Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 17; Box 17.4.
It’s time to find out how to calculate the Christoffel symbols. We start with their definition in terms of the basis vectors in some coordinate system:
We take the scalar product of this equation with another basis vector and use the definition of the metric tensor as :
In this equation the index is a dummy (being summed over), so only the indices , and are specified. We can cyclically permute these indices to generate two more equations:
Finally we can use the fact that
and multiply both sides of 11 by to get
This gives us a formula for explicitly evaluating Christoffel symbols:
This is a bit cumbersome to use as it requires finding the inverse metric tensor and has 3 sums over different derivatives.
Example As an example, we’ll work out for 2-D polar coordinates. The metric tensor and its inverse here are:
so the derivatives are
Having only one non-zero derivative helps a lot, since the only non-zero term on the RHS of 16 is . We’ll work out a couple of the symbols explicitly and then give the final result.
All of the other symbols are zero. The final results are
Using these in 1 gives the 4 derivatives of the basis vectors: