Does anybody have a better analogy of describing space-time curvature?

[Dropout]

The old 2D paper describing 3D curvature is a little lame because it uses gravity to describe gravity. You know, the little ball circling the 2D psuedo-black hole, well remove gravity and the ball would fly off the 2D paper, you can't use gravity to describe gravity. It would be like saying 1=1 because 1=1. Proving your answer with the question.

How else can you visualize shortened distances, even in 2D. Has anyone else been bothered by that before?

 

[Chronos]

I have no idea what you are talking about.

 

[Dropout]

http://www.pbs.org/wgbh/nova/time/images/blackhole.jpeg

Allot of textbooks put a ball in this diagram and ooh and awe about how it orbits around the gravity well. Well yeah, that's cause gravity in the 3rd and final spatial dimension is keeping the ball from flying off.

 

[wisp]

http://www.pbs.org/wgbh/nova/time/images/blackhole.jpeg

Allot of textbooks put a ball in this diagram and ooh and awe about how it orbits around the gravity well. Well yeah, that's cause gravity in the 3rd and final spatial dimension is keeping the ball from flying off.

If you roll two balls along the same line at different speeds, the one going faster should take a bigger orbit. But in this simplified 2D spacetime model they would follow the same path, so it's not really a good model.

The demo's I've seen using a stretched rubber sheet with a heavy mass in the middle, do use gravity to bend the path of the slower moving ball. So you'r right, without the Earth's gravity the demo would fail.

 

[JesseM]

http://www.bun.kyoto-u.ac.jp/~suchii/apple.html

The Parable of the Apple

We now have a number of popular expositions of the theory of general relativity, including Einstein's own version (1917) and such classics as Eddington's Space, Time and Gravitation (1920) and Geroch's General Relativity from A to B (1978). But to my knowledge (not extensive, I warn you!), Misner, Thorne and Wheeler's "Parable of the Apple" in Chapter 1 of their Gravitation (1973) is the best, in that it gives a basic and overall view and the analogy used goes far enough, giving clear hints on the nature of mathematical tools used in the theory.

...

The parable tries to explain the nature of gravitation in terms of the curvature of spacetime, and the two-dimensional curved surface (the curvature varies depending on locations) of the apple is used for this purpose. The following figure is adapted from their Figure 1.1.

http://www.bun.kyoto-u.ac.jp/~suchii/apple.jpg

The tale goes like this. One day a student, reflecting on the difference between Einstein's and Newton's views about gravity, noticed ants are running along the surface of an apple. Ants seemed to take a most economical path; wow, they are going along geodesics on this surface! But each geodesic may also be regarded as a path (world line) of a free particle on this surface (taken as a two-dimensional spacetime). Look at two ants going from the same spot on the top (near the dimple) into different directions; one goes down into the bottom of the dimple whereas the other goes around the dimple!

According to Newton, this is because of gravitation acting at a distance from a center of attraction. But according to Einstein, this is because of the local geometry of the surface at that spot, namely, because of the curvature of the spacetime there. But how do geometry and matter in the spacetime interact with each other? In brief, Einstein's geometrodynamics (according to his field equations) is "a double story of the effect of geometry on matter (causing originally divergent geodesics to cross) and the effect of matter on geometry (bending spacetime initiated by concentration of mass, symbolized by effect of stem nearby surface of apple)". Thus Einstein dispenses with any action-at-a-distance, and physics becomes simple only when analyzed locally. In a word, spacetime tells matter how to move, and matter tells spacetime how to curve (one of Wheeler's favorite phrases).

Note that if the ant was walking on a perfect sphere, the shortest path (geodesic) between two points would always be a section of the great circle (like a line of longitude, or like the equator) that passes through those points. The only difference in spacetime is that instead of a "geodesic" being the path with the shortest spatial length, it is instead the path through spacetime with the greatest proper time (ie time as measured by an observer who follows that path). For example, in the twin paradox the twin who moves inertially between two points in spacetime is the one following a geodesic, thus he will be older than another twin who takes a non-geodesic path between those points.

 

[chronon]

The old 2D paper describing 3D curvature is a little lame because it uses gravity to describe gravity. You know, the little ball circling the 2D psuedo-black hole, well remove gravity and the ball would fly off the 2D paper, you can't use gravity to describe gravity. It would be like saying 1=1 because 1=1. Proving your answer with the question.

How else can you visualize shortened distances, even in 2D. Has anyone else been bothered by that before?

Yes, it is a little lame. It fails to capture the crucial point that gravity is cause by curvature in spacetime, not just space. However there are a couple of problems with a more realistic visualisation.

1) Space time is 4 dimensional. To visualise a curved 4 dimensional manifold, I think you need to embed it into 8 dimensional Euclidean space. Its not that easy to visualise 8 dimensions. Even if you throw away a dimension you are still left with visualising 6 dimensions.

2) Time is different from space. The signature of the metric is - + + + (or + - - - if you prefer) rather than + + + + of 4 dimensional Euclidean space. This means that the normal ideas of rotating an object don't apply.

 

[selfAdjoint]

Levi-Civita taught us to express curvature in terms of "parallel transport of a (differentially small) vector". In a plane you can move a short line representing the vector around parallel to itself, and it always keeps pointing in the same direction. In a curved geometry, not so, you transport it around a simple closed curve and it comes back pointing in a different direction than it started in.

To show this, as I have posted before, you need a 20D stand-in for the 4-D curvature; I suggest an upended mixing bowl. You also need a vector stand-in; a small strip of cardboard with parallel sides will do. And finally something to make marks on the surface.

Draw a closed curve on the bowl, fairly large to make the directions easily visible. Place the cardboard strip across some part of the curve and make a mark along the left edge where it cuts the curve. Rock the strip a bit so the right edge cuts the curve and make another mark along that edge, showing its direction. Then move the strip so its left edge lies along the mark you made for the right edge. Make a new mark along the new position of the right edge, across the curve. Keep going this way till you move around the curve, and you will readily see the edges come back pointing differently. And this is entirely because of the curvature; the difference angle, in the limit, expresses the curvature!.

exercise for the lurker; use this technique to derive the formula for the coriolis acceleration.

 

[tdunc]

there's no such thing as free space curvature, by gravity or mass, however you want to put it, it just don't - It cant, its not fixed, no "fabric of space". So there's no question that an illustration like such is complete nonsense in the context of gravity, like you say they use gravity to describe gravity - [anyone that does not get what we are saying will slap there forehead so hard once they figure it out they will knock themselves out] Space-time curvature on the other hand can be interpreted to mean trajectory or path, whereby you throw an object forward and its path_through_space will curve down over_time.

You can, if you really want, use the term or idea of space curvature in the context of describing a physical object - its curved geometry- then illustrate various paths taken on it like selfAdjoint is doing and make all sorts of interesting comments, but (not recommended in the context of "space" as most people relate the word to)... know that "space" then refers to that physical objects geometry Or in a different context you could visualize a 'square section of space' that contains that object and therefore that section of space would have a curvature Within it - The Earth for example is a curvature within space. < GR then goes on to say that the mass of the Earth causes a curvature Of space... which is, wrong.. wrong.. wrong

bah, GR forum...

 

[pervect]

I'm not sure what the last rant was about, but according to GR gravity does curve space. While the primary effect of gravity is to curve space-time, space is also curved in the presence of large masses.

The effects of space-space-curvature are usually small, but they are important to explain why light bends twice as much near a mass as Newtonian theory predicts. Space curvature shows up as the length of the meter being dependent on one's depth in a gravity well.

 

[eNathan]

The old 2D paper describing 3D curvature is a little lame because it uses gravity to describe gravity. You know, the little ball circling the 2D psuedo-black hole, well remove gravity and the ball would fly off the 2D paper, you can't use gravity to describe gravity. It would be like saying 1=1 because 1=1. Proving your answer with the question.

How else can you visualize shortened distances, even in 2D. Has anyone else been bothered by that before?

Well, without getting mathematical, you must picture spacetime as an actual "entity" that can be changed, like water. Imagine what could happen when you put mass into this material, you can image, atleast, some effects taking place. Einstein has failed (in my opinion, just like Newton) to explain WHY relativity works the way it does, that is, in the respect that we cannot fully image 4D spacetime; But he has done a better job than Newton Anyway, an understanding of the mathematics can help you visualize it too, but my main point is to imagine spacetime as an actual entity, or substance, not just something invisible. And you also would have to drop the thoughs like "what is time and how can it slow down", as far as the slowing down of time simply imagine things moving slower. I am not saying you cannot imagine these things, I am just saying that many people have a hard time grasping the concept that time actually exists, and is not constant.