Is the universe a 3D manifold living in 4D space?

[pivoxa15]

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.

It seems that the universe is made out of 3D objects. So put all of them together (stars, black holes, galaxies etc) and you have the whole universe. It also means that locally in the universe (or each individual object in the universe) such as the earth, it is just a 3D object. 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 space which the whole universe must 'live' in would be R^4 which is the 4th dimension.

Is this a conventional view?

 

[hellfire]

There is a three dimensional hypersurface \Sigma for each value of the cosmological time. The cosmological time is a parameter that labels the elements of a one-parameter foliation of the spacetime manifold M = \mathbb{R} \times \Sigma. I think this is the conventional view.

 

[Flatland]

I've always been bothered by the question but no one has ever clearly explained. If gravity is the curvature of space, it must be curving into another "direction" correct? If our universe is finite but unbounded, doesn't that mean that our 3d space curves into the fourth dimension back onto itself kinda like a 4d sphere? But this totally contradicts quantum mechanics and especially string theory.

 

[pervect]

I've always been bothered by the question but no one has ever clearly explained. If gravity is the curvature of space, it must be curving into another "direction" correct? If our universe is finite but unbounded, doesn't that mean that our 3d space curves into the fourth dimension back onto itself kinda like a 4d sphere? .

You can visualize a curved manifold as a lower dimensional manifold embedded in a higher dimensional one - for instance, the surface of the Earth is a curved 2-d manifold embedded in a 3-d space.

However,one doesn't have to study curvature via an embedding. You can treat curvature intrinsically, as a property that an inhabitant of a space can measure without leaving the space.

See for instance any of the various articles on "intrinsic curvature" or "Gaussian curvature" (contrast to "extrinsic curvature").

Since we can't leave our universe, any higher dimensions than are represented in our universe are not observable by us. Therfore it is much cleaner to study curvature as it can be understood by measurements inside our universe, without getting into the metaphysics of hypothetical unobservable extra-universal entities.

 

[EL]

Note that according to String theory, the universe need to have extra spatial dimensions. However, as pervect said, curvature can be treated completely as an intrinsic property, and hence there's no need for embeddings until we observe any extra dimensions.

 

[Flatland]

if you treat it completely as an intrisic property, wouldn't it be pretty meaningless to say that space curves? Because saying that it curves implies that it must be curving into a direction. I find it much more comforting to think of our universe as some sort of 4d hypersphere and we occupy the "surface."

 

[yenchin]

if you treat it completely as an intrisic property, wouldn't it be pretty meaningless to say that space curves? Because saying that it curves implies that it must be curving into a direction. I find it much more comforting to think of our universe as some sort of 4d hypersphere and we occupy the "surface."

But considering embedding can sometimes obscure the intrinsic properties. I.e. you may be confused as to which properties are intrinsic and which are the consequences of the embeddings. A cylinder for example, has zero Gaussian curvature, i.e. it is flat. As far as the inhabitants on the surface is concerned, cylinder is flat and triangles add up to 180 degrees just like that on Euclidean plane. We say that there is an isometry from cylinder to a plane. When we embed cylinder in 3 dimensional space of course, we introduce another form of curvature, i.e. the mean curvature, which is the consequence of the introduction of the embedding, and clearly not an intrinsic property of the cylinder itself.

 

[Garth]

But considering embedding can sometimes obscure the intrinsic properties. I.e. you may be confused as to which properties are intrinsic and which are the consequences of the embeddings. A cylinder for example, has zero Gaussian curvature, i.e. it is flat. As far as the inhabitants on the surface is concerned, cylinder is flat and triangles add up to 180 degrees just like that on Euclidean plane. We say that there is an isometry from cylinder to a plane. When we embed cylinder in 3 dimensional space of course, we introduce another form of curvature, i.e. the mean curvature, which is the consequence of the introduction of the embedding, and clearly not an intrinsic property of the cylinder itself.

That is a good example. Inhabitants living in such a universe might observe the anisotropies in its CMB and conclude that their universe was spatially flat. Yet as it was finite they would notice the largest anisotropies were missing - there would be a low-l power deficiency...

Garth

 

[pervect]

if you treat it completely as an intrisic property, wouldn't it be pretty meaningless to say that space curves?

No - that's the whole point of (for instance) Gaussian curvature. It can be defined (even rigorously definied) in terms of measurrements that a Flatlander living on a plane can actually make with their Flatland rulers.

The Riemann curvature tensor of GR also measures intrinsic curvature.

Because saying that it curves implies that it must be curving into a direction. I find it much more comforting to think of our universe as some sort of 4d hypersphere and we occupy the "surface."

While this can be a useful visual aid, it's not necessarily a good idea to take it too seriously.

The mathematics of GR is based on intrinsic measures of curavature - specifically the Riemann curvature tensor. It is not actually necessary to envision the universe as actually being embedded in some higher dimensional manifold.

The basic issue is rather metaphysical - if you take your visulizations, based on your "comfort factor" too seriously, you start to imagine that hypothetical entities that are not observable and can never be observed must be "real".

These entities don't actually have to be "real" though, and in fact, since they can never be observed, they probably aren't. They arise and exist soley to make you "comfortable", not out of any logical necesity.

 

[setAI]

the visual geometry of spacetime is only an abstract mapping of relationships perceptible/relatable by the human brain- like color- the degrees of freedom that determine how the elements of a causal set can interact can be mapped as a hypersurface- but there is no need for an actual hypersurface in some 'real' Euclidean hyperspace- the causal set's relationships are connected as to be described as a hypersurface- but it is a virtual representation- an abstracted model of the ways in which information can be shared amongst the elements of a causal set

I can define a 1 dimensional linear space containing 10 atoms with 1 degree of freedom to communicate between them- a line of 10 atoms- but I can describe a causal set in which atom0 and atom9 are connected to each other just like all the others - forming a causal loop- now I can describe and perceive this set as a 1 dimensional surface curved through the second dimension and meeting itself into a 2-D circle so that we would say this is a closed 1 dimensional universe curved and closed back on itself as the 2 D hypersurface of a circle [i.e. 2-sphere]- however there is no such geometry in 'reality' this is simply our perceptual model of a looped causal set- we say that the 1 D universe is 'curved' through the 2nd dimension- but it isn't- all that has been defined is 1 degree of freedom between the ten atoms and that there is no first or last because they are connected in the same way- so any closed homogeneous connected set can be described as a closed hypersurface in n+1 dimensions where n is the number of degrees of freedom between elements of the set-