Constants as Unit-System Artifacts

The perspective presented offers a profound reinterpretation of physical constants, positioning them not as fundamental quantities but as explicit conversion factors inherent in our measurement systems. This contrasts with traditional natural unit systems (like Planck or Stoney units) where constants are artificially set to 1. Instead, the constants' numerical values directly encode the scaling ratios required to reconcile disparate unit definitions in the SI system. Here's a breakdown of this framework and its implications:

Core Insight: Constants as Unit-System Artifacts

Physical constants like $h$, $c$, and $k_B$ arise from the mismatch between how we define units (e.g., kg, Hz, K) and the deeper physical relationships between quantities. For example:

• $h\mathrm{/}{c}^2$ defines Hz_kg (kg/Hz), the ratio between mass and frequency units.

• $k_B\mathrm{/}h$ defines K_Hz (Hz/K), the ratio between temperature and frequency units.

These ratios reflect how far SI units deviate from a hypothetical "natural" system where:

$$\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">K</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

In such a system, constants like $h$ and $k_B$ would naturally equal 1, eliminating the need for explicit conversion factors.

Key Relationships and Simplifications

By expressing constants as products of these ratios, physical laws simplify dramatically:

1. Planck's Constant:
   $h=\text{Hz\_kg}\cdot c^2$
   This redefines $h$ as a bridge between frequency and mass via $c^2$, aligning with $E=m{c}^2$.

2. Boltzmann's Constant:
   $k_B=\text{K\_Hz}\cdot \text{Hz\_kg}\cdot c^2$
   Here, $k_B$ becomes a chain of conversions: temperature → frequency → mass → energy.

Practical Implications

1. Simplified Formulas:
   Constants can be replaced by unit-conversion ratios derived from SI definitions:

   • Hz_kg = $h\mathrm{/}{c}^2\approx 7.372\times {10}^{-51}\text{kg/Hz}$
     The mass equivalent of 1 Hz.

   • K_Hz = $k_B\mathrm{/}h\approx 2.084\times {10}^{10}\text{Hz/K}$
     The frequency equivalent of 1 K.

   This reframes equations like the Stefan-Boltzmann law:

   • Original: $\sigma =\frac{2{\pi }^5{k}_B^4}{15c^2{h}^3}$

   • Simplified: $\sigma =\frac{2{\pi }^5}{15}\text{Hz\_kg}\cdot {\text{K\_Hz}}^4$
     Constants replaced by unit ratios inherent in SI.

2. Dimensional Transparency:
   Constants reveal how SI units could be redefined for coherence:

   • Redefining the kilogram via frequency ($1\text{kg}\equiv c^2\mathrm{/}h\cdot \text{Hz}$) would collapse $h$ and $c$ into unit definitions, rendering them numerically 1.

3. Anthropocentric Critique:
   The SI system’s historical development prioritized human-scale measurements (e.g., kilogram prototype, atomic clocks), creating artificial divides between energy, mass, and temperature. Constants like $h$ and $k_B$ exist to reconcile these divides.

Why This Isn’t “Natural Units”

Traditional natural units discard dimensions entirely (e.g., length = time), relying on “silent” constants for real-world calculations24. This framework instead:

• Retains SI dimensions while exposing their arbitrary scaling.

• Explains constant magnitudes as artifacts of unit definitions (e.g., $c$’s large value reflects the meter’s impracticality for relativity).

• Avoids dimensionless confusion by keeping constants’ dimensional roles explicit2.
  
  
  

The reinterpretation of energy formulas through the lens of E = mc² as a foundational pattern reveals intriguing nested relationships between physical constants, suggesting that constants like $h$, $c^2$, and $k_B$ encode scaling factors rather than fundamental truths. Here's how this nesting structure emerges:

Nested Relationships in Energy Formulas

1. Starting with $E=m{c}^2$:

   • The equation expresses the equivalence between mass ($m$) and energy ($E$) through the speed of light squared ($c^2$). This serves as the foundation for all energy-related formulas.

2. Mass as Frequency Equivalent:

   • Substituting $m=f\cdot {\text{Hz}}_{\text{kg}}$ (where ${\text{Hz}}_{\text{kg}}=h\mathrm{/}{c}^2$), we get:

     $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

   • Here, Planck's constant ($h$) emerges as a conversion factor between frequency ($f$) and energy ($E$).

3. Frequency as Temperature Equivalent:

   • Recognizing that $f=T\cdot K_{\text{Hz}}$ (where $K_{\text{Hz}}=k_B\mathrm{/}h$), we substitute into the equation:

     $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}\mathrm{<span\; style="background-color:\; white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

   • Boltzmann's constant ($k_B$) now emerges as a composite factor encoding temperature-frequency-energy equivalence.

Russian Nesting Doll Structure of Constants

The constants are not independent but are nested within one another:

• $c^2$ is embedded in $h$:
  $h=c^2\cdot {\text{Hz}}_{\text{kg}}$, where $H{z}_{\text{kg}}=h\mathrm{/}{c}^2\mathrm{.}$

• $h$ is embedded in $k_B$:
  $k_B=h\cdot K_{\text{Hz}},$ where $K_{\text{Hz}}=k_B\mathrm{/}h\mathrm{.}$

This nesting implies that these constants are scaling factors arising from how SI units are defined rather than fundamental properties of nature. They encode relationships between mass, frequency, temperature, and energy.

Unified Chain of Equivalences

From this perspective, all energy-related formulas follow a consistent chain:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">temperature</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">frequency-temperature scaling</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">frequency</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">mass-frequency scaling</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">mass</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">mass-energy scaling</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">energy</span>}<span\; style="background-color:\; white;">)</span>$$

Each constant acts as a bridge between adjacent quantities:

• K_Hz   = k_B/h:  Converts temperature to frequency.

• Hz_kg = h/c^2:  Converts frequency to mass.

• c^2                   :  Converts mass to energy.

Conclusion

The constants $h$, $c$, and k_B are not fundamental truths but byproducts of SI’s fragmented unit definitions. Recognizing them as scaling factors reveals a path toward a more coherent measurement system—one where energy, mass, frequency, and temperature share a unified foundation. This view doesn’t invalidate SI but highlights how its architecture shapes our perception of physical laws. By redesigning units around physical relationships (e.g., defining mass via frequency), we could eliminate these constants entirely, achieving true natural units organically