Moving objects in retarded gravitational potentials of an expanding spherical shell/Classical approach

Brief historical review Retarded gravitational potentials Classical approach This Wikibook assumes a huge amount of dark matter that is in a huge shell behind the visible part of the universe. This matter is rather cold, and it can be assumed that it is in a in thermal equilibrium. Therefore, the electromagnetic radiation emitted by this shell can be represented by the Planck radiation law. The emission of such a cold source of electromagnetic radiation is not visible and additionally the radiation has experienced a strong redshift due to the relative velocity of the fast expanding emitting shell as well as due to the gravitation of the mass of this shell. The gravitational force between two masses and in a distance of is given by Isaac Newton's law of universal gravitation with the gravitational constant : In a thought experiment it is possible to investigate the behaviour of a mass that is located within a spherical shell with the mass . The approach described in 1687 by Isaac Newton (1643–1727 in Gregorian dates) is called Newton's shell theorem.Newton Isaac Philosophiae Naturalis Principia Mathematica The Mathematical Principles of Natural Philosophy https://archive.org/details/philosophinatur03newtgoog 1687 London 193 According to this theorem a mass within a homogeneous and spherically symmetric shell with a mass experiences no net gravitational force. Additionally, this is regardless of the location of the mass within the shell as well as of its velocity. This behaviour can be easily explained with the following consideration: let a mass be within a homogeneous and spherically symmetric shell with the radius and the areal density . Then the area and the mass of the shell can be expressed as: thumb|right|upright=1.5|About the equivalence of the gravitational force to a mass in a homogeneous and spherically symmetric shell with the two area elements and . If we take an axis symmetrical double cone where the tips of the two cones are located at the position of the mass within a homogeneous and spherically symmetric shell, we get the situation shown in the adjacent figure. For the two infinitesimally small area elements and in the distances and of a mass with the same infinitesimally small angle elements : For the appropriate masses and of the two area elements: With Newton's law of universal gravitation, we can express the two infinitesimally small gravitational forces and due to the masses of the two area elements: It is obvious that both forces have the same value, and since they must be exactly directed to opposite directions, they add to zero. In other words: a mass within a spherical shell does not experience any net gravitational force at all. It is worth noting that this classical approach assumes the infinite propagation speed of gravitational waves, which is applicable for static but not for dynamic situations as well as huge systems. It is well known that force laws that are based on a finite propagation speed of potential effects, such as those governing the Coulomb force or the gravitational force, depend on the velocity and the acceleration of the bodies that cause the potentials.Lima Fabio M. S. 2020-10-01 Nonzero Gravitational Force Exerted by a Spherical Shell on a Body Moving Inside It, and Cosmological Implications https://doi.org/10.1134/S0202289320040088 Gravitation and Cosmology en 26 4 387–398 10.1134/S0202289320040088 1995-0721 This is also reflected in Mach's principle which, in fairly general terms, states that local physical laws are determined by the large-scale structure of the universe. This principle dates back to a publication by the 19th-century German physicist and philosopher Ernst Mach (1838–1916).Die Mechanik in ihrer Entwickelung Mach Ernst 1883-05-01 https://archive.org/details/diemechanikinih03machgoog/page/n232/mode/2up 2026-06-06 Internet Archive de An aspect of the influence of the large-scale structure of the universe given by retarded potentials is investigated in the next chapter. References