Geodesic Maximality: Answers to Relativists' Questions

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In summary, the conversation discusses the concept of geodesic maximality in relation to extending solutions of Einstein's equations. It is mentioned that this concept implies that any timelike or spacelike curve approaching the border of a maximally extended spacetime would do so in an infinite amount of proper time. However, it is also pointed out that it is possible for a worldline to cover an infinite amount of distance in a finite amount of proper time through unbounded acceleration, but such a path is not a geodesic and would not be possible for physical objects to follow without being destroyed.
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Umaxo
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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:)
 
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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.
 
  • #3
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.
 

1. What is geodesic maximality?

Geodesic maximality is a concept in general relativity that refers to the idea that geodesics, the paths that objects follow in spacetime, should be as long as possible. This is based on the principle of least action, which states that objects will follow the path of least resistance.

2. How does geodesic maximality relate to general relativity?

Geodesic maximality is a fundamental principle in general relativity. It is used to describe the motion of objects in a curved spacetime, as predicted by Einstein's theory of gravity. According to general relativity, objects will follow the path of a geodesic, which is the longest possible path between two points in a curved spacetime.

3. What are some real-world applications of geodesic maximality?

Geodesic maximality has many applications in the field of astrophysics and cosmology. For example, it is used to study the motion of celestial bodies such as planets, stars, and galaxies. It is also used in the study of gravitational lensing, which is the bending of light around massive objects in space.

4. How is geodesic maximality different from other principles in general relativity?

Geodesic maximality is closely related to other principles in general relativity, such as the principle of equivalence and the principle of general covariance. However, geodesic maximality is unique in that it specifically focuses on the behavior of geodesics, while the other principles have broader applications in the theory.

5. Are there any criticisms or controversies surrounding geodesic maximality?

There have been some debates and criticisms surrounding geodesic maximality in the scientific community. Some argue that it is not a necessary or fundamental principle in general relativity, while others believe it is a crucial aspect of the theory. Additionally, some scientists have proposed alternative theories of gravity that do not rely on geodesic maximality. However, the principle remains a widely accepted and important concept in general relativity.

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