How is 3-space curved without a fourth spatial dimension?

[Robert100]

When I was studying physics in college I learned two contradictory ways to understand Einstein's theory of general relativity (GR).

The most common way was to visualize our three-dimensional space as being distorted by mass. Anything with mass created a distortion in spacetime. Just as a bowling ball placed on a 2-D rubber sheet creates a curvature in a third spatial dimension, mass in our 3-D universe creates a curvature in fourth spatial dimension.

The second way I learned was identical to the above, but with the following bizarre provision: We don't need a fourth spatial dimension for the 3D universe to be curved into. When I asked how this could even be possible, I was told that the math doesn't require it, but this was hardly an answer.

Lately I have been searching for an explanation of how anyone could believe the second conception (our 3-space is curved without any higher space for it to be curved into!?), yet have found nothing beyond flat assertions that the math allows this to be so.

Here is an example of a useless non-explanation, from two otherwise reliable authors:

"This balloon analogy should not be stretched too far. From our point of view outside the balloon, the expansion of the curved two-dimensional rubber is possible only because it is embedded in three-dimensional space. Within the third dimension, the balloon has a center, and its surface expands into the surrounding air as it infl ates. One might conclude that the expansion of our three-dimensional space requires the presence of a fourth dimension. But in Einstein’s general theory of relativity, the foundation of modern cosmology, space is dynamic. It can expand, shrink and curve without being embedded in a higher-dimensional space."

"Misconceptions About the Big Bang" Tamara Davis and Charles Lineweaver, March 2005

I urge readers here to read this article, and others by these same authors. In all cases where they explain away misconceptions about cosmology, they do a good job. They provide graphs and pictures, they give analogies, and generally make themselves understood. But what they do here is to totally gloss over the issue, and just assert that the math says so.

Worse, I have seen the same non-answer from other people, who otherwise write well.

Can anyone tell me what is going on here? Do most/all cosmologists really agree that GR predicts that there is NOT a fourth spatial dimension? If so, can someone point me to a useful explanation of what this means?


Robert

 

[Chris Hillman]

Recommend some good nontechnical books

When I was studying physics in college I learned two contradictory ways to understand Einstein's theory of general relativity (GR).

The most common way was to visualize our three-dimensional space as being distorted by mass. Anything with mass created a distortion in spacetime. Just as a bowling ball placed on a 2-D rubber sheet creates a curvature in a third spatial dimension, mass in our 3-D universe creates a curvature in fourth spatial dimension.

The second way I learned was identical to the above, but with the following bizarre provision: We don't need a fourth spatial dimension for the 3D universe to be curved into. When I asked how this could even be possible, I was told that the math doesn't require it, but this was hardly an answer.

If you took a real physics course :-/ you must misremember some important details.

We model the setting for physics, called spacetime, as a four-dimensional manifold. According to Wheeler's slogan for the EFE, "matter tells spacetime how to curve; spacetime tells matter how to move".

Now, we can always "embed" any manifold as a kind of curved hypersurface in some higher dimensional manifold (although we typically require more than just one extra dimension for four-manifolds!). In fact, there are always infinitely many ways to do this. In each embedding, we can define and study an "extrinsic curvature", but this tells how the hypersurface bends in the bigger space, and is not an "intrinsic feature" of the geometry of the small space, because it changes if we consider a new embedding.

(Incidently, embeddings in homogeneous but nonflat higher dimensional manifolds are currently popular.)

In constrast, Gauss and Riemann showed that manifolds have intrinsic curvature. This curvature does not depend upon any embedding in a higher dimensional flat space, so it is an intrinsic description of the internal geometry.

Lately I have been searching for an explanation of how anyone could believe the second conception (our 3-space is curved without any higher space for it to be curved into!?), yet have found nothing beyond flat assertions that the math allows this to be so.

So you are unwilling/unable to just learn the math? Fair enough, but then you should accept what people tell you about what the math of curved manifolds says. Be this as it may, try the book by Sklar listed at http://math.ucr.edu/home/baez/RelWWW/HTML/reading.html#phil

Here is an example of a useless non-explanation, from two otherwise reliable authors:

"Misconceptions About the Big Bang" Tamara Davis and Charles Lineweaver, March 2005

Nice paper--- maybe after reading Sklar you'll appreciate it more.

I urge readers here to read this article, and others by these same authors. In all cases where they explain away misconceptions about cosmology, they do a good job. They provide graphs and pictures, they give analogies, and generally make themselves understood. But what they do here is to totally gloss over the issue, and just assert that the math says so.

Worse, I have seen the same non-answer from other people, who otherwise write well.

Again, are you insisting that this be explained to you without using any mathematical reasoning? If so, that could be a big part of the problem. Be this as it may, you might try the book by Jeffrey Weeks, The Shape of Space, which makes a real attempt to explain manifold theory (including issues of global structure) using lots and lots of excellent pictures.

Can anyone tell me what is going on here? Do most/all cosmologists really agree that GR predicts that there is NOT a fourth spatial dimension?

I take it you now appreciate that this question rests upon a basic misconception, which is probably why others you have asked apparently just gave you a funny look and walked away.
Robert[/QUOTE]

 

[yenchin]

You know how to construct a cylinder from a square right? You identify one of the opposite sides, i.e. glue them up and form cylinder. Now of course by doing this you introduce a type of curvature.

But you can simply identify the sides topologically without really gluing them in 3 dimensional space. The geometry on this flat cylinder is the same as that on the "real cylinder" according to any flatlander who lives on them.

In other words, a cylinder is actually intrinsically flat: its (Gaussian) curvature is zero. This concept of curvature is what matters. The (scalar) curvature we introduced by gluing physically (by embedding in 3-space) is extrinsic.

And since we can only observe what is in the universe, it is the intrinsic curvature that we are interested in, and that is the information contains in the Riemann curvature tensor in GR formulation.