Einstein Relatively Easy

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 In this section, as an exercise, we will calculate the Christoffel symbols using polar coordinates for a two-dimensional Euclidean plan.


and given the fact that, as stated in Geodesic equation and Christoffel symbols

we are then ready to calculate the Christoffel symbols in polar coordinates. As we know from the definition of Christoffel Symbol or Connection coefficient, in 2 dimensional space, we have to find 2x2x2 = 8 connection coefficients, and only 6 distinct values because of the symmetry on the lower indices.

The eight Christoffel symbols to find are summarized in the two matrix below, with the symbols being symmetric on the lower index (meaning that the connection coefficients that are linked below by the blue arrow are equal).

Let's start by populating the four values of the first matrix with r as upper indice:



So finally we get the first matrix equal to:

Let's calculate now the four following coefficients, all with θ as upper indice:



The calculation for the last coefficient gives:

Finally the last four Christoffel symbols can be summarized as follow:





"The essence of my theory is precisely that no independent properties are attributed to space on its own. It can be put jokingly this way. If I allow all things to vanish from the world, then following Newton, the Galilean inertial space remains; following my interpretation, however, nothing remains.."
Letter from A.Einstein to Karl Schwarzschild - Berlin, 9 January 1916

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the 'old one'. I, at any rate, am convinced that He is not playing at dice."
Einstein to Max Born, letter 52, 4th december 1926

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