# Why not a flat but finite spatial universe?

1. Mar 26, 2014

In the Wiki article on the FLRW metric, http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric
it says "the universe is nearly an isotropic and homogeneous FLRW spacetime".

OK, so spacetime is globally flat, which implies that space is too. This is backed up by
http://www.isciencetimes.com/articl...ed-perfect-accuracy-infinite-flat-eternal.htm
which states
" The scientists [at the Apache Point Observatory] translated the data into a 3-D map of the universe. What they discovered is that the universe is 'flat'...."

In the same article the team leader David Schlegel says "...it's likely the universe extends forever in space..."

But in a 2001 interview http://www.esa.int/Our_Activities/S...ite_or_infinite_An_interview_with_Joseph_Silk
Prof. Joseph Silk pointed out that a spatially flat universe could be either infinite or finite (with the surface of a torus as an example of a flat finite space), but that the Planck satellite might be able to distinguish between the two possibilities.

So, has the Planck satellite findings or Apache Point Observatory data cleared this up? If so, how? If not, why does Schlegel (and other sites) say that the universe is "probably" infinite?

2. Mar 26, 2014

### Simon Bridge

You mean, how would we be able to tell one from the other?
Because the answer to the stated question is: "no reason".

The "global" topology of the Universe is an area of active research and there are no positive conclusions so far.
Wikipedia has a brief overview of a bunch of possibilities:
http://en.wikipedia.org/wiki/Shape_of_the_universe#Flat_universe

Both authors are speculating on their favorite models.

3. Mar 26, 2014

Thank you, Simon Bridge. Since, as you say, between the two possibilities for models of a flat universe, there is no reason to choose one over the other at the moment (and possibly never), it is curious that most articles on the Internet (sorry, this is a subjective impression) seem to favor the "spatially infinite" model. Whereas science is not a democracy, I wonder what the appeal of that model over the other one is.

4. Mar 26, 2014

### Bill_K

Simplicity. In a finite model, space would exhibit an apparent periodicity: you'd see the same galaxy repeated endlessly, in different equally spaced locations. The images of the galaxy would form a 3-D lattice.

But note that in such a model there is no way to preserve isotropy. Although the expansion rate could still be isotropic, the distance associated with the periodicity would necessarily be different in different directions. As a result, the universe would have a principal set of axes.

5. Mar 26, 2014

Thanks again, Bill_K. Very well explained. Fascinating!

6. Mar 26, 2014

### timmdeeg

In contrast to the infinite plane the torus is a non-trivial solution, perhaps that's why people don't prefer it.

Last edited: Mar 26, 2014
7. Mar 26, 2014

Thanks, timmdeeg. An interesting consideration.

8. Mar 26, 2014

### timmdeeg

Would this mean that the cosmological principle isn't preserved in the special case of the 3-torus, though being a FRW universe.

9. Mar 26, 2014

### George Jones

Staff Emeritus
Yes, a torus violates the cosmological principle because it is not isotropic.

It is possible for a spacetime that has an FLRW metric to have toroidal spatial sections, but such a spacetime usually is not considered to be an FLRW universe. The usual definition of an FLRW universe is a spacetime that has an FLRW metric, and that is spatially isotropic and homogeneous.

10. Mar 26, 2014

### timmdeeg

Ah, good to know. Thanks, George.

11. Mar 27, 2014

### Chalnoth

This is false. Space-time is most definitely curved. It is possible to select time slicings where all three spatial dimensions are flat, but there is still curvature between the space and time dimensions. We see this curvature as the expansion of our universe.

12. Mar 28, 2014

### Chronos

Even if space is infinite, there are regions that will remain forever unobservable. See; http://arxiv.org/abs/astro-ph/0310808, Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe, for discussion.

13. Mar 29, 2014

Thanks to all; a couple of replies:
Chalnoth: I stand corrected, that should have been that spacetime is asymptotically flat.

Chronos: good article, a summary of which should be included in standard physics courses.

14. Mar 29, 2014

### Mordred

Last edited: Mar 29, 2014
15. Mar 29, 2014

Mordred, thanks. The second article (or past post) is quite nice. It would have been even nicer if the last section (FLRW metric) had been expanded upon a bit.

As to the first, mistaken post: I started reading it before I saw your edit, and would just note the following point on that: http://cecelia.physics.indiana.edu/life/redshift.html

16. Mar 29, 2014

### Chalnoth

Not at all. If I remember correctly, the space-time curvature is proportional to $H^2$, which doesn't tend towards zero (It's been a while since I worked it out, but I'm sure it's at least proportional to some power of the expansion rate....).

17. Mar 29, 2014

### Mordred

glad you enjoyed it, the cutoff point on the 4D metric was due to length, to properly cover the 4D section for each geometric shape would of nearly doubled its length. However the metrics covered is usually sufficient to get the concept of universe geometry across.

18. Mar 29, 2014