# Time-orientability of Lorentzian manifolds

1. Aug 12, 2015

### loislane

A spacetime is said to be time-orientable if a continuous designation of wich 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?

2. Aug 12, 2015

### Orodruin

Staff Emeritus
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.

3. Aug 13, 2015

### loislane

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?

4. Aug 13, 2015

### Staff: Mentor

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.

5. Aug 13, 2015

### George Jones

Staff Emeritus

6. Aug 13, 2015

### Staff: Mentor

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).

7. Aug 13, 2015

### bcrowell

Staff Emeritus
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.

8. Aug 13, 2015

### loislane

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

9. Aug 13, 2015

### bcrowell

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

10. Aug 13, 2015

### loislane

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.

11. Aug 13, 2015

### loislane

Only in the pseudo-Riemannian case AFAIK.

12. Aug 13, 2015

### bcrowell

Staff Emeritus
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.

13. Aug 13, 2015

### PAllen

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

14. Aug 13, 2015

### Staff: Mentor

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.

Is there one for Godel spacetime?

15. Aug 13, 2015

### bcrowell

Staff Emeritus
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.

Last edited: Aug 13, 2015
16. Aug 13, 2015

### 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”

17. Aug 14, 2015

### PAllen

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

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

Last edited: Aug 14, 2015
18. Aug 14, 2015

### bcrowell

Staff Emeritus
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.

19. Aug 14, 2015

### bcrowell

Staff Emeritus
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.

Last edited: Aug 14, 2015
20. Aug 14, 2015

### Ben Niehoff

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.

21. Aug 14, 2015

### Staff: Mentor

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

22. Aug 15, 2015

### loislane

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.
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.