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

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In summary, the conversation discusses the concept of geodesic incompleteness in the context of Relativity and the time evolution of the universe. It is argued that Relativity does not require geodesics to extend indefinitely in the past and that the theory allows for geodesics to have a starting point in the past. The concept of singularity theorems, which prove this under certain assumptions, is also mentioned. Additionally, the idea that geodesic incompleteness does not depend on a choice of coordinates is discussed.
  • #1
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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  • #2
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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  • #3
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.
 
  • #4
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 beggining.
 
  • #5
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.
 
  • #6
Thanks. I will read it.
 
  • #7
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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  • #8
Tio Barnabe said:
he showed that geodesics could be incomplete in the past, arguing that Relativity doesn't require the universe to have a beggining.

This seems misstated. If geodesics are incomplete in the past, then the universe does have a beginning.
 
  • #9
PeterDonis said:
This seems misstated. If geodesics are incomplete in the past, then the universe does have a beginning.
Oh yes, sorry for that.
 

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

What is the concept of "incompleteness of geodesics as ##t \longrightarrow 0##"?

The concept of "incompleteness of geodesics as ##t \longrightarrow 0##" refers to the phenomenon in which the length of a geodesic (the shortest path between two points on a curved surface) approaches infinity as the parameter t approaches 0. This means that the geodesic cannot be extended to reach the point at t=0, and thus becomes "incomplete".

What causes the incompleteness of geodesics as ##t \longrightarrow 0##?

The incompleteness of geodesics as ##t \longrightarrow 0## is caused by the curvature of the surface on which the geodesic lies. As the parameter t approaches 0, the geodesic becomes more and more tightly curved, eventually reaching a point where it cannot be extended any further without increasing its length to infinity.

How does the incompleteness of geodesics affect our understanding of space and time?

The incompleteness of geodesics implies that there are limitations to our understanding of space and time, particularly in the context of curved spaces. It suggests that there may be points or regions in space where our current mathematical models break down and cannot accurately describe the behavior of particles or objects.

Are there any real-world applications of the concept of "incompleteness of geodesics as ##t \longrightarrow 0##"?

Yes, the concept of incompleteness of geodesics has practical applications in various fields such as physics, engineering, and computer graphics. For example, it is used in the design of curved structures such as bridges and tunnels, and in the development of algorithms for simulating the motion of objects on curved surfaces.

Is there ongoing research or debate surrounding the incompleteness of geodesics as ##t \longrightarrow 0##?

Yes, there is ongoing research and debate in the field of differential geometry and general relativity surrounding the concept of incompleteness of geodesics. This is because it has implications for our understanding of the structure of space and time, and its connection to gravity. There are also ongoing efforts to develop more precise mathematical models and techniques for studying and predicting the behavior of geodesics near t=0.

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