In this paper the gravitational field of a bounded and isolated material source is expressed explicitly in terms of the density, pressure, and other characteristics of the source in a scheme of successive approximations. Particular emphasis is given to the radiation zone and to the relation of the gravitational waves to the source. The paper is divided into two parts. In the first part, the approximation method is established for a general source. The components of the metric tensor in a coordinate system t, x, y, and z are written as an infinite sum of terms wfth each term being identified by the factor E (s = c-1), which the term contains. To describe the emission of gravitational energy, independent coordinates u, x, y, and I are chosen (instead of t, x, y, z), with n becoming the retarded time and x, y, the usual Cartesian coordinates at spatial infinity. The Ricci tensor is expanded in a power series of E and the gauge condition is determined by the requirement that the metric must reduce to the Minkowskian metric at spatial infinity. With the energy-momentum tensor given in general terms, the inhomogeneous field equations are written in the ith approximation, and the procedure for determining the news function and the rate of energy radiation is described. In the second part, the method is applied for a bounded source of perfect fluid. The energy-momentum tensor is expanded in powers of E, and the field equations are solved explicitly for the first four approximations (i = 1, 2, 3, 4) after the Minkowskian metric. The integrability conditions for the Einstein equations are derived. Finally, the field in the radiation zone is related to the source by calculating explicitly the first nonxero term of the news function and the rate at which energy is radiated in the form of gravitational waves.
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S. Persides (1971) studied this question.