Physical dimensions/units and universal constants: their invariance in special and general relativity

Abstract

The theory of physical dimensions and units in physics is outlined. This includes a discussion of the universal applicability and superiority of quantity equations. The International System of Units (SI) is one example thereof. By analyzing mechanics and electrodynamics, we are naturally led, besides the dimensions of length and time, to the fundamental units of action $\mathfrak h$, electric charge $q$, and magnetic flux $\phi$. We have $q\times \phi=\text{action}$ and $q/\phi=1/\text{resistance}$. These results of \emph{classical physics} suggests to look into the corresponding quantum aspects of $q$ and $\phi$ (and also of $\mathfrak h$): The electric charge occurs exclusively in elementary charges $e$, whereas the magnetic flux can have any value; in specific situations, however, in superconductors of type II at very low temperatures, $\phi$ appears quantized in the form of fluxons (Abrikosov vortices). And $\mathfrak{h}$ leads, of course, to the Planck quantum $h$. Thus, we are directed to superconductivity and, because of the resistance, to the quantum Hall effect. In this way, the Josephson and the quantum Hall effects come into focus quite naturally. One goal is to determine the behavior of the fundamental constants in special and in general relativity, that is, if gravity is thought to be switched off versus the case in the gravitational field.