Initial assuption: The universe is finite. An infinite universe would likely mean infinite matter because of the strange relationship between space and matter(as described by GR), and a universe with infinite mass/matter/energy poses serious theoretical problems and violates the conservation laws.

I'm sure this sounds fascinating: Imagine having a map of the universe, but not just an atlas collection containing every region of the universe but a small replica of the universe, analogous to what the a small globe of the world is to the Earth.

The first obvious question is: what is the shape of the universe? I' ll come back to this.

What would a "globe" of the universe look like? If we had every region mapped out we could certainly construct a small replica of the universe.

At this point, some might argue that you can't have a replica-map of the universe because for instance lets say the universe is one HUGE sphere, what is outside the sphere? That's the equivalent of saying what is outside the universe, and we could NOT know because we are inside of it.

Another argument is that lets for example Earth and our Solar System and the Milkyway galaxy is located is the near the center of the universe. Couldn't we just travel travel in a straight line from Earth outward, along a three dimensional radius, and after a long time reach the the outer boundary of the universe? It's as if the universe could really be modeled by a three-dimesional sphere, that would implicate that we would travel in space until we hit a crystal clear shell that prevented us from traveling any further in that direction, or that the universe has an edge where you can travel no more. This is obviously absurd and so my map of the universe represented by a three-dimentional sphere (or a compact/finite two-sphere) is flawed and will not work in representing the universe.

So far, I showed that you can't represent the universe in a small replica-map as a three-dimentional sphere (or a compact/finite two-sphere). Lucky we now have the mathematical tools of topology and this shell-edge problem can be easily fixed.

Instead of one compact 2-sphere representing the universe in our replica-map, it would be two compact 2-spheres joined at one point along the surface.

http://i159.photobucket.com/albums/t121/camilus23/shapeoftheuniverse2.jpg

Now by aligning the two-dimensional manifolds of both three-dimensional spheres, we can travel in any straight line indefinitely through a finite universe (due to the fact that the manifold of a 2-sphere has no boundary).

The implication of the previous statement is that to travel in a straight line all the way across the universe would land you back where you started.

If this theory is correct, then the possible shape of the universe is a 3-sphere, or a three-dimensional manifold that models a four-dimensional sphere(also known as a hypersphere).

Now the easiest way to picture the universe shaped as a hypersphere is the two connected three-dimensional spheres, and this means that if I am correct, than in the far future we will have replica-maps or "globes" of the universe that will look like two spinning connected spheres next to each other that align depending on direction of travel.

I agree.
What an excellent point!

Off-topic: (meaning me, not you)
I apologise for the delay, seems I can't override the mod for the post count for the image however people can click it to see it for now. You'll be able to sort the image out later once you've hit the magic post count however make sure you put forwards some discussion not just post numbers the locals seem to get a little irate.

Welcome to Sciforums btw, I hope you manage to prepare yourself for a lot of responsive posters that might or might not be cruel or kind or even on topic for that matter.

Btw, heres the image in it's full glory.

http://i159.photobucket.com/albums/t121/camilus23/shapeoftheuniverse.jpg

now the theory{that the universe is a 3-sphere) has been made at least theoretically possible by the recent proof of the Poincare conjecture, because in order to travel in a straight line indefinitely in our universe, the boundary manifolds must be simply connected, and the Poincare conjecture states:
Every simply connected compact 3-manifold (without boundary) is homeomorphic to a 3-sphere.

finite universe?
the universe can be mapped by probability and time measurements...with probability of objects being there in that time based on gravitation measurements...wobbling of stars...thermal/x-ray measurements. XYZ map with embedded time scales and probabilities of planets/stars/nebulas existing there from less than 1% to more than 99% probability.

I would call it Hypersphere map.

dragon, I see what you're saying but you are trying to map the universe onto four-dimensional space-time, or onto three-dimensional space relative to time.

I'm trying to map the universe spatially, and even though it is expanding, the 3-sphere is homeomorphic and can be stretched without disrupting continuity.

im sorry for being unclear, everytime i say 3-sphere, I mean a compact 3-sphere..

Camilus:

We can map the universe in our own centric reference frame, which is the easiest.

Millions of galaxies are already mapped and assigned 3-D coordinate positions [solid-angle, distance].

The distance is actually a red-shift measurement, caused by the galaxy's recession from earth in our 'centric' Milky Way reference frame, converted into distance.

We see some discrete galaxies at great red-shift corresponding to distances of roughly 8 billion light-years away [so we see the galaxy as it existed 8 billion years ago in our reference frame]. Most observed galaxies are much closer.

The most distant "object" that we can see is not actually a discrete object, but a continuous 'smear' of hot plasma that is relatively isotropic across the entire sky. Its redshift is roughly 1,000 [corresponding to a recessional velocity of about 0.9999991c]. Thus, the light that we see from that 'object' shows in the microwave frequencies, and is known as the Cosmic Microwave Background. More recent investigations reveal that it is actually slightly anisotropic, likely revealing the formation of even more distant galaxies than those nearer to us.

That plasma is matter separate and distinct from the matter of all of the visible galaxies we see in our telescopes. We cannot see beyond that 'plasma', though we can presume that there is even more matter, even hotter than the 'plasma'. That is the goal of the LHC, to create conditions that existed in the earliest stages of the Universe's development, where matter is much hotter than that distant plasma [in our reference frame - in its own reference frame that plasma has expanded and cooled, and likely formed galaxies].

I know what you are saying Walter but that reveals nothing about the shape and structure of the universe as a whole. I'm trying to look at the 'big picture', visualizing it as God must see it.

Tell me guys, do you believe the universe is infinite?

I know what you are saying Walter but that reveals nothing about the shape and structure of the universe as a whole. I'm trying to look at the 'big picture', visualizing it as God must see it.

Tell me guys, do you believe the universe is infinite?

yes

http://www.astro.princeton.edu/~mjuric/universe/all100.gif

yes
incredible. Then the model I described offers you no insight whatsoever into the secrets of the universe.

I believe the universe cant be infinite, because:

Initial assuption: The universe is finite. An infinite universe would likely mean infinite matter because of the strange relationship between space and matter(as described by GR), and a universe with infinite mass/matter/energy poses serious theoretical problems and violates the conservation laws.

An infinite universe would likely mean infinite matter because of the strange relationship between space and matter(as described by GR), and a universe with infinite mass/matter/energy poses serious theoretical problems and violates the conservation laws.

Why can't the universe be infinite? The WMAP data says that the universe can be flat, within error bars. If it's flat, then it is infinite, end of story.

Why can't the universe be infinite? The WMAP data says that the universe can be flat, within error bars. If it's flat, then it is infinite, end of story.

once again, I dont know if I'm just too advanced in the subject or you guys dont see what I'm seeing but listen, I'm trying to map the universe as a whole! what you are describing Ben is just local geometry within the universe, I'm focusing on the global geometry of the universe as a whole.

Based on analyses of the WMAP data, cosmologists during 2004-2006 focused on the Poincaré dodecahedral space (Poincaré homology sphere), but also considered horn topologies to be compatible with the data

as you can see, the WMAP data actually suggests the universe is probably spherical, but horn topologies also do not rule out the possibility of hyperbolic universe.

I believe the universe must be spherical, because even the extra six or seven space-like dimensions in String theory are all formulated with compact topology. and the following theorem "every simply connected compact 3-manifold (without boundary) is homeomorphic to a 3-sphere," makes the universe of string theory a hypersphere.

Note: compact means finite in topological terms.

do you really believe that the universe is a three-dimensional Euclidian space? extending in the XYZ directions forever?

as you can see, the WMAP data actually suggests the universe is probably spherical, but horn topologies also do not rule out the possibility of hyperbolic universe.

This is not right. The dodecohedral sphere thing is dead---it was a wild thought a few years ago.

I am telling you---the flatness parameter is called \Omega_r. If it is zero, then the universe is flat---i.e. topologically flat. Two parallel light rays never intersect. Flat in every sense of the word---including ``not a sphere'' flat. The WMAP data give a value of the flatness prameter that is something like 1.01 +/- 0.05 or something (greater than one means closed, like a sphere, less than one means open, and exactly one means flat).

So it is possible that we live in a flat and infinite universe, which poses absolutely no problems for conservation of energy whatsoever.

There's no reason to believe anything other than what is consistent with the experiments :)

There's no reason to believe anything other than what is consistent with the experiments :)

first you have to believe in something and that you test it with experiments to prove it.

Camilus,
"Now by aligning the two-dimensional manifolds of both three-dimensional spheres, we can travel in any straight line indefinitely through a finite universe (due to the fact that the manifold of a 2-sphere has no boundary)."

You lost me here. How is this so? Can you show how you would travel in a straight line indefinately?

The WMAP data give a value of the flatness prameter that is something like 1.01 +/- 0.05 or something (greater than one means closed, like a sphere, less than one means open, and exactly one means flat).

Are you sure? I thought the WMAP just showed the relative spacing between matter.

Camilus,
"Now by aligning the two-dimensional manifolds of both three-dimensional spheres, we can travel in any straight line indefinitely through a finite universe (due to the fact that the manifold of a 2-sphere has no boundary)."

You lost me here. How is this so? Can you show how you would travel in a straight line indefinately?

Well, not necessarily a straight line, more like a geodesic, and travel indefinitely similarly to how you can board a plane right now and fly in a straight line aound the earth indefinitely. The 'indefinite' is due to the fact that the sphere is unbounded, you'll never come to an edge where you can say "this is the end of the sphere."

but the real meaning behind the statement that "you can travel in a straight line (or geodesic) indefinitely" is that traveling in a straight line across the universe, you will never run into an edge where you can "go no more." That would be completely counter-intuitive.

Are you sure? I thought the WMAP just showed the relative spacing between matter.

Pretty sure. The number they measure is a relative amount of dark energy, dark matter, and baryonic matter. If these numbers all add up to one, then the universe is flat. If they add up to more than one, it is closed (a sphere), and if they add up to less than one, it is open.

but the real meaning behind the statement that "you can travel in a straight line (or geodesic) indefinitely" is that traveling in a straight line across the universe, you will never run into an edge where you can "go no more." That would be completely counter-intuitive.

No, the statement is that two parallel light rays will never intersect. This is what flat means. If you don't believe me, then look at the lines of longitude on a globe.

The longitudinal lines are a byproduct of the latitudinal lines. Pick any longitudinal line L and put a point P outside the line. There exist no line parallel to L that passes through point P. It's the basis of spherical geometry.

In a flat universe, two parallel lines will never intersect, but in a hyperspherical universe, geodesics will always intersect due to the curved nature of the compact 3-sphere.

longitudinal lines are not parallel, they only appear parallel because we draw them that way.

if two planes next to each other were flying around the Earth, if they both travel in a straight line, they will eventually meet. They must try to stay a certain distance from each other to not intersect. But that's not travelling in a straight line.

longitudinal lines are not parallel, they only appear parallel because we draw them that way.
They are parallel at the equator.

if two planes next to each other were flying around the Earth, if they both travel in a straight line, they will eventually meet. They must try to stay a certain distance from each other to not intersect. But that's not travelling in a straight line.
Exactly. That's how we know that the surface of the Earth is curved. If the surface were flat, then two straight trajectories that start parallel will stay parallel - that's what flat means (to paraphrase Ben, with camilus's understanding of "parallel").

from what I got, I thought Ben was saying parallel lines exist on spherical surfaces because he thinks since this model allows lines to travel in geodesics indefinitely, two travelling next to each other will never intersect.

No---this is exactly what I meant:

In a flat universe, two parallel lines will never intersect, but in a hyperspherical universe, geodesics will always intersect due to the curved nature of the compact 3-sphere.

If the universe is flat, and it is, within error bars, then it is infinite. Period.

what do you mean "within error bars"?

If it's not flat, then it's so close that we can't measure the difference.

Pete is right. Experimenters don't measure exact numbers, the measure ranges of numbers. The flatness parameter of the universe is something like 1.01 +/- 0.04 or something. I can't remember the exact numbers, but it's called \Omega or something like that.

If omega is 1, then the universe is flat, greater than 1 it is closed, less than 1 it is open (like a saddle).

you're saying a lot of 'if's though. I heard hyperbolic universes would be hard to detect, but what if \Omega is greater than 1, that would make the universe spherical; after all, you are saying +/-.

I'm afraid this is still being studied and researched.

camilus---

I'm sort of playing devil's advocate here. But there's no real way to know for sure what the right answer will be.

If you want to map the universe onto a three-sphere, go ahead. But realize that't you're only mapping our hubble volume. The boundary of the universe is at t=300,000 years after the big bang, when the CMB photons aren't trapped in the Compton scattering of electrons anymore. So really, we can't SEE any further back than that.

And even IF the universe is flat, we still have a hubble volume that is a three-sphere, whose boundary is the CMB spectrum measured by WMAP.

Finally, if the universe is closed, there's no guarantee that it looks like a four-sphere. Sure, it may be isomorphic to a three or four-sphere, just like the surface of my pen is ismorphic to a two sphere. That doesn't mean that mapping my pen onto a globe really DOES anything for me.

camilus---
Finally, if the universe is closed, there's no guarantee that it looks like a four-sphere. Sure, it may be isomorphic to a three or four-sphere, just like the surface of my pen is ismorphic to a two sphere. That doesn't mean that mapping my pen onto a globe really DOES anything for me.


That was the case, until Grisha Perelman's work on the theory of geometrization proved that if a three-manifold is simply-connected then it is homeomorphic to a 3-sphere.

and in order to avoid the edge problem, the universe must be simply-connected.

The whole aim of mapping the Universe is to find its shape and understand its curvature.

BTM:

If you want to map the universe onto a three-sphere, go ahead. But realize that't you're only mapping our hubble volume. The boundary of the universe is at t=300,000 years after the big bang, when the CMB photons aren't trapped in the Compton scattering of electrons anymore. So really, we can't SEE any further back than that.

Exactly so.

The farthest distance that we see are those last-scattered photons, which we see arriving from that distant portion of our universe that is the CMB emitter, red-shifted by 1,000 due to its high recessional velocity of about 0.9999991c Those last-scattered photons that were emitted from our neck of the woods are just now arriving at that CMB emitter, which anyone [i.e. E.T.] residing there [in a presumed evolved galaxy with solar systems and ET] would perceive as its CMB background, but seeing us only as we once looked in our earliest history, not what we have evolved into.

The matter within that Hubble volume, defined as a sphere [to the best of our measurements of the CMB] is finite. The matter beyond that Compton-scattering region [sometimes called an "opacity wall"], which we cannot yet see, would be infinite. I will leave the proof for you.

Regards,


Walter

PS - Would you be interested in working on compton scattering as a means of generating monochromatic, tunable "X-rays"? A friend of mine was working on that by shining a visible-light laser at a 30 MeV electron beam, with the compton scattering resulting in a discrete-energy beam of photons at about 30 KeV being back-scattered. By tuning the energy of the electron beam, you tune the energy of the back-scattered photon beam. He had a working model, that would be of great interest in the medical community.

The matter beyond that Compton-scattering region [sometimes called an "opacity wall"], which we cannot yet see, would be infinite. I will leave the proof for you.
Nice try :)

Which universe are you talking about? Their are more universes than galaxies!

The matter beyond that Compton-scattering region [sometimes called an "opacity wall"], which we cannot yet see, would be infinite. I will leave the proof for you.

Not if the universe is finite---i.e. closed.

That was the case, until Grisha Perelman's work on the theory of geometrization proved that if a three-manifold is simply-connected then it is homeomorphic to a 3-sphere.

Sure it is homeomorphic (I said isomorphic, but I used the wrong word). Homeomorphic just means that you can deform it into a three sphere. That means you can map it as a three sphere, as you said. But this may not be the best way to see it. Like my pen. I can map my pen onto a two sphere, but it may not tell me anything.

It is impossible to travel in a circle in the universe. The universe is growing everywhere at the same time. If you tried it would look like a 90degree cone if traveling at the speed of light. If looked from the outside it would look like a cone is growing outwards in a circular direction as time went on.

It seems to me(i'm probably wrong), he is trying to describe/map the universe as it stands now, not as it appears to us via the past(telescopes/peering at the universe from earth and mapping it).

it is homeomorphic (I said isomorphic, but I used the wrong word). Homeomorphic just means that you can deform it into a three sphere. That means you can map it as a three sphere, as you said. But this may not be the best way to see it. Like my pen. I can map my pen onto a two sphere, but it may not tell me anything.

That's because a pen has nothing interesting to tell, as well as we already know the shape of a pen. But if we were to map the universe onto a three-manifold, it would provide insight into its global shape and curvature.

Now the universe modeled by a compact three-sphere isnt my idea, I just realized the shell-edge problem can be resolved by a universe thats simply-connected.

That's because a pen has nothing interesting to tell, as well as we already know the shape of a pen. But if we were to map the universe onto a three-manifold, it would provide insight into its global shape and curvature.

No---once you've done that you've destroyed anything about the curvature that you once knew. Have you ever looked at a map of the world that's drawn on a piece of paper? When you map one shape onto another, you lose the information about what it originally looked like.

It seems to me(i'm probably wrong), he is trying to describe/map the universe as it stands now, not as it appears to us via the past(telescopes/peering at the universe from earth and mapping it).

No--you're right. And it CAN be done (it HAS been done). I just don't know what one could learn from it, other than how the galaxies are distributed.