In the current International System of Units (SI), the kilogram and the ampere are defined as base units.
However, this classification reflects a practical convention established for measurement convenience,
rather than a fundamental necessity of nature.
In this article, we argue that mass and electric charge are not fundamental quantities,
but derived quantities emerging from deeper space-time relations.
Furthermore, we demonstrate that the gravitational constant \(G\),
the vacuum permittivity \(\varepsilon_0\), and the vacuum permeability \(\mu_0\)
function as proportional constants introduced to preserve internal consistency
within an artificial unit system based on the kilogram and the coulomb,
rather than as intrinsic constants of nature (see References).
Historically, the unit of mass was defined first, and the unit of charge was later introduced independently.
Subsequently, in order to maintain consistency among physical equations,
the constants \(G\), \(\varepsilon_0\), and \(\mu_0\) were incorporated.
However, a re-examination of Newtonian gravitational potential energy
and Coulomb potential energy reveals that the kilogram and the coulomb
are fundamentally artificial constructions.
In the eras of Newton and Coulomb, both gravitational and electrostatic forces
were originally expressed as proportional relations.
Since the present study focuses on scalar dimensional comparison,
signs and vector notation are omitted.
\[ F_g \propto \frac{m_1 m_2}{r^2} \qquad F_e \propto \frac{q_1 q_2}{r^2} \]
These expressions share an identical structural form, suggesting a deep analogy between physical laws. The currently adopted formula for gravitational potential energy is given by:
Current Gravitational Potential Energy
\[ E = \frac{G\,m\,M}{r} \]
In the opening of A Treatise on Electricity and Magnetism (1873), James Clerk Maxwell noted that if mass were defined in terms of gravitational strength within an astronomical unit system, the dimensional formula of mass \(M\) would become \(L^3/T^2\).
Indeed, if the gravitational constant \(G\) is absorbed into the kilogram, the units of transformed masses \(m'\) and \(M'\) become \(m^3/s^2\). Since this study employs scalar dimensional analysis, signs are omitted.
Gravitational Potential Energy with \(G\) Absorbed
\[ E' = \frac{m' M'}{r} \]
Because the gravitational potential energy \(E'\) itself contains mass in the numerator, \(G\) is simultaneously absorbed. At this stage, the unit of mass becomes \(m^3/s^2\), and the unit of energy becomes \(m^5/s^4\). Applying the same dimensional reasoning to Coulomb potential energy, we begin from its standard form:
Current Coulomb Potential Energy
\[ E = \frac{k\,q\,Q}{r} \]
Just as the gravitational constant \(G\) was absorbed into the unit of the kilogram, one may formally absorb the square root of the Coulomb constant \(k\), an artificially introduced conversion constant, into the unit of the coulomb. In doing so, one obtains, at least formally, the expression corresponding to the Gaussian (cgs) unit system. However, a closer examination of the internal structure of the Coulomb constant \(k\) reveals that it can be decomposed as follows:
\[ k = \frac{1}{4\pi \varepsilon_0} = \frac{\mu_0\,c^2}{4\pi} \]
As shown on the right-hand side of the above equation, decomposition of the Coulomb constant makes clear that the speed of light \(c\) is embedded within it. Since \(c\) is a universal constant, it is not absorbed into the unit of charge in this study. Moreover, because the numerator of the vacuum permeability \(\mu_0\) contains mass dimension, the gravitational constant \(G\) is absorbed simultaneously.
Coulomb Potential Energy with Artificial Constants Absorbed
\[ E' = \frac{q' Q'\,c^2}{r} \]
From this framework, a relation between the elementary charge and the electron mass emerges:
\[ m_e' = e' c \beta \]
Here, \(\beta\) is a dimensionless constant defined as the square root of the ratio between gravitational and Coulomb potential energies. Furthermore, since the transformation factor from coulomb to \(m^2/s\) contains \(\sqrt{G}\) in its numerator, additional simplification becomes possible. By absorbing \(\beta^2\) into the kilogram in advance, the final transformation system consisting of the mass factor \(\mu\) and charge factor \(\varepsilon\) is obtained. Notably, through this procedure, the gravitational constant \(G\), which is the least precisely measured constant, disappears naturally from the formalism.
\[ \mu = \frac{\mu_0}{4\pi}\left(\frac{e\,c}{m_e}\right)^2 \]
\[ \varepsilon = \frac{\mu_0}{4\pi}\left(\frac{e\,c}{m_e}\right) \]
Accordingly, in the new unit system, the electron mass \(m_0\) and the elementary charge \(e_0\) are defined as:
\[ m_0 = \mu\,m_e \]
\[ e_0 = \varepsilon\,e \]
We refer to this new framework as the Space-Time (ST) Unit System. Within the ST unit system, physical structures that were obscure under the SI system become transparent. In particular, a remarkably simple relation emerges:
\[ m_0 = e_0\,c \]
For centuries, we have rarely questioned the assumption that the kilogram and the coulomb are fundamental units. Yet by reconsidering artificially introduced proportional constants and reconstructing the unit system from the electron level, previously hidden aspects of physical law become strikingly clear.
Reference
Note : This page serves as an introductory overview. For full derivations and systematic discussions, please refer to:
Space-Time Unit System for Unifying Gravitational Mechanics, Electromagnetism and Quantum Physics