A useful visual aid used in GR is the Flamm paraboloid.
http://en.wikipedia.org/wiki/File:Flamm.jpg
A triangle projected onto to the surface of the paraboloid, shows all those nasty non-Euclidean effects such as angles not adding up to 180 degrees.
Quote:
Originally Posted by NotPrinceHamlet
Say that I'm an ant that lives on this curved sphere.
I walk up to the vertex of the triangle at the north pole and I measure it - I find it to be 90 degrees - right?
I walk to the other two and I measure 90 degrees. Totalling 180 degrees.
Would I perceive that I was upsidedown at the south pole as compared to the north pole? I don't think so, as the space itself is curved, but if I was on a sphere in euclidian space, then I would perceive myself as upsidedown on the south pole.
So according to my ant brain, this triangle is flat, but with 270 degrees in it.
Does this sound right?
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Yes that's true if the ant and sphere are embedded in 3-D space, it will perceive an "upside down".
If on the other hand the ant is a 2-D organism on the surface of the sphere it is constrained in 2-D curved space. The triangle will be flat as the ant won't perceive the curvature of the sphere. The angles however will not total 180 degrees.
If you master tensor analysis you'll find the maths used in GR quite straightforward although the calculations can be rather laborious.
Regards
Steven