Where the Math Ends and the Universe Begins: A Natural Unit Scaling Framework

 J. Rogers, SE Ohio, 15 Mar 2025, 1333

Abstract

The fundamental constants of nature, such as the speed of light ($c$), Planck’s constant ($h$), and Boltzmann’s constant ($k$), are often treated as mere numbers in physical equations. However, these constants encode the precise scaling factors needed to convert between human-created unit systems (like the International System of Units, or SI) and the intrinsic scales of the universe. By isolating these scaling factors, we can separate the mathematical formalism we use to describe the universe from the physical reality that exists independently of our descriptions. This paper introduces a natural unit scaling framework that clarifies where the "math ends and the universe begins," revealing the intrinsic relationships between physical quantities while exposing the artifacts of our choice of units. This framework not only simplifies physical equations but also provides deeper insight into the nature of physical constants and the structure of the universe.

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1. Introduction

Physics is the study of the fundamental principles that govern the universe. To describe these principles, we use mathematical models and unit systems that allow us to quantify physical quantities like length, mass, time, and temperature. However, these models and unit systems are human constructs, while the universe operates according to its own intrinsic rules. The challenge lies in distinguishing between the mathematical scaffolding we build to describe the universe and the physical reality that exists independently of our descriptions.

In this paper, we introduce a natural unit scaling framework that achieves this distinction. By expressing physical equations in terms of scaling factors derived from fundamental constants, we isolate the artifacts of our unit systems from the intrinsic relationships between physical quantities. This framework reveals where the "math ends and the universe begins," providing a clearer understanding of the role of constants like $c$, $h$, and $k$ and the structure of physical laws.

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2. The Role of Fundamental Constants

Fundamental constants like $c$, $h$, and $k$ are central to modern physics. They appear in equations ranging from relativity ($c$) to quantum mechanics ($h$) to thermodynamics ($k$). Traditionally, these constants are treated as fixed numbers that bridge different physical quantities. For example:

• $c$ relates space and time ($E=m{c}^2$).

• $h$ relates energy and frequency ($E=hf$).

• $k$ relates energy and temperature ($E=kT$).

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However, these constants also encode the scaling factors needed to convert between human-created unit systems (like SI) and the intrinsic scales of the universe. For example:

• $c$ encodes the ratio between meters and seconds.

• $h$ encodes the ratio between joules and hertz.

• $k$ encodes the ratio between joules and kelvin.

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By recognizing these constants as conversion factors, we can separate the artifacts of our unit systems from the intrinsic structure of the universe.

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3.1. Scaling Factors

We define the following scaling factors to convert between human-defined units (SI) and the intrinsic scales of the universe:

1. Mass-Energy Conversion Factor:

   $$\text{<span style="background-color: white;">kg\_J</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">(units: J/kg)</span>}$$

   This factor converts mass (in kilograms) to energy (in joules), as seen in the relation $E=m{c}^2$.
   
   

2. Frequency-Mass Conversion Factor:

   $$\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\text{<span style="background-color: white;">(units: kg/Hz)</span>}$$

   This factor converts frequency (in hertz) to mass (in kilograms), reflecting the frequency-mass equivalence in quantum mechanics.
   
   

3. Temperature-Frequency Conversion Factor:

   $$\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">k</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\text{<span style="background-color: white;">(units: Hz/K)</span>}$$

   This factor converts temperature (in kelvin) to frequency (in hertz), bridging thermodynamics and quantum mechanics.

Using these scaling factors, we can redefine the fundamental constants $h$ and $k$ in terms of their intrinsic roles:

4. Planck's Constant ($h$):

   $$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">kg\_J</span>}\text{<span style="background-color: white;">(units: J/Hz)</span>}$$

   Here, $h$ is expressed as the product of the frequency-mass conversion factor ($\text{Hz\_kg}$) and the mass-energy conversion factor ($\text{kg\_J}$). This highlights its role in connecting frequency to energy ($E=hf$).
   
   

5. Boltzmann's Constant ($k$):

   $$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">kg\_J</span>}\text{<span style="background-color: white;">(units: J/K)</span>}$$

   Here, $k$ is expressed as the product of the temperature-frequency conversion factor ($\text{K\_Hz}$), the frequency-mass conversion factor ($\text{Hz\_kg}$), and the mass-energy conversion factor ($\text{kg\_J}$). This highlights its role in connecting temperature to energy ($E=kT$).
   
   

These definitions make it clear that $h$ and $k$ are not just arbitrary constants—they are composite scaling factors that encode the relationships between frequency, mass, temperature, and energy. By expressing them in this way, we separate the intrinsic physics from the artifacts of our unit systems.

3.2. Example: Thermal de Broglie Wavelength

The thermal de Broglie wavelength, which describes the average wavelength of particles in a gas at thermal equilibrium, is traditionally written as:

$${\mathrm{<span\; style="background-color:\; white;">\lambda </span>}}_{\text{<span style="background-color: white;">th</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{\sqrt{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}\mathrm{<span\; style="background-color:\; white;">k</span>}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Using the natural unit scaling framework, we can rewrite this as:

$${\mathrm{<span\; style="background-color:\; white;">\lambda </span>}}_{\text{<span style="background-color: white;">th</span>}}<span\; style="background-color:\; white;">=</span>$$

Here, $f_T$ and $f_m$ are the intrinsic frequency scales associated with temperature and mass, respectively. The factor of $c$ arises from our choice of units, while the dependence on $\sqrt{f_T}$ and $\sqrt{f_m}$ reflects the intrinsic physics.

3.3. Example: Stefan-Boltzmann Law

The Stefan-Boltzmann law, which describes the total energy radiated by a black body, is traditionally written as:

$$\mathrm{<span\; style="background-color:\; white;">\sigma </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">5</span>}}{\mathrm{<span\; style="background-color:\; white;">k</span>}}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}{\mathrm{<span\; style="background-color:\; white;">15</span>}{\mathrm{<span\; style="background-color:\; white;">h</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Using the natural unit scaling framework, we can rewrite this as:

$$\mathrm{<span\; style="background-color:\; white;">\sigma </span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">5</span>}}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">K\_Hz</span>}}^{\mathrm{<span\; style="background-color:\; white;">4</span>}}}{\mathrm{<span\; style="background-color:\; white;">15</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

This simplification removes the explicit dependence on $c$, $h$, and $k$, revealing the intrinsic relationship between energy, frequency, and temperature.

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4. Where the Math Ends and the Universe Begins

The natural unit scaling framework clarifies the boundary between the mathematical formalism we use to describe the universe and the physical reality that exists independently of our descriptions. By isolating the scaling factors, we can distinguish between:

• Real Physics: The intrinsic relationships between physical quantities (e.g., $E=hf$, $\lambda =h\mathrm{/}p$).

• Artifacts of Unit Scaling: The factors that arise from our choice of units (e.g., the numerical values of $c$, $h$, and $k$).

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This distinction reveals where the "math ends and the universe begins," providing a clearer understanding of the structure of physical laws.

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5. Implications and Applications

The natural unit scaling framework has several important implications:

• Simplification of Equations: By removing unnecessary clutter, the framework makes physical equations more elegant and easier to work with.

• Deeper Insight into Constants: The framework reveals the deeper role of constants like $c$, $h$, and $k$ as bridges between human-scale measurements and the intrinsic scales of nature.

• Philosophical Clarity: The framework highlights the relational nature of physical reality, emphasizing that the universe operates according to its own rules, independent of our mathematical models.

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6. Conclusion

The natural unit scaling framework provides a powerful tool for understanding the relationship between mathematical formalism and physical reality. By isolating the scaling factors encoded in fundamental constants, we can separate the artifacts of our unit systems from the intrinsic structure of the universe. This framework not only simplifies physical equations but also reveals where the "math ends and the universe begins," offering deeper insight into the nature of physical constants and the structure of the universe.

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References

• Planck, M. (1900). "Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum." Annalen der Physik.

• Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper." Annalen der Physik.

• Boltzmann, L. (1877). "Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung." Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften.