Curvature and Riemann Tensor
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As Eddington nicely explained, Einstein's field equations can be constructed from components of Riemann curvature tensor. In a nutshell, the tensor G Eddington mentioned (sometimes called "Einstein tensor") is a sort of average of the Riemann curvature over all directions. Thus Riemann curvature is the basic notion for expressing gravitational fields; and although the expression of Riemann curvature tensor is different depending on our choice of a coordinate system, this curvature is an invariant quantity. A sphere has a definite (positive) curvature, and it is the same whatever coordinate system you may choose, and likewise an Euclidean plane is flat (zero curvature), independent of any coordinate system. Thus although metric is different in different coordinate systems, the curvature characterized in terms of metric is an invariant quantity.
Here is a brief illustration of this important notion of Riemann curvature, without getting into the mathematical definition of tensor (which may be quite tedious for ordinary people, including philosophers). First let us see a familiar example of two-dimensional curved surface, a sphere. (Any two geodesics on the sphere cross. Why? Because of its curvature. This is quite analogous to "attracting force". See Curvature within the Earth.)
For any point on a two-dimensional surface, there are two circles that fit best at that point; if these two circles are on the opposite side of the surface, the curvature is negative.
So far, we defined curvature using circles outside of the surface itself. But this is not essential; there is an intrinsic method for defining curvature.
Last modified, July 23, 2002. (c) Soshichi Uchii
suchii@bun.kyoto-u.ac.jp
