Three-Dimensional Formulation of Gravitational Null Fields. I

Abstract

The four-dimensional normal hyperbolic Riemannian space is represented as a direct product of a three-dimensional space and a timelike line. The null gravitational field is defined in a manner analogous to that of electromagnetic field. It is shown that in this way three types of gravitational null fields can be characterized. We call them gravitational null fields of types A, B, and C. We find, as necessary and sufficient conditions, that the gravitational field be null field of types A and B, respectively. It is also shown that these null fields admit null vectors in accordance with the properties of gravitational radiation fields.