Exploring the Milne Universe in 2D: Rindler Spacetime Comparison

[alialice]

I'm studying the Milne Universe in two dimensions. The metric is similar to that of Rindler spacetime, but with time and space inverted.
$ds^{2}=-dt^{2}+t^{2}dx^{2}$
The Carter Penrose diagram of this spacetime would be the same of Rindler spacetime?

 

[HallsofIvy]

The "Milne universe"? Is that where Winnie ther Pooh lives?

 

[Dickfore]

The "Milne universe"? Is that when Winnie ther Pooh lives?

The Wiki

 

[alialice]

Apart from Winnie the Pooh, anyone has an idea on what to do?

 

[Dickfore]

Apart from Winnie the Pooh, anyone has an idea on what to do?

I don't. What have you done? What's a Carter-Penrose diagram? What's a Rindler spacetime?

 

[PAllen]

The Milne universe I've familiar with is just a funny coordinate system on flat, topologically trivial spacetime. Carter-Penrose diagrams, such as I understand them, deal only with coordinate independent geometry and topology of the manifold. Given these definitions (which may not be yours - please clarify) - the Carter-Penrose diagram would just be that for Minkowski spacetime.

 

[alialice]

I don't. What have you done? What's a Carter-Penrose diagram? What's a Rindler spacetime?

If you don't know what are they, you can't help me, I'm sorry!

The Milne universe I've familiar with is just a funny coordinate system on flat, topologically trivial spacetime. Carter-Penrose diagrams, such as I understand them, deal only with coordinate independent geometry and topology of the manifold. Given these definitions (which may not be yours - please clarify) - the Carter-Penrose diagram would just be that for Minkowski spacetime.

I think my definitions are the same of you, I don't think there are others...! (I studied Wald and Townsend)
I conclude that the CP diagram of Milne is the same of part of Minkowski in two dimensions.

 

[George Jones]

Sorry, I don't know much about this, so my post doesn't help much. I am familiar with the Milne universe, but I have never really worked with Penrose diagrams. There is a small bit on this in the book Physical Foundations of Cosmology by Mukhanov.

"Region I in Figure 2.7 corresponds to a future light cone which can also be covered by Milne coordinates. The Milne conformal diagram is geometrically similar to the Minkowski one, though it is four time smaller.

Problem 2.10 Draw the conformal diagram for the Milne universe and verify this last statement."

This is part of a section on conformal diagrams.

I have a copy of the book, but the relevant pages (52, 53) are available at Google Books.

 

[alialice]

Sorry, I don't know much about this, so my post doesn't help much. I am familiar with Milne universe, but I have never really worked with Penrose diagrams. There is a small bit on this in the book Physical Foundation of Cosmology by Mukhanov.

Perfect, thank you!

 

[alialice]

I am familiar with Milne universe, but I have never really worked with Penrose diagrams.

Please George, do you think that t=0 is a true cosmological singularity?
Metric $ds^2=dt^2 -t^2dx^2$ with the constraint t>0

 

[bcrowell]

Please George, do you think that t=0 is a true cosmological singularity?
Metric $ds^2=dt^2 -t^2dx^2$ with the constraint t>0

I'm not George, but the answer to your question is no. Your spacetime is flat for t>0, so there is no singularity for t>0.

At t=0 what you have is an apparent change of signature which a sensible physicist will not panic and interpret as follows. If you start with flat spacetime described by the usual coordinates, and then carry out a certain singular change of coordinates, you get coordinates in which the metric looks like the form you're talking about. The change of signature is then an artifact of your bad choice of coordinates. Whether this is the *right* interpretation is something that the standard formalism of GR can't answer, since the standard formalism of GR can't deal properly with changes of signature.

Here is a discussion of a very similar example: http://www.lightandmatter.com/html_books/genrel/ch06/ch06.html#Section6.4 Your example basically describes a universe in which rulers systematically conspire to change their lengths over time; such a change is unobservable. My example is one in which clocks systematically conspire to change their rates over time, and it's similarly unobservable.

 

[alialice]

Your spacetime is flat for t>0, so there is no singularity for t>0.

Thank you Ben.
I know that the spacetime is flat, and it doesn`t present any "curvature" singularity. But by "cosmological" singularity I mean a singularity such as that of a Friedmann-Lemaitre-Robertson-Walker metric for the scale factor a(t)→0.
Is Milne universe espanding in time? I think yes because the scale factor is t. So for t=0 I aspect a cosmological singularity.
The Milne metric I consider now is
$ds^{2}=-dt^{2}+t^{2}dx^{2}$

(even previously I wrote a metric with signs interchanged in referring to the book of Mukhanov)

 

[bcrowell]

I know that the spacetime is flat, and it doesn`t present any "curvature" singularity. But by "cosmological" singularity I mean a singularity such as that of a Friedmann-Lemaitre-Robertson-Walker metric for the scale factor a(t)→0.

The definition of singularity you want to use is not standard and not useful. If you insist on that definition, then the question of whether a given spacetime has a singularity depends on the coordinates arbitrarily chosen to describe it.

 

[hazha]

Hello alialice

I am also studying Milne Universe in two dimensions, so If you have time i will be happy to communicate.

 

[hazha]

compact milne universe

I am doing a research about the compact Milne Universe in two dimensions Using special relativity.
such that (t,θ) goes to (t,θ+β).

the metric is ([ds][/2]=[dt][/2]+t[dθ][/2]

and i am trying to find this compactification parameter (β) and age of the universe.

appreciate anyone who can recommend any article and research paper regarding this kind of model.