A New Dual System For The Fundamental Units, including and going beyond the newly revised SI

Abstract

We propose a new system for the fundamental units, which includes and goes beyond the present redefinition of the SI, by choosing also $c=\hbar=1$. By fixing $c=c_\circ $m/s = 1, $\hbar=\hbar_\circ $ Js = 1 and $\underline{μ_\circ}=μ_\circ $N/A$^2$ = 1, it allows us to define the metre, the joule, and the ampere as equal to (1/299 792 458) s, $(1/\hbar_\circ = .948 ... \times \ 10^{34})\ {\rm s}^{-1}$ and $\sqrt{μ_\circ \rm N}= \sqrt{μ_\circ c_\circ / \hbar_\circ}\ {\rm s}^{-1}= 1.890...\times 10^{18}\ {\rm s}^{-1}$. It presents at the same time the advantages and elegance of a system with $\hbar = c = \underline{μ_\circ} = \underline{ε_\circ } = k = N_A = 1\,$, where the vacuum magnetic permeability, electric permittivity, and impedance are all equal to 1. All units are rescaled from the natural ones and proportional to the s, s$^{-1}$, s$^{-2}$, ... or just 1, as for the coulomb, ohm and weber, now dimensionless. The coulomb is equal to $\sqrt{μ_\circ c_\circ / \hbar_\circ}= 1.890... \times 10^{18}$, and the elementary charge to $e=1.602...\times 10^{-19} {\rm C} = \sqrt{4πα}=.3028... $ . The ohm is equal to $1/μ_\circ c_\circ$ so that the impedance of the vacuum is $Z_\circ = 376.730... Ω=1$. The volt is $ 1/ \sqrt{μ_\circ c_\circ \hbar_\circ}\ {\rm s}^{-1} = 5.017... \times\ 10^{15}\ {\rm s}^{-1}$, and the tesla $c_\circ $V/m = $ \sqrt{{c_\circ^3}/{μ_\circ\hbar_\circ}}\ {\rm s}^{-2} = 4.509... \times 10^{32}\ {\rm s}^{-2}$. The weber is $ 1/ \sqrt{μ_\circ c_\circ \hbar_\circ} = 5.017... \times\ 10^{15}$. $\ K_J =483\,597. \ $... GHz/V $= e/π= $ .09639..., and $R_K = 25\,812.\ ...\ Ω=1/2α\simeq 68.518$. One can also fix $e$ = 1.602 176 634 $\times 10^{-19}$ C, at the price of adjusting the coulomb and all electrical units with $μ_\circ=4π\times 10^{-7}η^2$ where $η^2, \propto α$, is very close to 1.

26 pages, 2 tables