The Quantum of Relative Mass: A New Framework for Photons and Fundamental Constants

 J. Rogers, SE Ohio, 23 Jan 2025, 0124

Abstract

This paper introduces the concept of the Quantum of Relative Mass, a dynamic, frequency-dependent property of photons that arises from their interaction with spacetime. By redefining Planck’s constant ($h$) as a quantum of relative mass rather than a quantum of energy, we establish a unified framework that connects the effective mass of a photon ($m$), the speed of light ($c$), and the gravitational constant ($G$). This framework provides a deeper understanding of the interplay between quantum mechanics, relativity, and gravity, and offers a pathway toward unifying these fundamental theories.

---

1. Introduction

The traditional interpretation of Planck’s constant ($h$) as a quantum of energy has been a cornerstone of quantum mechanics since its inception. However, this perspective treats energy as the fundamental quantity, with mass being a secondary property derived via $E=m{c}^2$. In this paper, we propose a paradigm shift: reinterpreting $h$ as a quantum of relative mass, where the effective mass of a photon ($m$) is a dynamic, frequency-dependent property that scales with its interaction with spacetime. This framework reveals deep connections between $h$, $c$, and $G$, and provides a unified description of photons and fundamental constants.

---

2. Definitions and Key Concepts

2.1. The Quantum of Relative Mass ($m$)

The quantum of relative mass is defined as:

$$\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

where:

• $h$ is Planck’s constant ($6.62607015\times 10^{-34}\text{J}\cdot \text{s}$),

• $c$ is the speed of light ($2.99792458\times 10^8\text{m/s}$).
  $c$m is the quantum of relative mass  ($2.99792458\times 10^8\text{m/s}$).
  

  
  

This quantity has units of $\text{kg}\cdot \text{s}$, representing a mass-time property. This is the quantum of relative mass that has always been encoded into h along with two powers of c.  To obtain a pure mass, $m$ must be scaled by frequency ($f$):

$$\text{<span style="background-color: white;">Observed mass</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">f</span>}$$

2.2. Frequency Scaling

The effective mass of a photon is not a fixed property but rather a dynamic one that scales with frequency. This effective mass is scaled by its interactions with space time through its worldline. This reflects the photon’s energy and momentum, which are also frequency-dependent:

• At 1 Hz, the effective mass is $m=\frac{h}{c^2}$.

• For a photon of frequency $f$, the effective mass is $m\cdot f$.

• 
  

---

3. Fundamental Equations

3.1. Planck’s Constant ($h$)

Planck’s constant is redefined in terms of the quantum of relative mass:

$$\mathrm{<span\; style="background-color:\; white;">h</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

This equation shows that $h$ encodes the effective mass of a 1 Hz photon ($m$) and the speed of light ($c$).

3.2. Energy of a Photon ($E$)

The energy of a photon is expressed in terms of its effective mass:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}$$

This generalizes the traditional equation $E=h\nu$, where $\nu$ is the frequency.

3.3. Momentum of a Photon ($p$)

The momentum of a photon is proportional to its effective mass:

$$\mathrm{<span\; style="background-color:\; white;">p</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}$$

3.4. Gravitational Constant ($G$)

The gravitational constant is expressed in terms of the quantum of relative mass and the Planck mass ($m_P$):

$$\mathrm{<span\; style="background-color:\; white;">G</span>}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

where:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}<span\; style="background-color:\; white;">=</span>\sqrt{\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">c</span>}}{\mathrm{<span\; style="background-color:\; white;">G</span>}}}$$

This equation reveals that $G$ is not an independent constant but rather a derived quantity that emerges from the interplay between $m$, $c$, and $m_P$.

---

4. Implications of the Framework

4.1. Unification of Constants

The framework reveals a deep connection between $h$, $c$, and $G$, suggesting that these constants are not independent but rather interrelated through the quantum of relative mass ($m$). This provides a pathway toward unifying quantum mechanics and general relativity.

4.2. Dynamic Nature of Mass

The effective mass of a photon is a dynamic property that scales with frequency, reflecting the photon’s energy and momentum. This aligns with general relativity, where energy and momentum contribute to spacetime curvature.

4.3. Physical Interpretation of $h$ and $G$

The framework provides a physical interpretation of $h$ and $G$ as quantities that emerge from the effective mass of a photon and its interaction with spacetime. This shifts our understanding of these constants from abstract mathematical entities to fundamental properties of the universe.

4.4 The Primacy of Mass

1. Mass as a Fundamental Property:
   
   

   • Mass is a direct physical property that describes the interaction of an object with spacetime. It is measurable, observable, and tied to the curvature of spacetime in general relativity.
     
     

   • In this framework, the effective mass of a photon ($m$) is a dynamic, frequency-dependent property that arises from the photon’s energy and momentum. This mass is not a rest mass but rather a manifestation of the photon’s interaction with spacetime.

   • 
     

2. Planck’s Constant ($h$) as a Composite Quantity:

   • 
     

   • Planck’s constant ($h$) is traditionally treated as a fundamental constant, but in your framework, it is revealed to be a composite quantity:
     
     

     $$\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">h</span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">=</span></span>\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">m</span></span>}<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">\cdot </span></span>{\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">c</span></span>}}^{\mathrm{<span\; style="background-color:\; white;"><span\; style="font-family:\; georgia;\; font-size:\; x-small;">2</span></span>}}$$

   • 
     Here, $h$ is built from the effective mass of a photon ($m$) and the speed of light ($c$). This means $h$ is not fundamental but rather a derived quantity that encodes the relationship between mass and spacetime.

3. 
   

4. Why Mass is More Fundamental:
   
   

   • Mass is a pure physical property that can be directly linked to the curvature of spacetime and the behavior of particles. It is not dependent on other constants or units. The quantum of relative mass has a specific value at a specific frequency
     
     

   • In contrast, $h$ is a mathematical construct that combines mass, length, and time. While it is useful for calculations, it obscures the underlying physical reality of mass.
     
     

---

5. Discussion

The Quantum of Relative Mass framework challenges the traditional view of $h$ as a quantum of energy and instead treats it as encoding a quantum of relative mass. This reinterpretation provides a unified description of photons and fundamental constants, offering new insights into the interplay between quantum mechanics, relativity, and gravity. Future work will explore the experimental implications of this framework.

---

6. Conclusion

The Quantum of Relative Mass framework is a profound and elegant reimagining of the relationships between $h$, $c$, $G$, and the effective mass of a photon. By redefining $h$ as a quantum of relative mass and showing how this mass scales with frequency, we establish a consistent and insightful framework for understanding the interplay between quantum mechanics, relativity, and gravity. This work has the potential to revolutionize our understanding of physics and bridge the gap between these fundamental theories.

---

References

1. 
   Planck, M. (1901). "On the Law of Distribution of Energy in the Normal Spectrum." Annalen der Physik.

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