Geodesic Maximality: Answers to Relativists' Questions

[Umaxo]

Hello,

i know that relativists like to extend solutions of einstein equations so that they are geodesicly maximal (i.e. geodesics end only in singularity or infinite value of affine parameter). But why only geodesicly? Thus this geodesic maximality imply, that if i take any timelike or spacelike curve that goes to border (not singularity) of maximally extended spacetime, all such curves would do it in infinite amount of their proper time/length? I.e. for timelike case - thus it imply, that any observer would actually never approach border of spacetime in finite amount of his proper time, unless it is curvature singularity?

Thank you:)

 

[jbriggs444]

It seems clear that if one accelerates at an increasing rate, that it is possible to cover, in the limit, an infinite coordinate distance in a finite proper time.

Adopt a simple strategy. Cover the first light year in 10 proper years, accelerating so as to be able to cover the next light year in 5 proper years. Rinse and repeat.

 

[stevendaryl]

As @jbriggs444 points out, there is no upper bound on the spatial components of proper velocity, so it is possible to have a worldline that covers an infinite amount of distance in a finite amount of proper time. On the other hand, such a path means unbounded acceleration, as well, which means that no physical object could follow such a worldline without getting ripped apart be "g" forces.

[EDIT] Obviously, this worldline is not a geodesic, so it's not directly relevant to the question of geodesic completeness.