May the space be flat and infinite or curved and finite?

Question

May the space be flat and infinite or curved and finite? Personally I cannot explain myself a infinite object and how eventually to describe it but on the other hand a curved and finite space should require extra dimensions and there should not be any possibility of stright velocity vectors.

Answer 1

Side note: As already pointed out in the comments, curvature of spacetime in general relativity is intrinsic, i.e. does not require extra spatial dimensions.

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As described in the Wikipedia article Shape of the universe, there are multiple aspects which describe the geometry of the universe, e.g.

1. Boundedness (whether the universe is finite or infinite)
2. Flat (zero curvature), hyperbolic (negative curvature), or spherical (positive curvature)
3. Connectivity: how the universe is put together, i.e., simply connected space or multiply connected space.

Let's briefly discuss what is known about each of these.

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1. Boundedness

There are two aspects here to consider. First, whether there is some sort of physical border at which the universe ends or not (Such a border might be comparable to the end of a disc which simply ends there). Second, the actual size of the universe.

One might be tempted to say that if there isn't a border, then the universe must be infinite. However, it could also be similar to a sphere – its surface has no border but is still finite in size.

So what do we know about our universe? We can only observe some parts of it (the observable universe) so if there was a border beyond that, we wouldn't be able to tell. However the FLRW metric which is used in the standard model of cosmology starts with the assumption that the universe is homogenous and isotropic. If there were any border, this place would be different from the other places in the universe, thus violating the two conditions.

Thus, according to our current understanding, the universe is either infinite or "looped" in itself (like the sphere).

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2. Curvature

One can measure the large-scale curvature of the universe by evaluating whether the angles in triangles add up to 180°. If yes, spacetime is said to be flat. If the angles add up to more or less than 180°, spacetime has positive curvature (spherical geometry) or negative curvature (hyperbolic geometry), respectively.

So far, no large-scale curvature has been detected, implying that the universe is flat. However, it could also be the case that it is simply not large enough to be measured with current technology. Nevertheless, as far as I know, most physicist believe the universe to be flat.

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Now let's get to your question. Could space be flat and infinite or curved and finite? To both, the answer is yes. The universe could, as explained above be flat and infinite or have spherical geometry (curved and finite, but no border).

What would it mean if the universe were infinite? I personally found the most helpful description that it means that there is no upper bound on the distance between two objects. In a closed (finite) universe, there is such an upper bound. The actual distance between two objects however cannot be infinite.

However, as pointed out in the comments, the universe could also be flat and finite (no border though) which brings us two the third point.

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3. Connectivity

I am not very familiar with connectivity, but here is what I think I have understood.

The universe could be flat and looped in itself (i.e. finite, but no boundary). While a curved finite and universe without border can relatively simply be understood by the sphere analogy, a flat "looped" universe is a bit more complicated.

A somewhat intuitive explanation is the following:

It can be obtained by "gluing" the three pairs of opposite faces of a cube, where being "glued" can be intuitively understood to mean that when a particle moving in the interior of the cube reaches a point on a face, it goes through it and appears to come forth from the corresponding point on the opposite face

This is then called a 3-torus. Thus curvature is not necessarily required for a finite universe without border. However, as far as I know, there is no evidence for the universe having the topology of such a 3-torus.

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Summary

• The universe probably does not have a physical border since this would violate the assumptions of homogeneity and isotropy
• No large-scale curvature has been detected so far, thus the universe is probably flat
• The universe could either be flat and infinite or flat and finite ("looped"), the latter for example through the shape of a 3-torus
• If the large-scale curvature simply has not been detected yet but is there, the universe could have spherical (i.e. finite in size, but no border) or hyperbolic geometry

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Related reading: Is the universe finite or infinite?

Answer 2

"May the space be flat and infinite or curved and finite?"

I am not sure what you mean by "may".

One definition might be: Is it theoretically possible to create a mathematical model for the universe with an arbitrary set of basic constraint assumptions?

Another definition might be: Among professional cosmologists, what is the currently most popular universe model, and what are its constraint assumptions.

I am assuming you chose the second definition. In this case the most popular model assumes a universe that at a large scale (but a scale smaller than the size of the observable universe) it is homogeneous and isotropic.

See https://en.wikipedia.org/wiki/Cosmological_principle for definition.

With these constraints the universe may be either infinite or finite. If flat it will be infinite. If not flat it may be either 3D-hyper-spherical and finite, or it may be 3D-hyperbolic and infinite.

Based on a lot of observation data it appears that the universe is either flat, or very close to being flat. If the latter, it can be either infinite or finite. The key relevant model variable is Omega-K representing curvature density.

Flat requires Omega-K = 0. Nearly flat means 0 < |Omega-K| << 1.

If Omega-K < 0 then the corresponding finite universe is 3-D hyper-spherical.

If Omega-K > 0 then the corresponding infinite universe is 3-D hyperbolic.

Answer 3

The entirety of space may actually include any combination of the four characteristics ("flat", "infinite", "curved", "finite") mentioned by the question's originator.

A plane's the limit of a ball (a spherical "solid", but "solid" in the sense of having three spatial dimensions, and not in the sense of necessarily being completely filled with matter) of radii perhaps extremely long, with the limit being defined by observablility: It would not necessarily comprise any actual or physical limit of space, which could be infinitely remote and would almost certainly remain outside our currently observable region, quite possibly forever. (Beings of a civilization older than our own, &/or on different scales of space and time, might find that limit beyond, or in the horizon of, a region that may not actually have remained observable by us.)

When I'm saying "observability", I'm referring to that extent to which the optical (biological &/or cybernetic) or magnification equipment in use (by whatever beings might be making the observations concerned) either might, or might not, be adequate for differentiating curved surfaces in space from any and all surfaces adjoining it, amid the particulate matter (dust, planets, stars) prevalent in their locality. The "simple" approach of estimating spacetime curvature thru any such measurements might be extremely difficult in practice: A problem hypothetically visualized, in this approach, would be the possibility that the concentrations of energy involved in the necessary magnification might, given mass/energy equivalence, tend to collapse observed objects into "mini" black holes (for which no evidence has, nevertheless, been found). It might require some modification of Einsteinian relativity, in an application of the Einstein-Cartan Theory (which assigns a tiny spacelike extent to fermions) that was developed by Einstein in collaboration with the mathematician Cartan in 1929, a few years after the discovery of particulate spin, and was somewhat refined by Sciama and Kibble a few decades later.

As the formalism used in the extremely complex ECT (or ECSK Theory) is not widely known, I should mention the reported fact that it reduces to General Relativity (Einstein's 1915 theory) in vacuum, for the convenience of anyone wanting to assess the realistic applicability of my answer: Given the fact that General Relativity had been modified, after its 1915 publication, by Einstein's adoption of a cosmological constant, I should mention that some alteration in the cosmological role of energy (whose particles remain pointlike in ECT) might, at least in the view of the renowned Laura Mersini-Houghton, be accomplished by abandoment of that standard assumption of cosmological homogeneity that was mentioned in the answer previous to my own. The inhomogeneous viewpoint was brought out through her use of it, within recent years, in the explanation of a void in the Cosmic Microwave Background Radiation.