posted
A little background. Ok, so we have 3 spatial dimensions in which we live. And time can be thought of as a fourth dimension. And the dominant version of string theory posits 11 spatial dimensions (7 of them wrapped up tightly into Calabi–Yau spaces). And of course, topology allows one to play with n-dimensional space, for very many n. The 3-d equivalent of a mobius strip is a Klein bottle, where the "twist" occurs in 4-d space. Anyone modeling multivariable systems may use n-dimensions. All these things are cool mathematical puzzles, tools, as well as (in the case of string theory, or physics) models of looking at the universe.

Yet what I have never seen is a definitive statement regarding the existence of a real 4th spatial dimension. Most of the time, it (in particular) is treated as some theoretical exercise- a cool part of topology and more abstract maths. Now normally, that wouldn't bother me. Nothing wrong with that.

But one thing occurs to me and I can't seem to figure it out.

The general theory of relativity (which, in my mind, is a proven model) maintains that the force of gravity is due to the "warping" of space-time (I know I am generalizing here). We've all seen the rubber-sheet visuals, with the massive planet or sun in the center causing the warping. But of course, these can only be mere illustrative representations, because space-time is not, in fact, a 2-d plane that is being warped in a 3rd spatial dimension. It is a 3 (or 4, if you're factoring in time)-d volume that is being bent.

If that is the case, does that not prove the existence of a 4-d space? Doesn't the warping/bending have to occur in a higher spatial dimensional. Mobius strips are 2-d surfaces twisted in the 3rd spatial dimention. Klein bottles are 3-d volumes pushed "through" a 4-d space. The shifting of any n dimensional space has to occur in, at the least, the next higher spatial dimension, right? So if space-time is being warped (which I believe is correct) then the warping has to be occurring in a higher spatial dimension, right? So given the correctness of the relativity model, at least a 4-d space has to exist. Am I missing something or is that right? (And don't want to get into proposed n-branes or anything. I'm not talking about speculation from string theory or anything. I'm talking about what has been actually verified so far- relativity.)

4d space exists, right?

Can someone help me with this? Is there something wrong with my reasoning? Am I missing something? It's just something I've wondered about.
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posted
Warping in the way you're talking about is only an analogy. The distortion is "like" distortion in a rubber sheet in the sense that things tend to follow the distortions, but it isn't like it in the sense that the distortion requires a further dimension to be distorted into. For instance, a gas can have varying densities, but there's no requirement there be a fourth dimension for its densities to vary "into".

So, no, gravity as a distortion of space-time does not prove the existence of a fourth spatial dimension.
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posted
It used to be thought that there must be an infinite number of dimensions. A one dimensional line can only exist, for example, when embedded in a plane. A two dimensional plane can only exist when embedded in solid (3D) space. By analogy, 3D space must be embedded in 4D hyperspace, and so on, ad infinitum. I'm not sure why that's not the current view. It makes sense to me.
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posted
That's not the current view because the reasoning is very flawed.

quote:A one dimensional line can only exist, for example, when embedded in a plane.

First, this is not self evident. In particular, it is not at all clear that any one dimensional lines exist -- we can draw things we call "one dimension", but they all have three dimensions.

Similarly for the "two dimensional plane".

That is, our only actual sample of a truly N-d thing is the three dimensional (edit: as far as we can observe) world, and I certainly don't see it embedded in any four dimensional space.
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posted
Btw there is no guarantee that a 3-D manifold can be embedded in R^4. The known bound is R^6.

The image I work with when thinking of the intrinsic geometry of a manifold is of walking around a room filled with honey of varying density, so at some places it is easy to move, at others very difficult. The dimension is simply the dimension of Euclidean space I feel like I'm in at a particular position in the room.
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posted
Interesting. Ok, that makes sense. It's like the question "if the universe is finite, what's on the other side of the edge of the universe." There is no other side- there is no space for there to be another side. Or what happened before the Big Bang. There is no 'before'. Time, as we know it began at the that moment.

It's confusing physical metaphors with the mathematical reality.
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