Reconstructing Physical Constants: Unit Scaling Factors and the Equivalence of Mass, Energy, Frequency, and Temperature

 J. Rogers, 06 Mar 2025, 1735

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Abstract

The traditional view of physical constants like $c$, $h$, and $k$ as fundamental quantities obscures their true nature as unit scaling factors that convert between different units in our measurement system. By isolating these scaling factors—$k{g}_E=c^2$, $H{z}_{kg}=\frac{h}{c^2}$, and $K_{Hz}=\frac{k}{h}$—we reveal that mass, energy, frequency, and temperature are equivalent manifestations of the same underlying physical reality. This paper demonstrates how these scaling factors are not derived from the constants but are instead simple unit scaling factors that define them. We show that this framework simplifies the laws of physics, eliminates unnecessary clutter, and highlights the deep unity of physical quantities.

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1. Introduction

The traditional view of physics treats constants like $c$ (speed of light), $h$ (Planck’s constant), and $k$ (Boltzmann’s constant) as fundamental quantities that define the fabric of the universe. However, these constants are better understood as unit scaling factors that arise from the way we’ve chosen to measure physical quantities. By isolating these scaling factors, we can show that mass, energy, frequency, and temperature are equivalent and interconnected through simple unit conversions.

This paper begins with the traditional view of $E=m{c}^2$, $E=hf$, and $E=kT$ and demonstrates how to isolate the unit scaling factors $k{g}_E$, $H{z}_{kg}$, and $K_{Hz}$. We then show how these factors are not derived from the constants but are instead actual factors that define them. Finally, we demonstrate how this framework simplifies physical laws and reveals the equivalence of mass, energy, frequency, and temperature.

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2. The Traditional View

The traditional view of physics treats the following equations as fundamental:

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

2. Energy-Frequency Relation: $E=hf$

3. Energy-Temperature Relation: $E=kT$

4. 
   

These equations are often interpreted as revealing deep truths about the universe, but they are better understood as unit conversions between different units of our measuring system.

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3. Isolating Unit Scaling Factors

By isolating the unit scaling factors, we can rewrite the traditional equations in a way that highlights their role as unit converters.

3.1 Energy-Mass Scaling Factor ($k{g}_E=c^2$)

The equation $E=m{c}^2$ can be rewritten as:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">k</span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

where $k{g}_E=c^2$ is the mass-to-energy scaling factor. This factor converts mass ($kg$) to energy ($E$) and has units of $m^2\mathrm{/}{s}^2$.

3.2 Frequency-Mass Scaling Factor ($H{z}_{kg}=\frac{h}{c^2}$)

The equation $E=hf$ can be rewritten as:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">k</span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

where $H{z}_{kg}=\frac{h}{c^2}$ is the frequency-to-mass scaling factor. This factor converts frequency ($Hz$) to mass ($kg$) and has units of $kg\cdot s$.

3.3 Temperature-Frequency Scaling Factor ($K_{Hz}=\frac{k}{h}$)

The equation $E=kT$ can be rewritten as:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">k</span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

where $K_{Hz}=\frac{k}{h}$ is the temperature-to-frequency scaling factor. This factor converts temperature ($K$) to frequency ($Hz$) and has units of $Hz\mathrm{/}K$.

It is important to note that these scaling factors are not derived from the constants, they are just isolated from those constants. These scaling factors have always existed as factors inside the constants, this was just not recognized.  When we define the constant using these isolated factors we can then use that definition to simplify formulas so we can see exactly how these formulas have always been working, even when it looked like abstract energy ratios in the traditional monolithic view of the frameworks. 

These are the exact unit scaling factors for temperature, mass, and length scales to achieve natural units. This demonstrates that "fundamental constants" are artifacts of unit systems, not intrinsic properties of nature. These scaling factors provide a roadmap to simplify equations by aligning measurement scales with physical equivalence.

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4. Constructing Constants from Scaling Factors

The traditional constants $h$ $k$ are not fundamental—they are composite quantities built from the scaling factors $k{g}_E$, $H{z}_{kg}$, and $K_{Hz}$:

1. Planck’s Constant ($h$):

   $$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">k</span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

2. Boltzmann’s Constant ($k$):

   $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">k</span>}{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">E</span>}}$$

These equations show that the constants are not derived from the scaling factors—they are defined by them.

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5. Equivalence of Mass, Energy, Frequency, and Temperature

The scaling factors $k{g}_E$, $H{z}_{kg}$, and $K_{Hz}$ reveal the equivalence of mass, energy, frequency, and temperature. For example:

• Mass and Energy: $E=m\cdot k{g}_E$

• Frequency and Energy: $E=f\cdot H{z}_{kg}\cdot k{g}_E$

• Temperature and Energy: $E=T\cdot K_{Hz}\cdot H{z}_{kg}\cdot k{g}_E$

These relationships show that mass, frequency, and temperature are different manifestations of the same underlying physical reality.

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6. Application to Planck’s Law

The traditional form of Planck’s law for blackbody radiation is:

$$\mathrm{<span\; style="background-color:\; white;">D</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}$$

Using the scaling factors, this can be rewritten as:

$$\mathrm{<span\; style="background-color:\; white;">D</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}$$

This reformulation eliminates the explicit appearance of $h$, $k$, and $c$ and reveals the underlying simplicity of the relationship between frequency and temperature.

The Einstein Bose relationship becomes very interesting as a ratio between 3 different mathematically equivalent ratios.  The ratios:

$$\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}}{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">,</span>\frac{\mathrm{<span\; style="background-color:\; white;">f</span>}}{{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">,</span>\frac{\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}\mathrm{<span\; style="background-color:\; white;">z</span>}}}{\mathrm{<span\; style="background-color:\; white;">T</span>}}$$

are all numerically the same because the scaling factors ($h$, $k$, $K_{Hz}$) cancel out, leaving a dimensionless ratio. This shows that the ratios of energy, frequency, and temperature are equivalent. What happens is just converting the numerator and denominator to the same units, not what unit it is. 

This is important because looking at this formula in terms of the domain (temperature or frequency ratios) it is working with is more physically meaningful than converting out of that domain to an abstract energy. 

By setting each scaling factor to 1 individually, you show how the constants naturally reduce to unity:

1. :

   • The speed of light becomes dimensionless, unifying space and time scales.

   • E = mc^2 reduces to:

     $$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.</span>}$$

   • Planck’s constant reduces to:

     $$h={\text{Hz}}_{\text{kg}}\cdot 1.$$

   • Boltzmann’s constant reduces to:

     $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.</span>}$$

2. :

   • This sets Planck’s constant to:

     $$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.</span>}$$

   • Boltzmann’s constant becomes:

     $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.</span>}$$

3. :

   • Boltzmann’s constant simplifies further:

     $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">1.</span>}$$

At this point, all constants ($c,h,k$) reduce to unity, demonstrating that they are not fundamental but rather emerge from our choice of unit systems.

This step-by-step process highlights that your scaling factors ($c^2,h\mathrm{/}{c}^2,k\mathrm{/}h$) are precisely the ratios required to transition from SI units (or other systems) into natural units where all physical constants vanish into dimensionless unity:

•  = k/h.

This means that natural units are not "special" or "fundamental" in themselves—they simply reflect a system where all unit scaling factors have been normalized to unity.

This framework reveals that what we call "constants" (e.g., $c,h,k$) are nothing more than conversion factors between human-defined units like meters, kilograms, seconds, and kelvin. When we scale these units appropriately:

• The constants disappear because they were never fundamental—they only existed as bridges between mismatched units in our measurement system.

• Physical laws simplify because they reflect the underlying equivalence of mass, energy, frequency, and temperature. All the properties can be seen as equivalent. 

This insight reframes the search for "fundamental constants" in physics. Instead of treating them as immutable properties of the universe, we recognize them as artifacts of our measurement conventions.

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8. Conclusion

By isolating the unit scaling factors $k{g}_E$, $H{z}_{kg}$, and $K_{Hz}$, we’ve shown that the traditional constants $h$, $k$, and $c$ are not fundamental—they are composite quantities built from these factors. This framework reveals the equivalence of mass, energy, frequency, and temperature and simplifies the laws of physics by eliminating unnecessary clutter.

The scaling factors are not derived from the constants—they are actual factors that define them. This perspective shifts the focus from searching for "fundamental constants" to understanding how our human-constructed measurement systems shape the way we describe physical laws. It’s a powerful reminder that the universe is simple and unified, and that the apparent complexity is often an illusion created by our measurement systems.

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References

1. Einstein, A. (1905). "On the Electrodynamics of Moving Bodies." Annalen der Physik.

2. Planck, M. (1901). "On the Law of Distribution of Energy in the Normal Spectrum." Annalen der Physik.

3. Boltzmann, L. (1877). "On the Relationship between the Second Fundamental Theorem of the Mechanical Theory of Heat and Probability Calculations Regarding the Conditions for Thermal Equilibrium." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften.