Condition for a spacelike surface to be achronal

[PAllen]

TL;DR Summary

It is conjectured that being locally spacelike plus geodesically complete (I.e. nonsingular in GR terms) implies achronicity.

A hypersurface being spacelike (a local condition - every tangent to the surface being spacelike) does not preclude that points on it cannot be causally connected (one is in the future or past light cone of the other). A classic example is a spacelike spiral surface. Typically, for foliating a spacetime, we want the additional property of being achronal - no point on the surface is in the past or future light-cone of another. The achronal condition is global, rather than local. In another thread where this distinction was discussed, I came up with a hypothesis about this that I would like to get feedback on.

First, I am restricting to the case of smooth 3+1 dimensional pseudo-riemannian manifolds. Then my conjecture is that if a 3d spacelike hypersurface of such a manifold is geodesically complete, then it is achronal. This seems to relate two completely different types of global constraints on the hypersurface, and I have no ideal how to try to prove it. My only evidence is that every example of non achronal spacelike hypersurface I have found in the literature is not geodesically complete (and cannot be made so by any method I can see, while remaining everywhere spacelike).

Any thoughts on this would be highly appreciated.

 

[martinbn]

What about a two dimensional (1+1) cylinder and a space-like spiral? It is not achronal but it is complete.

 

[PAllen]

What about a two dimensional (1+1) cylinder and a space-like spiral? It is not achronal but it is complete.

Thanks, but a 1-d 'surface' is a truly degenerate case. That's why I explicitly posited 3+1 manifold, with 3-d spacelike surfaces. Even the generalization of the spiral to 2+1 d, with 2-d spiral surface is a problem, as far I can analyze it. All ways to make it geodesically complete (that I could find), made it no longer spacelike in some region.

 

[PeterDonis]

geodesically complete (I.e. nonsingular in GR terms)

"Nonsingular" is not the term I would choose, since while "singularity" in GR is taken to include geodesic incompleteness, it is also taken to include at least some invariant increasing without bound as the finite limit of the affine parameter along some geodesic is approached. That will not be the case with, say, a spacelike spiral surface; all invariants will remain finite and bounded along any geodesic that is not complete within the surface (because any spacelike geodesic will exit the surface into the surrounding spacetime somewhere).

if a 3d spacelike hypersurface of such a manifold is geodesically complete, then it is achronal

I suspect that at least some kind of causality condition will need to be imposed to make this true. For example, I suspect that there are geodesically complete spacelike 3-surfaces in Godel spacetime that are not achronal (but Godel spacetime does not satisfy any of the causality conditions in Hawking & Ellis, since it has closed timelike curves, so imposing any causality condition as a premise would exclude Godel spacetime).

 

[PAllen]

I suspect that at least some kind of causality condition will need to be imposed to make this true. For example, I suspect that there are geodesically complete spacelike 3-surfaces in Godel spacetime that are not achronal (but Godel spacetime does not satisfy any of the causality conditions in Hawking & Ellis, since it has closed timelike curves, so imposing any causality condition as a premise would exclude Godel spacetime).

This is the key point. I don't see any clear reason for the conjecture to be true, so a counterexample is desirable. Even without full details, can you describe your idea from Godel spacetime a bit more? Wait, I get the idea. In Godel spacetime there is likely no such thing as an achronal spacelike surface. So, yes, to make the conjecture more substantive, I would have to add that the spacetime has no CTC.

Thanks.

 

[PeterDonis]

In Godel spacetime there is likely no such thing as an achronal spacelike surface.

Quite possibly not. I haven't had a chance to try an actual computation, though.

 

[cianfa72]

In another thread where this distinction was discussed, I came up with a hypothesis about this that I would like to get feedback on.

Maybe you were talking about this thread spacetime distance between spacelike related events. Events (points) in red on that spacelike hypersurface are actually timelike separated.

As definition of geodesically complete I take the following: the exponential map at each point P is defined on $T_pM$, the entire tangent space at P.

 

[PeterDonis]

As definition of geodesically complete I take the following

Where are you getting this definition from?

 

[cianfa72]

Where are you getting this definition from?

From here -- first sentence.

 

[PeterDonis]

From here -- first sentence.

Ah, ok. This is a different formulation than I'm used to seeing, but I see how it works. The key point is that the exponential map being defined for a vector in the tangent space requires that you can extend the geodesic "pointing" in the direction of that vector through the manifold to arbitrary values of its affine parameter. The latter property (extendibility) is the way I am used to seeing geodesic completeness defined.

 

[cianfa72]

A point related to this topic, also discussed in a recent thread. In relativity (SR & GR) If two events are not timelike or null separated (i.e. there is not a timelike or null geodesic connecting them) then they are spacelike separated (i.e. it does exist a spacelike geodesic connecting them). Right ?

 

[martinbn]

A point related to this topic, also discussed in a recent thread. In relativity (SR & GR) If two events are not timelike or null separated (i.e. there is not a timelike or null geodesic connecting them) then they are spacelike separated (i.e. it does exist a spacelike geodesic connecting them). Right ?

Not always. Take Minkowski remove a square and take two points near oposite sides.

 

[cianfa72]

Not always. Take Minkowski remove a square and take two points near opposite sides.

Is it somehow related to the geodesically complete property ?

To me the case you pointed out looks like the punctured plane.

 

[PAllen]

Is it somehow related to the geodesically complete property ?

To me the case you pointed out looks like the punctured plane.
View attachment 288430

Yes, this type of case is ruled out by geodesic completeness. The current modified conjecture is that: in a causal spacetime, if a 3 surface is geodesically complete (per the induced 3 metric) and spacelike, then it is achronal.

definitions: https://en.wikipedia.org/wiki/Causality_conditions

 

[cianfa72]

So, coming back to my post #11, in case of spacetime not geodesically complete we cannot claim for sure that two arbitrary given events are actually timelike, null or spacelike separated, I believe.

For example two events not timelike or null separated (namely there is not a timelike or null geodesic of 4d spacetime connecting them) are not necessarily spacelike separated (namely there is not a spacelike geodesic of 4d spacetime connecting them too).

 

[PAllen]

So, coming back to my post #11, in case of spacetime not geodesically complete we cannot claim for sure that two arbitrary given events are actually timelike, null or spacelike separated, I believe.

For example two events not timelike or null separated (namely there is not a timelike or null geodesic of 4d spacetime connecting them) are not necessarily spacelike separated (namely there is not a spacelike geodesic of 4d spacetime connecting them too).

Yes, as you've defined things, what you say is true for spacetimes with geodesic incompleteness.

 

[cianfa72]

Yes, as you've defined things, what you say is true for spacetimes with geodesic incompleteness.

I gave those definitions from this thread. It seemed reasonable to include in the definition the requirement to be timelike/null/spacelike geodesics respectively not just timelike/null/spacelike curves.