Curvature

> negative curvature
What the hell is that ? Curvature is curvature. There are concave and convex forms, is that what you mean ?

>>7953392
The angles of a triangle on a sphere is the same no matter if you are looking outside than inside.

>>7953395
of course, I hadn't thought of that.
>>7953394
"positive" and "negative" are the terms that are used the most in terms of the shape of the universe. "negative" curvature is usually describes as a saddle shape

just found an article which is actually helpful in imagining it
http://www.geom.uiuc.edu/docs/education/institute91/handouts/node21.html

>>7953394
The typical example of negative curvature is the hyperbolic space (think "small triangles") whereas the typical example of positive curvature is the sphere ("fat triangles")

U wot m8? The universe is flat until proven otherwise. Too many variables at work to state otherwise.

>>7953392
>Is negative curvature actually concave in one axis and convex in another like a saddle?
yes, for a 2D chape you get the curvature by multiplying the curvature along 2 different axis, so negative m,eans they curve in 2 different directions.

File: cone_curvature.png (10KB, 614x309px) Image search: [Google]

10KB, 614x309px

>>7953394
>>7953403
I think the cone example works better, to explain positive and negative anyway.

Take a cone,draw a line from the point of the cone to the base. Put an equitorial circle anywhere on the cone that goes around the tip and through this line. Now take a vector, and move it around the circle.

If you cut along the line you've made, and flatten the cone, you will see that the vector has changed angle. The difference between the original angle and the new angle is, essentially, the curvature.

For a sphere, you end up with a larger angle, you must add, so, positive curvature. For a cone, the new angle will be smaller, you must subtract, hence negative curvature.

>>7953710
Cones are positive curvature.
The negative curvature equivalent of a cone would be to cut a piece of paper from a point to the edge and tape in a triangular piece.

File: cone_curvature.png (8KB, 614x309px) Image search: [Google]

8KB, 614x309px

>>7953710
Corrected version of your arrows.

Intuitively (and this can be made rigorous) :

Positive curvature is where nearby parallel lines (geodesics) curve towards each other, and negative curvature is when they curve away.

Naturally in zero curvature space (Euclidean) parallel lines stay at the same distance from each other.

>>7954797
Cont..

A cone is actually *zero* gaussian curvature.

It is (except at its point) locally isometric to flat space. (since you can bend paper into it).

Gauss' Theorema Egregimum (or however you spell it) says then that it must have the same curvature as the paper.

A way to see this, as in the above post:
Two concentric circles about the center stay the same distance apart all of the way around.

>>7954814
The curvature is zero everywhere but the tip of the cone. At that point, the curvature is a delta distribution and positive.

>Two concentric circles about the center stay the same distance apart all of the way around.
This is also true for a sphere.

>>7954833
>>Two concentric circles about the center stay the same distance apart all of the way around.
>This is also true for a sphere.
That said, it is correct that there's no curvature between the two circles -- but that argument doesn't show it. You'd have to compare the curvature of the two circles.

>>7954845
Or better, in the spirit of >>7954797, draw two geodesics. Circles about the point of a cone are not geodesics.

Spacetime curves in an extra dimension. #dealwithit

>>7955287
That works fine for two dimensional surfaces and can be a great visualization tool, but you'd need a rather large number of extra dimensions to reproduce 4D general relativity. Most people just think of it as replacing the parallel postulate.

>>7955295

Spacetime warps in relation to another dimension. Travel through curved space and you move through XYZT+1 dimension.

>>7955313
I literally just told you why that's wrong.

>>7955313
To elaborate, the Riemann tensor has 20 independent components; the extrinsic curvature of a 4D surface in one extra dimension only has 10. So you'd need more than one extra dimension to even reproduce GR at one point.

>>7955340

>jargonese

>>7955357
Calling "Riemann tensor" jargonese in a general relativity discussion is literally like going into a discussion of Newtonian mechanics and calling words like "force" or "acceleration" jargonese.

>>7955340
I thought it was only 16 independent in GR

>>7955313
>what is gauss's theorema egregium
>what is intrinsic geometry
>what do you mean curvature isnt a bowling ball in a trampoline

>>7953392

There are two seemingly distinct ways to describe a notion of curvature for a manifold (if the word bothers you, replace it with surface):

a) The way the manifold bends in itself (Gaussian curvature)
b) The way the manifold bends as seen from a higher dimension space (Curvature relating to the Shape operator: a mapping of the unit normal vector to the manifold to the corresponding unit vector on a sphere)

Curvature as stated in a) is a property intrinsic (i.e. depends only on the coefficients of the first fundamental form, a fact that has already been mentioned in the form of the Theorema Egregium), while curvature in terms of b) is an extrinsic property, meaning that a surrounding space comes into play.

There are three model spaces for curvature:
a) Zero Gaussian curvature
b) +1 Gaussian curvature
c) -1 Gaussian curvature

So, in the case c) (the saddle surface) when you speak of convexity, you are attributing properties of the surrounding space (therefore extrinsic) to a notion that is intrinsic (the Gaussian curvature).

To get negative curvature into intrinsic perspective, consider the Lobatsevsky model: a closed disk on the plane, minus its center,
equipped with a metric that renders the distance of any point from the center equal to infinity. Then, near the circumference, the space is almost flat, and Euclidean notions of Geometry apply. But as we approach the center, distances between points become very large, and so 'straight lines' are curves that give negatively curved triangles their distinct shape.

>>7955313
stop talking about things you know nothing about

>>7955527
<3 hyperbolic geometry. It's what makes the sky work.