Is our universe a 4D object?

[Strange]

There is a very lucid discussion of this issues here: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_variable/index.html

 

It states what I have been maintaining. That both intrinsic and extrinsic curvature require a higher dimension to accommodate that curvature.

 

Nope. It doesn't. It draws a clear distinction between extrinsic curvature which does require a higher dimension to describe it and intrinsic curvature (which doesn't).

[MigL]

Here's an example...

 

Consider a computer monitor on which you're playing Pac-Man.

As you move your Pac-Man towards the right side of the screen, it re-appears on the left hand side. All on a 2dimensional space.

This is because every point on the right hand side of the map has a one-to-one correspondence, or identifies, with every point on the left hand side,

This is an example of INTRINSIC curvature, Topologically, it is certainly curvature, but it doesn't require an embedding dimension.

 

It is not EXTRINSIC, which it would be if the Pac-Man was physically required to go behind the screen and move to the opposite side.

This 'depth' would be an embedding dimension.

[Strange]

A similar analogy I thought of would the use of "bump mapping" in 3D graphics. This gives 2D surfaces a 3D appearance by modifying the surface normal at each point. This makes it look as if the height varies even though there is no deviation in that dimension.

[Rumfrd]

From this site:

 

The first is most familiar to us, extrinsic curvature. It arises whenever we have a surface that curves into a higher dimension. We have seen many examples. One of the simplest arises when a flat sheet of paper is bent or rolled up into a cylinder. A more interesting case arises when the surface is dome-like, such as a hemisphere.

In this last case of the hemisphere, the curvature of the surface into the higher dimension is associated with a failure of ordinary Euclidean geometry in the surface of the sphere. This failure of Euclidean geometry arises fully within the surface; it is a manifestation ofintrinsic curvature. To summarize:

[MigL]

Whether intrinsic or extrinsic, Euclidian, or 'flat' geometry fails when curvature is involved.

In Euclidian geometry a triangle's angles always add up to 180 deg, and Pythagoras can be used to measure distances.

In any 'curved' geometry this does not hold.

A triangle's angles add to more than 180 deg on a positively curved surface such as a sphere.

And less than 180 deg on a negatively curved surface like the inner part of a doughnut ( torus ).

 

Reminds me of a joke/riddle...

 

A hunter sets off from his camp due South.

After travelling 50 mi he turns due West.

After travelling 50 mi he shoots a bear.

Then he turns North and, after travelling 50 mi is back at his camp.

What color is the bear ?

[elfmotat]

A hunter sets off from his camp due South.

After travelling 50 mi he turns due West.

After travelling 50 mi he shoots a bear.

Then he turns North and, after travelling 50 mi is back at his camp.

What color is the bear ?

 

White.

[MigL]

Darn !

Your avatar gives it away !

[ajb]

That both intrinsic and extrinsic curvature require a higher dimension to accommodate that curvature.

Intrinsic does not require any embedding.

 

The definion of a smooth manifold and a metric (we do have a more general notion of curvature for tangent bundle connections or even connection in/on fibred manifolds), do not require any embedding into a higher dimensional space.

Edited May 2, 2016 by ajb

[michel123456]

 

 

Wouldn't time qualify as a forth spatial dimension in your analogy?

 

 

No spatial dimensions are used to measure geometric objects.

 

There are 3 spatial dimensions one time dimension.

But in cosmology the 3 spatial dimensions increase through time. No?

[Flatland]

Close. The balloon analogy is intended to show that a 2D surface does not have an edge or a center, but can still expand.

 

Unfortunately, we see the balloon expanding in 3D space. There isn't any equivalent for the way the universe expands. However, someone has suggested that you can view the radius of the balloon (the fourth dimension) as time and so you see the surface expanding as it moves through time. That is close the the way the universe is described in general relativity.

 

A finite but unbounded Universe can be thought of as the surface of a hypersphere. I don't understand why that analogy is never used.

[Strange]

 

A finite but unbounded Universe can be thought of as the surface of a hypersphere. I don't understand why that analogy is never used.

 

 

Isn't that exactly that the balloon analogy attempts to do? (without introducing the idea of a hypersphere, which might be just as confusing as the thing that the analogy is trying to describe)

[Flatland]

Yeah but the balloon analogy does it pretty poorly. From the layman's perspective the balloon is still expanding a way from a "center." If you explain that our Universe should only be thought of as the surface of this expanding balloon then you might as well explain what a hypersphere is otherwise the analogy wouldn't make sense to a layman. In other words, go into more detail.

[Strange]

Yeah but the balloon analogy does it pretty poorly.

 

 

I agree.

[Prophet12A]

My passing comment---Perhaps it would be better/evident once you understood the center of your expansion(universe) can be found when flipped, and, you think of it as contraction (not expansion)...and imagine the universe/matter is contracting at an 'ever increasing rate'. When thought of as contraction rather than expansion at an ever increasing rate, the rest becomes explainable/obvious/observable. Trying to explain expansion at an ever increasing rate or imagining an 'expanding universe' in 4D is like ''looking for nothing in the dark''....and when 'they-popular expansionism' find something it always seems to be 'nothing' again and again.

[Mordred]

This last post is based on a misconception that expansion or contraction requires a center. Which is simply wrong.