What does it mean for the universe to be globally flat,

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Discussion Overview

The discussion revolves around the implications of the universe being globally flat, spherical, or hyperbolic. Participants explore how these geometric properties relate to the universe's expansion, potential recollapse, local geometry, average energy density, and the total mass of the universe.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants propose that the universe's geometry (flat, closed, or open) is an intrinsic property that influences its topology and fate regarding expansion or recollapse.
  • It is suggested that without dark energy, open and flat models will expand forever, while a closed model will recollapse, but the presence of dark energy complicates this relationship.
  • Participants discuss how local geometry, such as the ratio of a circle's circumference to its diameter, varies with the universe's shape, with closed models exhibiting positive curvature, open models negative curvature, and flat models zero curvature.
  • Some argue that the average energy density determines the geometry of space, with different densities leading to different geometric outcomes (closed, open, or flat).
  • There is a claim that if the universe is spherical, its mass will be finite, while in open or flat models, it may be infinite.
  • One participant challenges the assertion that open or flat universes cannot be finite, questioning the criteria for defining finiteness in relation to particle count and age.
  • Another participant defines an infinite universe as one where new points can always be found farther away, suggesting that such a universe would contain an infinite amount of matter if homogeneous and isotropic.
  • There is a humorous exchange regarding the implications of an infinite universe on visibility and perception, with references to mirrors and video recordings.

Areas of Agreement / Disagreement

Participants express differing views on the finiteness of open and flat universes, with some asserting that these cannot be finite while others challenge this notion. The discussion remains unresolved regarding the criteria for defining an infinite universe.

Contextual Notes

Participants highlight the complexity of defining geometric properties and their implications, noting that assumptions about homogeneity and isotropy may affect conclusions about finiteness and mass.

cesiumfrog
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What does it mean for the universe to be globally flat,

What does it mean for the universe to be globally flat, spherical or hyperbolic?

How does it relate to whether the universe will recollapse, or never stop expanding, or expand at an accelerating rate?

How does it relate to local geometry, eg. the ratio of a lab-measurable circle's circumference to diameter? Is this the same as positive vs. negative intrinsic curvature?

How does it relate to the "average energy density" (I understand +ve cosmological constant is equivalent to a uniform distribution of "dark energy", or "material" with negative mass)? How about the total mass in the universe?
 
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cesiumfrog said:
What does it mean for the universe to be globally flat, spherical or hyperbolic?
Space may be flat, closed or open. This is an intrinsic geometric property of space. If it is homogeneous and isotropic, these three posibilities lead to three diferent topologies or (global) shapes of the universe. You can read more about this here: http://en.wikipedia.org/wiki/Shape_of_the_Universe

However, note that the term "globally hyperbolic" may have a different meaning. It is very often used to talk about a global property of space-time, rather than a global property of space. Space-times that are are globally hyperbolic are predictable with well posed initial value problems.

cesiumfrog said:
How does it relate to whether the universe will recollapse, or never stop expanding, or expand at an accelerating rate?
If there is no dark energy, then the open and flat models will expand forever and the closed will recollapse. If there is dark energy, then this relation between geometry of space and fate of the universe does not exist. A closed universe with enough dark energy may also expand forever.

cesiumfrog said:
How does it relate to local geometry, eg. the ratio of a lab-measurable circle's circumference to diameter? Is this the same as positive vs. negative intrinsic curvature?
In the closed model the circumference of a circle would be less than π times the diameter. In the open model it will be greater and in the flat model it will be equal. In the closed model space has a positive intrinsic curvature, in the open model negative and in the flat model zero.

cesiumfrog said:
How does it relate to the "average energy density" (I understand +ve cosmological constant is equivalent to a uniform distribution of "dark energy", or "material" with negative mass)?
The energy density determines the geometry of space. If the density is greater than the critical then space is closed, if less then it is open, if equal then it is flat. Dark energy has positive energy density, but unlike ordinary energy it exerts a negative pressure. Pressure does not contribute to the geometry of space, but in general to the geometry of space-time and influences the expansion of space.

cesiumfrog said:
How about the total mass in the universe?
If its shape is spherical then the mass will be finite. In the other two cases it will be infinite.
 
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hellfire said:
If its shape is spherical then the mass will be finite. In the other two cases it will be infinite.
Could you explain to me how you make the claim that an open or flat universe cannot possibly be finite?
 
MeJennifer said:
Could you explain to me how you make the claim that an open or flat universe cannot possibly be finite?
There is no way for an open or flat space to curve and close globally to a finite shape. There may be shapes that are flat and finite but then these are not completely homogeneous and isotropic. An example may be the 3-torus that contains an euclidean metric, but it contains also special directions that can be identified.
 
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hellfire said:
There is no way for an open or flat space to curve and close globally to a finite shape.
I understand that. But closure does not seem to be the only criterium for it being infinite.

Do you consider a universe that is not closed with a countable number of particles infinite just because it is not closed?
Do you consider a universe that is not closed with a given age infinite just because it is not closed?
 
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OK, this obviously depends on what you understand with infinite universe. With infinite I mean spatially infinite, where you can find always a new point that is farther away than another. I think this is the usual meaning. If such an universe contains an homogeneous and isotropic distribution of matter, then the amount of matter or particles is infinite.
 
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In an infinite universe you can never see yourself looking at the back of your head; no matter how big your telescope is.
 
Chronos said:
In an infinite universe you can never see yourself looking at the back of your head; no matter how big your telescope is.
True (except perhaps for exotic local curvatures) but that can even be the case in certain closed universes.
 
Chronos said:
In an infinite universe you can never see yourself looking at the back of your head; no matter how big your telescope is.

Take two mirrors, one on front of you, and the other behind you. Have someone video tape you doing it. Then pop the tape into the VCR and hit PLAY. :rolleyes:
 

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