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:

We can now use the symmetry of the Christoffel symbols to solve for by swapping indices in 7 and 8 to get

We can now add 6 to 10 and subtract 9 to get

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:

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gwrowePost authorPingback: Ricci tensor for a spherically symmetric metric: the worksheet | Physics Pages

gwrowePost authorPingback: Schwarzschild metric: the Newtonian limit & Christoffel symbol worksheet | Physics Pages

Collin WittCan you please clarify step (3). What happened from step to (2) to (3). Looks like a derivative? The funny part is I understand the rest from that point.

gwrowePost authorIt’s just the product rule:

Collin WittYea, I figured out that it’s the derivative on the first term of the RHS side and then if you subtract you get what is above in step (2). Thanks for clarifying, it was not immediately clear.