'Direction' of space-time curvature ?

[mjones32]

Hi, I'm new here. I want post a specific question that's been rattling around in my head.

Basically, if you consider the curvature of 3 dimensional space into a 4'th dimension due to gravitational field, has anyone considered the 'direction' of that curvature ?

If you think about the curvature of a one dimensional bowstring, by an archers hand or the familiar bowling ball on a trampoline analogy, there is a distinct direction and physical reason for the direction in the curvature that occurs.

By analogy, it seems clear that there must be a 'direction' in which gravity bends 3-dimenional space. It also seems to me that it is always the 'same' direction - otherwise the curvature in space due to the moon could cancel out that generated by the earth. So would that be a universal phenomenon? Could that be demonstrated in some way ?

So what is the 'direction' - is it forward and backward in time ? Or some other spatial 'direction' - but how would the universe know which way to curve ?

Appreciate if someone can shine some light in this area,

Martin

 

[Passionflower]

In layman's terms:

Actually in GR it is not a 3 but a 4 dimensional spacetime that is curved. In this spacetime each different point, which is a location at a particular point in time, can influence the curvature (based on some kind of difficult formula). This influence can differ from point to point. In the most simple cases you could embed a four dimensional curved spacetime in 5 dimensions but in even slightly more difficult cases the number of dimensions increases rapidly. However thinking of curved spacetime as embedded in some higher dimension is only used by few people as it has no computational benefits. Most prefer to treat spacetime as a curved 4 dimensional surface.

 

[Mentz114]

The mathematical object called curvature in general relativity is multi-valued and contains information about slopes ( gradients).

If you were standing on a smooth knoll you might observe that the slope down the knoll is different depending on which way you look. It turns out that the shape of the knoll ( ie its deviation from flatness) can be described by lots of gradients of gradients.

This is a simplification of course, and it becomes even less accurate when we go to 4 dimensions.

So "has anyone considered the 'direction' of that curvature ?" can be answered with a resounding 'yes'.

 

[cesiumfrog]

Basically, if you consider the curvature of 3 dimensional space into a 4'th dimension due to gravitational field, has anyone considered the 'direction' of that curvature ?
If you think about the curvature of a one dimensional bowstring, by an archers hand or the familiar bowling ball on a trampoline analogy, there is a distinct direction and physical reason for the direction in the curvature that occurs.

First just emphasizing Passionflower's point: "curvature of 3 dimensional space into a 4'th dimension due to gravitational field" is wrong. It's curvature of a 4 dimensional space-time into roughly an 8 dimensional space, and the kind of "physical reason" you ask for is what string/brane theory speculates about.

But normally in GR we don't even think about that ~8 dimensional space, because GR is expressed in differential geometry, the mathematics of cartography. Imagine if aborigines had produced detailed maps of the Australian outback, and measured the distances along many routes therein. From this they could mathematically deduce that the Earth has a radius of curvature of 6000 km. But imagine if they had never found a mountain tall enough to actually see (over the scrub) the curvature directly: they would never be able to tell whether the shape of their Earth is like the outside of an emu egg shell (spherical) or like the inside of an emu egg shell (dished, bowl-like). And moreover, it wouldn't matter, because using diff.geom. they have a formula to go on calculating the distance along every possible route (over the 2D intrinsically-curved surface) without ever needing to know how the Earth is embedded (whether in extra dimensions it is the outside of a sphere, or the inside, or the edge of a donut or Klein bottle, for example).

 

[haael]

@mjones32:
There are 2 types of curvature: external and internal. You are talking of an external curvature, GR is about internal.

External curvature, as you say, is a deformation of an n-dimensional manifold in a direction orthogonal to the manifold - that is, in an (n+1) dimension.
Internal curvature is a deformation in a direction parallel to the manifold. It does not require additional dimensions.

Let's make an experiment: get some thin rubber sheet (cut it from a baloon), draw some lines on it and put it on a flat table. Then start stretching the sheet, but don't take it from the table. The lines will get curved, indicating that the sheet itself is curved. But you did it all in two dimensions. Deformation of 2-dimensional sheet in 2 dimensions - that is an internal curvature.

"Direction" of an internal curvature is described by a Riemann tensor.

 

[cesiumfrog]

Let's make an experiment: get some thin rubber sheet (cut it from a baloon), draw some lines on it and put it on a flat table. Then start stretching the sheet, but don't take it from the table. The lines will get curved, indicating that the sheet itself is curved. But you did it all in two dimensions. Deformation of 2-dimensional sheet in 2 dimensions - that is an internal curvature.

I don't like that analogy, I think you're conflating the "extrinsic-curvature vs. intrinsic-curvature" distinction with "curvature vs. coordinate transformation". That is, you've altered the coordinate system over your table (and perhaps even changed the stress distribution) but it still has Euclidean geometry.

 

[Passionflower]

I don't like that analogy, I think you're conflating the "extrinsic-curvature vs. intrinsic-curvature" distinction with "curvature vs. coordinate transformation". That is, you've altered the coordinate system over your table (and perhaps even changed the stress distribution) but it still has Euclidean geometry.

I was thinking exactly the same thing when I read haael's posting.

 

[mjones32]

Thanks for the responses folks. I clearly was thinking of 'extrinsic' curvature - where can I learn more about the I guess intrinsic curvature , eveidently that's what I need to get my head round. Any good links ?

 

[mjones32]

Hmmm - so after looking further into the thiis, I think my original question remains valid.

If we take an examples of flatlanders living in 2 dimensions, plus time, if their space was subject to a curvature into a higher dimension, the flatlanders would be able to detect that parallel lines would converge or diverge and hence speculate on the existence of an N+1 dimension.

Now a being that existed in 3 dimensions plus time, could see the curvature of their space but also see that the curvature was convex or concave in a specific direction.

Am I being incredibly obtuse , but are all the answers telling me other things about curved spaces and not addressing this very specific point ?

 

[my_wan]

Yes, not only can the 'intrinsic' curvature differ, concave can become convex for another normal if accelerated observer.