Natural Constants Determined to High Precision from Boltzmann’s Constant and Avogadro’s Number—A Challenge to Experiments and Astrophysical Observations to Match the Precision of the Results

In this investigation, we explore previously unknown relations between natural constants by taking the following steps: (1) We discard Dirac’s constant ℏ from the universal man-made constants of physics, which we redefine in terms of Planck’s constant h. (2) Working in the SI system of units, we determine Newton’s gravitational constant G from Boltzmann’s constant kB and the elementary charge e, recognizing the entropy of matter as their common underlying characteristic. (3) By comparing the mass of 1 mole of electrons to the h-defined Planck mass MP, we deduce nature’s own molar constant (≃0.1 mol) that contains a `reduced Avogadro number’ ℵA=NA/fA of particles, where NA is Avogadro’s number and fA≃10 is the associated Avogadro factor. (4) From the new effective gravitational constant G★≡4πϵ0G, where ϵ0 is the vacuum permittivity, we obtain MOND’s universal constant A0 and its critical acceleration a0, recognizing the Newtonian source of gravity as the common underlying characteristic and repudiating the need for a principle of equivalence of masses. (5) We derive the gravitational coupling constant αg solely from ℵA. (6) We adopt the measured value of the h-defined fine-structure constant (FSC) α and the value of αg (or, equivalently, nature’s ℵA), and we determine the relative ratio βg=αg/α precise to 10 significant digits. (7) We derive the strong relative ratio βs=αs/α directly from the Avogadro factor fA. (8) We determine the coupling constants of weak and strong interactions (αw and αs, respectively) in terms of the FSC α. (9) The relation αw=α leads to a determination of the mass of the W boson mW from the measured values of α and the reduced Fermi constant GF0. (10) Using the Planck mass as a principal constant (MP=ℵAme, where me is the electron mass), we obtain new classical definitions of h,α, and the Compton radius rc; and we reformulate in a transparent, geometrically clear way several important QED equations, as well as the extended Planck system of units itself. We discuss the implications of these results, and we pave a way forward in exploring the unification of the fundamental forces of nature.