# Milne universe

1. Jul 11, 2012

### 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?

2. Jul 11, 2012

### HallsofIvy

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

Last edited: Jul 17, 2012
3. Jul 11, 2012

### Dickfore

The Wiki

4. Jul 13, 2012

### alialice

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

5. Jul 13, 2012

### Dickfore

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

6. Jul 13, 2012

### 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.

7. Jul 13, 2012

### alialice

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

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.

8. Jul 13, 2012

### George Jones

Staff Emeritus
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.

Last edited: Jul 16, 2012
9. Jul 14, 2012

### alialice

Perfect, thank you!

10. Jul 16, 2012

### alialice

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

11. Jul 16, 2012

### bcrowell

Staff Emeritus
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 [Broken] 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.

Last edited by a moderator: May 6, 2017
12. Jul 17, 2012

### alialice

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)

13. Jul 17, 2012

### bcrowell

Staff Emeritus
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.

14. Aug 28, 2013

### hazha

Hello alialice

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

15. Aug 28, 2013

### 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 recomend any article and research paper regarding this kind of model.