Does a bounded universe require more dimensions?

[Vast]

Suppose the universe is bounded and not infinite, would it require more dimensions for the universe to curve back upon itself? Like most, I find it impossible to picture 3 dimensional space having a boundary.

 

[Garth]

Suppose the universe is bounded and not infinite, would it require more dimensions for the universe to curve back upon itself? Like most, I find it impossible to picture 3 dimensional space having a boundary.

So does everybody else; that is why applying the Cosmological Principle to a closed universe (Friedmann model with $\Omega$ > 1) gives you a finite unbounded hyper-surface. Analogous to the 2D surface of the Earth.

Garth

 

[Chronos]

The universe is not 3 dimensional, it is 4 dimensional. Try to imagine that having an edge!

 

[Vast]

So does everybody else; that is why applying the Cosmological Principle to a closed universe (Friedmann model with $\Omega$ > 1) gives you a finite unbounded hyper-surface. Analogous to the 2D surface of the Earth.

Well yes, omega greater than 1 gives a positively curved spherical geometry, but wouldn’t that also be a closed bounded universe? not unbounded? This may depend on what you mean by an unbounded hypersurface, I’m not exactly clear on what a hypersurface is, whether it’s a 4 manifold, or some other geometry?

 

[Vast]

The universe is not 3 dimensional, it is 4 dimensional. Try to imagine that having an edge!

Yes, that’s much more complicated, but how do we include the time dimension into the 3 spatial dimensions with regards to a bounded universe?

 

[Garth]

Well yes, omega greater than 1 gives a positively curved spherical geometry, but wouldn’t that also be a closed bounded universe? not unbounded? This may depend on what you mean by an unbounded hypersurface, I’m not exactly clear on what a hypersurface is, whether it’s a 4 manifold, or some other geometry?

I use the term "hypersurface" as it is used in differential geometry and topology, to mean a surface of one-dimension less than the manifold being considered. So as Chronos said as space-time is four dimensional the hypersurface I was referring to is three dimensional space. This means we have to foliate, or 'slice' 4D space-time in a particular way to obtain this hypersurface.

That is where the Cosmological principle (CP) comes in. At any event in the universe you can slice up space-time in one particular way so that the universe is isotropic and homogeneous.

This is the frame of reference of an observer co-moving with the Surface of Last Scattering of the CMB.

Now if $\Omega$ > 1, and the CP holds, then that hypersurface is a hypersphere, a 3D equivalent of the 2D spherical surface of the Earth. It is finite but unbounded, it has no edges or boundaries. Go far enough and you will end up where you started! (Actually it is not possible to circumnavigate the universe because it is expanding too quickly, you would have to go faster the light.)

I hope this helps.

Garth

 

[Vast]

Thanks for clearing that up Garth, I had misunderstood what you meant by unbounded, I see that an unbounded surface is analogous to a spherical surface. Not that I can visualize them, but as I understand, there are different http://www.mathsci.appstate.edu/~sjg/class/1010/wc/geom/universe2a.html such as a 2-torus, 3-torus, etc.

 

[Chronos]

Taking another stab at it.

In a 4 dimensional universe, time [the 4th dimension] defines the 'edge' of the universe. So the only way to reach the edge, from any point in space, is to travel in excess of the 'speed of time'. This is impossible by our current understanding of physics - and logic.

 

[Vast]

In a 4 dimensional universe, time [the 4th dimension] defines the 'edge' of the universe. So the only way to reach the edge, from any point in space, is to travel in excess of the 'speed of time'. This is impossible by our current understanding of physics - and logic.

Is it correct to say the edge of the observable universe is the edge of the universe? The light from a distant object takes a certain amount of time to traverse the distance between the object and the observer, and taking into consideration the expansion of the universe, that object has now moved even further from us. In short, its current position is further than its observed position.

 

[Chronos]

I think it is more accurate to view every object in the universe as residing on the edge of its observable universe. I think this is almost certainly true in any universe that includes a t dimension and finite speed of light.

 

[George Jones]

Is it correct to say the edge of the observable universe is the edge of the universe?

This is usually called a (cosmological) horizon.

I think that it is more correct to call singularities edges.

Singularities occur, for example, at the Big Bang and in black holes. Originally, these singularities were defined by the blowing up of mass/energy density and curvature, but the work of Penrose, Hawking, Geroch used a concept of singularity that is more like edge of (or a tear or rip in) spacetime.

If we ever get a quantum theory of gravity, we may find that these "edges" are smoothed away, but nobody knows for sure.

Regards,
George

 

[Chronos]

A lot of people are working on that, George. I agree singularities are mathematical artifacts. Similar examples exist in the history of science . . . e.g., the ultraviolet catastrophe. IMO, science is on the verge of making the next great leap.

 

[Thor]

Suppose the universe is bounded and not infinite, would it require more dimensions for the universe to curve back upon itself? Like most, I find it impossible to picture 3 dimensional space having a boundary.

APPLAUSE!
If space was not infinite then for any point in the universe there must exist another point within a finite distance at which motion in any direction would not increase the distance between the two.
When you find this, I will then give credence to a finite Universe, till then, keep looking.:)

 

[Vast]

If space was not infinite then for any point in the universe there must exist another point within a finite distance at which motion in any direction would not increase the distance between the two.

I don’t see how that follows: Isn’t an expanding surface of a sphere finite and yet the points are moving away from each other?

 

[Thor]

I don’t see how that follows: Isn’t an expanding surface of a sphere finite and yet the points are moving away from each other?

At whatever moment of time you consider and regardless of the speed of the expansion the "next" increment of distance would not exist until the FOLLOWING moment.

And what is the universe "expanding" into except space which already exists?

If the universe was 'created' from a point of singularity then unless it expanded at an infinite rate or for an infinite amount of time, it must be finite in nature. It is; however, very simple to show the premise of creation is contrary to logic -


The existence of nothing ostensibly requires no justification, so most popular theories of Universal origin begin with a primal void. At the beginning of time a transformation must have occurred which brought forth the material presence of the cosmos. The Theory of Singularity - or Big Bang - envisions a Universe cast from the bowels of some spontaneous cosmic eruption. But that would require the pre-existence of a spawning force - the very presence of which would violate the original assertion that nothing existed. And if all which exists was created, then whatever sired the Universe must, too, have been created by some predecessor which, in turn, must have been predated by a limitless procession of ancestry. The endless cycle of chicken-and-the-egg redundancy which results from any cause and effect approach to the enigma of existence implies no logical beginning.

Supernatural versions of creation sidestep the issue of redundancy by declaring that whatever created the Universe was not subject to the laws of nature. Of course when the rules of reality are suspended anything is possible, even the absurd. And if one such exemption can be conceded, so can others - without limit.

Conditions or states of being change during the process of cause and effect. But existence is not a condition or a state of being, it is the phenomenon of being, itself. Before something can change, before something can act or be acted upon it must first exist. And if being is required in order for change to occur then cause and effect is a function of existence. This is, of course, the antithesis of the premise that existence is a function of cause and effect - the product of creation.

If the existence of the universe is not the result of a nascent event, then why would you suppose it is expanding?

Given a finite number of moving particles randomly vectored at random velocities within a finite volume, eventually all collisions which could occur WILL occur - within a finite period of time. Many of those collisions may occur outside of the original volume, but they will still take place within a finite period and within a finite distance. Once all collisions have occurred, all particles will eventually reach the boundary of the initial volume and be moving away from each other.

It is small wonder the bodies within the infinitesimal portion of the Universe we can detect with our technology seem to be moving away from each other. The default assumption seems to be this is due to a 'ballooning' of the Universe from a point of singularity, but the above scenario explains the phenomenon equally as well.

 

[Garth]

And what is the universe "expanding" into except space which already exists?

Incorrect. In the understanding of GR it is space itself that expands, the galaxies etc. are carried along with it. It doesn't have to expand into anything.

If the universe was 'created' from a point of singularity then unless it expanded at an infinite rate or for an infinite amount of time, it must be finite in nature.

Incorrect. In the Friedmann models an infinite flat or hyperbolic universe also expands from a singularity. However, you cannot say anything about the volume of that initial singularity.


Garth

 

[Thor]

Incorrect. In the understanding of GR it is space itself that expands, the galaxies etc. are carried along with it. It doesn't have to expand into anything. Incorrect. In the Friedmann models an infinite flat or hyperbolic universe also expands from a singularity. However, you cannot say anything about the volume of that initial singularity.
Garth

I'm sure Mssrs. Einstein and Friedmann were fine and intelligent gentlemen; however, when an abstract theory conflicts with logic it is usually logic which turns out to be correct.

Change IS a function of existence, not visa versa. Their original premise, i.e. that the Universe was created, is blatantly flawed - see post. If you can refute anything in the post concerning creation, please enlighten me.

Thanks

 

[Nacho]

Did the original question ever get answered? I take it to be: For a 3 spatical dimension Universe that is finite and unbounded to expand, does that require a 4th spatial dimension within which to expand?

 

[Garth]

Did the original question ever get answered? I take it to be: For a 3 spatical dimension Universe that is finite and unbounded to expand, does that require a 4th spatial dimension within which to expand?

No, it does not need an extra dimension. Curvature, and the spatial expansion of a universe with curvature, can all be completely described intrinsically by the behaviour of 'marker' points within it relative to each other. The angles and distances between such 'marker' points are described by the metric and the Riemannian and Ricci tensors derived from the metric.

However to visualise what is going on you may find it more convenient to embed such a hyper-surface in a flat static higher dimensional manifold.

Garth

 

[Nacho]

Garth,

However to visualise what is going on you may find it more convenient to embed such a hyper-surface in a flat static higher dimensional manifold.

I see. I don't know if this question is answerable or even makes any sense, but I'll ask it anyway:

The GR expressions you use describing a 3 spatial dimension Universe, are they equivalent to what would be used in 4 spatial dimension math to describe a 3 spatial dimension "bubble" within it?

 

[Garth]

The GR expressions you use describing a 3 spatial dimension Universe, are they equivalent to what would be used in 4 spatial dimension math to describe a 3 spatial dimension "bubble" within it?

Mathematically you can embed the 3D space in a higher dimensional manifold, the question is whether these higher dimensions have any physical significance.

If you need higher dimensions for a testable string, brane or GUT theory then they would take on physical significance.

The 'GR expressions' are actually 4D space-time descriptions, which for our convenience we 'slice up' or 'foliate' into 3D space hyper-surfaces separated by time.

However, how you slice up that cake is observer dependent, different co-existing observers, moving relatively to each other, would disagree on which events lie in their respective 'futures' and 'pasts'.

Garth