Time-orientability of Lorentzian manifolds

[loislane]

A spacetime is said to be time-orientable if a continuous designation of which timelike vectors are to be future/past-directed at each of its points and from point to point over the entire manifold. [Ref. Hawking and Israel (1979) page 225]

I want to make sure what conditions must hold in order for a given Lorentzian manifold to be time-orientable,i.e.for the existence of a nonvanishing timelike vector field at each point in the manifold, to express the above in different words.
Such a timelike vector field can easily be seen to exist for those Lorentzian manifolds like Minkowski spacetime for instance, with a timelike Killing vector field, but absent such symmetry, for singular spacetimes(say FRW) there is a lack of continuity at singularities where the vector field vanishes that should prevent the existence of a nowhere vanishing timelike vector field. Or under what circumstances is this avoided?

 

[Orodruin]

You can divide the tangent bundle of the manifold into a timelike and a spacelike bundle. The manifold time orientable if the timelike vector bundle has a trivial first Stiefel-Whitney class.

 

[loislane]

You can divide the tangent bundle of the manifold into a timelike and a spacelike bundle. The manifold time orientable if the timelike vector bundle has a trivial first Stiefel-Whitney class.

This definition just restates what a time orientable manifold is. I asked how is the continuity requirement dodged in singular manifolds. Just by defining the manifold leaving out the singularities?

 

[PeterDonis]

Just by defining the manifold leaving out the singularities?

Yes. But that isn't a "dodge"; it's part of the definition of a manifold. The "singularities" you are referring to--for example, the "initial singularity" in an FRW solution--are not actually part of the manifold. The precisely correct way to describe what's going on would be to say that curvature invariants increase without bound as a finite affine parameter is approached along particular geodesics; but that finite value of affine parameter is never actually reached within the manifold.

 

[George Jones]

time-orientability doesn't directly depend on the metric structure, only on the topology

Are you sure about this? It seems to me that this can't possibly be true.

 

[PeterDonis]

Time-orientability is a topological property.

This can't be true. Minkowski spacetime and Godel spacetime both have topology $R^4$, but the former is time orientable and the latter is not (it can't be because it contains closed timelike curves).

 

[bcrowell]

Thanks, George Jones and PeterDonis, for the corrections. I've deleted #5, which had wrong content.

If I'm not still mistaken, orientability is a purely topological notion, but now that I think about it, time-orientability obviously can't be, because without the metric, you can't say whether a particular direction is timelike.

This can't be true. Minkowski spacetime and Godel spacetime both have topology $R^4$, but the former is time orientable and the latter is not (it can't be because it contains closed timelike curves).

I don't see how the existence of CTCs indicates a lack of time-orientability. For instance, suppose I take a spacelike strip of Minkowski space and wrap it around into a cylinder. It's still got the same time orientation, but now it has CTCs.

I do think that the existence of a Penrose diagram for a spacetime indicates that it's time-orientable.

 

[loislane]

If I'm not still mistaken, orientability is a purely topological notion,

It's not. It depends.on the differentiable structure.

 

[bcrowell]

It's not. It depends.on the differentiable structure.

OK, but not on the metric, I would think...?

 

[loislane]

Yes. But that isn't a "dodge"; it's part of the definition of a manifold. The "singularities" you are referring to--for example, the "initial singularity" in an FRW solution--are not actually part of the manifold.

I see. There is not much choice left anyway, the alternative is dismissing the singular manifold as pathological like is tipically done in Riemannian geometry.

 

[loislane]

OK, but not on the metric, I would think...?

Only in the pseudo-Riemannian case AFAIK.

 

[bcrowell]

Only in the pseudo-Riemannian case AFAIK.

If we're just talking about orientability (not time-orientability), then I don't think it makes any difference whether the manifold is Riemannian or semi-Riemannian, because the metric is irrelevant.

 

[PAllen]

Thanks, George Jones and PeterDonis, for the corrections. I've deleted #5, which had wrong content.

If I'm not still mistaken, orientability is a purely topological notion, but now that I think about it, time-orientability obviously can't be, because without the metric, you can't say whether a particular direction is timelike.
I don't see how the existence of CTCs indicates a lack of time-orientability. For instance, suppose I take a spacelike strip of Minkowski space and wrap it around into a cylinder. It's still got the same time orientation, but now it has CTCs.

I do think that the existence of a Penrose diagram for a spacetime indicates that it's time-orientable.

Extreme example: Godel spacetime is time orientable but has CTC through every point of the manifold:

http://www.math.nyu.edu/~momin/stuff/grpaper.pdf

 

[PeterDonis]

I don't see how the existence of CTCs indicates a lack of time-orientability. For instance, suppose I take a spacelike strip of Minkowski space and wrap it around into a cylinder. It's still got the same time orientation, but now it has CTCs.

Yes, I see how with a topology like this you can have time orientability and CTCs. I'm still going to have to think about Godel spacetime some more, in view of PAllen's post; I'm not sure I see how $R^4$ topology and CTCs are compatible with time orientability.

I do think that the existence of a Penrose diagram for a spacetime indicates that it's time-orientable.

Is there one for Godel spacetime?

 

[bcrowell]

Is there one for Godel spacetime?

Dunno. I think the existence of a Penrose diagram should be sufficient for time orientability, but it's certainly not necessary. A Penrose diagram only works for a conformally flat metric.

 

[samalkhaiat]

Whether or not a spacetime $(M^{4} , g_{ab})$ is time- or space-orientable depends not just on its Lorentz metric, $g_{ab}$, but also on the topology of the underlying manifold $M^{4}$. In fact one can show that any non-compact manifold can be the underlying manifold for a time-orientable spacetime and also for a space-orientable spacetime. Also, if $M^4$ is not simply connected, then there exists a spacetime $(M^{4} , g_{ab})$ which is not time-orientable, and one which is not space-orientable.

These issues are investigated thoroughly by R. Geroch and G. T. Horowitz in:

“General Relativity” An Einstein Centenary Survey, Edited by S. Hawking & W. Israel: Chapter 5 “Global structure of spacetimes”

 

[PAllen]

A more authoritative source confirming the time orientability of Godel spacetime is the following (by Malament, in book edited by Ashtekar):

https://books.google.com/books?id=izr6CAAAQBAJ&pg=PA188&lpg=PA188&dq=time+orientable+Godel&source=bl&ots=UnGBPYumip&sig=cQyBXmrwIf3J3DkNCJHlr4iFqTU&hl=en&sa=X&ved=0CD4Q6AEwBzgKahUKEwjW766Wh6jHAhXFFT4KHYX1AdI#v=onepage&q=time orientable Godel&f=false

The following makes the (to me) somewhat surprising claim that any simply connected manifold with any Lorentzian metric is time orientable (possibly unstated technical assumptions are made, e.g. smooth 4-d manifold; differentiability of the metric, etc.):

http://www.pitt.edu/~jdnorton/teaching/2675_time/handouts/Godel's Argument for the Ideality of Time.pdf

 

[bcrowell]

The following makes the (to me) somewhat surprising claim that any simply connected manifold with Lorentzian metric is time orientable:

This doesn't seem so surprising to me. The fact that you can even put a Lorentzian metric on it already tells you something nontrivial about its topology. For instance, a sphere isn't compatible with a Lorentzian metric, the reason for which I think is probably similar to the idea of the hairy ball theorem.

Wikipedia's article "Orientability" has a nice characterization of time orientability, which is that if two observers meet up at event A, agree on the direction of time at A, and also meet at B, then they also agree on the direction of time at B. The two world-lines, taken together, form a closed curve. If the manifold is simply connected, then it seems like you could contract the curve to a point, and then you wouldn't even *locally* be able to define past and future, which can't be the case.

 

[bcrowell]

It also turns out a compact manifold is compatible with a Lorentzian metric if and only if it has Euler characteristic zero: http://mathoverflow.net/a/47446/21349 . So for example a Klein bottle is compatible with a Lorentzian metric, but a sphere is not.

It would be interesting to know whether there are manifolds that admit a Lorentzian metric, but do not admit a time-orientable one. For example, I wonder if a Klein bottle admits a time-orientable metric.

 

[Ben Niehoff]

This can't be true. Minkowski spacetime and Godel spacetime both have topology $R^4$, but the former is time orientable and the latter is not (it can't be because it contains closed timelike curves).

As Orodruin pointed out, it's a topological property of the subbundle of timelike vectors in the tangent bundle $TM$. The particular splitting of $TM$ into timelike, spacelike, and null vectors is what is known as the causal structure, which is more general the metric structure.

 

[PeterDonis]

The particular splitting of $TM$ into timelike, spacelike, and null vectors is what is known as the causal structure, which is more general the metric structure.

Understood. I'm still trying to wrap my mind around how it works for Godel spacetime, but that's a separate issue.

 

[loislane]

As Orodruin pointed out, it's a topological property of the subbundle of timelike vectors in the tangent bundle $TM$.

To avoid further confusion: Although the Stiefel-Whitney class is indeed a topological invariant of real vector bundles, the tangent bundle itself is in the smooth manifolds category so we are talking about topological information of smooth manifolds. A topological property is something that is preserved by homeomorphisms. If you define orientability in terms of the tangent bundle, I can see no straight-forward way to speak properly about it only in terms of topological properties since homeomorphisms are not necessarily differentiable. Certain manifolds(like 4-manifolds) have multiple incompatible differentiable structures. I think one has to use the homology theory to define orientability as a topological property only rather than tangent bundles. On the other hand, one can start from a topological property like simply-connectedness to derive time-orientability of the Lorentzian manifold with that topology.

The particular splitting of $TM$ into timelike, spacelike, and null vectors is what is known as the causal structure, which is more general the metric structure.

Yes, that would be the conformal class up to scale, but I would add that the orientability of the subbundles is unrelated to this causal structure. Yes we have the distinction between time-oriented and a space-oriented that comes from the indefinite Lorentzian metric signature. But it turns out that physically only the concept of total-orientation(both time- and space-oriented) can be discerned. See:“General Relativity: An Einstein Centenary Survey" by S. Hawking & W. Israel, pages 228-229.