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

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 (Phys. Rev. Lett.; 11:237 (1963)) and Tomimatsu-Sato (Phys. Rev. Lett.; 29:1344 (1972)) 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. One of the two equations is solved explicitly for the case delta = 5 and efficient algorithms are presented which make it possible to perform such calculations by hand. The metric and Ernst (Phys. Rev.; 167:1175 (1968) and J. Math. Phys.; 15:1409 (1974)) potentials assume simple functional forms on the symmetry axis. Actually, this three-parameter family of asymptotically flat solutions is shown to be contained in a family of unphysical solutions with six non-trivial parameters, one of which is the familiar NUT parameter. (author)