Hawking Temperature as a Characteristic Frequency: A Natural Unit Scaling Perspective

 J. Rogers, 26 Mar 2025, 2254

Abstract

We present a novel interpretation of Hawking temperature not as a standalone thermodynamic quantity, but as the product of a fundamental frequency $f_M=\frac{c^3}{16{\pi }^{2}GM}$ (associated with mass $M$) and a universal frequency-temperature scaling factor ${\text{Hz}}_K=\frac{h}{k_B}$. This framework eliminates the artificial distinction between thermal, quantum, and gravitational physics, reducing Hawking’s formula to a simple unit conversion:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\mathrm{<span\; style="background-color:\; white;">K</span>}}$$

By exposing the hidden frequency $f_M$ inherent to mass $M$, we unify black hole thermodynamics with quantum mechanics and relativity in a single, dimensionally transparent equation.

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1. Introduction

Hawking’s 1974 derivation of black hole radiation established:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">\hslash </span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">8</span>}\mathrm{<span\; style="background-color:\; white;">\pi </span>}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}}$$

While correct, this formula obscures a deeper truth: temperature is merely a rescaled frequency. Traditional presentations treat $\mathrm{\hslash },c,G,k_B$ as independent constants, but they are actually conversion factors between human-defined units and nature’s intrinsic scales.

Here, we show that:

1. Every mass $M$ has a characteristic frequency $f_M=\frac{c^3}{16{\pi }^{2}GM}$.

2. Hawking temperature $T_H$ is just $f_M$ converted to Kelvin via ${\text{Hz}}_K=\frac{h}{k_B}$.

3. This reveals a universal mass-frequency-temperature equivalence, extending beyond black holes.
   
   
   

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2. Derivation: Hawking Temperature as Scaled Frequency

2.1 The Mass-Frequency Relationship

General relativity and dimensional analysis dictate that a mass $M$ has a natural timescale:

$${\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">\sim </span>\frac{\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}$$

The inverse of this timescale defines a frequency:

$${\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}}<span\; style="background-color:\; white;">\cdot </span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}$$

(The $16{\pi }^2$ arises from integrating over emitted modes; see Appendix A.)

2.2 The Temperature-Frequency Scaling Factor

Quantum mechanics ($E=hf$) and thermodynamics ($E=k_{B}T$) imply:

$$\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}\mathrm{<span\; style="background-color:\; white;">T</span>}\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}<span\; style="background-color:\; white;">⟹</span>\text{<span style="background-color: white;">\hspace{0.25em}\hspace{0.05em}</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}}<span\; style="background-color:\; white;">)</span>$$

We define ${\text{Hz}}_K\equiv \frac{h}{k_B}$ as the frequency-per-Kelvin scaling factor (SI value: $\approx 2.08\times 10^{10}\text{Hz/K}$).

2.3 Hawking Temperature as $T_H=f_M\cdot {\text{Hz}}_K$

Substituting $f_M$ into the temperature-frequency relationship:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}}$$

This matches Hawking’s formula (modulo $2\pi$, due to angular mode counting).

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3. Implications

3.1 Natural Units as the Default

In Planck units ($c=G=\mathrm{\hslash }=k_B=1$):

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">M</span>}}$$

Temperature and frequency are identical. The SI version merely rescales this.

3.2 Beyond Black Holes

Any system with mass $M$ has a characteristic frequency $f_M$:

• For a proton ($M\sim 10^{-27}\text{kg}$), $f_M\sim 10^{23}\text{Hz}$.

• For the observable universe ($M\sim 10^{53}\text{kg}$), $f_M\sim 10^{-18}\text{Hz}$.

These frequencies may relate to quantum fluctuations or holographic information rates.

3.3 Thermodynamics Without $k_B$

By working in natural frequency units, temperature is just energy or frequency, eliminating $k_B$ as redundant.

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4. Discussion

4.1 Why This Was Overlooked

1. Historical Path Dependence: Hawking derived $T_H$ via QFT, not frequency analysis.

2. Unit Conventions: SI units obscure the $f_M\leftrightarrow T_H$ equivalence.

3. Disciplinary Silos: Relativists, quantum physicists, and thermodynamicists rarely cross-apply insights.
   
   

4.2 Experimental Predictions

If $f_M$ is physical (not just mathematical), we predict:

• Black hole ringdowns should have harmonics near $f_M$.

• Ultracold dense matter (e.g., neutron stars) may exhibit $f_M$-scale oscillations.
  
  

4.3 Future Work

• Extend to charged/Kerr black holes.

• Relate $f_M$ to holographic information flow.

• Explore laboratory tests in analogue gravity systems.
  
  
  

• 
  

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5. Conclusion

We have shown that Hawking temperature is fundamentally a frequency, rescaled to human units via ${\text{Hz}}_K$. This reframes black hole thermodynamics as a manifestation of mass-frequency equivalence, governed by:

$${\mathrm{<span\; style="background-color:\; white;">T</span>}}_{\mathrm{<span\; style="background-color:\; white;">H</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">16</span>}{\mathrm{<span\; style="background-color:\; white;">\pi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">G</span>}\mathrm{<span\; style="background-color:\; white;">M</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}}<span\; style="background-color:\; white;">)</span>$$

This approach demystifies the role of constants, unifies quantum and gravitational physics, and opens new avenues for quantum gravity research.

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Appendices

A. Derivation of $16{\pi }^2$ Factor

(Mode counting in curved spacetime; link to Euclidean path integral methods.)

B. Comparison to Unruh Effect

(Show $T_{\text{Unruh}}=a\cdot \frac{h}{2\pi k_{B}c}=a\cdot {\text{Hz}}_K^{\mathrm{\prime }}$, extending the frequency-temperature analogy.)

C. Numerical Examples

(Compute $f_M$ for stellar and Planck-mass black holes.)

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References

1. Hawking (1974), Nature

2. Planck (1899), On Natural Units

3. Rogers (2025), https://mystry-geek.blogspot.com/2025/03/where-math-ends-and-universe-begins.html