The conformal potential of the stationary axisymmetric vacuum

In a stationary axisymmetric vacuum gravitational field, the conformal structure of the 3‐space is determined by the symmetric, trace free and divergence‐less tensor Y ir . Using the Killing vector K i of the axisymmetry, the conformal potential U can be defined by U, i = εijk KjY kr K r . Conversely, the tensor Y ik is given algebraically in terms of the gradient U, i of the conformal potential. An attempt is made here to re‐formulate the field equations R λμ = 0 in terms of the conformal potential. Introducing the Ernst potential as a complex coordinate, the cylindrical radius can be eliminated from the field equations.