Relationship between the gravitational constant and several quantum mechanical fundamental constants

摘要

By assuming a three-body system for the hydrogen atom—electron, proton, and spherical Planck mass-and assuming that the electron and the proton are bare point charges, the following question arises: How far from the center of the spherical Planck mass will the electron have to orbit such that its orbital velocity, arising from the gravitational force between the spherical Planck mass and the electron instead of the electrostatic force between the proton and the electron, is equal to the orbital velocity of an electron in the ground state of the Bohr hydrogen atom? The result of the analysis of this equivalence is the derivation of a relationship between the gravitational constant (G) and several quantum mechanical fundamental constants and the 26 space-time dimensions defined in string theory.