. Is that the kind of thing you have in mind?

Chris Hillman

No, not really. I probably should have been a little bit more specific. Let's take a, say, five-dimensional spacetime $\mathcal{N}$. I construct this spacetime so that it has the special property that, for some "standard" Lorentzian spacetime $\mathcal{M}$, we can say $\mathcal{N}=\mathcal{M}\times S^1$. Now let me propose that the compact dimension in $\mathcal{N}$ is spatial, so that we can think of $\mathcal{N}$ as being essentially the same spacetime we encounter in Kaluza-Klein theory. I then have two questions about this spacetime:

(1) Does the presence of a compact spatial dimension in $\mathcal{N}$ present any obstacle to foliating it by spacelike hypersurfaces so that we have $\mathcal{N}\simeq\Sigma\times S^1\times I$, for $I\subseteq\mathbb{R}$?

(2) If not, does anybody have a link to where this has been done in the literature? I find it hard to believe that somebody hasn't published something on this question, but can't seem to find anything.

 

[josh1]

Hi shoehorn,

Firstly, a foliation of spacetime is simply a choice of time coordinate with respect to which the hypersurfaces are surfaces of simultaneity. The topology of these hypersurfaces doesn`t matter. They can be noncompact or compact or some product that includes both types, which is the case in your own example MxS. So consider some particle in your example spacetime with M being three dimensional. To completely specify it`s position on the hypersurface at a specific time t requires both the coordinates [x(t),y(t),z(t)] for it`s position on M, and the coordinate s(t) of it`s position on the circle S. Thus the coordinates on MxS at time t are [x(t),y(t),z(t),s(t)] so that t parametrizes the foliation.

 

[coalquay404]

Hi shoehorn,

Firstly, a foliation of spacetime is simply a choice of time coordinate with respect to which the hypersurfaces are surfaces of simultaneity.

I'm going to be brutal here: the above statement is utterly, completely, woefully incorrect. A foliation of a spacetime $\mathcal{M}$ is not simply some clever "choice of time coordinate" that gives you surfaces of simultaneity. A foliation of a spacetime is a very precise process whereby one can topologically identify the spacetime with, for example, $\Sigma\times I$, where $\Sigma$ is some spatial three-surface. There are, of course, other possibilities such as null foliations, but since in GR the most common one is as a foliation by three-surfaces that's the one I'll deal with here.

More precisely, the existence of a foliation is the same as saying that there exists some one-parameter family of embeddings $\varphi_t:\Sigma\to\mathcal{M}$, $t\in\mathbb{R}$ such that $\Sigma\times I$ is diffeomorphic to $\mathcal{M}$. Admittedly, the use of the subscript $t$ on the embedding is suggestive of some notion of "equal time hypersurfaces" but this is of course a matter of choice.

The topology of these hypersurfaces doesn`t matter. They can be noncompact or compact or some product that includes both types, which is the case in your own example MxS.

Again, this is incorrect. Please, please, please, if this is your understanding of foliations of a spacetime, take a look at the (voluminous) literature. The review articles of Isham and Thiemann should be particularly suited to disabusing you of your confusion regarding the allowable topologies of the hypersurfaces and their relation to the topology of the spacetime.

 

[josh1]

I'm going to be brutal here: the above statement is utterly, completely, woefully incorrect. A foliation of a spacetime $\mathcal{M}$ is not simply some clever "choice of time coordinate" that gives you surfaces of simultaneity. A foliation of a spacetime is a very precise process whereby one can topologically identify the spacetime with, for example, $\Sigma\times I$, where $\Sigma$ is some spatial three-surface. There are, of course, other possibilities such as null foliations, but since in GR the most common one is as a foliation by three-surfaces that's the one I'll deal with here.

More precisely, the existence of a foliation is the same as saying that there exists some one-parameter family of embeddings $\varphi_t:\Sigma\to\mathcal{M}$, $t\in\mathbb{R}$ such that $\Sigma\times I$ is diffeomorphic to $\mathcal{M}$. Admittedly, the use of the subscript $t$ on the embedding is suggestive of some notion of "equal time hypersurfaces" but this is of course a matter of choice.

Hi coalquay404,

Your remarks hold for the general notion of a foliation independent of any metrical structure on the space being foliated. In particular you draw attention only to the topological and differentiable structure. But we are talking about solutions of the gravitational field equations of general relativity so the additional stipulation that the hypersurfaces be surfaces of simultaneity is really just a statement about the physics. However...

Again, this is incorrect. Please, please, please, if this is your understanding of foliations of a spacetime, take a look at the (voluminous) literature. The review articles of Isham and Thiemann should be particularly suited to disabusing you of your confusion regarding the allowable topologies of the hypersurfaces and their relation to the topology of the spacetime.

Certainly solutions of the gravitational field equations with positive and vanishing cosmological constants can be foliated by surfaces of simultaneity that are compact and noncompact respectively. But solutions with negative cosmological constants aren`t necessarily globally hyperbolic and may not admit the kind of foliations I was talking about, so perhaps I should've been more careful. It just seemed like shoehorn thought that compact components present some special problem with respect to foliations and I just wanted to get the message across that this wasn`t so. Sorry if I made your blood boil.

 

[coalquay404]

In response to the OP, the following paper appeared on the ArXiv this morning and might be of use:

http://arxiv.org/abs/gr-qc/0612118