9. Christoffel Symbol

Now that we have some idea on the concept of metric tensor, we now move on to another very important concept in the curvilinear system (the Christoffel symbol). Let us consider the curved coordinate system $(u_{1}, u_{2}, u_{3}…. u_{j})$ and their respective direction tangent vector be represented by $\vec{e_{i}}$. We must remember that the direction vector (basis vector) is not a constant, i.e. it is position dependent quantities. But, what if we differentiate the basis vector with respect to the coordinate axes? What can we assume about the answer? Well, we can still say that the differentiated quantity is still a vector of the same basis vector, but what about the coefficient of the vector. The coefficient is what we call as the Christoffel Symbol.

I.e. mathematically we can write as follows,

This then is the Christoffel symbol. In the above definition we can calculate the coefficient by comparing a coefficient to the left hand side of the equation. However, this method in general is not the method used to calculate the Christoffel symbol. In general we find the Christoffel symbol from the metric of the given coordinate system.

We now derive the relation between the metric tensor and the Christoffel symbol. We know the relation of the metric tensor and the unit tangent vector $e_{1}, e_{2}, e_{3}….$ is as follows,

We now differentiate ${g_{pq}}$ with respect to the coordinate axes; we then have the following,

                                                     

                    

                                                                                                       $(1)$

 This is the relation of the Christoffel to the metric tensor. Notice that we can write the Christoffel solely in terms of the metric tensor. If we cycle through the indices ${{p,q,k}}$ we then have different relation for the metric tensor and the Christoffel symbol. In particular two of the following,                          

                                                                                                     $(2)$

                                                                                                    $(3)$

Now adding $(2),(3)$ and subtracting $(1)$ we have,                              

Where, $\partial_{p}$ implies a partial derivative with respect to $p^{th}$ coordinate axes. And also contracting the metric in the left side of the equation we then have the Christoffel symbol as generally seen in many text books as,                                 

We have thus found the relationship between the Christoffel symbol and the metric tensor. 

Some notes can be found here..Warwick,Gatech, I hope this helps..