Is the Universe a 3D Surface of a 4D Hypersphere?

[Sheyr]

Please follow my deduction and tell me where am I wrong, or maybe I’m not?

1) the cosmological principle:
"Viewed on sufficiently large distance scales, there are no preferred directions or preferred places in the Universe"

it says that there is no center of the universe, or in other words – the centre is everywhere, like on a sphere, balloon, circle etc.

2) the Hubble law:
"The galaxies are all receding from us if we go to large enough distances; the velocity of recession is proportional to the distance from us."

That means that the expansion of the universe, viewed form everywhere, is like blowing the balloon up. And the SURFACE of this balloon is 3D!

If the universe is a 3D surface of a sphere so the sphere should be a 4D hypersphere i.e. a sphere of 4 SPATIAL dimensions?

S.

 

[matt grime]

Firstly you assert there is no centre and then say the centre is everywhere. If there is no centre how can it be everywhere?

Anyway, your dimension count is wrong.

the surface of a sphere is two dimensional, it is not a 3 dimensional object.

 

[daveb]

The expansion of the surface of a balloon appears to our sense to expand in 3 spatial dimensions, but to a 2 dimensional creature living on the "universe" of the surface of the balloon, only 2 dimensions are required for its universe to expand. Mathematically, only 2 coordinates are required to specify a specific point on the surface since the radial coordinate (the one we perceive but the 2D creature doesn't) is unnecessary. Similarly, we only need 3 spatial dimensions for our universe to expand and 3 coordinates. Hope that explains it to your satisfaction. So, it's possible that we are embedded in 4 spatial dimensions, but since we can't really verify this (as far as I know) why include the extra dimension?

 

[selfAdjoint]

There may soon be evidence (or not) of higher dimensions. Lisa Randall in her new book Warped Passages describes a universe where our spacetime is on the boundary, not of a sphere, but of a hyperbolic geometry in higher dimensions. Working out the physics of this she thinks there may be particles moving in that higher space, generally called "the bulk", and by the holographic principle (physics in the bulk is exactly desribed by physics on the boundary) we should see images of these particles moving in spacetime, with their higher dimensional momentum appearing to us as other quantum numbers. She can describe the behavior of these new particles and is hopeful the LHC will be able to see them. They would be striking and not like anything in the standard model.

 

[Sheyr]

Firstly you assert there is no centre and then say the centre is everywhere. If there is no centre how can it be everywhere?

Anyway, your dimension count is wrong.

the surface of a sphere is two dimensional, it is not a 3 dimensional object.

Thank you Matt,
My language wasn’t precise. Sorry, I’m neither mathematician nor fluent English speaker. I meant that our universe acts like a 3-sphere (a 3 dimensional hypersphere). If yes, than I ask if this is the evidence that the universe is in fact a 4 dimensional object - the object of 4 spatial dimensions?

 

[Sheyr]

There may soon be evidence (or not) of higher dimensions. Lisa Randall in her new book Warped Passages describes a universe where our spacetime is on the boundary, not of a sphere, but of a hyperbolic geometry in higher dimensions. Working out the physics of this she thinks there may be particles moving in that higher space, generally called "the bulk", and by the holographic principle (physics in the bulk is exactly desribed by physics on the boundary) we should see images of these particles moving in spacetime, with their higher dimensional momentum appearing to us as other quantum numbers. She can describe the behavior of these new particles and is hopeful the LHC will be able to see them. They would be striking and not like anything in the standard model.

Great, do you know any papers, links to such theories based on 4 dimensional universe?

 

[matt grime]

Thank you Matt,
My language wasn’t precise. Sorry, I’m neither mathematician nor fluent English speaker. I meant that our universe acts like a 3-sphere (a 3 dimensional hypersphere). If yes, than I ask if this is the evidence that the universe is in fact a 4 dimensional object - the object of 4 spatial dimensions?

You are still off. Firstly remember that the surface of a balloon is an analogy, it is an idealized thing. Thus this idealized 2-d surface is in no sense a 3-d object. Correspondingly, it does not make sense to state it is a 3-sphere and then say it is a 4-dimensional object. It might be the boundary of a 4-dimensional object but that is not the same thing at all.

 

[selfAdjoint]

Great, do you know any papers, links to such theories based on 4 dimensional universe?

As far as I know, Lisa Randall's papers online are all technical, here's the latest one: http://www.arxiv.org/abs/hep-th/0512247.

Her book, Warped Passages is really the best (i.e only) nontechnical introduction to the area.

 

[pivoxa15]

What about stating that the universe is a 3D manifold existing or 'living' in 4D space?
Note: I have ignored time as a dimension and only considered the spatial dimensions.

I originally made this post in the Cosmology Section but have had 0 replies in a few days so since this thread is directly related to my question I will add it here.

I only learned the meaning of a manifold recently and in the most elementary terms but I thought that I might link it with this example. The definition I use is: A subset M part of R^n is a k-dim manifold if locally M is the graph of a function from R^k->R^n-k.

It seems that the universe is made out of 3D objects (by this I mean not 3D look alikes such as a 2D sphere but real physical objects that you see everyday such as a marble or rock). Put all of the objects in the universe together (stars, black holes, galaxies etc) and you have the whole universe. This means that locally in the universe (or each individual object in the universe) such as the earth, mountains, computers, are all 3D objects. We ourselves are 3D objects as well. Every object in the universe can be fully described mathematically by mapping R^3->R which is the graph of any 3D object. So the minimum dimension in which the whole universe is contained would be R^4, the 4th spatial dimension. Hence the universe is a 3D (manifold) and possibly contained in 4D.

It might be the case that our universe, the 3D manifold might be contained in higher dimensions of space. If in R^5 space than everything in our universe would map R^3->R^2 which would make things more complicated, not that it isn't already.

 

[BornLogician79]

3D real + 1D conceptual time
multiuniverse, expand and contract through black holes.
Universe walls are dropoff in CNP / foundation, all matter energy pulled apart. Simular to how the ocean floor drops in elevation?
Not perfect wall drop off, in the fashion of the waves against the shoreline, although 3D?

 

[matt grime]

What about stating that the universe is a 3D manifold existing or 'living' in 4D space?

There is no real way to tell what ambient dimension a manifold lives in if it lives in any at all; this isn't a well defined notion. It's hard to even tell what a manifold looks like from its defining information, and it is only recoverable upto homeomorphism.

The definition I use is: A subset M part of R^n is a k-dim manifold if locally M is the graph of a function from R^k->R^n-k.

Your definition is wrong. There is nothing in the definition that asserts M is a subset of R^n a priori. You can see that just by thinking of the circle, it is embedded (submersed, whatever) in R^2 and R^3 (and any R^n for n larger than 2) so there is no unique ambient space. There will of course be one of unique smallest dimension. I think that any smooth manifold of (local) dimension d can be embedded in R^{2d}or perhaps R^{2d+2} or something like that: you ought to check. The Klein bottle for instance is locally 2-d but cannot be embedded in R^3 (but it can be in R^4).

In any case, these days we've removed the (unnecessary) crutch of thinking of it a priori as a subspace of R^n.

A manifold is a space with a collection of open sets, locally homeomorphic to R^n, with, usually, smooth coordinate transfroms (at leats C^2 at any rate), the so-called patching information.

Every object in the universe can be fully described mathematically by mapping R^3->R which is the graph of any 3D object. So the minimum dimension in which the whole universe is contained would be R^4, the 4th spatial dimension. Hence the universe is a 3D (manifold) and possibly contained in 4D.

I can't make that make sense. What has some mapping to R to do with anything? What is the graph of a 3-dimensional object? I know what the graph of a function is, f:X --->Y has graph {(x,f(x)} < XxY, but not an object.

 

[pivoxa15]

Your definition is wrong. There is nothing in the definition that asserts M is a subset of R^n a priori. You can see that just by thinking of the circle, it is embedded (submersed, whatever) in R^2 and R^3 (and any R^n for n larger than 2) so there is no unique ambient space. There will of course be one of unique smallest dimension. I think that any smooth manifold of (local) dimension d can be embedded in R^{2d}or perhaps R^{2d+2} or something like that: you ought to check. The Klein bottle for instance is locally 2-d but cannot be embedded in R^3 (but it can be in R^4).

In any case, these days we've removed the (unnecessary) crutch of thinking of it a priori as a subspace of R^n.

A manifold is a space with a collection of open sets, locally homeomorphic to R^n, with, usually, smooth coordinate transfroms (at leats C^2 at any rate), the so-called patching information.

What about if I add the word 'Let' in front of the my definition?
'Let a subset M be part of R^n. Then M is a k-dim manifold if locally M is the graph of a function from R^k->R^n-k.'
So I have defined M to be part of R^n instead of thinking apriori that it is.

I can't make that make sense. What has some mapping to R to do with anything? What is the graph of a 3-dimensional object? I know what the graph of a function is, f:X --->Y has graph {(x,f(x)} < XxY, but not an object.

Sorry, what I meant was that any physical hence 3D object can be portrayed as a graph of a mathematical function, f:R^3->R. Although it may be possible that lower or higher dimensional objects exist in our universe but I am not aware of them.



So you think describing the universe as a some sort of manifold is not
good? How should we describe it given our limited experimental evidence?

 

[matt grime]

How did you decide that I said that we shouldn't use manifolds? Your method of a priori assuming you know what the ambient space is is problematic, both mathematically and philosophically.

 

[pivoxa15]

One thing for sure is that there is no natural law which states that everything in nature must conform to mathematics. All the maths does is model nature and sometimes it is very effective (i.e. QED). So it seems handy to model the universe as a 3D manifold since we live in a universe and everything around us seem 3D.

 

[selfAdjoint]

One thing for sure is that there is no natural law which states that everything in nature must conform to mathematics. All the maths does is model nature and sometimes it is very effective (i.e. QED). So it seems handy to model the universe as a 3D manifold since we live in a universe and everything around us seem 3D.

Works just as well with a 4D manifold since every "event" around us seems to have a (relative) time coordinate as well as three (relative) space coordinates. The corner of Main and Elm streets (elev above sea level 220 ft) is an abstraction; if you actually go to that corner you will be there at some time.