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Tensors and Relativity: Chapter 6

# Covariant derivatives and Christoffel symbols

In Minkowski spacetime with Minkowski coordinates (ct,x,y,z) the derivative of a vector is just since the basis vectors do not vary. In a general spacetime with arbitrary coordinates, with vary from point to point so Since is itself a vector for a given it can be written as a linear combination of the bases vectors: The 's are called Christoffel symbols [ or the metric connection  ]. Thus we have: so Thus we can write where Let us now prove that are the components of a 1/1 tensor. Remember in section 3.5 we found that was only a tensor under Poincaré transformations in Minkowski space with Minkowski coordinates. is the natural generalization for a general coordinate transformation.

Writing , we have: Now therefore so we obtain: Now using , and we obtain: so We have shown that are indeed the components of a 1/1 tensor. We write this tensor as It is called the covariant derivative  of . Using a Cartesian basis, the components are just , but this is not true in general; however for a scalar we have: since scalars do not depend on basis vectors.

Writing , we can find the transformation law for the components of the Christoffel symbols . This is just We can calculate the covariant derivative of a one- form by using the fact that is a scalar for any vector : We have Since and are tensors, the term in the parenthesis is a tensor with components: We can extend this argument to show that 