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Talk:General theory of relativity (view source)
Revision as of 02:32, November 15, 2009
, 02:32, November 15, 2009→Idea for presenting this
::::: HA! Yeah, I'd say we can do better than that for [[curvature]]. ;-) If it were me writing it (and I'm glad it's not; thank you) from the perspective of a physicist, the point I'd emphasize is that it is ''not'' possible to put a consistent set of Cartesian coordinates onto a curved surface. If the surface is curved, then it cannot be flattened by a coordinate transform, period. So you ''have'' to deal with the curvature. That's the point I emphasize when I do "differential geometry for dummies." But of course, your article won't be for budding physicists, it'll be for budding mathematicians or anybody who's interested. So I'll support whatever choice you make. --[[User:KSorenson|KSorenson]] 20:46, 14 November 2009 (EST)
::::: HA! Yeah, I'd say we can do better than that for [[curvature]]. ;-) If it were me writing it (and I'm glad it's not; thank you) from the perspective of a physicist, the point I'd emphasize is that it is ''not'' possible to put a consistent set of Cartesian coordinates onto a curved surface. If the surface is curved, then it cannot be flattened by a coordinate transform, period. So you ''have'' to deal with the curvature. That's the point I emphasize when I do "differential geometry for dummies." But of course, your article won't be for budding physicists, it'll be for budding mathematicians or anybody who's interested. So I'll support whatever choice you make. --[[User:KSorenson|KSorenson]] 20:46, 14 November 2009 (EST)
(Unindent) My take on this, and what I'm thinking of doing for my work on this article (feel free to steer me in a different direction if you disagree) is:
*There are flat coordinate systems. (''We'' know this as Christoffel symbols = 0, but that's beyond our scope for the article.) When dealing with spacetime, if you are in a flat coordinate system, you don't feel any "fictitious forces" (accelerations). Your system is intertial.
*There are curved coordinate systems. (Nonzero Christoffel symbols, geodesics are hairy.) When dealing with spacetime, you feel fictitious forces.
*There are flat manifolds (Riemann=0.)
*There are curved manifolds (Riemann !=0.)
*A flat manifold may have flat or curved coordinate systems. In physics, we could be in a rotating or accelerating frame of reference. But we can always find a flat coordinate system. That is, the curvature of the coordinate system can be "transformed away" (for example, by stepping off the amusement park ride.)
*A curved manifold has '''NO''' flat coordinate systems. At least not globally. You can't step outside.
Then we say that gravity is really just a fictitious force (equivalence principle), and it arises because the spacetime manifold itself is curved. It can't be globally transformed away.
Obviously, I come at this from a physicist's perspective (though my major was in math.) It sounds as though Mark is planning the mathematicians' approach for the curvature article. The two articles will make interesting, and excellent, reading.
[[User:PatrickD|PatrickD]] 21:32, 14 November 2009 (EST)