Distorted, structure (3D) vs warped, fabric (2D)

[Boffin]

Even Einstein said that it was important to be able to visualize concepts. Words are important tools we use to visualize concepts. Aren't present words used in relativity like warped, bending, fabric, and sheet confusing? These are words we generally use to depict 2 dimensional concepts. Wouldn't it be better to use words like distorted and structure? Some scientists have used these 3 dimensional words but they aren't popular in relativity? I wonder why? Oh, and are there better words we can use than what I've mentioned to depict 3 dimensional concepts?

Sure we can use the concept of a sheet to represent the path of a photon traveling at right angles through a gravitational field. But the sheet concept doesn't make sense when the photon travels towards the center of the mass.

Somewhat related threads
Curved space/ spacetime http://www.physicsforums.com/showthread.php?t=104508
General relativity question http://www.physicsforums.com/showthread.php?t=101629
How to imagine distorted space http://www.physicsforums.com/showthread.php?t=75511
Bending of Space and Time http://www.physicsforums.com/showthread.php?t=40907

[chroot]

Hint: words like "warped, bending, fabric, and sheet" are not used in the theory of relativity at all.

Such words are only used in "explanations" of the theory designed to give laypeople -- those without the mathematical skill to appreciate the equations -- an idea of how the theory works. The rubber sheet analogy is just an analogy.

- Warren

[Boffin]

Hint: words like "warped, bending, fabric, and sheet" are not used in the theory of relativity at all. Such words are only used in "explanations" of the theory designed to give laypeople -- those without the mathematical skill to appreciate the equations -- an idea of how the theory works. The rubber sheet analogy is just an analogy.
Sure, I thought I implied that. But are you saying that you learn new concepts totally from equations? Without any words to describe the concept or even an introduction? A book totally just equations and nothing else? Do you follow equations with no conceptial visualization of how they are applied in pictorial or spatial form?

[chroot]

It's impossible to directly visualize a 4D space. Most mathematicians use visualization in lower-dimensional situations to gain a geometric understanding of a concept before extending it to higher dimensions.

In most books on relativity, graphics are used to illucidate many small concepts and mathematical relationships, but no one actually attempts to visualize 4D spacetime.

- Warren

[Mortimer]

I think that quite often 3D analogies work fine and are accurate for situations where certain parameters do not change anyway in one or more of the dimensions, like e.g. an object falling radially towards a black hole. For radial distance in such case the rubber sheet analogy indeed gives a correct impression of the relation dR=(1-2m/r)^{-1/2}dr.

[Boffin]

I think that quite often 3D analogies work fine and are accurate for situations where certain parameters do not change anyway in one or more of the dimensions, like e.g. an object falling radially towards a black hole.

Good point. It's really 4D that's hard to verbalize, visualize, or draw on paper in graphical terms. We can use graphical tools such as lighter and darker to depict lesser or greater gravitational fields. For many situations, we can eliminate time temporarily from the thought process.

[chroot]

Most of the important features of the mathematics of GR can be just as easily explained with low-dimension examples like 2D spacetime diagrams. Generalizing to higher dimensions doesn't change the math at all, and the concepts make sense in any number of dimensions.

- Warren