Understanding 3D Space Bending in General Relativity

[p.tryon]

General relativity tells us that space-time bends. In what direction does/can 3D space bend? If its bending is non-directional, in what sense does it bend? Can you suggest any analogies that would help student's understand this phenominon as they learn relativity?

 

[NWH]

The best analogy is probably to think of a bowling ball resting on a sheet of cloth. The bowling ball's presence causes the cloth to bend and warp, telling other objects around it how to move. To give you an analogy better than that is beyond my knowledge... I'd imagine space can bend and warp in many different ways, whether we realize it or not is another question...

Actually, here's another one in regards to worm holes... Imagine taking a piece of paper and drawing a line between two dots, one on each side of the sheet. Now imagine folding the sheet of paper in half and connecting the two dots together, the distance between the dots in higher dimensional space is now shorter than the distance the line had to cover across the sheet of paper.

 

[DaveC426913]

General relativity tells us that space-time bends. In what direction does/can 3D space bend? If its bending is non-directional, in what sense does it bend?

Space bends within itself.

Imagine stacking a dozen tennis rackets together, face-to-face. The criss-crossing layers of strings - together with the stack - will form an orthagonal grid in 3 dimensions. Let's say string are all 1cm distance. Travel 1cm in any orthagonal direction (lengthwise, width-wise or even up/down-through-the-stack) and you weill arrive at another string.

Let's pretend all the rackets are strung with steel guitar strings.

Now, put a magnet in the centre of the stack of rackets. The sections of string closest to the magnet will all pull toward it, curving in all 3 dimensions away from straight. The sections farther from the magnet will remain almost straight.

If two ants were to walk along two strings next to each other, each walking at, say, 1cm per second, the one nearer the centre of the stack would have a longer distance to walk than his partner on the next string over. No matter which orientation the ants pick, (lengthwise, width-wise or up/down-through-the-stack) they will notice this discrepancy in their distances.

 

[OB 50]

Space bends within itself.

Imagine stacking a dozen tennis rackets together, face-to-face. The criss-crossing layers of strings - together with the stack - will form an orthagonal grid in 3 dimensions. Let's say string are all 1cm distance. Travel 1cm in any orthagonal direction (lengthwise, width-wise or even up/down-through-the-stack) and you weill arrive at another string.

Let's pretend all the rackets are strung with steel guitar strings.

Now, put a magnet in the centre of the stack of rackets. The sections of string closest to the magnet will all pull toward it, curving in all 3 dimensions away from straight. The sections farther from the magnet will remain almost straight.

If two ants were to walk along two strings next to each other, each walking at, say, 1cm per second, the one nearer the centre of the stack would have a longer distance to walk than his partner on the next string over. No matter which orientation the ants pick, (lengthwise, width-wise or up/down-through-the-stack) they will notice this discrepancy in their distances.

That's a very good analogy to illustrate the experience of traveling through curved space, but I don't think it necessarily demonstrates space bending within itself. Perhaps I'm reading a bit too much into a simplified analogy, so if that's the case, I apologize in advance.

It seems that if we accept the premise that the strings themselves represent the possible paths through space, there is no way to bend them without introducing an outside force - something not located on a path defined by a string. If the 3-D space represented by the grid of strings does not exist within some higher dimensional space, then anything capable of acting on a string would have to be located within the confines of the strings themselves for the analogy to be valid.

I suppose my real concern is that for 3-D space to bend within itself, and not through a higher dimension, it would need to be a closed set of elements not dependent or even capable of interacting with an outside force. Are the individual elements within such a closed set capable of altering the structure of the set itself? Can they even exert forces or move in a "direction" required for such an alteration?

 

[Naty1]

Spacetime bends in a concave manner toward mass, or its equivalent, energy...as the sheet analogy mentioned above. A mobius strip is one way to illustrate space is not so simple as we usually think it to be. It's also useful to mention that we are not even positive there are only three dimensions of space and one of time...string theory seems to suggest, for example, ten space and one time dimension...

The wormhole suggestion above is another good illustration showing how to think in different ways.

It's also helpful to consider that in cosmology (planets, univers,etc) most of space is pretty flat ...until you get in the area of singularities like black holes and perhaps the big bang singularities...

In FABRIC of the Cosmos, Brian Greene has an excellent sketch on page 61 (figure 3.7) illustrating constant velocity as a straight line in space time; uniform circular motion as a spiral corkscrew through spacetime and uniform acceleration as a parabola like curve.

A related concept is that we all move thru spacetime at a constant speed "c". If stationary, all our speed "c" is through time; if we move thru space then some of our speed is diverted from time (hence time passes more slowly) and we can only divert speed from time to space up to a maximum "c"...

And here's a recent post I made for further thought about what's curved and what is not: Is Spacetime Really Curved? https://www.physicsforums.com/showthread.php?t=308426

Finally, don't forget about geodesics...Wikipedia has a decent introduction...

 

[workmad3]

My normal way of 'visualising' is to think of both the interior and exterior perspective. Take a strip of paper and imagine you are walking along it (with a force holding you to it's surface and giving you a 'down' direction). This will appear flat as you walk along it even if the physical paper is bending in some way. If the strip is small enough or you move fast enough then you will be able to notice the curvature easily. If the strip is huge and you move slowly, you could wander forever and never notice it was curving (but you could make observations that would show it was).

Now take the view of the paper from the exterior. It is obviously either flat or curved. This unfortunately is not a view we can take with space as we are constrained to the 'surface', but we can make the observations I mentioned earlier (light bending, relativistic effects, etc) and satisfy ourselves that space is 'curved' in some appropriate sense.

 

[DaveC426913]

That's a very good analogy to illustrate the experience of traveling through curved space, but I don't think it necessarily demonstrates space bending within itself.

??

But in my analogy, the space got bent in the first few paragraphs, before I introduced the ants. The space was bent even if I hadn't introduced the ants traveling through it.

 

[p.tryon]

Thank you all for the thought provoking replies that helps. I am teaching relativity next and have never been taught it propely myself- but I am finding it fascinating to learn about it now! More questions to come...

 

[OB 50]

??

But in my analogy, the space got bent in the first few paragraphs, before I introduced the ants. The space was bent even if I hadn't introduced the ants traveling through it.

I wasn't really taking the ants into consideration at all. The distinction I'm trying to make is pretty subtle, but I think it's important.

Maybe this model doesn't describe our universe, but it describes a universe, and its rules should remain internally consistent.

If the strings represent all possible paths through space-time, then nothing existing in that same space-time can be located anywhere other than a string. That's why I ignored the ants. They can't walk on the strings. They would have to be some aspect of the strings themselves to exist completely within the model you're imagining.

Same thing with the magnet. It has to be located "in" a string, otherwise it is analogous to a higher dimensional force, suggesting that the paths defined by the strings are being bent through a higher dimensional space, not within the space they define.

I'm also assuming that there are infinite strings, there are no literal "gaps" between them, and it's not really important how they are arranged. The more I think about it, your model seems to represent the strings as existing in 3-D space, but I understood it to mean that they were space. Maybe that's where the disconnect lies.

 

[DaveC426913]

If the strings represent all possible paths through space-time, then nothing existing in that same space-time can be located anywhere other than a string.

...

The more I think about it, your model seems to represent the strings as existing in 3-D space, but I understood it to mean that they were space.

Yeah, it's just a grid in 1cm increments. Nothing's stopping points existing between strings.

Maybe I'm answering the wrong question. I assumed you were trying to figure out how the 2-D funnel - which requires a 3rd dimension to distort into - is analagous to a 3D bend - without postulating a 4th dimension for it to distort into.

 

[OB 50]

Yeah, it's just a grid in 1cm increments. Nothing's stopping points existing between strings.

Maybe I'm answering the wrong question. I assumed you were trying to figure out how the 2-D funnel - which requires a 3rd dimension to distort into - is analagous to a 3D bend - without postulating a 4th dimension for it to distort into.

That's almost what I'm asking. I don't dispute that 3D space can be bent within itself, but how can we (existing in 3D space) tell the difference between that and distortion through a 4th dimension. Wouldn't it appear exactly the same to us?

 

[A.T.]

General relativity tells us that space-time bends. In what direction does/can 3D space bend? If its bending is non-directional, in what sense does it bend?

Extrinsic curvature needs a direction to bend into. But intrinsic curvature doesn't. Spacetime doesn't bend, it is warped. Distances between coordinates are distorted. You could say spacetime is tighter packed in some areas.

Can you suggest any analogies that would help student's understand this phenominon as they learn relativity?

Check out the links in this post:
https://www.physicsforums.com/showpost.php?p=2046692&postcount=4