The Kerr solution: A geometrical construction

Abstract

We derive an expression for the metric tensorη ij of Minkowski space-time in terms of a co-ordinate system adapted to a timelike world-tube and a shear-free, twisting congruence of null geodesicsk i emanating into the future from every event on the world-tube. By requiring the tube to represent the history of a rigid, unaccelerated, nonrotating, axially symmetric ellipsoid (in a technical sense) the null congruence is found to coincide with the null congruence of the Kerr solution which is tangent to the multiple Debever-Penrose direction of that solution. If one takes the Minkowski metric tensor as a «background» for a Kerr-Schild metric,η ij +2H k i k j , the Kerr solution is readily obtained by solving Einstein's vacuum field equations for the only remaining unknown—the scalar «potential»H. We prove that if the world-tube is static (unaccelerated and rigid) and has spatially compact normal sections, which are not necessarily axially symmetric, then the background geometry is flatif and only if the normal sections are axially symmetric ellipsoids. In so far as our work emphasizes the role of the properties of the basic world-tube in our construction, it may be considered as complementing the recent work of Newman and Winicour and of Vaidya on the same topic.