Self-conjugate gravitational fields. I

Abstract

By splitting the curvature tensor R hijk into three 3-tensors of the second rank in a normal co-ordinate system, self-conjugate empty gravitational fields are defined in a manner analogous to that of the electromagnetic field. This formalism leads to three different types of self-conjugate gravitational fields, herein termed as types A, B and C . The condition that the gravitational field be self-conjugate of type A is expressed in a tensor form. It is shown that in such a field R hijk is propagated with the fundamental velocity and all the fourteen scalar invariants of the second order vanish. The structure of R hijk defines a null vector which can be identified as the vector defining the propagation of gravitational waves. It is found that a necessary condition for an empty gravitational field to be continued with a flat space-time across a null 3-space is that the field be self-conjugate of type A. The concept of the self-conjugate gravitational field is extended to the case when R ij # 0 but the scalar curvature R is zero. The condition in this case is also expressed in a tensor form. The necessary conditions that the space-time of an electromagnetic field be continued with an empty gravitational field or a flat space-time across a 3-space have been obtained. It is shown that for a null electromagnetic field whose gravitational field is self-conjugate of type A , all the fourteen scalar invariants of the second order vanish.