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GR - curve (in)completeness & (in)extendibility

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(This question failed to get any traction at Physics StackExchange; I have improved the precision slightly for this site.)

This question is being updated with corrections based on comments etc. so as to develop an accurate analysis of all four combinations of inextendible & incomplete - until such time as one is posted as an answer.

I am seeking clarification of the distinction between completeness of geodesics/extendibility of curves in GR spacetimes (Confirm: not the *geodesic completeness of a spacetime* but the completeness etc. of an individual geodesic.)

I think it's the use of the negative *in*extendible that may be confusing me.

I am happy with past/future inextendibility for a causal curve as being without a past/future endpoint as defined by Wald (*General Relativity*, 1984, p193) for a spacetime manifold $M$, paraphrased as,

Let $\gamma$ be a causal curve, then $\gamma$ has a *future endpoint* $p\in M$ if for every neighbourhood $O$ of $p$ there exists a $t_{0}$ such that  $\gamma(t)\in O\  \forall t>t_{0}$, and similarly for a *past endpoint*. [Because $M$ is Hausdorff it can have at most one future and one past endpoint.]

Some specific cases:

Consider a causal geodesic $\lambda$ through the (arbitrary) origin of spacetime (Lorentzian manifold, usual structure...). Some cases specifically refer to Minkowski space, others are general.

Case 1 - Inextendible + Incomplete

Initial Incorrect Text

Remove the point $\vec{0}$ from $\lambda$; $\lambda$ is now in two geodesic pieces; the piece $\lambda(t), t>0$ is past inextendible and the piece $t<0$ is future inextendible (each is an open set and so has no endpoints).

With the Minkowski metric, both pieces seem incomplete, since the affine parameter does not span $(-\infty, \infty)$.

Correction (Arnold Neumaier)

Applying Wald's definition both pieces are not inextendible because there is no requirement that the endpoint be on the curve. The curves are therefore not inextendible but they are incomplete, like Case 2.

An example of Inextendible and Incomplete is therefore sought.

Case 2 - Not Inextendible + Incomplete
If, instead of a point, a closed interval on $\lambda$ is removed, both pieces are extendible (they have endpoints) but they also incomplete.

Case 3 - Not Inextendible + Complete
$M$ not Minkowski, but with a metric that "pushes $p$ off to infinite distance along the curve".

Inextendible = no endpoint, but as Wald says , "the endpoint need not lie on the curve, i.e., there need not exist a value of $t$ such that $\lambda(t) = p$." so if the affine parameter takes $\lambda$ arbitrarily close to $p$ as it tends to an $\infty$, the geodesic would be complete and not inextendible.

(But definitely $p\in M$ and not not one of those philosophically troublesome "missing points" :)

Case 4 - Inextendible + Complete
$M$ Minkowski without excisions, $\lambda$ is infinitely long in both directions: affine parameter spans $(-\infty,\infty)$ and there are no endpoints.

Questions: is the analysis complete and correct? (I don't think so, but am unable to proceed further.) If not, can you provide clarification - and what's the picture for spacelike geodesic/curves?

asked Oct 25, 2016 in Theoretical Physics by Julian Moore (40 points) [ revision history ]
edited Oct 25, 2016 by Julian Moore

Since according to Wald's definition, an endpoint need not be part of the curve, removing the origin from a geodesic containing gives two resulting geodesics that have 0 as an endpoint, unlike what you claim in case 1.

@arnoldneumaier. Good point, thx. I will amend the question accordingly. It then leaves me missing an example of Inextendible + Incomplete. Any thoughts?

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