Planck’s Law as a Dimensional Rotation in Unit Space : The Fundamental Mass-Frequency-Temperature Relationship

Abstract
Planck’s Law of blackbody radiation is traditionally expressed using constants ($h$, $k$, $c$), this conversion to abstract energy that obscure its true physical meaning. By reformulating the law in terms of fundamental ratios—mass per frequency ($\text{Hz\_kg}$) and temperature per frequency ($\text{K\_Hz}$)—we demonstrate that the underlying physics is an invariant equivalence between two properties that are just different measures of the same underlying thing, revealing a purely geometric relationship between oscillation modes and thermal scaling. This work demonstrates that Planck’s Law is not inherently about "energy quantization" but rather a direct consequence of mass-frequency equivalence and temperature-frequency synchronization in natural unit space.

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1. Introduction

Planck’s Law, derived in 1901 to resolve the ultraviolet catastrophe, introduced the "quantum of action" ($h$) and tied thermal radiation to Boltzmann’s constant ($k$). However, these constants are artifacts of human unit choices (SI’s kg, m, s, K) rather than fundamental physics. We show that:

• The law’s core is a frequency-mass-temperature relationship, independent of $h$, $k$, or $c$.

• The "quantum" $h$ is redundant, derivable from $\text{Hz\_kg}=m\mathrm{/}f$.

• Temperature is a frequency scale ($\text{K\_Hz}=T\mathrm{/}f$), not an independent thermodynamic quantity.

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2. The Fundamental Ratios

2.1 Mass-Frequency Equivalence ($\text{Hz\_kg}$)

$$\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">m</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">7.372</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">51</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg/Hz</span>}<span\; style="background-color:\; white;">,</span>$$

where $m$ is the mass-equivalent of an oscillation at frequency $f$. This replaces Planck’s constant:

$$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">(SI crutch)</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

2.2 Temperature-Frequency Equivalence ($\text{K\_Hz}$)

$$\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">T</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}<span\; style="background-color:\; white;">\approx </span>\mathrm{<span\; style="background-color:\; white;">4.799</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">11</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">K/Hz</span>}<span\; style="background-color:\; white;">,</span>$$

where $T$ is the thermal frequency scale. Boltzmann’s constant $k$ is redundant:

$$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">(SI crutch)</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

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3. Planck’s Law in Dimensional rotation in Unit Space Form

The spectral radiance $B(f,T)$ reduces to:

$$\mathrm{<span\; style="background-color:\; white;">B</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2</span>}{\mathrm{<span\; style="background-color:\; white;">f</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz\_kg</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>\text{<span style="background-color: white;">K\_Hz</span>}<span\; style="background-color:\; white;">)</span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">.</span>}$$

Key Features:

• No $h$ or $k$: Replaced by simple direct dimensional rotation in unit space :
  $\text{Hz\_kg}$ and $\text{K\_Hz}$.

• No $c$: Irrelevant for spectral distributions in frequency space.

• Physical Meaning:

  • $f^3\text{Hz\_kg}$: Mass-density spectrum, scaled by $f^3$.
    This rotates the units and value of one  frequency from the frequency axis into a mass axis in unit space.

  • $e^{f\mathrm{/}(T\cdot \text{K\_Hz})}$: Thermal occupation in natural frequency units.
    This rotates the unit and value of the temperature axis to the frequency axis in unit space.  But the exponent can simultaneously be seen as the dimensionless ratio of temperature, frequency, or energy and this is mathematically and algebraically identical. 

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4. Derivation from First Principles

4.1 Mass-Frequency Duality

Oscillators emit mass-equivalent radiation ($m=\text{Hz\_kg}\cdot f$), not "energy quanta."

4.2 Thermal Scaling

Temperature $T$ sets the critical frequency $f_T=T\cdot \text{K\_Hz}$, where modes saturate.

4.3 Spectral Density

The $f^3$ term arises from phase-space volume in frequency units, requiring no reference to $c$.

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5. Implications

5.1 Demystifying "Quantum" Effects

• $E=hf$ is a unit conversion (Hz → J), not evidence of quantization.

• Blackbody spectra reflect mass-frequency distributions, not "energy."

5.2 Eliminating Redundant Constants

• $h$, $k$, and $c$ are coordinate artifacts because we rotate dimensional axis of equivalent properties to differently scaled units of measure, not laws of nature.

• All physics can be expressed via $\text{Hz\_kg}$, $\text{K\_Hz}$, c, and $e$ (charge). These are the definitions of the units of measure of mass, temperature, length, and charge versus time. 

5.3 Experimental Validation

• Predictions match data exactly when $\text{Hz\_kg}$ and $\text{K\_Hz}$ are derived from SI.

• No "adjustable parameters"—just unit rescaling.

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6. Conclusion

Planck’s Law is fundamentally a rotation in unit space, connecting:

$$\text{<span style="background-color: white;">mass</span>}<span\; style="background-color:\; white;">\leftrightarrow </span>\text{<span style="background-color: white;">frequency</span>}<span\; style="background-color:\; white;">\leftrightarrow </span>\text{<span style="background-color: white;">temperature</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

By seeing that  $h$, $k$, and $c$are fundamentally just unit scaling, we separate the artifacts of our own unit system from the law’s direct physical equivalence between T~f~m~E.  

This reformulation:

1. Unifies radiation, thermodynamics, and quantum mechanics as just artifacts of our unit system.

2. Clarifies that "quantization" is a rotation of dimensional axis in unit space artifact.

3. Requests that we consider how this simple view of physics affects our current interpretations of constants that they are mysterious fundamental numbers in the universe.

References

• Planck, M. (1901). On the Theory of the Energy Distribution Law of the Normal Spectrum.

• (The prior work on PUCS and unit rescaling on this blog and on github:
  

        https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System ).