Null field solutions of the wave equation and certain generalizations

Abstract

The ordinary wave equation in 3+1 dimensions ⧠φ=0, ⧠≡−∂2/∂t2+∂2/∂x2+∂2/∂y2 +∂2/∂z2 admits null field solutions, characterized by ∇φ⋅∇φ=0, ∇φ⋅∇φ≡−(∂φ/∂t)2+(∂φ/∂x)2 +(∂φ/∂y)2+(∂φ/∂z)2 with ∇φ≠0. It is shown that the general null field solution can be obtained from a knowledge of the ‘‘time-transported’’ solutions, i.e., those solutions of the form φ=t−ψ(x,y,z), where ψ satisfies both Laplace’s equation and the eikonal equation in a Euclidean space. We obtain all second-order scalar wave equations of form f(φ,φ;i;i, φ;i; j φ;i; j)=0 (in arbitrary dimension and involving a single potential function φ) for which the above technique applies. These equations are shown to be equivalent to the family of quasilinear third-order equations ∇φ⋅∇(⧠φ)+K(⧠φ)2=0, where K is a constant. Some null solutions of these equations are considered, and related to previous works. The results are applied to determine all shear-free hypersurface-orthogonal null geodesic congruences in Minkowski space–time, and some brief comments are made on complex solutions and on more general wave equations.