How should we convey curved space-time?

**Vast** · Nov4-06, 01:00 PM

When trying to visulaize curved space, in popular books and science documentaries we seem to always see the 2D rubber sheet analogy.[JPG] I was wondering, is this the best way to convey the idea? This is a computer generated rendition[JPG] taken from a book I have, and although it may be a little harder to draw an analogy from it, I think it's a better visual representation of curved space-time. Is there a good reason people always use the 2D verion rather than the 3D version? What are your thoughts?

**pervect** · Nov4-06, 01:07 PM

I rather like the "ants on an apple" approach, ala MTW's "Gravitation". See for instance http://www.bun.kyoto-u.ac.jp/~suchii/apple.html.

**pess5** · Nov5-06, 02:39 AM

Originally Posted by Vast

When trying to visulaize curved space, in popular books and science documentaries we seem to always see the 2D rubber sheet analogy.[JPG] I was wondering, is this the best way to convey the idea? This is a computer generated rendition[JPG] taken from a book I have, and although it may be a little harder to draw an analogy from it, I think it's a better visual representation of curved space-time. Is there a good reason people always use the 2D verion rather than the 3D version? What are your thoughts?

3d is better than 2d, but 2d conveys the point in the simplest manner. Another way of depicting it, would be with alot of dots showing spacetime density based on proximity to the surface of the body. Kinda like they do for electron clouds. I haven't seen a depiction like this though.

pess

**dicerandom** · Nov5-06, 04:58 AM

Originally Posted by pervect

I rather like the "ants on an apple" approach, ala MTW's "Gravitation". See for instance http://www.bun.kyoto-u.ac.jp/~suchii/apple.html.

Of course the surface of the apple is really just a closed rubber sheet

When trying to work out (or explain) various curved-space concepts I usually fall back to the surface of a sphere. It's a simple curved space that I'm very familiar with and can easily visualize myself walking around on in order to "experience" a Christoffel symbol or the Riemann curvature tensor or whatever else happens to be bothering me at the time.

**robphy** · Nov5-06, 07:33 AM

To convey "curved space-time", I think you need more than a "spatial" [embedding-]diagram.
http://www.rpi.edu/dept/phys/Courses...acetimeAJP.pdf
http://icsip.elte.hu/astro/hun/konyv...5_pa_blaga.pdf
http://arxiv.org/abs/gr-qc/9806123

**Lelan Thara** · Nov6-06, 08:54 AM

Is it actually misleading to say "spacetime is curved"?

When spacetime is represented in two dimensions, it's understandable to say gravity "curves" this planar representation of reality in a third dimension.

But curvature must happen in n+1 spatial dimensions for any object of n dimensions - right?

So four dimensional spacetime would have to be curved in a fifth dimension, right?

How does one conceive of an unbounded volume that is "curved"?

I think this is one of those instances where physicists understand mathematically what they mean by "curved spacetime", but the English phrase is misleading to the layman.

Would it not be more accurate to say "spacetime is variably dense"? That in the presence of gravity, spacetime contains "more inches per inch"?

This description lets us imagine gravitational fields as spheres, black holes as spheres, which is closer to reality. As it is, laymen tend to latch on to visualizations that are actually inaccurate, such as black holes being funnels.

And I believe the "variably dense" description doesn't change much, if anything - it can still describe gravitational lensing and other relativistic effects.

Do you think we should abandon the "spacetime is curved" linguistic convention altogether?

**dicerandom** · Nov6-06, 09:57 AM

The misconception is easy enough to clear up, just explain that the natural extension of "curved" into arbitrary spaces is that straight lines which were parallel at some point deviate from one another. This also provides a good example of curvature near the Earth's surface, which people might think of as some abstract concept. Explain that if you drop two masses off a tall tower, one slightly above the other, the distance between them will increase at an increasing rate as they fall. Thus, formerly parallel lines (the world lines of the masses) deviate away from one another near the Earth's surface.

**robphy** · Nov6-06, 10:07 AM

One must make the distinction between "intrinsic curvature" and "extrinsic curvature". GR deals with the "intrinsic curvature" of spacetime.

**agenttrotter** · Dec15-07, 09:01 PM

whoa... you just really helped me out by posting this:

http://img506.imageshack.us/img506/5...acetimelu3.jpg

Believe it or not, I've been searching for this image for quite some time. This image has been inside my head, but I couldn't produce it. The 2-d representation of curved space-time is incorrect, but neccesary to teach people the concept.

this 3-d version is the true model. This is what Einstein saw in his mind's eye.

most people like the analogy of the ant walking on some curved surface. it's technically wrong! THE ANT NEEDS TO BE SWIMMING!! It's not just the surface that bends, but everything around the ant.

thank you for posting this image. I don't know why more people don't use it. In fact, this problem GOES BEYOND SPACETIME. people are always using 2-d models to depict things that should be in 3-d.

for example, waves. Some students have actually been convinced that electromagnetic waves and sound waves travel just like they're depicted- as wiggly lines that move forward like an eel swimming. They don't realize that drawings of waves are 2-d, but that all waves radiate outwards in all directions, 3-d.

**agenttrotter** · Dec15-07, 09:09 PM

on another note-

Diagrams of black holes usually depict a FUNNEL of infinite depth.

what does the black hole diagram look like in this 3-d version (http://img506.imageshack.us/img506/5...acetimelu3.jpg)

does anyone know of a program that lets you change the strength (or mass) of the center object causing the distortion? So you could model a small object, a big object, and a maximum density object?

**A.T.** · Dec16-07, 04:01 AM

Originally Posted by Vast

When trying to visulaize curved space, in popular books and science documentaries we seem to always see the 2D rubber sheet analogy.[JPG] I was wondering, is this the best way to convey the idea?

Not really. Showing balls rolling down into dimples is like explaining gravity with gravity. Ants walking only straight forward on this rubber sheet would be better. Unlike the balls they take the same way around a bulge, as they would take around a dimple (of the same form).

Originally Posted by Vast

This is a computer generated rendition[JPG] taken from a book I have, and although it may be a little harder to draw an analogy from it, I think it's a better visual representation of curved space-time.

I gives you an idea about curved space, not about curved space-time. My favorite way to understand the later is presented here in the 2nd chapter. There is also an interactive version here (requires flash9 plugin)

Originally Posted by Vast

Is there a good reason people always use the 2D version rather than the 3D version? What are your thoughts?

One method to visualize the curvature of an manifold, is to embed it in a higher dimensional manifold. So in 3D you can only visualize the curvature of 2D-surfaces. But that's enough for one space dimension and the time dimension.

**DaleSpam** · Dec16-07, 07:41 AM

Originally Posted by Lelan Thara

Would it not be more accurate to say "spacetime is variably dense"? That in the presence of gravity, spacetime contains "more inches per inch"?

I think the word "curved" actually conveys the theory more accurately. "Dense" is not a geometric description, but "curved" is. GR treats gravitation as a geometric phenomenon.

**Pendragon42** · May23-08, 10:28 AM

I don't know much about space-time, but do you think it could be a 'fourth' dimension?

**Pendragon42** · May23-08, 01:22 PM

If anyone's interested (I doubt it) I have done a presentation on space time. I put some screenshots on it on here. I'd like you to tell me what you think.

**Pendragon42** · May23-08, 01:27 PM

Attached

**MeJennifer** · May23-08, 09:54 PM

Originally Posted by Vast

When trying to visualize curved space, in popular books and science documentaries we seem to always see the 2D rubber sheet analogy.[JPG] I was wondering, is this the best way to convey the idea? This is a computer generated rendition[JPG] taken from a book I have, and although it may be a little harder to draw an analogy from it, I think it's a better visual representation of curved space-time. Is there a good reason people always use the 2D version rather than the 3D version? What are your thoughts?

Keep in mind that visualizing curved spacetime is not just a dimensional problem. An additional problem is that spacetime manifolds are Lorentzian which are considered pseudo-Riemannian manifolds. For instance the arc lengths of curved paths in Lorentzian manifolds are shorter than the cord lengths. Which is never the case for non-complex Riemannian manifolds.