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

[cesiumfrog]

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?

 

[hellfire]

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.

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.

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.

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.

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.

 

[MeJennifer]

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?

 

[hellfire]

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.

 

[MeJennifer]

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?

 

[hellfire]

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.

 

[Chronos]

In an infinite universe you can never see yourself looking at the back of your head; no matter how big your telescope is.

 

[MeJennifer]

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.

 

[kmarinas86]

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.