The Gravitational Force on Charges and the Electric Force on Masses, Two Extremely Weak Action-Reaction Pairs, and the Geometric-Mean Relations of the Fundamental Constants of Nature

If the gravitational and electromagnetic forces have a common origin, then the combined force field must be capable of acting on both masses and charges. Newton’s gravitational law and Coulomb’s law describe special cases of interactions in the combined field, but cross-forces on to charges by the gravitational-field component and on to masses by the electric-field component have not been previously explored. We derive such action-reaction pairs of cross-forces in this work. The field constant introduced in these forces is the geometric mean GK of the well-known constants G (Newton’s gravitational constant) and K=(4πε0)−1 (Coulomb’s constant). This geometric-mean relation implies that the cross-forces F× of the combined field are extremely weak between electrons (although stronger than gravity Fg) as compared to the Coulomb forces Fe, which explains why these forces (F×=FgFe) have not been detected in experiments. The new coupling GK is not the only example of a geometric mean of known constants that produces a new functional constant. We explore many such cases across physics disciplines, and we analyze the geometric means that appear in various natural contexts.