Complete Analytic Extension of the Symmetry Axis of Kerr's Solution of Einstein's Equations

Abstract

The 2-dimensional metric on the symmetry axis of the Kerr solution is examined and it is shown that in the form usually given it is incomplete when ${a}^{2}<~{m}^{2}$. The method developed by Kruskal for completing the Schwarzschild solution is adapted to the distinct cases ${a}^{2}<{m}^{2}$ and ${a}^{2}={m}^{2}$. In each case a singularity-free metric is obtained which is periodic with respect to a timelike coordinate, and which is shown to be a complete analytic extension. The generalization to the full 4-dimensional Kerr solution is discussed, and finally the questions of uniqueness and causality are considered.