Incompleteness of geodesics as ##t \longrightarrow 0##

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Discussion Overview

The discussion revolves around the nature of geodesics in the context of relativity, particularly focusing on whether geodesics can have a starting point in the past and the implications of geodesic incompleteness. Participants explore theoretical aspects and references related to singularity theorems and their interpretations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question how to demonstrate that relativity allows for geodesics to have a starting point in the past.
  • There is a suggestion that geodesics might asymptotically approach a time coordinate of 0 without ever reaching it, raising questions about the implications of such behavior.
  • A reference to a paper by Alan Guth is mentioned, which discusses the incompleteness of geodesics in the past.
  • Another participant cites the work of Borde, Guth, and Vilenkin, indicating that their findings support the idea of incomplete geodesics in the past.
  • One participant notes that singularity theorems, proven by Hawking and Penrose, provide conditions under which geodesics can be shown to have a starting point in the past.
  • There is a contention regarding the interpretation of geodesic incompleteness and its implications for the universe having a beginning, with some participants asserting that incompleteness suggests a beginning, while others challenge this interpretation.

Areas of Agreement / Disagreement

Participants express differing views on the implications of geodesic incompleteness and whether it necessitates a beginning of the universe. The discussion remains unresolved regarding the interpretation of these concepts.

Contextual Notes

Participants reference various theoretical frameworks and papers, indicating that the discussion is dependent on specific assumptions and interpretations of relativity and singularity theorems.

Tio Barnabe
The ##t## in the title states the time evolution of our universe.

How does one show that Relativity doesn't require geodesics extending indefinetely in the past? That is, how does one show that the theory allows for geodesics to have a starting point in the past? Is it hard to show?
 
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Tio Barnabe said:
How does one show that Relativity doesn't require geodesics extending indefinetely in the past? That is, how does one show that the theory allows for geodesics to have a starting point in the past?
It is not clear to me that those two are the same. For instance, it might be the case that, tracing a geodesic back, the time coordinate asymptotically approaches 0 but never attains it. That would satisfy the first statement but not the second.
 
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I first saw a discussion like this one in a paper by Alan Guth. Unfortunately, I missed the link for it. Maybe someone knows what paper I'm referring to.
 
the time coordinate asymptotically approaches 0 but never attains it.
Actually, he showed that geodesics could be incomplete in the past, arguing that Relativity doesn't require the universe to have a beginning.
 
Perhaps the following:
A. Borde, A.H. Guth and A. Vilenkin,
Inflationary Spacetimes Are Incomplete in Past Directions, Phys. Rev. Lett. 90, 151301 (2003)
I think this link doesn't require a journal subscription.
 
Thanks. I will read it.
 
Tio Barnabe said:
how does one show that the theory allows for geodesics to have a starting point in the past?

The theorems that prove this, given certain assumptions, are called "singularity theorems" and were proved by Hawking, Penrose, and others in the late 1960s and early 1970s. The Wikipedia article has a decent, if brief, discussion of the actual conditions and methods by which the theorems were proved (although the discussion earlier in that article leaves quite a bit to be desired):

https://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems#Nature_of_a_singularity

andrewkirk said:
it might be the case that, tracing a geodesic back, the time coordinate asymptotically approaches 0 but never attains it.

The definition of geodesic incompleteness is that a geodesic cannot be extended to arbitrary values of its affine parameter. This property is independent of any choice of coordinates.
 
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Tio Barnabe said:
he showed that geodesics could be incomplete in the past, arguing that Relativity doesn't require the universe to have a beginning.

This seems misstated. If geodesics are incomplete in the past, then the universe does have a beginning.
 
PeterDonis said:
This seems misstated. If geodesics are incomplete in the past, then the universe does have a beginning.
Oh yes, sorry for that.
 

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