We close with some additional topics of interest both to general relativity, and to the program of “advanced classical mechanics”. First, because it is of the most immediate astrophysical interest, we discuss qualitatively the Kerr solution for the exterior of a spinning, spherically symmetric mass. The Kerr metric is stated, and its linearization compared to the closest electromagnetic analogue: a spinning sphere of charge. Using this comparison, we can interpret the two parameters found in the Kerr metric (when written in Boyer–Lindquist coordinates) as the mass and angular momentum (per unit mass) of the source.

The Kerr metric is nontrivial to derive using our Weyl method, so we are content to verify that it is a solution to Einstein's equation in vacuum. Some of the physical implications of the Kerr solution are available in linearized form, but the more interesting and exotic particle motions associated with the geodesics are easier to explore numerically (see for an exhaustive analytical treatment). For this reason, we include a brief discussion of numerical solutions to the geodesic ODEs that arise in the context of the Kerr space-time. In some ways, a hands-on approach to these trajectories can provide physical insight, or at the very least, predictions of observations that are interesting and accessible.

Finally, given the work we have done on variational methods and geometry, we are in good position to understand extremization in the context of physical area minimization.