General Relativity/Christoffel symbols

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Definition of Christoffel SymbolsEdit

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 DerivativesEdit

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

 

 

Analogously, for tensors:

 

Calculation of Christoffel SymbolsEdit

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.