The two classical vacuum solutions of Einstein's field equations and the geometry they describe: the static spherically symmetric Schwarzschild(-Droste) metric obtained via Birkhoff's theorem, its presentation in several coordinate charts, the distinction between a coordinate singularity at the event horizon and the physical curvature singularity, Kruskal-Szekeres extension and Penrose diagrams, the charged Reissner-Nordstroem case, the interior solution with the Tolman-Oppenheimer-Volkoff equation, and the stationary axisymmetric Kerr metric reached through the Papapetrou line element and the Ernst equation, with its ring singularity, Cauchy and event horizons, ergosurfaces, ergoregion frame dragging, the Penrose process, black-hole thermodynamics, multipole moments and the Kerr-Newman and uniqueness results.
An invited review (96 pages, 17 figures; also published as Int. J. Mod. Phys. D 24 (2015) 1530006) that builds the two best-known exact solutions of Einstein's vacuum field equation up from Newtonian…