I What is the nature of space and time in deep empty regions of the cosmos?

[Satyam]

When we talk about the universe expanding, we are not normally talking about an increase in the radius of the observable universe. Rather we are talking about the fact that all of the [large scale] things in the universe are getting farther apart.

If the universe as a whole is infinite, then it does not have a defined volume. So it is not correct to talk about its volume increasing or decreasing.

There is no such domain. It is not that sort of expansion. It is not expanding to fill some existing empty space. It is simply expanding. Distances are getting greater. There is no need consider our universe as somehow embedded within a higher dimensional space in order to describe its expansion.

Edit...

You had invoked the balloon analogy, pointing out that when the balloon expends, it displaces air. This is a good example of what it means to have a lower-dimensional space embedded in a higher dimensional space. The surface of the balloon is a two dimensional space. We picture it embedded in a pre-existing three dimensional space. We do that because it is easy to imagine.

But there is no requirement for the three dimensional space to actually exist. One can describe all of the relevant properties of a two dimensional surface with a spherical topology without ever considering it to exist within a three dimensional space. One can do it with a two dimensional coordinate system (like latitude and longitude). The trick is to use a distance metric that is different from the euclidean $\sqrt{x^2 + y^2}$ one. [One also has to split it up into multiple patches -- that's what we call a manifold].

Same for our four-dimensional space-time. We can describe the relevant properties in terms of a metric rather than in terms of some euclidean hyper-space within which it is hypothetically embedded.

I thought we do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper like an 'infinite' sheet of paper and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus i.e like a doughnut. The surface of the torus is also spatially flat, but it is finite. So we have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

 

[jbriggs444]

I thought we do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper like an 'infinite' sheet of paper and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus i.e like a doughnut. The surface of the torus is also spatially flat, but it is finite. So we have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

Again, that is not a flat Universe. That is a flat spatial slice of a universe. I did not make a claim one way or the other about the size of such a slice.

 

[Dale]

Sir on what reference we are saying that it is homogeneous. In order to understand FLRW Model. It is important to grasp why it is considered homogeneous?

The frame where it is homogenous and isotropic is called the comoving frame or comoving coordinates. Those are the standard coordinates for cosmology.

Regarding if it is important: I would say yes. The assumption of homogeneity and isotropy greatly constrains the form of the possible solutions. That is what makes it one of the few analytically tractable spacetime models.

 

[PAllen]

Isotropy everywhere implies homogeneity everywhere. However homogeneity (everywhere) does not imply isotropy. Consider a geometrically flat cylinder. It is homogeneous but not isotropic.

 

[Satyam]

Again, that is not a flat Universe. That is a flat spatial slice of a universe. I did not make a claim one way or the other about the size of such a slice.

The surface of the torus is termed as flat and surface is a slice of 3d as it is 2d. The point on which I want to drag your attention is that the universe can be in the form of torus and as there are no ends to it , it can be termed as endless universe. But being endless doesn't mean it is infinite.
We don't know yet the universe is finite or infinite.

If the universe as a whole is infinite, then it does not have a defined volume. So it is not correct to talk about its volume increasing or decreasing.

Is this correct to term universe as infinite and then making a statement based on that..?

 

[Dale]

The point on which I want to drag your attention is that the universe can be in the form of torus

Do you have a reference for this? I am skeptical that a toroidal universe is compatible with observation.

 

[PAllen]

A universe with toroidal spatial slices would not be isotropic. This would have observational consequences (and could not be described by an FLRW metric). (I see @Dale made a similar point).

 

[Satyam]

Do you have a reference for this? I am skeptical that a toroidal universe is compatible with observation.

Yes Sir
https://en.m.wikipedia.org/wiki/Shape_of_the_universe
Here I will tell you the headings in which you can look into this matter more quickly.
•Shape of the observable universe.
•Global universe structure

 

[Dale]

Yes Sir
https://en.m.wikipedia.org/wiki/Shape_of_the_universe
Here I will tell you the headings in which you can look into this matter more quickly.
•Shape of the observable universe.
•Global universe structure

Nothing on that page says that a toroidal universe is consistent with observation. In fact, it says:

"The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat,[7] but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space[8][9] and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by 2-dimensional lattice).[10] "

A torus is not listed as one of the other possible shapes consistent with the data, despite its various mathematical properties being mentioned several times in the article. I think that your point in post 35 is incorrect. A torus-shaped universe is not consistent with the data.

 

[PeterDonis]

A torus is not listed as one of the other possible shapes consistent with the data

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes. For one thing, I have seen references given in previous PF threads to papers stating that a finite, closed 3-sphere universe is not completely ruled out by the data, just highly unlikely; but a finite closed 3-sphere is not one of the possibilities listed in that paragraph.

(Quite frankly, to me the two possibilities other than "infinite and flat" that are listed seem considerably more esoteric to me than a 3-sphere, so the fact that those esoteric possibilities were listed and the 3-sphere was not makes me give less credibility to the article as a whole.)

A flat 3-torus with a large enough finite volume would be indistinguishable from a flat infinite Euclidean 3-space given the finite age of the universe, so I don't see how it could be ruled out completely; the most we could do from the data would be to set a lower bound on the finite volume of the 3-torus.

 

[Dale]

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes.

Sure, but it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe. I had requested such a reference to support the claim made and this is not one.

 

[PeterDonis]

it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe.

Yes, agreed.

 

[Satyam]

Nothing on that page says that a toroidal universe is consistent with observation. In fact, it says:

"The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat,[7] but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space[8][9] and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by 2-dimensional lattice).[10] "

A torus is not listed as one of the other possible shapes consistent with the data, despite its various mathematical properties being mentioned several times in the article. I think that your point in post 35 is incorrect. A torus-shaped universe is not consistent with the data.

Sure, but it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe. I had requested such a reference to support the claim made and this is not one.

Sir I'm not making a claim but just asking that could it be possible. As I have read it in some articles.
Here I will post the lines from the article of Wikipedia which makes me think this way.
If we assume a finite universe then possible considerations Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.
In mathematics, there are definitions for a closed manifold (i.e., compact without boundary) and open manifold (i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.
Here it is being said as in FLRW model the universe is considered to be without boundries in which case compact universe could describe a universe that is a closed manifold..

 

[Satyam]

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes. For one thing, I have seen references given in previous PF threads to papers stating that a finite, closed 3-sphere universe is not completely ruled out by the data, just highly unlikely; but a finite closed 3-sphere is not one of the possibilities listed in that paragraph.

As stated in the introduction, investigations within the study of the global structure of the universe include:

•Whether the universe is infinite or finite in extent
Whether the geometry of the global universe is flat, positively curved, or negatively curved
Whether the topology is simply connected like a sphere or multiply connected, like a torus[14]
A finite closed 3-sphere is one of the possibilities listed in that paragraph.
However there is also an interview given by Joseph silk who was Head of Astrophysics, Department of Physics, University of Oxford, United Kingdom http://www.esa.int/Science_Explorat...ite_or_infinite_An_interview_with_Joseph_Silk

 

[Dale]

Sir I'm not making a claim but just asking that could it be possible.

Ah ok. It may just be a language issue. You said “the universe can be in the form of torus” which in English is a statement of fact rather than “can the universe be in the form of torus” which would be the corresponding question.

So as a question I don’t know of any analysis of current cosmological evidence in terms of a torus. There may be some literature on the topic but I am not aware of it.

However, as a moderate Bayesian I have a preference for parsimonious models. If the evidence equally supports a simply-connected geometry and a toroidal geometry then I would use the simply-connected model and would consider the non-simply connected geometry to not be supported by the data.

 

[PeterDonis]

As stated in the introduction

That Wikipedia article is not a good reference, even more so than the average Wikipedia article. If you look at the non-mobile version, you will see that a number of statements in the article are disputed.

there is also an interview

He says basically what I said in post #40:

A flat 3-torus with a large enough finite volume would be indistinguishable from a flat infinite Euclidean 3-space given the finite age of the universe

Note, however, that a flat 3-torus as a model requires more assumptions than the standard infinite flat FRW universe, so if the data is equally consistent with both, the flat 3-torus gets ruled out by Occam's razor. If we ever find evidence in favor of a flat 3-torus, that would be different, but we haven't.

 

[Satyam]

Ok sir I have understood it as there are no evidence till yet in support of the flat 3 torus and we have to make more assumptions in order to consider this shape. Data is consistent for the one which requires minimum assumptions.
So do we know the shape of the universe yet , if we do then of what shape it is?

 

[Dale]

So do we know the shape of the universe yet , if we do then of what shape it is?

The shape is the FLRW shape. It doesn’t have an English name, but the shape is specified by the FLRW metric. The closest English word would be trumpet-shaped, but that isn’t exactly right. Hence the need for the math.

 

[PAllen]

A flat 3 torus would have the same FLRW metric as for infinite flat spatial slices, but this metric would be over one chart of several that are joined by transition functions to specify the topology. So the difference is not the metric over some region, but that by assumption of homogeneity and isotropy, the metric is global, and one chart covers the whole manifold. Whereas, for the 3 torus, you have several charts with transition functions, but the metric in each chart can be arranged to be same as the FLRW metric. This atlas of charts for the flat 3-torus violates isotropy (which is always assumed to mean isotropy everywhere, in cosmology), but as @PeterDonis noted, for a big enough torus up to the current age of the universe, it could be that this anisotropy is not yet observable (i.e. the causal past from present day Earth can be contained in one chart).