# How should we convey curved space-time?

• Vast
In summary, the 2D image that is often used to represent curved space-time is incorrect. The 3D version is a better representation because it shows the density of spacetime based on proximity to the surface. Another way of depicting it would be with a lot of dots showing spacetime density based on proximity to the surface of the body.
Vast
When trying to visulaize curved space, in popular books and science documentaries we seem to always see the http://www.metaresearch.org/cosmology/images/rubber%20sheet%20analogy.jpg" [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?

Last edited by a moderator:
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 .

Last edited by a moderator:
Vast said:
When trying to visulaize curved space, in popular books and science documentaries we seem to always see the http://www.metaresearch.org/cosmology/images/rubber%20sheet%20analogy.jpg" [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 a lot 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

Last edited by a moderator:
pervect said:
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.

Last edited by a moderator:
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 let's 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?

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.

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

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

http://img506.imageshack.us/img506/5015/spacetimelu3.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.

Last edited by a moderator:
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 let's 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?

Last edited by a moderator:
Vast said:
When trying to visulaize curved space, in popular books and science documentaries we seem to always see the http://www.metaresearch.org/cosmology/images/rubber%20sheet%20analogy.jpg" [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).

Vast said:
This is a http://img506.imageshack.us/img506/5015/spacetimelu3.jpg" [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 http://fy.chalmers.se/~rico/Theses/tesx.pdf"

Vast said:
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.

Last edited by a moderator:
Lelan Thara said:
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.

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

Presentation??

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.

#### Attachments

• 1c.GIF
10.6 KB · Views: 599
• 2c.GIF
110.9 KB · Views: 627
• 3c.GIF
72.3 KB · Views: 608
More screenshots

Attached

#### Attachments

• f1.GIF
5 KB · Views: 390
• dimensions.GIF
97.5 KB · Views: 637
• conclude.GIF
23.6 KB · Views: 598
Vast said:
When trying to visualize curved space, in popular books and science documentaries we seem to always see the http://www.metaresearch.org/cosmology/images/rubber%20sheet%20analogy.jpg" [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.

Last edited by a moderator:
Pendragon42 said:
I don't know much about space-time, but do you think it could be a 'fourth' dimension?

well no as space-time is a field. It uses "length" "width" and "depth" the 3 obvious dimensions but also uses time as the 4th dimension. It wouldn't really work for something which needs 4 dimensions to exist to be a dimension in its own right. When Lelan Thara was talking about a "fifth" dimension earlier i believe he meant that people often misconcieve spacetime to be literally "curved" which would require an extra dimension.

Hello,

I really have a problem with the "sheet of rubber" visualisation.

If the space is curved like it is often shown, all the planets would roll down this "sheet of rubber" towards the Sun and not continue to circle around.

That is, if they were on a flat bit of the rubber sheet (i.e. the bit that is outside of the influence of the sun and hence flat, i.e. not angled/declined towards the Sun) the planets would keep going straight, just as if the Sun wasn't there, and if they were on a declined plane, angled down towards the sun, they would roll towards it.

dicerandom said:
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.
Same here. I usually think of time increasing to the north and space increasing to the east, and then I find that most of the basic concepts of curved spacetime become clear.

Garderp said:
I really have a problem with the "sheet of rubber" visualisation.
Hi Garderp, welcome to PF.

I think you will find that most people here agree with you, the rubber sheet analogy is very poor for many reasons. The main reason that it is poor is that it only depicts curved space, and most gravitational effects are due to curvature in the time dimension. But there are other reasons that it is a bad analogy. I recommend simply ignoring it.

Garderp said:
Hello,

I really have a problem with the "sheet of rubber" visualisation.

If the space is curved like it is often shown, all the planets would roll down this "sheet of rubber" towards the Sun and not continue to circle around.

That is, if they were on a flat bit of the rubber sheet (i.e. the bit that is outside of the influence of the sun and hence flat, i.e. not angled/declined towards the Sun) the planets would keep going straight, just as if the Sun wasn't there, and if they were on a declined plane, angled down towards the sun, they would roll towards it.

They'd only roll straight down the dip in the rubber sheet if they started off with no velocity relative it, or with velocity directed entirely towards the centre of the dip. If there was no friction, and they had the right initial velocity, they could circle around the mouth of the funnel, just as the real planets orbit the sun.

Vast said:
When trying to visulaize curved space, in popular books and science documentaries we seem to always see the http://www.metaresearch.org/cosmology/images/rubber%20sheet%20analogy.jpg" [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?

The rubber sheet analogy has its limitations and is probably a better depiction of Newtonian gravity than spacetime, but it is still a better visualisation than the 3D visualisation from your book which is essentially meaningless and not at all useful for anything other than looking pretty in a "coffee table" book. Like the rubber sheet analogy the 3D depiction does only shows the curvature of space and not the curvature of time and is therefore not a depiction of spacetime. It also seems to imply that gravitational length contraction happens equally in the vertical and horizontal directions which is not true. Gravitatioanal length contraction only happens in the vertical direction. Imagine a teacher showing that visualisation to some students on an introductory course to spacetime. How would he answer the following questions from the students? 1) How does the visualisation demonstrate the bending of light paths in a gravitational field? 2) How does the visualisation demonstrate gravitational time dilation? 3) How does the visualisation demonstrate precession of orbits? 4) What does it demonstrate? The answers would be 1) Not sure, 2) It doesn't, 3) Ummmmm 4) Don't know... It's pretty though..

Would the students leave feeling they had a firm grasp on spacetime?

The visualisations linked to by Robphy and A.T. are much better.

Last edited by a moderator:
I saved the following explanation of spacetime curvature from Dr. Greg in which he replied to a posted question from me. Note the relationship of "curvature" to intertial frames, accelerating frames, and gravitational effects.

It's pretty obvious after reading this that any simple representation of "curvature" will be necessarily incomplete...an approximation.

What we call the "curvature of spacetime" has a technical meaning; the equations that describe it are very similar to the equations that describe, say, the curvature of the Earth's surface in terms of latitude and longitude coordinates, or any other pair of coordinates you might choose. This "curvature" need not manifest itself as a physical curve "in space".

For the rest of this post let's restrict our attention to 2D spacetime, i.e. 1 space dimension and 1 time dimension, i.e. motion along a straight line. …
In the absence of gravitation, an inertial frame corresponds to a flat sheet of graph paper with a square grid. If we switch to a different inertial frame we "rotate" to a different square grid, but it is the same flat sheet of paper. (The words "rotation" and "square" here are relative to the Minkowski geometry of spacetime, which doesn't look quite like rotation to our Euclidean eyes, but nevertheless it preserves the Minkowski equivalents of "length" (spacetime interval) and "angle" (rapidity).)

If we switch to a non-inertial frame ([an accelerated observer] but still in the absence of gravitation), we are now drawing a curved grid, but still on the same flat sheet of paper. Thus, relative to a non-inertial observer, an inertial object seems to follow a curved trajectory through spacetime, but this is due to the curvature of the grid lines, not the curvature of the paper which is still flat.

When we introduce gravitation, the paper itself becomes curved. (I am talking now of the sort of curvature that cannot be "flattened" without distortion. The curvature of a cylinder or cone doesn't count as "curvature" in this sense.) Now we find that it is impossible to draw a square grid to cover the whole of the curved surface. The best we can do is draw a grid that is approximately square over a small region, but which is forced to either curve or stretch or squash at larger distances. This grid defines a local inertial frame, where it is square, but that same frame cannot be inertial across the whole of spacetime.

So, to summarize, "spacetime curvature" refers to the curvature of the graph paper, regardless of observer, whereas visible curvature in space is related to the distorted, non-square grid lines drawn on the graph paper, and depends on the choice of observer.

## 1. What is curved space-time?

Curved space-time is a concept in physics that describes the curvature of space and time caused by massive objects. It is a fundamental component of Einstein’s theory of general relativity, which explains how gravity works.

## 2. How does curved space-time affect our daily lives?

Curved space-time has a very small effect on our daily lives, as it is only noticeable when dealing with massive objects like planets and stars. However, it plays a crucial role in our understanding of the universe and how gravity operates.

## 3. How do we visualize curved space-time?

Curved space-time is a difficult concept to visualize, as it involves the combination of three-dimensional space and one-dimensional time. One way to visualize it is to think of a trampoline with a heavy object placed in the center, causing the surface to curve. Objects placed on the trampoline will roll towards the center due to the distortion of the fabric, similar to how gravity works in curved space-time.

## 4. How can we convey curved space-time to non-scientists?

Explaining curved space-time to non-scientists can be challenging, but it can be done by using analogies and visual aids. Comparing it to the trampoline example or using other relatable examples can help people understand the concept better. It is also essential to use simple language and avoid technical jargon.

## 5. What are some real-world applications of curved space-time?

Curved space-time has several practical applications, including GPS technology, which relies on the precise measurement of time and space. It is also crucial in understanding the behavior of black holes and other astronomical objects. Additionally, the concept of curved space-time is essential in the development of future space travel and exploration.

• Special and General Relativity
Replies
27
Views
4K
• Special and General Relativity
Replies
1
Views
1K
• Special and General Relativity
Replies
18
Views
2K
• Special and General Relativity
Replies
23
Views
3K
• Cosmology
Replies
11
Views
1K
• Special and General Relativity
Replies
18
Views
6K
• Special and General Relativity
Replies
25
Views
7K
• Special and General Relativity
Replies
1
Views
1K
• Beyond the Standard Models
Replies
4
Views
2K
• Special and General Relativity
Replies
4
Views
2K