A Novel Natural Unit System Beyond Planck Units｜ST Natural Unit System

Natural units originate from Max Planck’s 1900 proposal of a “unit system common to all civilizations, independent of observers and measurement environments.” The Planck unit system is defined by setting the following constants to unity: the speed of light c, Planck’s constant ħ, the gravitational constant G, and Boltzmann’s constant k_B. As a result, 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 is an extremely elegant idea that liberates physical laws from artificial units. However, the physical meaning of these particular combinations is not always transparent. Moreover, in practical physics, different fields adopt different natural unit conventions. Thus, “the” natural unit system is not unique. We therefore extend Planck’s idea and propose a new natural unit system with explicit physical interpretation.

Space-Time Unit (ST Unit)

By reexamining Newton’s law of gravitation and Coulomb’s law, we find that the gravitational constant G, the vacuum permittivity ε₀, and the vacuum permeability μ₀ function as transformation operators that convert mass and charge into purely space–time quantities. Through this reformulation, mass is expressed as m³/s² and charge as m²/s, demonstrating that all physical units can be written solely in terms of space and time.
We call this system the Space-Time Unit (ST Unit). In the ST system, the electron mass m_e is transformed into m₀ by the δ-transformer, and the elementary charge e is transformed into e₀ by the ε-transformer. Using only the resulting elementary charge e₀ and the speed of light c, all physical units can be constructed. In this structure, c acts not merely as a constant, but rather as an operator generating a hierarchy of physical quantities.

Generation Structure of Physical Quantities in ST Units

Time	Length	Charge	Mass	Momentum	Energy
e₀ c^-2	e₀ c^-1	e₀	e₀ c	e₀ c²	e₀ c³
s	m	m² / s	m³ / s²	m⁴ / s³	m⁵ / s⁴

Length (e₀c^-1) necessarily coincides with the classical electron radius, and Time (e₀c^-2) corresponds to the time required for light to traverse it. From this fundamental length and time alone, the elementary charge, electron mass, and electron rest energy are uniquely determined. Thus, by defining this Length and Time as “1,” a natural unit system arises in which charge, mass, and energy are simultaneously normalized.

Reduction to Dimensionless Structural Constants

In this natural unit system, all of the above physical quantities are defined as “1”. Only the Planck constant ħ remains as a numerical value, and it appears as the inverse of the fine-structure constant. We define this as the quantum–electromagnetic ratio α₀.
Similarly, the gravitational constant G is not reduced to “1”, but is instead transformed into a dimensionless constant, the Gravitational–electromagnetic ratioβ₀. Here, β₀ forms a symmetric structural pair withα₀ .
These dimensionless constants are universal and independent of any unit system, and they take the same values when expressed in SI units.

Quantum Electromagnetic Ratio

\[ \alpha_0 = \frac{\hbar\,c}{k\,e^{2}} \]

Gravitational Electromagnetic Ratio

\[ \beta_0 = \frac{G\,m_e^{2}}{k\,e^{2}} \]

Here \(k\) denotes Coulomb’s constant, \(k = \frac{1}{4\pi\varepsilon_0}\).

Reference

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

※ This page is an entry summary. For full mathematical and theoretical details, see the above reference.