General Relativity/Christoffel symbols

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Definition of Christoffel Symbols[edit]

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[edit]

From above, one can obtain the relations between covariant derivatives and regular derivatives:

Analogously, for tensors:

Calculation of Christoffel Symbols[edit]

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.