A Novel Natural Unit System Beyond Planck Units

Philosophy of Natural Units

Natural units originate from the idea proposed by Max Planck in 1900: a system of units that is independent of civilization, observers, and measurement environments, and common to all of nature. The Planck unit system defines the speed of light c, Planck's constant $\hbar$, the gravitational constant G, and Boltzmann's constant kB all as "1", so that the Planck length, Planck time, Planck mass, and Planck temperature all become "1".

\[ \ell_{\mathrm{P}}=\sqrt{\frac{\hbar G}{c^{3}}} \] \[ t_{\mathrm{P}}=\sqrt{\frac{\hbar G}{c^{5}}} \] \[ m_{\mathrm{P}}=\sqrt{\frac{\hbar c}{G}} \] \[ T_{\mathrm{P}}=\sqrt{\frac{\hbar c^{5}}{G\,k_{\mathrm{B}}^{2}}} \]

This was an extremely beautiful idea, aiming to detach physical laws from artificial human-defined units. At the same time, however, the physical meaning of these particular combinations of constants is not necessarily transparent. Moreover, in actual physics, what is set equal to "1" depends on the field, and natural unit systems are not uniquely defined. Inheriting the spirit of Planck units, we propose a new natural unit system that explicitly incorporates physical interpretation.

Space-Time Unit (ST Unit System)

In the historical development of physics, the kilogram and the coulomb were defined first, and in order to maintain the consistency of physical laws, the gravitational constant $G$, the vacuum permittivity $\varepsilon_0$, and the vacuum permeability $\mu_0$ were introduced afterward. However, when the roles of $G$, $\varepsilon_0$, and $\mu_0$ appearing in Newton's law of gravitation and Coulomb's law are reexamined dimensionally, it becomes clear that the kilogram and the coulomb are in fact derived space-time units: \[ \mathrm{kg} \;\rightarrow\; \mathrm{m^3/s^2}, \qquad \mathrm{C} \;\rightarrow\; \mathrm{m^2/s}. \] We collectively refer to this system of units as the Space-Time (ST) unit system (see References).

Transformation from SI Units to ST Units

In the ST unit system, the kilogram and the coulomb are transformed by the $\mu$-converter and the $\varepsilon$-converter, respectively. Transforming the SI electron mass $m_e$ and elementary charge $e$ into the ST electron mass $m_0$ and elementary charge $e_0$, we obtain

\[ m_0 = \mu m_e, \quad \mu = \frac{\mu_0}{4\pi}\left(\frac{e c}{m_e}\right)^2 \] \[ e_0 = \varepsilon e, \quad \varepsilon = \frac{\mu_0}{4\pi}\frac{e c}{m_e} \]

In the ST framework, the gravitational constant $G$, which originally carries the kilogram in its dimensional structure, no longer appears explicitly. Instead, it is recast as the dimensionless parameter $\beta_0$.

$$ \beta_0 = \frac{G}{\mu} $$

Likewise, the vacuum permittivity $\varepsilon_0$ and vacuum permeability $\mu_0$ are fully determined by geometric factors together with the defined constant $c$. For this reason, the ST system introduces no additional independent symbols for them.

$$ \varepsilon_0 \, \frac{\varepsilon^2}{\mu} = \frac{1}{4\pi c^2}, \qquad \mu_0 \, \frac{\mu}{\varepsilon^2} = 4\pi $$

Dimensionless Structure of the ST Framework

Within the ST formulation, the dimensionless parameter $\beta_0$, together with the inverse fine-structure constant $\alpha_0$, governs the relationship between distinct physical regimes. In this context, $\beta_0$ characterizes the relative strength between gravitational and electromagnetic interactions.

Gravitational-Electromagnetic Coupling Ratio

$$ \beta_0 = \frac{G m_e^2}{k e^2} $$

Here $k$ represents Coulomb's constant, defined as $ k = \frac{1}{4\pi\varepsilon_0} $.

By contrast, symmetry considerations lead to the interpretation of the inverse fine-structure constant $\alpha_0$ as a measure of the quantum-to-electromagnetic scale relation.

Quantum-Electromagnetic Ratio

$$ \alpha_0 = \frac{\hbar c}{k e^2} $$

Planck Constant in the ST Representation

Although $\alpha_0$ could, in principle, be incorporated directly into $\hbar$, the ST natural unit construction keeps them distinct in order to maintain structural balance between the gravitational-electromagnetic and quantum-electromagnetic ratios. Under this convention, Planck's constant in the ST framework takes the form:

$$ \hbar_0 = \hbar \,\mu\,\alpha $$

Hierarchical Generation of Physical Quantities

In the ST unit structure, every physical quantity can be generated from only two fundamental elements: the elementary charge $e_0$ and the speed of light $c$. Here, $c$ functions not merely as a numerical constant, but as a scaling operator that organizes physical quantities into successive dimensional layers.

Quantity	Expression	Unit
Time	$ e_0 c^{-2} $	$ \mathrm{s} $
Length	$ e_0 c^{-1} $	$ \mathrm{m} $
Charge	$ e_0 $	$ \mathrm{m^2 / s} $
Mass	$ e_0 c $	$ \mathrm{m^3 / s^2} $
Momentum	$ e_0 c^{2} $	$ \mathrm{m^4 / s^3} $
Energy	$ e_0 c^{3} $	$ \mathrm{m^5 / s^4} $

The fundamental length $e_0 c^{-1}$ coincides with the classical electron radius $r_0$, while the corresponding time scale $e_0 c^{-2}$ represents the interval required for light to traverse that radius. From these two primitives, the speed of light $c$, the elementary charge $e_0$, the electron mass $m_0$, Planck's constant $\hbar_0$, and the rest energy $E_0 = m_0 c^2$ arise in a unified manner.

Constant	Expression	Unit
$ c $	$ r_0 / t_0 $	$ \mathrm{m / s} $
$ e_0 $	$ r_0^{2} / t_0 $	$ \mathrm{m^2 / s} $
$ m_0 $	$ r_0^{3} / t_0^{2} $	$ \mathrm{m^3 / s^2} $
$ \hbar_0 $	$ r_0^{5} / t_0^{3} $	$ \mathrm{m^5 / s^3} $
$ E_0 = m_0 c^2 $	$ r_0^{5} / t_0^{4} $	$ \mathrm{m^5 / s^4} $

Construction of the ST Natural Unit System

By assigning the fundamental length $r_0$ and fundamental time $t_0$ the normalized value of unity, one obtains a natural unit system in which the speed of light, elementary charge, electron mass, Planck’s constant, and electron rest energy simultaneously reduce to unity.

Why the Classical Electron Radius is a Natural Choice

Selecting the classical electron radius $r_0$ as the reference length reveals that principal physical constants can be expressed systematically as powers of the inverse fine-structure constant.

Quantity	Value
Classical electron radius	$ 1 $
Compton length	$ 2\pi \alpha_0 $
Bohr radius	$ \alpha_0^{2} $
Inverse Rydberg constant	$ 4\pi \alpha_0^{3} $

In this formulation, fundamental constants retain their physical meaning while numerically simplifying to unity. Physical laws then depend only on geometric relations and the dimensionless parameters $ \alpha_0 $ and $ \beta_0 $.

Next Theme : Electron-Positron Annihilation and Pair Creation : Equation relating mass and charge

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

Quantity	Expression	Unit
Time	\( e_0 c^{-2} \)	\( \mathrm{s} \)
Length	\( e_0 c^{-1} \)	\( \mathrm{m} \)
Charge	\( e_0 \)	\( \mathrm{m^2 / s} \)
Mass	\( e_0 c \)	\( \mathrm{m^3 / s^2} \)
Momentum	\( e_0 c^{2} \)	\( \mathrm{m^4 / s^3} \)
Energy	\( e_0 c^{3} \)	\( \mathrm{m^5 / s^4} \)

Constant	Expression	Unit
\( c \)	\( r_0 / t_0 \)	\( \mathrm{m / s} \)
\( e_0 \)	\( r_0^{2} / t_0 \)	\( \mathrm{m^2 / s} \)
\( m_0 \)	\( r_0^{3} / t_0^{2} \)	\( \mathrm{m^3 / s^2} \)
\( \hbar_0 \)	\( r_0^{5} / t_0^{3} \)	\( \mathrm{m^5 / s^3} \)
\( E_0 = m_0 c^2 \)	\( r_0^{5} / t_0^{4} \)	\( \mathrm{m^5 / s^4} \)

Quantity	Value
Classical electron radius	\( 1 \)
Compton length	\( 2\pi \alpha_0 \)
Bohr radius	\( \alpha_0^{2} \)
Inverse Rydberg constant	\( 4\pi \alpha_0^{3} \)