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:
Thus we can write
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:
so we obtain:
Now using , and we obtain:
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 :
Since and are tensors, the term in the parenthesis is a tensor with components: