Understanding full blown differential geometry is not something one does overnight. If you've got a good background in linear algebra and vector calculus, and differential equations and some other stuff I can't think of right now
, you'll be good to go. If not, you need to learn the prerequisites or it will be complete greek.
This is what is called intrinsic curvature. It's about describing a curved space without resorting to any higher spaces. Indeed in this picture, a flat space is just one with zero curvature.
But using the embedding picture can help you get started at seeing how it works. Then you take the step of realizing you don't need the higher space and all you need is a certain description of "the curvature".
Consider the 2D surface of a sphere in 3D dimensions, or say some infinite surface, a hyperbolic one or anything you can imagine, say some sinusoidal waving thing that extends to infinity.
Looking at it that way lets you see how various mathematical objects work. But these curved 2D surfaces have invariant properties that are indepedent of the higher space. You don't need it.
The notion of curvature is defined as how "straight" lines deviate from straight.
IOW, if we have two parallel lines in a local neigborhood, do they converge or diverge if extended globally. In flat space, parallel lines never interesect. If the curvature is positive, such as on the sphere, those lines will converge. If the curvature is negative, those lines will diverge, get farther apart with length.
Now, the curvature at point is describing, in a differential sense, how all possible parallel lines through a point are deviating from parallel with respect to each other. Mathematically, that requires a tensor in the general case. In a plane, with simple curves contained there, the curvature can be described by a single scalar number. But in the general case of arbitrary dimensions, it takes a tensor. Note above, I spoke of positive and negative curvature. In simple cases, that tensor can "collapse" into something that can be spoken of as a scalar, but not always.
Use the embedding space as a launching point in these simpler cases, you can see how this curvature thing can be defined in terms of things that work in the familiar flat Euclidean ways you are familiar with.
Then you realize you don't need the higher embedding space -- that curvature tensor thing can tell you all you need to know in terms of coordinates defined on the curved space itself. Know that, and you know the space without resort to a higher embedding space.
Now, the Einstein FIeld Equations, the governing equations of General Relativity relate the curvature of space-time to the mass-energy-momentum content of that space-time.
-Richard
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Originally Posted by Arcane
