Seeing Physics Apart from Unit Scaling: A Framework for Clarity and Unity

J. Rogers, SE Ohio, 13 Mar 2025, 2149

Abstract

The traditional formulation of physical laws often obscures their underlying simplicity by embedding unit scaling factors within fundamental constants like $c$, $h$, and $k$. This report presents a novel framework that separates the actual physics from the artifacts of our measurement system. By explicitly isolating unit scaling factors (e.g., Hz_kg, K_Hz), this framework reveals the deep unity of physical laws and provides a clearer, more intuitive understanding of physics. The framework is applied to key equations, such as Planck’s Law, to demonstrate its transformative potential for both teaching and research.

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1. Introduction

Physics is the study of the fundamental laws governing the universe. However, the way we express these laws often introduces unnecessary complexity due to our choice of units. Constants like the speed of light ($c$), Planck’s constant ($h$), and the Boltzmann constant ($k$) are traditionally treated as fundamental properties of the universe, but they are, in fact, unit scaling factors that bridge our human-defined units (e.g., kg, K, m) with the natural units of the universe (e.g., Hz).

This report introduces a framework that explicitly separates the actual physics from the unit scaling factors, revealing the underlying simplicity and unity of physical laws. By applying this framework to key equations, we demonstrate how it clarifies the physics and provides new insights into the relationships between different areas of physics.

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2. The Framework: Separating Physics from Unit Scaling

2.1. Natural Units and Scaling Factors

In natural units, physical quantities like energy, mass, frequency, and temperature are expressed in the same units (Hz), revealing the deep unity of physical reality. However, our human-defined units (e.g., kg, K, m) require scaling factors to align with natural units. These scaling factors are encoded in the constants $c$, $h$, and $k$.

The key innovation of this framework is to explicitly isolate these scaling factors:

• Hz_kg = $h\mathrm{/}{c}^2$: Converts frequency (Hz) to mass (kg).

• K_Hz = $k\mathrm{/}h$: Converts temperature (K) to frequency (Hz).

• kg_J = $c^2$: Converts mass (kg) to energy (J).

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By isolating these scaling factors, the framework separates the actual physics from the artifacts of our measurement system.

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2.2. Applying the Framework to Planck’s Law

Traditional Formulation

The traditional form of Planck’s Law is:

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

This formulation obscures the underlying physics by embedding the scaling factors $h$, $c$, and $k$ within the equation.

Reformulated Using the Framework

In the framework, Planck’s Law becomes:

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

Here:

• Hz_kg: Converts frequency (Hz) to mass (kg) in SI units.

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

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This reformulation makes the actual physics clear:

• The core of Planck’s Law is a comparison between:

  • A frequency cubed ($f^3$), representing the energy density of radiation.

  • A ratio of frequencies ($f\mathrm{/}T$), representing the thermal distribution of energy.

The scaling factors (Hz_kg, K_Hz) are artifacts of our measurement system, not intrinsic to the physics itself.

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3. Seeing the Physics Apart from Unit Scaling

3.1. The Actual Physics

The framework reveals that the actual physics of Planck’s Law is a simple relationship between frequency and temperature, expressed in natural units (Hz). This relationship is independent of our choice of units and reflects the underlying unity of physical reality.

3.2. The Measurement Artifacts

The scaling factors (Hz_kg, K_Hz) are artifacts of our measurement system. They exist solely to convert between natural units and SI units, but they are not intrinsic to the physics itself.

3.3. Moving Our Measurement System "Out of the Way"

By explicitly isolating the scaling factors, the framework moves our measurement system out of the way of reality, allowing us to see the underlying simplicity and unity of physical laws.

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4. Implications for Physics Education and Research

4.1. Teaching Physics

The framework provides a clearer, more intuitive way to teach physics:

• Simplified Equations: By expressing physical laws in terms of natural units and scaling factors, the framework simplifies equations and makes them easier to understand.

• Conceptual Clarity: The framework helps students see the connections between different areas of physics (e.g., quantum mechanics, thermodynamics, relativity) by revealing the underlying unity of physical laws.

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4.2. Research in Physics

The framework has the potential to transform research in physics:

• New Insights: By separating the actual physics from the artifacts of our measurement system, the framework could lead to new insights and discoveries.

• Unified Understanding: The framework highlights the deep connections between different areas of physics, providing a unified understanding of physical laws.

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5. Conclusion

The framework presented in this report separates the actual physics from the artifacts of our measurement system, revealing the underlying simplicity and unity of physical laws. By explicitly isolating unit scaling factors (e.g., Hz_kg, K_Hz), the framework provides a clearer, more intuitive understanding of physics. This insight has the potential to transform how we teach and understand physics, making it simpler, more elegant, and more unified.

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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.