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

[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?

 

[andrewkirk]

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.

 

[Tio Barnabe]

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.

 

[Tio Barnabe]

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.

 

[andrewkirk]

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.

 

[Tio Barnabe]

Thanks. I will read it.

 

[PeterDonis]

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

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.

 

[PeterDonis]

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.

 

[Tio Barnabe]

This seems misstated. If geodesics are incomplete in the past, then the universe does have a beginning.

Oh yes, sorry for that.