Why does a universe with a spherical geometry have to be finite?

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Discussion Overview

The discussion revolves around the concept of a universe with spherical geometry and its implications for finiteness. Participants explore the nature of finite versus infinite geometries, particularly in the context of one-dimensional and two-dimensional analogs, and how these concepts relate to cosmological models.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants express confusion about why a spherical universe is considered finite, contrasting it with flat and hyperbolic universes which are described as infinite.
  • One participant proposes that a one-dimensional universe shaped like a ring must be finite, as it has a defined circumference, and questions if an infinite circumference can exist.
  • Another participant extends this reasoning to a two-dimensional universe, suggesting that the surface of a sphere must also have finite area.
  • Some participants discuss the properties of spherical geometries, noting that in a positively curved space, the sum of angles in a triangle exceeds 180 degrees and that parallel lines do not exist.
  • There is mention of a theoretical model of a spherical universe that could be infinite yet bounded, referencing galaxies receding at high velocities, though this model is challenged by another participant who points out inconsistencies with observed redshifts.
  • One participant introduces the idea of fractal surfaces, questioning whether such surfaces could be infinite, but acknowledges that in practical terms, real-world surfaces would reach a finite limit.
  • Concerns are raised about the analogy of parallel lines on a globe, with one participant noting that while they can draw parallel lines on a globe, the geometric principles suggest they should eventually intersect.
  • Another participant reflects on the mathematical properties of circles, specifically the infinite decimal expansion of π, and questions its implications for the concept of circumference.

Areas of Agreement / Disagreement

Participants express differing views on the finiteness of spherical universes, with some agreeing on the properties of finite geometries while others introduce competing models and challenge existing explanations. The discussion remains unresolved regarding the implications of these geometries in cosmology.

Contextual Notes

Participants highlight limitations in their understanding of geometric concepts and the implications of curvature on parallel lines. There is also a recognition of the complexity of cosmological models and the need for clarity in definitions and assumptions.

Jack_O
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I have read that "A universe with a spherical geometry is called a closed universe because a universe with this geometry must be finite" but even after looking up different sources i cannot find an decent explanation of it is finite. I know that a flat universe is just an unbound 3d grid that goes on forever and is therefore infinite, a hyperbolic universe is also infinite because it stretches on forever (i can't really visualize this).

I can't understand a why a spherical universe has to be finite. Would appreciate it if someone could shed some light on this for me.
 
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Jack_O said:
...
I can't understand a why a spherical universe has to be finite. Would appreciate it if someone could shed some light on this for me.

I need to know a little bit how you think and what you mean by your words. So please help me out and tell me something.

Let's imagine a one-dimensional universe that has the geometry of a ring. A circle.
I think this must be finite, length is the appropriate measure in one dimension, I think any particiular ring universe that you choose must have some finite length (called the circumference). Do you agree?

If not, please show me an example of some particular ring with infinite circumference.
============

After that, let's talk about a 2D universe that is the surface of a ball. I think any example of such a thing must have a finite area (the appropriate measure for a 2D sphere surface).
Do you agree, or can you describe some definite fixed sphere with infinite surface area?
 
Ok i think what you are saying is that a 1D ring universe is finite because if you go all the way around the ring you end up where you started after traveling some finite distance, ie bound to the ring?

What I don't understand is why definition of the ring universe doesn't explicitly state that the ring has to have a finite diameter and therefore finite circumference? Or is that just implicit because it uses a bound shape, the circle?

I guess my example of a ring with infinite circumference is one with infinite area and diameter.
 
When you read about a universe with a spherical geometry they are referring to a universe that obeys the geometry of a sphere, albeit a four dimensional sphere. The sum of the angles of triangles add up to more than 180 deg and there are no parallel lines. This would be a finite yet unbounded universe.

There is another kind of spherical universe discussed by Einstein that is infinite yet bounded. If as we look into space we see that the farther away a galaxy is the faster it is receding from us, and if the universe is old enough, there would be galaxies receding at velocities approaching the speed of light. Because they would be contracted in the direction they are receding, an infinite number of them could be squeezed into the space at the limit of the universe. So far, the available evidence about our universe doesn't fit either of these descriptions.
 
Jack_O said:
I guess my example of a ring with infinite circumference is one with infinite area and diameter.

Thanks. That gives me a better idea of your world of discourse, style of thinking etc.
 
skeptic2 said:
There is another kind of spherical universe discussed by Einstein that is infinite yet bounded. If as we look into space we see that the farther away a galaxy is the faster it is receding from us, and if the universe is old enough, there would be galaxies receding at velocities approaching the speed of light. Because they would be contracted in the direction they are receding, an infinite number of them could be squeezed into the space at the limit of the universe. So far, the available evidence about our universe doesn't fit either of these descriptions.

This part of your post does not make sense, Skep.
Most of the galaxies which we now observe are receding at rates greater than the speed of light. Anything with a redshift bigger than 1.4. And there is a whole lot of stuff visible with redshift > 1.4.

It doesn't cause any sort of "contraction". You may be thinking of the special relativity speed limit which doesn't apply to recession rates.

There is a good SciAm article called "Misconceptions about the Big Bang". I have a link to it in my signature (small print at bottom of post). It is the princeton.edu link. The author is Charles Lineweaver (a world class cosmologist by the way :smile:)
 
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Marcus, what you are saying is true. This theory is not my theory. I remember studying it many years ago in college and spent a little time searching for it but couldn't find it. I didn't intend to imply that that theory represents our current understanding of cosmology, in fact I think I said that.
 
marcus said:
I need to know a little bit how you think and what you mean by your words. So please help me out and tell me something.

Let's imagine a one-dimensional universe that has the geometry of a ring. A circle.
I think this must be finite, length is the appropriate measure in one dimension, I think any particiular ring universe that you choose must have some finite length (called the circumference). Do you agree?

If not, please show me an example of some particular ring with infinite circumference.
============

After that, let's talk about a 2D universe that is the surface of a ball. I think any example of such a thing must have a finite area (the appropriate measure for a 2D sphere surface).
Do you agree, or can you describe some definite fixed sphere with infinite surface area?


how about a fractal surface?
 
TalonD said:
how about a fractal surface?

Well, how about one? Would you like to make some point?

I was talking about the 2D surface of a 3D ball, something with a definite geometry.
If you want to talk about some other geometric object, some different surface, you could define it.
 
  • #10
skeptic2 said:
There is another kind of spherical universe discussed by Einstein that is infinite yet bounded. If as we look into space we see that the farther away a galaxy is the faster it is receding from us, and if the universe is old enough, there would be galaxies receding at velocities approaching the speed of light. Because they would be contracted in the direction they are receding, an infinite number of them could be squeezed into the space at the limit of the universe. So far, the available evidence about our universe doesn't fit either of these descriptions.
This is just the standard description using different spacetime coordinates. It is probably best understood by looking at the diagrams on Ned Wright's cosmology tutorial:
http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH
 
  • #11
marcus said:
Well, how about one? Would you like to make some point?

I was talking about the 2D surface of a 3D ball, something with a definite geometry.
If you want to talk about some other geometric object, some different surface, you could define it.

I was just thinking how a 2d surface of a 3d sphere could be infinite, so I thought fractals.. infinite recursions of a surface texture but that's just mathmatical, in the real world you would reach a finite limit. Of course if you are talking a pure geometric sphere then it would have no surface texture and would be finite.

one thing I kind of take issue with and it's probably due to a lack of understanding on my part. It has been said that in postivly curved space there can be no parralel lines, that they would eventually intersect. Yet I can draw parallel non intersecting lines all the way around a globe.
Actually I understand the concept, I'm just being nitpicky, I think the globe and poles analogy is useful only to a point. like the balloon analogy.

but I like a previous posters comment about a circle or sphere with infinite diameter. That would be as easy to imagine as an infinite 2d plane.

'edit'.. another quick thought. since the circumference of a circle is calculated based on PI and since PI has infinite decimal fraction then is the circumference of a circle really finite?




Hope all that's not too off topic.
 
  • #12
Well, the question is much deeper I believe

We observe a spacetime curvature, but curvature does not define the tolopogy of the space. For example, we can not say if we are inside a sphere or semi-sphere (half-sphere, not sure what term is correct. It has the same curvature as sphere everywhere, but its volume is 1/2 of the sphere) .
 
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  • #13
TalonD said:
'edit'.. another quick thought. since the circumference of a circle is calculated based on PI and since PI has infinite decimal fraction then is the circumference of a circle really finite?

Pi is an irrational number which is why it has non repeating decimal that goes on forever, it's magnitude is definitely finite though, for instance pi is a smaller number than 3.2

Back on topic... A euclidean (3D uniform grid) universe can become infinite but only after being allowed an infinite amount of time to expand. Why couldn't a spherical (3d universe bound to the 4D equivalent of a sphere?) universe become infinite after being allowed to expand forever?

I think i might be going round in circles (no pun intended) in my head trying to understand this concept.
 
  • #14
Jack_O said:
A euclidean (3D uniform grid) universe can become infinite but only after being allowed an infinite amount of time to expand.

No.
At first, finite object can not become infinite because of the expansion.
If our Universe is infinite, then it is infinite from the very beginning.
Euclidean universe is infinite from the very beginning (if there are no edges)
 

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