General Relativity/Christoffel symbols
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Definition of Christoffel Symbols
Consider an arbitrary contravariant vector field defined all over a Lorentzian manifold, and take at , and at a neighbouring point, the vector is at .
Next parallel transport from to , and suppose the change in the vector is . Define:
The components of must have a linear dependence on the components of . Define Christoffel symbols :
Note that these Christoffel symbols are:
- dependent on the coordinate system (hence they are NOT tensors)
- functions of the coordinates
Now consider arbitrary contravariant and covariant vectors and respectively. Since is a scalar, , one arrives at:
Connection Between Covariant And Regular Derivatives
From above, one can obtain the relations between covariant derivatives and regular derivatives:
Analogously, for tensors:
Calculation of Christoffel Symbols
From , one can conclude that .
However, since is a tensor, its covariant derivative can be expressed in terms of regular partial derivatives and Christoffel symbols:
Rewriting the expression above, and then performing permutation on i, k and l:
Adding up the three expressions above, one arrives at (using the notation ):
Multiplying both sides by :
Hence if the metric is known, the Christoffel symbols can be calculated.