Newton's modern formula of gravity and some properties of the structure of the Universe in the post-Newtonian approximation - Karpenko, Ivan (International Science Journal of Engineering & Agriculture, 2025)
Newton's formula for gravity is presented in the post-Newtonian (PN) approximation. It is shown that for the case of a concentrated (point) mass, the gravitational potential of a point does not depend on the mass and is equal to half the square of the speed of light (), going to zero at infinity. For a mass uniformly distributed in space, an arbitrary point has zero potential and tends to () at infinity. It is also shown that the gravitational potential at the center point of a spherically symmetric sphere is equal to (), and not one and a half times the potential of its surface. And this value does not depend on either the radius of the sphere or its mass. The property of limited long-range action of a static gravitational field in a flat Universe is substantiated. Its meaning: from any point (galaxy) of the Universe, the field strength increases to a distance , where the gravitational potential - the geometric radius, and then decreases to zero at infinity. This point corresponds to the experimentally established fact that 5-6 billion years ago, outer space began to expand at an abnormally increasing speed. The nature of the force that leads to the formation of cosmic webs - voids (giant cavities) and filaments (filamentous clusters of galaxies) in the space of the Universe is investigated. The study is based on the fact that according to the PN approximation, a spherically symmetric material shell, in contrast to the conclusions of Newton's theorie of gravity, creates a gravitational field in the internal volume of a hollow sphere. It is shown that in a universe with dispersed matter, masses located inside a cavity with a lower density of matter than in the surrounding environment are subjected to a directed gravitational influence from masses located outside the cavity. The latter tend to move the mass located inside the cavity towards the cavity wall. Therefore, areas with reduced density values ​​are transformed into inlets. And the substance that left them forms clusters in the form of galactic filaments (fibers) on the periphery of the inlets. That is, there is a density fragmentation of the volume of the Universe - the formation of a volumetric cosmic web or a cellular density structure. It is shown that in a static Universe, only masses adjacent to the cavity, which are at a distance of no more than three radius of the spherical cavity, have such an effect. Thus, around the cavity at a distance from its center, the matter that migrated from the cavity forms a shell of increased density with an inner radius and an outer radius .