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How to calculate Christoffel symbols in polar coordinates?
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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.
Feb 22, 2016 · Christoffel symbol exercise: calculation in polar coordinates part I. Details. Category: General Relativity. Created: 22 February 2016. Last Updated: 01 May 2023. Hits: 57661. In this section, as an exercise, we will calculate the Christoffel symbols using polar coordinates for a two-dimensional Euclidean plan.
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
Oct 26, 2016 · Christoffel Symbols for Spherical Polar Coordinates. Ask Question. Asked 7 years, 6 months ago. Modified 2 years 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;
Nov 2, 2019 · Try it for polar coordinates in the plane, knowing those Christoffel symbols and also $\Gamma^\theta_{r\theta} = \Gamma^\theta_{\theta r} = 1/r$, all others $0$. (Amusingly, this is actually an exercise in my differential geometry text, linked in my profile.) $\endgroup$
3 Christoffel Symbols of Flat Space-TimeinSphericalCoordinates Say we have a Minkowski space-time with euclidean co-ordinates x =(t,x,y,z), which has metric, gab = 0 B B B B B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 C C C C C A (7) =)ds2 =gab dxa dxb =dt 2 dx2 dy 2 dz2 (8) Let’s now look in spherical coordinates x˙ = (t 0,ˆ, ,˚). We have the ...
