The Goldberg--Kerr Approach to Lorentz Covariant Gravity

The approach to asymptotic electromagnetic fields introduced by Goldberg and Kerr is used to study various aspects of Lorentz Covariant Gravity. Retarded multipole moments of the source, the central objects of this study, are defined, and a sequence of conservation equations for these are derived. These equations are used extensively throughout the paper. The solution of the linearized Einstein equation is obtained in terms of the retarded moments for a general bound source, correct to $O(r^{-4})$. This is used to obtain the peeling–off of the linearized field, and to study the geometric optics approximation for the field and for the energy-momentum pseudotensor of the field. It is shown that the energy-momentum 4-vector splits into the total radiated 4-momentum' and the bound 4-momentum of the source’, similar to the case of the electromagnetic field. In the case of a source which has only retarded pole, dipole and quadrupole moments, a decomposition into arbitrary functions of a null coordinate is obtained which allows comparison with the solutions for linearized gravity obtained by other authors.