[math-fun] "space time vortex around earth"

The "Kerr metric" is the GR metric generated by a spinning mass (black hole, actually; for a nontrivial source like the Earth or Sun exact solution is not known). Gravity probe B attempts to measure this extremely precisely near earth, and the fact it agrees with the predictions of GR is confirmation of GR in a new and gratifying way. Also, it was extremely brilliant engineering to be able to build this experiment (and actually, it failed, but was rescued also in brilliant way). Open problem: devise a reasonably realistic model of a mass, such that outside it you get Kerr metic, and inside it you get something else (which both are to be exact solutions of GR please).