COMPLETE SET OF ANALYTIC SOLUTIONS FOR THE GEODESIC EQUATION IN PLEBAŃSKI–DEMIAŃSKI SPACE-TIMES

It has been shown that the Hamilton-Jacobi equation corresponding to the geodesic equation in a Petrov type D space-time is separable and, thus, integrable. All Petrov type D space-times are exhausted by the Plebański-Demiański electrovac solutions with vanishing acceleration of the gravitating source. Here we present the analytical integration of the geodesic equations in these space-times. Based on the general solution we discuss the special cases of geodesics in Taub-NUT and Kerr-de Sitter space-times. We define observables and also address the issue of geodesic incompleteness.