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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!