Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 8; Problem 8.7.
This is an example of the geodesic equation in a 2-d curved space (with two space dimensions). We have a parabolic bowl with equation
where and is a constant. We use the two coordinates (as defined here) and the azimuthal angle . We’ve already looked at this example and found that the metric is
Using the geodesic equation for the two coordinates, we get
The second equation can be integrated once to give
where is a constant. Substituting this into the first equation and working out the first term, we get
Integrating these equations directly isn’t easy, but if we look at the special case of a curve defined by the condition for some constant (that is, a curve starting at the bottom of the bowl and going directly up the side of the bowl), then and from the metric:
Inserting this into the above, we get
This will be equal to zero if , which also satisfies the condition . Thus these radial curves are in fact geodesics.