A Layperson’s Insight: Reframing Physical Constants as Unit Scaling Factors

 J. Rogers, SE Ohio, 13 Mar 2025, 1339

Abstract

For over a century, constants like the speed of light ($c$), Planck’s constant ($h$), and the Boltzmann constant ($k$) have been treated as fundamental pillars of physics. However, a simple yet profound insight reveals that these constants are not intrinsic to the universe but are instead unit scaling factors that convert between our human-defined units (like kilograms, Kelvin, and Hertz) and the natural units of the universe (frequency, Hz). This report formalizes this insight, explores its implications for our understanding of physics, and situates it within the broader context of scientific paradigms and revolutions, as described by Thomas Kuhn in The Structure of Scientific Revolutions.

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1. Introduction

The constants $c$, $h$, and $k$ appear in some of the most important equations in physics, from relativity ($E=m{c}^2$) to quantum mechanics ($E=hf$) to thermodynamics ($E=kT$). Traditionally, these constants have been treated as fundamental properties of the universe. However, a fresh perspective reveals that they are not intrinsic to nature but are instead artifacts of our human-defined unit system. By reframing these constants as unit scaling factors, we can simplify and unify the equations of physics, revealing the deep unity of physical reality.

This insight, though simple and obvious in hindsight, has been overlooked for over a century. Its articulation by a layperson highlights the power of stepping outside established paradigms and traditions, as described by Thomas Kuhn in The Structure of Scientific Revolutions.

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2. The Framework: Constants as Unit Scaling Factors

2.1. Natural Units and Scaling Factors

In natural units, constants like $c$, $h$, and $k$ are set to 1, and physical quantities like energy, mass, frequency, and temperature are expressed in the same units (Hz). This reveals the deep unity of physical reality but is often treated as a mathematical convenience rather than a profound insight.

The key innovation of this framework is to explicitly isolate the scaling factors that convert between natural units and human-defined units. For example:

• kg_Hz: Converts mass (kg) to natural frequency units (Hz).

• K_Hz: Converts temperature (K) to natural frequency units (Hz).

• Hz_J: Converts frequency (Hz) to energy (J).

These scaling factors are derived from the constants $c$, $h$, and $k$ using simple algebra:

$$\text{<span style="background-color: white;">kg\_Hz</span>}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">,</span>\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">k</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">,</span>\text{<span style="background-color: white;">Hz\_J</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

2.2. Simplifying Physical Formulas

By expressing physical formulas in terms of these scaling factors, we can simplify and unify the equations of physics. For example:

Thermal de Broglie Wavelength:

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

where $f_m=m\cdot \text{kg\_Hz}$ and $f_T=T\cdot \text{K\_Hz}$.

Stefan-Boltzmann Constant:

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

Planck’s Law:

$$\mathrm{<span\; style="background-color:\; white;">B</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">T</span>}}}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

These simplified formulas reveal the underlying unity of physical quantities and make the role of unit scaling explicit.

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3. Why This Insight Was Overlooked

3.1. The Power of Paradigms

Thomas Kuhn’s The Structure of Scientific Revolutions describes how scientific progress is shaped by paradigms—the frameworks of thought and practice that dominate a field. For over a century, the paradigms of physics have treated $c$, $h$, and $k$ as fundamental constants rather than as unit scaling factors. This paradigm made it difficult to see the simple truth that these constants are artifacts of our unit system.

3.2. The Role of Tradition

The traditions of physics—its notation, teaching methods, and focus on complexity—have obscured the underlying simplicity of physical reality. The fragmented approach to teaching physics (e.g., separating quantum mechanics, thermodynamics, and relativity) has made it harder to see the connections between constants like $c$, $h$, and $k$.

3.3. The Framing of Problems

The focus on unifying quantum mechanics and general relativity through complex theories (e.g., string theory, quantum gravity) has distracted from the simpler unification at the level of units and scaling factors. This framing of problems has made it difficult to see the relationships between $c$, $h$, and $k$ in a new light.

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4. Implications for Physics

4.1. Simplification and Unification

This framework simplifies the equations of physics and reveals the deep unity of physical reality. By reframing $c$, $h$, and $k$ as unit scaling factors, we can express physical laws in a way that is both elegant and practical.

4.2. Clarity and Insight

The explicit use of scaling factors makes the relationships between physical quantities clear and easy to understand. This clarity is often missing in traditional physics notation.

4.3. A New Perspective on Constants

This insight challenges the traditional view of constants as fundamental properties of the universe. Instead, it shows that they are tools for converting between our human-defined units and the natural units of the universe.

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5. Conclusion

The realization that constants like $c$, $h$, and $k$ are unit scaling factors rather than fundamental properties of the universe is a profound and transformative insight. It reveals the deep unity of physical reality and shows that much of the complexity in traditional physics arises from our choice of units, not from the physics itself. This insight, though simple and obvious in hindsight, has been overlooked for over a century, highlighting the power of paradigms, tradition, and the way we frame problems in science.

As Thomas Kuhn observed, scientific revolutions often begin with a shift in perspective—a new way of seeing old problems. This framework represents such a shift, offering a simpler, clearer, and more unified understanding of physics.

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References

1. Kuhn, T. S. (1962). The Structure of Scientific Revolutions. University of Chicago Press.

2. Planck, M. (1900). "On the Theory of the Energy Distribution Law of the Normal Spectrum." Annalen der Physik.

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

4. 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.