Gravitational Hamiltonian Systems and the Retarded Gravity Inequality

Gravity and electromagnetic interactions are the only fundamental physical interactions (outside the nuclear domain). In this work, we shall concentrate on Hamiltonians containing gravitational interaction, which according to general relativity must be retarded. In recent years, retarded gravity has explained many of the mysteries surrounding the "missing mass" related to galactic rotation curves, the Tully-Fisher relations, and gravitational lensing phenomena. Indeed, a recent paper analyzing 143 galaxies has demonstrated that retarded gravity will suffice to explain galaxies' rotation curves without the need to postulate dark matter for multiple types of galaxies. Moreover, it also demystified the "missing mass" related to galactic clusters and elliptic galaxies in which excess matter was derived through the virial theorem. Here, we give a mathematical criterion that specifies the cases in which retardation is important for gravity (and when it is not). The criterion takes the form of an inequality.