Spacetime distance between spacelike related events

[Nugatory]

@cianfa72
Given the discussion so far, with some caveats raised, can you more precisely reformulate your question?
(Otherwise a vague question will continue to raise all sorts of comments, without any clear direction.)

That's a fair point, but I must confess that I've been somewhat enjoying seeing all the subtleties that are involved here.

 

[PeterDonis]

Given a spacelike hypersurface with no stronger "acasual" condition, does always exist a spacelike geodesic that lives on it joining two given events belonging to that hypersurface ?

If you consider the hypersurface as a manifold by itself, without looking at how it is embedded in the 4-d spacetime, yes, there will always exist a spacelike geodesic within the hypersurface that joins any two points in it. But if you look at how the hypersurface is embedded in the 4-d spacetime, that same curve might not be a geodesic, and there might not be a spacelike geodesic joining a given pair of points that are in the hypersurface.

 

[PAllen]

Given a spacelike hypersurface with no stronger "acasual" condition, does always exist a spacelike geodesic that lives on it joining two given events belonging to that hypersurface ?

Yes, with the caveat that it is a geodesic of the induced metric on the hypersurface, and it is the exceptional special case that it is a geodesic of the spacetime. It will, however be a spacelike curve in the overall spacetime. The spacetime geodesic joining some points may even be timelike without additional restrictions.

Consider a really simple analog in pure Riemannian geometry. The geodesics within a 2-sphere with standard embedding in Euclidean 3-space are clearly not geodesics of the Euclidean space.

 

[robphy]

In the punctured plane [here, an acausal surface in Minkowski spacetime],
is there a [spacelike] geodesic from A to B?

 

[PeterDonis]

the punctured plane

I'm not sure this is a valid manifold, since it will include a boundary around the hole in the center and no coordinate chart is possible that can cover the boundary points.

A plane punctured by a single point would be a valid manifold since the single point left out does not preclude having a valid coordinate chart.

 

[robphy]

I'm not sure this is a valid manifold, since it will include a boundary around the hole in the center and no coordinate chart is possible that can cover the boundary points.

A plane punctured by a single point would be a valid manifold since the single point left out does not preclude having a valid coordinate chart.

(Is there a restriction to use a single coordinate chart?)

Is an infinite-plane with a point removed
topologically equivalent to an infinite-plane with a disk and its disk-boundary removed?

 

[PeterDonis]

Is there a restriction to use a single coordinate chart?

It's not a question of having to use a single coordinate chart; it's perfectly acceptable to have a manifold that can only be covered by an atlas of multiple charts. (All of the sphere manifolds are examples.)

The problem with the "punctured plane" with a finite sized hole is that there is no valid coordinate chart that can cover the boundary points, if those points need to be included in the manifold. Although now that I come to think of it, I suppose one could consider the manifold itself to not include the boundary points, but only to approach them as a limit. That would make it a valid manifold.

Is an infinite-plane with a point removed
topologically equivalent to an infinite-plane with a disk and its disk-boundary removed?

Yes, I think so. See my "although" above.

 

[PAllen]

In the punctured plane [here, an acausal surface in Minkowski spacetime],
is there a [spacelike] geodesic from A to B?
View attachment 267759

Ok, but that is a surface with a removable singularity, in the standard GR definition of geodesic incompleteness. Thus, one would need to add nonsingular spacelike hypersurface to guarantee that you you could connect any two points by a geodesic of the induced metric. Of course, this is a case of victory by definition, because if you can’t the surface is ipso facto singular by the GR definition.

In fact, I was wondering whether the spiral hypersurface could be extended to avoid geodesic incompleteness, but I couldn’t decide one way or the other by analysis that I could come up with.

 

[robphy]

Ok, but that is a surface with a removable singularity, in the standard GR definition of geodesic incompleteness. Thus, one would need to add nonsingular spacelike hypersurface to guarantee that you you could connect any two points by a geodesic of the induced metric. Of course, this is a case of victory by definition, because if you can’t the surface is ipso facto singular by the GR definition.

So, my point to the OP is that one needs to specify restrictions..
or else all of these issues that are allowed in GR
when one is not restricted to "simple regions of a plane" crop up.Possibly enlightening...

From https://www.google.com/search?q=penrose+techniques+of+differential+topology+in+relativity

 

[cianfa72]

So, my point to the OP is that one needs to specify restrictions..
or else all of these issues that are allowed in GR
when one is not restricted to "simple regions of a plane" crop up.

As said in post #25, I believe the main issue has to do with the *geodesic completeness* of the spacelike hypersurface endowed with the (positive definite) induced metric from 4D spacetime metric.

 

[PeterDonis]

I believe the main issue has to do with the *geodesic completeness* of the spacelike hypersurface endowed with the (positive definite) induced metric from 4D spacetime metric

What is the issue with geodesic completeness? Are you asking if it is possible to pick out spacelike hypersurfaces that are not geodesically complete? Of course it is. But why would you want to?

 

[PAllen]

What is the issue with geodesic completeness? Are you asking if it is possible to pick out spacelike hypersurfaces that are not geodesically complete? Of course it is. But why would you want to?

Well, @robphy points out you need to specify this if you don’t want cases like a point or ball removed.

A question I have is whether there is an example of geodesically complete spacelike 3-surface embedded in a pseudoriemannian manifold that is not achronal. I am having trouble, for example, seeing how to extend the spiral surface example to be geodesically complete.