A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. II. Separable solutions

Abstract

For pt.I see ibid., vol.11, no.12 (1978). The techniques of the preceding paper are applied to several cases where the gamma equation may be solved by separation of variables in the form, gamma = gamma 1( rho )+ gamma 2( tau ), where gamma 1( rho ) is either zero or a very simple function and gamma 2( tau ) satisfies an ordinary differential equation of the fourth order. Among the exact solutions constructed are the full six-parameter family of generalised Tomimatsu-Sato solutions, the rotating Curzon solution, the Kinnersley-Kelley solution and a class of solutions recently found by Ernst. Two new classes of solutions are presented as well as several new particular solutions expressible in closed form. All stationary axisymmetric vacuum metrics with a non-trivial second-rank Killing tensor whose components do not depend on the ignorable co-ordinates, phi and t, are derived. This problem reduces to finding separable solutions of the dual of the gamma equation of the form, e2 gamma -2u=R( rho , tau )(f( rho )+g( tau )), in four special co-ordinate systems, ( rho , tau ), where R( rho , tau ) is a prescribed simple function. A comparison is made with the canonical Schrodinger separable metric forms of Carter.