Decomposing Fundamental Constants: Unit Scaling Factors as the Bridge Between Human Measurement and Natural Units

J. Rogers, SE Ohio, 07 Mar 2025, 1746 

Introduction:

Fundamental constants like Planck's constant ($h$) and Boltzmann's constant ($k$) are often treated as intrinsic properties of the universe. However, this theory proposes that these constants are not fundamental but rather composite scaling factors that convert between different physical quantities in our human-defined measurement system. By decomposing $h$ and $k$ into simpler unit scaling factors, we reveal a deeper pattern of equivalence between energy ($E$), mass ($m$), frequency ($f$), and temperature ($T$). This approach not only simplifies our understanding of these constants but also shows how they encode the relationships between our units and the natural units of the universe.

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1. Energy-Mass Equivalence ($E=m{c}^2$):

• Scaling Factors:

  $${\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">m</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">/</span>}{\text{<span style="background-color: white;">s</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">(mass-to-energy scaling)</span>}$$$${\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\text{<span style="background-color: white;">(energy-to-mass scaling)</span>}$$

• 
  

• Implications:

  • $E=m{c}^2$

    E=mc2

    is not just a physical law but a unit conversion between mass and energy.

  • The speed of light ($c$) acts as a scaling factor that bridges our human-defined units of mass and energy.

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2. Frequency-Mass Conversion ($E=hf$):

• Scaling Factors:

  $${\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\mathrm{<span\; style="background-color:\; white;">E</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">s</span>}\text{<span style="background-color: white;">(frequency-to-mass scaling)</span>}$$$${\text{<span style="background-color: white;">kg</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}}\text{<span style="background-color: white;">(mass-to-frequency scaling)</span>}$$

• 
  

• Implications:

  • 
    

  • Planck's constant ($h$) is not fundamental but a composite scaling factor that converts frequency to mass.

  • This reveals the equivalence between energy, frequency, and mass, hidden within $h$.

  • 
    

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3. Temperature-Frequency Conversion ($E=kT$):

• Scaling Factors:

  $${\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">k</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}\text{<span style="background-color: white;">(temperature-to-frequency scaling)</span>}$$$${\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">K</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{\mathrm{<span\; style="background-color:\; white;">k</span>}}\text{<span style="background-color: white;">(frequency-to-temperature scaling)</span>}$$

• 
  

• Implications:
  
  

  • Boltzmann's constant ($k$) is not fundamental but a composite scaling factor that converts temperature to frequency.

  • This reveals the equivalence between energy, temperature, and frequency, hidden within $k$.

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4. Redefining $h$ and $k$:

• Definitions:

  $$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

• 
  

• Implications:

• 
  

• $h$ and $k$ are not monolithic constants but composite scaling factors built from simpler unit conversions.

• Each unit has a scaling to another unit.

• In natural units frequency is the same as in our current unit definitions.

• This decomposition reveals the hidden structure of our measurement system.

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5. Energy Calculations:

• Equations:

  $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

• 
  

• Implications:
  
  

• Each constant shows the chain of unit conversions to calculate the energy from the given property.

• The property is stepped one unit at a time, becoming the next property. 

• This shows the equivalence between temperature, frequency, mass, and energy.

5. Mass to Energy equivalence is encoded in both h and k:

• Equations:

  $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\text{<span style="background-color: white;">kg</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

• Implications:

• The same pattern of equivalence ($E=m{c}^2$) is present in $h$ and $k$, but it is hidden within the "black box" of these constants.

• This shows that energy, mass, frequency, and temperature are different manifestations of the same underlying reality.

• All these properties scale 1:1 with each other. 

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6. Scaling to Natural Units:

What these individual unit scalings define is the scaling we have to do to our units of measure to actually mathematically reach natural units from our existing units of measure.

• Steps:

  1. Scale the meter by $c$: $\text{New meter}=\text{meter}\mathrm{/}c$. This makes kg_E = 1.

  2. Scale the kilogram by ${\text{Hz}}_{\text{kg}}$: $\text{New kilogram}=\text{kilogram}\times {\text{Hz}}_{\text{kg}}$. This makes Hz_kg = 1.

  3. Scale the kelvin by ${\text{K}}_{\text{Hz}}$: $\text{New kelvin}=\text{kelvin}\mathrm{/}{\text{K}}_{\text{Hz}}$.  This makes  ${\text{K}}_{\text{Hz}}$= 1.
     
     

• Result:

  • After scaling, $c=h=k=1$, and $E=m=f=T$ in natural units.

  • 
    

• Implications:

  • The values of $h$ and $k$ are not fundamental but artifacts of our measurement system.

  • Natural units reveal the intrinsic equivalence of energy, mass, frequency, and temperature.

  • These individual unit scaling factors have the value they need to convert to natural units

  • This means that the constants have the value they do as a results of the individaul unit scaling units that define them. 

Conclusion:

This theory shows that fundamental constants like $h$ and $k$ are not intrinsic properties of the universe but composite scaling factors that encode the relationships between our human-defined units and the natural units of the universe. By decomposing these constants into simpler unit scaling factors, we reveal a deeper pattern of equivalence between energy, mass, frequency, and temperature. This approach simplifies our understanding of physics, demystifies the role of constants, and highlights the arbitrary nature of our measurement system. Ultimately, it suggests that the so-called "fundamental constants" are not fundamental at all—they are emergent properties that arise from the mismatch between our units and the intrinsic scales of the universe.