Maximal Analytic Extension of the Kerr Metric

TLDR

For low angular momentum (a² < m²), the maximal extension of Kerr consists of an infinite chain of Einstein‑Rosen bridges linked by successive horizons. The study generalizes Kruskal’s transformation to the Kerr metric to construct its maximal analytic extension and examines key properties such as equatorial geodesics. By extending Kruskal’s transformation to Kerr spacetime and analyzing equatorial geodesics, the authors build the maximal analytic extension. The extension can be reduced to four asymptotically flat sheets via suitable identifications, and geodesic completeness is shown for all trajectories that do not strike the ring singularity.

Abstract

Kruskal's transformation of the Schwarzschild metric is generalized to apply to the stationary, axially symmetric vacuum solution of Kerr, and is used to construct a maximal analytic extension of the latter. In the low angular momentum case, a2 &amp;lt; m2, this extension consists of an infinite sequence Einstein-Rosen bridges joined in time by successive pairs of horizons. The number of distinct asymptotically flat sheets in the extended space can be reduced to four by making suitable identifications. Several properties of the Kerr metric, including the behavior of geodesics lying in the equatorial plane, are examined in some detail. Completeness is demonstrated explicitly for a special class of geodesics, and inferred for all those that do not strike the ring singularity.