Two manifold points have shortest path {geodesic, tensor}| between them. Geodesic is straightest possible direction between two points.
metric
Quadratic differential linear metric forms can measure geodesic length: ds^2. Geodesic length is sum from points i = 1 to i = m, and from points j = 1 to j = m, of g(i, j) * du(i) * du(j). Coefficients g(i, j) = Du(i) / Du(j), where D are partial derivatives and u are coordinates.
geometry
Geodesic metric defines surface geometry at manifold points.
linear
Using only local operations allows geodesic to be linear.
operator
Geodesic metric operates on vectors to give squared lengths. Squared length can be greater than zero {space-like vector}, less than zero {time-like vector}, or equal to zero {light-like vector}.
space-time
In four-dimensional space-time, particles move along maximum spatial-length lines, which is the shortest possible time as measured in particle reference frame. In flat space-time, geodesics are straight lines. On spheres, geodesics are on great circles.
Mathematical Sciences>Calculus>Vector>Tensor>Kinds
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Date Modified: 2022.0224