Understanding Positive and Negative Intrinsic Curvature in General Relativity

[WannabeNewton]

I'm not resorting to higher dimensions. If you punch the trampoline and hold the punch, such that you deform the flat surface of the trampoline locally into a hemisphere, then you have induced curvature in that local region (a sphere has intrinsic curvature). Then if you slowly retract your hand, the local region will slowly go back to its original flat form.

We can describe asymptotic flatness of space-time mathematically but you can't easily picture it. This is not a problem as the math takes care of everything. The curvature tensors used in GR are all intrinsic measures of curvature; it's built into the theory.

 

[A.T.]

you are quick to say it "flattens", but what does that even mean unless you are visualizing spacetime as being embedded in a higher dimension?

Try this picture:

From: http://scienceblogs.com/startswithabang/2011/10/04/discover-the-fate-of-the-unive/

The triangles are created by connecting three points of the surface, by shortest possible paths within the surface.

- Within the plane (zero intrinsic curvature) the angle sum of the triangle is 180°.
- Within the surface of positive intrinsic curvature the angle sum of the triangle greater than 180°.
- Within the surface of negative intrinsic curvature the angle sum of the triangle less than 180°.

Note that these are entrielly intrisic measures, that can be performed by 2D creatures living within the 2D Surfaces, without any reference to the 3rd dimension, or embedding within a flat 3D space. Flattening intrinsically means that the triangle angle sum goes towards 180°.

 

[ktx49]

^nice, an excellent way of looking at things without resorting to using a "direction" or a higher dimension...and excuse my ignorance, but GR deals with positive intrinsic curvature, right?

so flattening does somewhat imply a direction to the curvature...or at least an orientation of the observer?

for example if we look at the images posted by A.T. above, we can easily imagine that 2D creatures are not necessarily limited to being on the upward-facing(visible side)...they could be living on the surface inside that sphere...or the underside of the hyperbolic saddle.

i feel like positive and negative intrinsic curvature are 2 sides of the same coin...ie. the same relationship as up/down or left/right. where you have one you inherently should have the other...even if its not observable.

 

[Nugatory]

for example if we look at the images posted by AT above, we could imagine that 2D creatures are not necessarily limited to being on the upward-facing visible side...they could be living on the surface inside that sphere.

i feel like positive and negative intrinsic curvature are 2 sides of the same coin...ie. the same relationship as up/down or left/right. where you have one you inherently should have the other...even if its not observable.

Not necessarily. You could imagine the surface of the sphere to be a very thin sandwich of two very thin stretchable sheets... With our two-dimensional flatlanders constrained to live between the sheets. That's actually closer to the mathematical concept of a two dimensional manifold than the single two-sided surface you're imagining.

However, there comes a point when the analogies won't tell us any more and we have to start working with the mathematical formalism - and we're getting pretty close to that point.

 

[WannabeNewton]

so flattening does somewhat imply a direction to the curvature...or at least an orientation of the observer?

No. Intrinsic curvature is independent of any orientation. It is extrinsic curvature which depends on the orientation.

I must agree with Nugatory that there comes a point when you just need the math to tell right from wrong. Classical pictures can only help you for so long (speaking from experience).

 

[A.T.]

and excuse my ignorance, but GR deals with positive intrinsic curvature, right?

It deals with both. Within a mass you have positive, in the vacuum nearby negative spacetime curvature.

so flattening does somewhat imply a direction to the curvature...or at least an orientation of the observer?

No.

for example if we look at the images posted by A.T. above, we can easily imagine that 2D creatures are not necessarily limited to being on the upward-facing(visible side)...they could be living on the surface inside that sphere...or the underside of the hyperbolic saddle.

No. They live within the 2D surface, not on either side of it. The 2D surfaces represents one 2D layer, not two layers/sides.

i feel like positive and negative intrinsic curvature are 2 sides of the same coin

No. The side is irrelevant. Even if the creatures would live on both sides, they would still measure the same intrinsic curvature on either side.