The motion of the system of two bodies and their center of mass in an inhomogeneous environment

In this paper, a material system consisting of two spherically symmetric bodies of comparable masses located inside a gas-dust ball with a spherically symmetric distribution of the density of the medium in it is considered. After choosing the corresponding energy-momentum tensor from the Einstein field equations using the Einstein-Infeld approximation procedure, the metric of the corresponding space-time, the gravitational field created by the «two-body – medium» system are found, and then the equations of motion of the bodies and their center of mass are obtained in Newton’s and post-Newtonian approximations of the general theory of relativity. It is proved that in the case of the indicated density of the medium, the following effect should exist already in the Newtonian approximation. The center of mass of two bodies shifts at a variable speed, although it was at rest in the void. This situation is a consequence of the fact that the two-body-medium system is not closed. For the first time, formulas for calculating the displacement value, which is proportional to the density of the medium in the center of the gas-dust ball and the 5th degree of the distance between the bodies, are derived. Therefore, at large distances between bodies, their center of mass has large displacements (it can reach several million kilometers per revolution of bodies around their center of mass). If the masses of the bodies are equal, their center of mass is at rest if it is at rest in the void.