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- The Christoffel symbols can be computed from the derivative of the metric. But the flat metric is all constants - 1's and 0's. So taking these derivatives make the Christoffel symbols all identically 0.
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Generally, whenever a Christoffel symbol Γ k ij is zero, it means that the i:th basis vector remains constant in the k:th direction with respect to a change in the j:th coordinate.
A corollary to this property is that it is impossible to find a coordinate system in which all elements of Christoffel symbol are zero at a point, unless lower indices are symmetric. This property was pointed out by Albert Einstein and Erwin Schrödinger independently.
The Christoffel symbols can be computed from the derivative of the metric. But the flat metric is all constants - 1's and 0's. So taking these derivatives make the Christoffel symbols all identically 0.
Oct 22, 2022 · So if you've picked Cartesian coordinates, the christoffel symbol is zero; every Poincare transformation keeps it zero. If you've picked spherical coordinates, the christoffel symbol is non-zero, and stays that way under Poincare transformation. [3]
From what we've said so far, it's clear that for the Christoffel symbols to be zero identically, the basis vectors must not change as we go from point to point. This means that we won't introduce any spurious changes to our vector fields by not accounting for change of basis.
Oct 16, 2017 · No. General Relativity makes no difference between gravitational and inertial ('fictitious') forces, and in accelerated frames of references, the Christoffel symbols will be non-zero even in spacetimes we would think of as free of gravity.
We can easily see that it reproduces the usual notion of straight lines if the connection coefficients are the Christoffel symbols in Euclidean space; in that case we can choose Cartesian coordinates in which = 0, and the geodesic equation is just d 2 x /d = 0, which is the equation for a straight line.
