I Spacetime distance between spacelike related events

[cianfa72]

TL;DR Summary

How to define the spacetime 'distance' between events belonging to a spacelike hypersurface

Hi,

in general relativity I'm aware of the spacetime 'distance' between two timelike related events is maximized by the free falling timelike path (zero proper acceleration) joining them.

Consider now a couple of events belonging to a spacelike hypersurface (AFAIK it is an hypersurface with the feature that all the 'directions' belonging to it are actually spacelike in the tangent space at each point).

How is defined in this case the maximum (or minimum ?) spacetime 'distance' between a couple of events belonging to it ?

Thanks in advance !

 

[anuttarasammyak]

ds^2=c^2dt^2-dx^2-dy^2-dz^2= -dl_0^2 <0
where $dl_0$ is (infinitesimal) minimum or proper length measured in the IFR where the two events take place simultaneous $dt=0$.

 

[cianfa72]

Just to be clear: we are talking about GR with events A and B not infinitesimally separated.
I assumed the following definition: two events are said spacelike separated if there exist no timelike or null path joining them.

 

[Ibix]

You can define spacelike geodesics, if that's what you're asking. As far as I'm aware you use the geodesic equation and simply impose the condition that $g_{\mu\nu}\frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}$ have the appropriate sign for your metric signature.

 

[Nugatory]

@cianfa is asking about the extremal spacelike curve that lies in a given spacelike hypersurface containing the two events. That’s not the same thing as the spacelike geodesic between the two events (although it may be more useful if the hypersurface was chosen to correspond to some interesting/relevant simultaneity convention).

 

[kent davidge]

Why not take the hypersurface with its induced metric and coordinates and simply extremize the distance between two points lying on it?

 

[kent davidge]

Summary:: How to define the spacetime 'distance' between events belonging to a spacelike hypersurface

How is defined in this case the maximum (or minimum ?) spacetime 'distance' between a couple of events belonging to it ?

By doing what I said, you will find that the curve is a spacelike geodesic. However it may not be a geodesic in the full space (i.e., spacetime).

 

[robphy]

Just a comment:
one probably needs to be specific about restrictions or conditions.

Here are two events on a spacelike surface that are timelike-related.

You can exclude events on this spacelike surface that causally-related to the lower-event A
by drawing the boundary and interior of the light-cone of A and excluding events in the intersection.

 

[PAllen]

Just a comment:
one probably needs to be specific about restrictions or conditions.

Here are two events on a spacelike surface that are timelike-related.
View attachment 267590
You can exclude events on this spacelike surface that causally-related to the lower-event A
by drawing the boundary and interior of the light-cone of A and excluding events in the intersection.

I assumed the OP meant a geodesic within the hypersurface, with induced metric (though they didn't necessarily know the terminology or techniques). In that case, there is nothing to exclude - there is a spacelike spiral geodesic of the induced metric between your two points.

 

[robphy]

I assumed the OP meant a geodesic within the hypersurface, with induced metric (though they didn't necessarily know the terminology or techniques). In that case, there is nothing to exclude - there is a spacelike spiral geodesic of the induced metric between your two points.

[strike]Yes, there is a geodesic on the spacelike surface I gave.[/strike]
hmmm... I’m not so sure now.

But, I raised the issue because
the title says

Spacetime distance between spacelike related events

but the summary in the original post says

How to define the spacetime 'distance' between events belonging to a spacelike hypersurface

and later the original poster says

Just to be clear: we are talking about GR with events A and B not infinitesimally separated.
I assumed the following definition: two events are said spacelike separated if there exist no timelike or null path joining them.

So, my post is to suggest that the question should be clarified
since two events on a spacelike surface are not necessarily spacelike-related.

 

[PeterDonis]

two events on a spacelike surface are not necessarily spacelike-related

Yes, this is something a lot of people don't stop to think about when they think about spacelike surfaces.

The technically correct term for a surface such that any two points of the surface are spacelike separated from each other is, IIRC, "acausal".

 

[cianfa72]

@cianfa is asking about the extremal spacelike curve that lies in a given spacelike hypersurface containing the two events.

That was my original question based on an incorrect assumption (see below).

since two events on a spacelike surface are not necessarily spacelike-related.

Ah that was my incorrect understanding !

The technically correct term for a surface such that any two points of the surface are spacelike separated from each other is, IIRC, "acausal".

ok good point, I was not aware about this definition.

I assumed the following definition: two events are said spacelike separated if there exist no timelike or null path joining them.

Does this definition make sense for a couple of spacelike separated events ?

What about instead the following definition of spacelike hypersurface: a 3-dimensional hypersurface such that at each point (event) the normal vector (in the attached tangent space of course) is timelike ?

 

[vanhees71]

That's the usual definition of a spacelike hypersurface, i.e., a 3D hypersurface such that the normal vector is timelike everywhere. You can then take such a spacelike hyersurface as a surface of simultaneity and then define the spatial distances for points on the hypersurface by the length of geodesics conneting the point in the induced spatial metric. This is of course not always possible for any two points but usually only locally. In general you can even synchronize clocks of a set of observers along a spacelike curve only locally. If you have a closed spacelike curve usually you cannot synchronize the clocks of these observers.

One famous example is the rotating disk in Minkowski spacetime. This was discussed also recently in this forum at length.

A nice discussion about this problem is in Landau and Lifshitz vol. II.

 

[cianfa72]

You can then take such a spacelike hyersurface as a surface of simultaneity and then define the spatial distances for points on the hypersurface by the length of geodesics connecting the point in the induced spatial metric. This is of course not always possible for any two points but usually only locally

Why not ? Assuming a connected manifold all the curves (geodesics or not) that lie in the given spacelike hypersurface are themselves spacelike, thus why can't we find out an extremal spacelike curve joining the two given events belonging to the hypersurface ?

 

[vanhees71]

I think if you already have a spacelike hypersurface, there's no problem since this defines a hypersurface of simultaneity, but it's not always globally possible to find one!

 

[PeterDonis]

Does this definition make sense for a couple of spacelike separated events ?

Your definition of "spacelike separated events" is fine.

What about instead the following definition of spacelike hypersurface: a 3-dimensional hypersurface such that at each point (event) the normal vector (in the attached tangent space of course) is timelike ?

The helical spacelike surface @robphy described satisfies this definition.

 

[PeterDonis]

if you already have a spacelike hypersurface, there's no problem since this defines a hypersurface of simultaneity

No, it doesn't. The helical spacelike surface that @robphy described is a counterexample. A hypersurface of simultaneity must be acausal, which is a much stronger condition than just being a spacelike hypersurface.

 

[vanhees71]

True. One must restrict oneself to small enough neighborhoods around a point. Finally only local observabes are uniquely defined.

 

[cianfa72]

By the way I believe that any spacelike hypersurface equipped with the induced metric from the pseudo-riemman spacetime metric is actually positive definite.

 

[PAllen]

By the way I believe that any spacelike hypersurface equipped with the induced metric from the pseudo-riemman spacetime metric is actually positive definite.

Yes, this is true without exception.

 

[pervect]

My impression was and is that it sufficient (and necessary) to specify a closed space-like hypersurface to do what the OP wants.

If the surface isn't closed, the difficulties alluded to arise. This is not a frivolous point, because the rotating disk is a physical example that can generate non-closed surfaces. The issue of closed vs non-closed surfaces can arise even in small regions.

The methodology, once one has a closed surface, is, as others have mentioned, to use a projection operator to induce a 3-metric on the surface from the 4-metric of the space-time. Once one has the induced 3-metric, one has at least a local notion of distance. Usually textbooks stop with having defined a metric, and don't go on to discuss how one gets distance from a metric over extended regions.

I tend to think of distance over an extended region of a space with a metric as the length of the shortest geodesic connecting them. Though as I said, this isn't from a textbook.

However, this is all from memory, I don't have a reference handy that formally defines a closed space-like hypersurface, for instance.

 

[PeterDonis]

I don't have a reference handy that formally defines a closed space-like hypersurface

I'm not even sure what informal definition you are using. Can you elaborate? Or give some specific examples of closed spacelike 3-surfaces?

 

[PAllen]

I don’t think Minkowski spacetime admits any closed spacelike 3- surfaces.

The requirement I’ve seen for a foliation is acausal, as mentioned by others in this thread.

 

[PeterDonis]

I don’t think Minkowski spacetime admits any closed spacelike 3- surfaces.

If "closed" means "manifold without boundary", yes, I believe that's correct. The only fairly well known spacetime I'm aware of that admits closed spacelike 3-surfaces in this sense is spatially closed FRW spacetime.

 

[cianfa72]

The methodology, once one has a closed surface, is, as others have mentioned, to use a projection operator to induce a 3-metric on the surface from the 4-metric of the space-time. Once one has the induced 3-metric, one has at least a local notion of distance.

Which is the problem with this kind of hypersurface ? We have a spacelike 3D hypersurface with an (induced) metric that's positive definite on it.

The issue you are referring to has something to do with the 'geodesic completeness' --- see Hopf–Rinow theorem ?

 

[PAllen]

If "closed" means "manifold without boundary", yes, I believe that's correct. The only fairly well known spacetime I'm aware of that admits closed spacelike 3-surfaces in this sense is spatially closed FRW spacetime.

A trivial example that is flat but not topologically trivial is the flat 3-torus X R, with Minkowski metric everywhere. But, of course, this is not what is normally meant by Minkowski space.

 

[cianfa72]

Which is the problem with this kind of hypersurface ? We have a spacelike 3D hypersurface with an (induced) metric that's positive definite on it.

The issue you are referring to has something to do with the 'geodesic completeness' --- see Hopf–Rinow theorem ?

Any though about this?

 

[PeterDonis]

Which is the problem with this kind of hypersurface ?

There might not be one. See below.

We have a spacelike 3D hypersurface with an (induced) metric that's positive definite on it.

Yes. But, as has been pointed out, not all hypersurfaces that meet this condition also meet the much more stringent condition, the "acausal" condition, that every pair of events in them are spacelike separated. Whether that is a "problem" or not depends on what you are trying to do with the hypersurface.

 

[robphy]

@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.)

 

[cianfa72]

@cianfa72
Given the discussion so far, with some caveats raised, can you more precisely reformulate your question?

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 ?

 

[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.