Understanding Inextendible Worldlines in Manifolds: Examples and Applications

[Matterwave]

Hi, I've been reading Wald's treatment of the causal structure of manifolds and I would like to confirm my understanding of the concept of inextendible worldlines because Wald gives a conceptual definition, but no examples, and I think much better if I have some concrete examples.

Wald defines a future end point p (a point in M, the manifold) of a world line γ(t) as a point such that for every neighborhood O of p there is a parameter t_0 such that γ(t>t_0) is a member of O.

He then defines a future inextendible worldline as a worldline without a future end point.

I wanted to make sure my intuitive understanding of this is correct.

So, it seems to me, that there are several cases in which a future end point could arise:
1) I simply defined my curve to go from a≤t≤b (i.e. it ranges only over a finite range of parameter), so that the curve just "ends" artificially due to me not defining it for greater ranges.
2) The curve is defined for -∞<t<∞, but there is some sort of horizon which forms on the manifold (not a singularity) such that my curve never moves beyond some point p in the manifold. An example of this would be in an exponentially expanding FLRW space-time where there's a horizon formed so that my light signals can never go farther than some coordinate distance (comoving distance) r, even though it travels a proper distance of ∞ (as time goes to infinity of course).

These are the only 2 cases I can think of right now for there being a future end point. I'm sure there are others, so please enlighten me. If I am mistaken, also please correct me. I'm not 100% sure if the worldline of the light signal in scenario 2) is "extendible", so, if someone would confirm that with me, that would be great.

Now, the cases where the worldline is inextendible seem to me to be:
1) The worldline is closed (i.e. it's a loop). You keep going in circles as t increases, so you never converge to a point.
2) The worldline "moves off to infinity". In other words, the worldline never ends at any point on the manifold, and just keeps going forever.
3) The worldline hits a singularity. Because the singularity point is not defined on our manifold, if our worldline "converged" towards the singularity, it's convergence point is not on our manifold and therefore the worldline is inextendible.

These are the 3 cases I can think of for an inextendible world line. Again, if I'm mistaken, please correct me, etc.

I'm trying to iron this out because this is important for the singularity theorems. Thanks!

 

[Matterwave]

Surely somebody knows o.o help...

 

[atyy]

New to me too, but Wald explicitly gives your 3 cases as being inextendible on p193, just before his formal definition.

 

[Matterwave]

If you are referring to what I think you are referring to, I did read that. He says something like "We need to distinguish between a curve which loops, or hits a singularity versus a curve which just ends because we didn't define it to go further" (I'm paraphrasing). I thought this comment was sufficiently disconnected that I couldn't draw conclusively the conclusions I wanted.

Additionally, I am more concerned with my first 2 cases of an extendible world-line. I would like to know since this affects the formal analysis of the causal structure of, say, the FLRW metric.

 

[atyy]

On my reading, I would guess that an infinite curve that avoids a horizon is inextendible. Googling it seems that an inextendible curve can have a horizon http://books.google.com/books?id=t0D_YmZpTucC&dq=inextendible+curve+horizon&source=gbs_navlinks_s, p394.

 

[PeterDonis]

2) The curve is defined for -∞<t<∞, but there is some sort of horizon which forms on the manifold (not a singularity) such that my curve never moves beyond some point p in the manifold.

I don't think this qualifies as a future end point, at least not the way I think you are thinking of it. The point p is a point in *spacetime*, not just space. Using your example:

An example of this would be in an exponentially expanding FLRW space-time where there's a horizon formed so that my light signals can never go farther than some coordinate distance (comoving distance) r, even though it travels a proper distance of ∞ (as time goes to infinity of course).

The point p is not "at some comoving distance r from the origin"; that would be a *line*, specifically a comoving worldline at radial coordinate r. The point p would have to be some specific point *on* that line; but it should be evident that a single point on that line can't possibly be a future endpoint for any comoving worldline, since p must lie in a particular comoving spacelike slice, and any comoving worldline will "pass" any given slice and move arbitrarily far away from it, in spacetime, as t increases.

 

[Matterwave]

Well, Wald specifically said that the point didn't have to lie on the line, so I wasn't entirely sure about that example.

I'm not quite sure about your argument Peter, can you rephrase it maybe?

Now that I think about it, it must be that the points farther than the horizon distance from point p are causally disconnected from the point p. In that sense, there cannot be any time-like or causal line which connect the two.

I guess this should be a case for saying that if worldlines have an infinite range of (any) affine parameter, then they are inextendible? We cannot make an affine reparameterization such that the affine parameter range is infinite in one case and finite in another case right.

So, then, the only possibility for an "extendible" world line would be just one in which I defined the curve to range only over some finite range of affine parameter?

This horizon example is really confusing me.

 

[atyy]

My guess is that any geodesic with infinite range of affine parametrization is inextendible. But it is possible to have inextendible geodesics with finite range of affine parametrization. Those are the ones he calls "incomplete" (p215).

For the horizon case, I was thinking why not just take the Rindler observer? I imagine he's got to be inextendible.

 

[PeterDonis]

Well, Wald specifically said that the point didn't have to lie on the line, so I wasn't entirely sure about that example.

I can see a future endpoint of a worldline not necessarily lying "on" the line (strictly speaking, a future singularity isn't "on" the worldlines that go into it). But it's not the fact that the point isn't on the line that I was using for my argument.

I'm not quite sure about your argument Peter, can you rephrase it maybe?

Any FRW spacetime can be foliated by a set of spacelike hypersurfaces that are orthogonal to the worldlines of "comoving" observers. Any point p (remember, a point in spacetime, not just space) must lie in one of those spacelike hypersurfaces; and since each spacelike hypersurface can be labeled with its comoving time (the time coordinate t of the FRW spacetime, which is also the proper time of all the comoving observers), the point p must have some specific time coordinate, call it t_p. But since the worldline you hypothesize in your case #2 goes to t -> infinity, such a worldline will get arbitrarily far to the future of t_p, and therefore from any point in the hypersurface of time t_p, including p. So p can't be a future endpoint of any such worldline.

You will see that the argument I just gave is very general; it proves that there are *no* future endpoints in FRW spacetime at all. This should not be a surprise since FRW spacetime is known to be geodesically complete.

The scenario you appear to be visualizing is something like this: I have some worldline that p is on, and I have some other worldline that "asymptotes" to the first worldline as t -> infinity. But asymptoting to the worldline that p is on is *not* sufficient for p itself to be a future endpoint. Even if p's worldline is a horizon for the other worldline, that doesn't make p itself a future endpoint, because there will always be parts of the other worldline that are arbitrarily far to the future of p. They may get closer and closer to p's *worldline*, but they will still be arbitrarily far to the future of p itself.

Now that I think about it, it must be that the points farther than the horizon distance from point p are causally disconnected from the point p. In that sense, there cannot be any time-like or causal line which connect the two.

Yes, that's the definition of "horizon" in this context. (I say "in this context" because this horizon is two-way, so to speak; if I am on one side and you are on the other, neither of us can causally communicate with the other. Some horizons, like the horizon of a black hole, are "one-way"; observers inside can't causally communicate with observers outside, but the reverse is *not* true; observers outside *can* causally communicate with observers inside.)

I guess this should be a case for saying that if worldlines have an infinite range of (any) affine parameter, then they are inextendible? We cannot make an affine reparameterization such that the affine parameter range is infinite in one case and finite in another case right.

I agree, yes.

So, then, the only possibility for an "extendible" world line would be just one in which I defined the curve to range only over some finite range of affine parameter?

Or you could be in a weird spacetime that had "holes" or "edges" in it--actual boundaries where geodesics come to an abrupt halt. Points on the boundary would be future endpoints. Much of the literature about the causal structure of spacetimes seems to center on finding the right conditions to rule out such pathological cases; as an outsider looking in, I have to remind myself of this because these kinds of spacetimes would not even occur to me as examples to consider since they are obviously "unphysical". But to make the physics rigorous, the experts have to consider these kinds of possibilities so they can be sure they know what conditions are required to exclude them.

 

[Matterwave]

Or you could be in a weird spacetime that had "holes" or "edges" in it--actual boundaries where geodesics come to an abrupt halt. Points on the boundary would be future endpoints. Much of the literature about the causal structure of spacetimes seems to center on finding the right conditions to rule out such pathological cases; as an outsider looking in, I have to remind myself of this because these kinds of spacetimes would not even occur to me as examples to consider since they are obviously "unphysical". But to make the physics rigorous, the experts have to consider these kinds of possibilities so they can be sure they know what conditions are required to exclude them.

For this to work, the manifold must compact (closed? I suck at topology, forgive me) near these holes/edges right. Otherwise, the geodesics would move towards a point not on the manifold, and one could consider these geodesics inextendible (and therefore incomplete) in that case, then, and we can consider those "holes" singularities, right.

 

[PeterDonis]

For this to work, the manifold must compact (closed? I suck at topology, forgive me) near these holes/edges right.

The boundary points are part of the manifold; I believe "compact" is the correct term in this case.

 

[bobc2]

...Or you could be in a weird spacetime that had "holes" or "edges" in it--actual boundaries where geodesics come to an abrupt halt. Points on the boundary would be future endpoints. Much of the literature about the causal structure of spacetimes seems to center on finding the right conditions to rule out such pathological cases; as an outsider looking in, I have to remind myself of this because these kinds of spacetimes would not even occur to me as examples to consider since they are obviously "unphysical". But to make the physics rigorous, the experts have to consider these kinds of possibilities so they can be sure they know what conditions are required to exclude them.

You could create such a "unphysical" world line if you could create an elementary particle from nothing at t-0, then make it disappear at 1 sec (a 186,000 mi world line). Physical law would not allow this. Perhaps there are some world lines of this type in the quantum field that extend less than a Plank length (but how would you ever know?). (I know... someone will say that Planck length does not apply to the 4-D interval.)