Trying to understand the Shape of the universe.

[dubsed]

Please note that I am not trying to forward any type of personal theory. I am only trying to understand generally accepted physics.

I have heard the universe described as the surface of a balloon. Inflating the balloon is the expansion of the universe and everything move away from everything. No edges nice and neat.

I would suppose that if you extend the balloon to a fourth dimension then you would have space as the surface of a hypersphere. If this is not the correct way to look at it then please explain what is.

The questions bellow are only pertinent if a 4d hypersphere is the correct way to view the universe.

Given enough resources and ignoring the expansion of the universe. Is it possible to circumnavigate the universe (go in one direction for long enough and end up where you started?) What would be the path equation for such a trip look like?

If the universe is a 4d hypersphere then I assume that by definition it is curved? How would someone at the surface be able to measure the curvature from a fixed point?

 

[bcrowell]

I would suppose that if you extend the balloon to a fourth dimension then you would have space as the surface of a hypersphere. If this is not the correct way to look at it then please explain what is.

You can visualize it embedded in a 5-dimensional space, but that's not a necessary part of the theory. We also don't know whether it's spatially closed or open.

Given enough resources and ignoring the expansion of the universe. Is it possible to circumnavigate the universe (go in one direction for long enough and end up where you started?)

http://en.wikipedia.org/wiki/Observable_universe#The_Universe_versus_the_observable_universe

If the universe is a 4d hypersphere then I assume that by definition it is curved? How would someone at the surface be able to measure the curvature from a fixed point?

In practice this is done by looking at the angular scale of the fluctuations in the cosmic microwave background. In principle you could build a triangle out of laser beams and measure deviations from the Pythagorean theorem.

 

[marcus]

Please note that I am not trying to forward any type of personal theory. I am only trying to understand generally accepted physics.

I have heard the universe described as the surface of a balloon. Inflating the balloon is the expansion of the universe and everything move away from everything. No edges nice and neat.

I would suppose that if you extend the balloon to a fourth dimension then you would have space as the surface of a hypersphere. If this is not the correct way to look at it then please explain what is.

The questions bellow are only pertinent if a 4d hypersphere is the correct way to view the universe.

Given enough resources and ignoring the expansion of the universe. Is it possible to circumnavigate the universe (go in one direction for long enough and end up where you started?) What would be the path equation for such a trip look like?

If the universe is a 4d hypersphere then I assume that by definition it is curved? How would someone at the surface be able to measure the curvature from a fixed point?

I appreciate very much the spirit of patient questioning, and the focus on understanding conventional mainstream cosmology! There are a handful of well-qualified people that currently check in, and I hope several will answer. Also I hope you keep up your questioning in this vein!

I'll offer my personal unauthoritative take on the issues. EDIT: Now I see B.Crowell has already replied!

The balloon is just an analogy, not a description. We actually don't know the overall topology or the overall average curvature. I think the visual analogy here:
http://www.astro.ucla.edu/~wright/Balloon2.html
would be useful even if the real universe were average zero curvature and infinite spatial volume.

Mathematically it makes some sense to think about a 3D hypersphere embedded in 4D Euclidean space (time not shown in the picture, this is just an instantaneous snapshot.)
But we don't know that a surrounding 4D space exists.

We only know the geometry of space from inside. We can't step outside and view it in some larger context. For us all existence is concentrated in this world. There is no way to "point your finger" towards the inside or outside of the balloon surface---to speak using that analogy.

We can only measure geometry from the inside, measure curvature analogously to checking to see if the angles of a triangle add up to 180, or that the area of a large sphere increases as the square of radius----say by counting galaxies at larger and larger distance and assuming uniform distribution. Crowell mentioned one of the methods that is used (angular scale of CMB temp fluctuations.)

Curvature has been estimated by WMAP, but the best they can do is a 95% confidence interval which indeed contains zero---but would also be consistent with a slight positive or negative curvature.

IF the universe were in fact spatially closed, say a hypersphere, (which we don't know) and IF you could stop expansion, then with enough time you could circumnavigate. One of the WMAP papers indicated a minimum circumference at present of about 600 billion lightyears (95% confidence minimum). That was after 5 years of data. I think the minimum would be larger now. After 7 years of data the picture seems slightly flatter. If you want links to stuff, say.

αβγδεζηθικλμνξοπρσςτυφχψω...ΓΔΘΛΞΠΣΦΨΩ∏∑∫∂√ ∧± ÷←↓→↑↔~≈≠≡≤≥½∞ ⇐⇑⇒⇓⇔∴∃ℝℤℕℂ⋅

 

[dubsed]

By spatially open vs closed do you mean finite vs infinite?

 

[dubsed]

Can you explain or provide a link that explains how the CMB shows curvature? I thought the CMB only showed temperature variation which could be used to infer an initial matter distribution?

 

[bcrowell]

Can you explain or provide a link that explains how the CMB shows curvature? I thought the CMB only showed temperature variation which could be used to infer an initial matter distribution?

http://www.lightandmatter.com/genrel/

See subsection 8.2.9.

By spatially open vs closed do you mean finite vs infinite?

I think there are several equivalent ways of defining it, and that's one of them.

 

[Kevin_Axion]

Well, I thought by shape they are always talking about the global geometry of the universe, such as euclidean, hyperbolic, and elliptic, That's how I think about it.

 

[dubsed]

How is a homogeneous infinite universe possible? Wouldn't that require the mass and energy in the universe to be infinite as well?

Wouldn't an infinite universe also mean that statistically all things that could have already happened?

In an infinite universe wouldn't you eventually reach a size that no matter how low the density you would eventually hit a Schwarzschild radius? Or would the observable limit prevent that from happening?

 

[dubsed]

From what I read in your book It looks like the CMB is used to determine curvature by looking for changes in the CMB at different positions in Earth's orbit (different angles). How good is the assumption is it that most of the differences in the data actually come from the CMB and not a closer unknown source? Additionally how reliable would that method of determining curvature be as the parallax method is only good for several to tens of thousands of ly. or am I misunderstanding?

 

[bcrowell]

From what I read in your book It looks like the CMB is used to determine curvature by looking for changes in the CMB at different positions in Earth's orbit (different angles).

No, it doesn't have anything to do with the Earth's orbit.

In an infinite universe wouldn't you eventually reach a size that no matter how low the density you would eventually hit a Schwarzschild radius? Or would the observable limit prevent that from happening?

A Schwarzschild solution has a center of spherical symmetry. A cosmological solution doesn't.

How is a homogeneous infinite universe possible? Wouldn't that require the mass and energy in the universe to be infinite as well?

FAQ: How does conservation of energy apply to cosmology?

It doesn't. General relativity doesn't have a conserved scalar mass-energy that can be defined in any spacetime.[MTW 1973] It only has certain definitions that work in special cases, including stationary spacetimes and asymptotically flat spacetimes. Cosmological solutions aren't stationary or asymptotically flat. Therefore there is no way to define the total energy of the universe (regardless of whether the universe is spatially finite or infinite). There is not even any way to define the total energy of the observable universe. There is no way to say whether or not energy is conserved during cosmological expansion.

MTW: Misner, Thorne, and Wheeler, Gravitation, 1973. See p. 457.

 

[Chalnoth]

Please note that I am not trying to forward any type of personal theory. I am only trying to understand generally accepted physics.

I have heard the universe described as the surface of a balloon. Inflating the balloon is the expansion of the universe and everything move away from everything. No edges nice and neat.

I would suppose that if you extend the balloon to a fourth dimension then you would have space as the surface of a hypersphere. If this is not the correct way to look at it then please explain what is.

The questions bellow are only pertinent if a 4d hypersphere is the correct way to view the universe.

Given enough resources and ignoring the expansion of the universe. Is it possible to circumnavigate the universe (go in one direction for long enough and end up where you started?) What would be the path equation for such a trip look like?

If the universe is a 4d hypersphere then I assume that by definition it is curved? How would someone at the surface be able to measure the curvature from a fixed point?

Well, there are actually more ways for it to wrap back on itself than simply being a hypersphere. It could be a donut shape, for instance (a torus). This shape is geometrically flat, but still wraps back on itself.

You can also have a negatively curved space that wraps back on itself, but this isn't possible to visualize.

In the end, we just don't know what the global geometry of our universe is, simply because the universe is significantly larger than our observable region. All that we do have are lower limits on the overall size, and upper limits on the curvature of our observable region. The rest is hidden from us.