New family of exact stationary axisymmetric gravitational fields generalising the Tomimatsu-Sato solutions

Abstract

A new three-parameter family of exact solutions of the stationary axisymmetric vacuum Einstein equations, which represent rotating bounded sources, are presented. This family contains the solutions of Kerr and Tomimatsu-Sato as special cases, and may be regarded as a generalisation of the latter to arbitrary continuous delta parameter. The final form of the metric depends on two ordinary differential equations of the second order. When delta is not an integer, these equations define unfamiliar transcendental functions for which rapidly converging series expansions of several types are available. When delta is an integer, the solutions are polynomial or rational functions of spheroidal coordinates and define the discrete Tomimatsu-Sato series, for which those authors give the cases delta =1, 2, 3, 4.