A new formulation of the field equations for the stationary axisymmetric vacuum gravitational field. I. General theory

TLDR: In this paper, a new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented, and three methods are given for the construction of the full metric from e2γ.

Abstract: A new formulation of the stationary axisymmetric vacuum gravitational field equations which is substantially different from the well known formulations of Lewis and Ernst is presented. The basic variable is e2γ = -g11g44 and satisfies a field equation of the fourth differential order which may be interpreted as the condition that a certain 2-space has constant curvature, K = -1. The principal motivation is that for many known solutions and all known asymptotically flat (non-static) solutions, e2γ takes a much simpler functional form than either the metric coefficients, g44, g34 and g33, or the Ernst potentials, E and ξ . Three methods are given for the construction of the full metric from e2γ. A duality principle is invoked to provide a very similar field equation for the metric coefficient, e2γ-2u = -g11.