Finding the Kerr metric of a rotating black hole

The formulation of General Relativity by Albert Einstein in 1915 was one of the greatest advances of modern physics. It describes the dependence of the structure of space-time on the distribution of matter, and the converse effect of this space-time structure on matter distribution. Despite the overwhelming clarity of its foundation and the elegance of its basic equations, it has proved to be very difficult to find exact analytical solutions of the Einstein equations. Moreover, of all the exact solutions which are known, only a limited class seem to have a real physical meaning. Among them are the famous solutions of Schwarzschild and Kerr for black holes, and the Friedman solution for cosmology. Although the "simple" solution for a static, spherically symmetric black hole (static vacuum solution with spherical symmetry and central singularity) was found by Schwarzschild shortly after Einstein's publication of his equations, nearly 48 years were to elapse before Kerr discovered the vacuum solution for the stationary axisymmetric rotating black hole. In the paper below, a heuristic way of finding the Kerr solution is presented, revealing its simplicity and elegance by using an alternative presentation of its metric. Technically, the calculations are done in the language of differential forms. All algebraic calcultions were done with the Mathematica package Difform.m.

The paper "A heuristic way of finding the metric of a rotating black hole", Am. J. Phys. 65, 1997, 897-902.

The corresponding Mathematica 3.0 notebook kerr.nb

The Mathematica 2.2.4 package Difform.m