Spacetime distance between spacelike related events

[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.