An Explicit Determination of the Non-self-adjoint Wave Equations that Satisfy Huygens’ Principle on Petrov Type III Background Space-times

Abstract

It is shown that the validity of Huygens’ principle for the non-self-adjoint wave equation on a Petrov type Ill space-time implies that the space-time is conformally related to one in which every repeated null vector field of the Weyl tensor is recurrent. It is further shown that, given a certain mild assumption imposed on the covariant derivative of the Weyl curvature spinor, there are no Petrov type III space-times on which the non-self-adjoint scalar wave equation satisfies Huygens’ Principle.