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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