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A nice visual way to see how these Christoffel symbols can be interpreted is by considering the Christoffel symbols in polar coordinates. In a polar coordinate system, there are two coordinate-axes, r and θ (r being the “radial” axis and θ the “angular” axis) and every point can be labeled by an r-coordinate and a θ-coordinate.
The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the name Christoffel symbols is reserved only for coordinate (i.e., holonomic) frames. However, the connection coefficients can also be defined in an arbitrary (i.e., nonholonomic) basis of tangent vectors u i by
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Oct 26, 2016 · Christoffel Symbols for Spherical Polar Coordinates. Ask Question. Asked 7 years, 7 months ago. Modified 2 years, 1 month ago. Viewed 34k times. 13. If we are given a line element; ds2 = dr2 +r2dθ2 +r2sin2θdφ2 d s 2 = d r 2 + r 2 d θ 2 + r 2 s i n 2 θ d φ 2. We can easily then see that the metric and the inverse metric are;
This gives us a formula for explicitly evaluating Christoffel symbols: Gm ij= 1 2 gml @ jg il+@ ig lj @ lg ji (17) This is a bit cumbersome to use as it requires finding the inverse metric tensor gmland has 3 sums over different derivatives. As an example, we’ll work out Gm ij for 2-D polar coordinates. The metric tensor and its inverse here ...
Acceleration in non-inertial coordinates requires consideration of non-vanishing Christoffel symbols, which describe the tangential transport and occur in the geodesic equation when computing the shortest or longest path between two points. These Christoffel symbols are similar to tensors of rank three, yet they are not tensorial objects itself.
flat space-time metric in euclidean coordinates to the spherical coordinate system. Note here that using the covariant metric, that is with indices on the bottom, then we must have the euclidean coordinates in terms of the spherical coordinates. Applying gmn =Aa m A b n gab we get, gmn =Aa m A b n gab (24) = 0 B B B B B @ 1 0 0 0 0 cos sin ...
