Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 17; Box 17.6, Problem P17.2.

We’ve seen that the Christoffel symbols in terms of the metric are given by

This expression can be cumbersome to work with, since it involves calculating the inverse metric tensor and doing a lot of sums to find each Christoffel symbol. Often an easier way is to exploit the relation between the Christoffel symbols and the geodesic equation.

The geodesic equation is (where a dot above a symbol means the derivative with respect to ):

The following equation is formally equivalent to this:

The method for calculating the Christoffel symbols is to work out the terms in 2, divide through by and then compare the result term by term with 3. By doing this we are able to read off the as the coefficients of in 2.

**Example** * We can use this technique to work out the for the Schwarzschild (S) metric, which is *

First, take in 2. Since the S metric doesn’t depend on for all elements. Further, since the S metric is diagonal, is restricted to , so the equation becomes

Since there are 2 non-zero derivatives, so this equation expands to

By comparing this with 3 we can read off the symbols:

Here, we’ve used the symmetry of the Christoffel symbols. Because no other terms appear in the equation, all the other are zero, so the complete set is, where the rows are labelled , , and from top to bottom, and the columns the same order from left to right:

Now consider . This time, one of the does depend on so we will get a contribution from the term. We get

Again, we can read off the symbols to get

For things get a bit messier since all four terms depend on . We get

Comparing terms, we get

Finally, for the metric is again independent of so the situation is a lot simpler:

The symbols are

### Like this:

Like Loading...

*Related*

Pingback: Maxwell’s equations in cylindrical coordinates | Physics pages

Pingback: Covariant derivative in semi-log coordinates | Physics pages

Pingback: Christoffel symbols in sinusoidal coordinates | Physics pages

Pingback: Schwarzschild metric: acceleration | Physics pages

Pingback: Covariant derivative of a vector in the Schwarzschild metric | Physics pages

Pingback: Riemann tensor for an infinite plane of mass | Physics pages

Pingback: Riemann tensor in 2-d polar coordinates | Physics pages

Pingback: Riemann tensor in 2-d curved space | Physics pages

Pingback: Riemann tensor in the Schwarzschild metric | Physics pages

Pingback: Riemann tensor is 2-d flat space | Physics pages

Pingback: Riemann tensor in a 2-d curved space | Physics pages

Pingback: Ricci tensor and curvature scalar for a sphere | Physics pages

Pingback: Einstein equation for an exponential metric | Physics pages

Pingback: Christoffel symbols for a general diagonal metric | Physics pages

Avishek DusoyeI find this technique pretty cool. At the same time, i found a typo, in equation (17), the coefficient of r_dot|r_dot should be of power of -1, instead of -2, as we multiply equation (16) by (1-2GM/(r^2)) throughout on both sides. Cheers take care.

gwrowePost authorFixed now. Thanks.

TobiasGreat + elegant method! In equation 8 and 10, should the phi dot r dot and the theta dot r dot be gone?

gwrowePost authorFixed now. Thanks.

EnriqueThis method is very interesting. By the way, in equations 10 and 11 there’s a Christoffel symbol that has r, theta, I think it should be theta,phi instead. Cheers.

gwrowePost authorFixed now. Thanks.