A new approach of the stationary axisymmetric vacuum S(A) solutions

Abstract

We revisit axisymmetric stationary vacuum solutions of the Einstein equations, like we did for the cylindrical case [J. Math. Phys. 41, 7535 (2000)]. We explicitly formulate the simplest hypothesis under which the S(A) solutions, or axisymmetric Lewis solutions can be found and demonstrate that this hypothesis leads to a linear relation between the potentials. We show that the field equations still can be associated to the motion of a classical particle in a central field, where an arbitrary harmonic χ function plays the role of time. Three classes of solutions are obtained without the need of invoking the Papapetrou class. They depend on two real parameters, and the potentials are functions of χ only. The new approach exempts the need of complex parameters. We interpret one of the parameters as related to the vorticity of the source.