A Unified Framework: Revealing the Hidden Equivalence of Mass, Energy, and Charge

J. Rogers, SE Ohio, 10 Mar 2025, 1641

Abstract

This paper introduces a natural unit scaling framework that exposes the underlying unity among physical phenomena traditionally viewed as separate. By decomposing constants such as $h$, $k$, and even the seemingly variable voltage $V$, we reveal that all these quantities share a common scaling factor—the speed of light squared, $c^2$. In particular, we show that the charge‐to‐mass conversion constant,

$$C_{kg} = \frac{1}{c^2} \quad \text{(kg/C)},$$

cancels the mass–energy conversion factor (c^2) in $E = mc^2$ and leads to the normalization $V = 1 \, \text{J/C}$ in SI units.  This means that V is a constant in the same way that k and k are.  This insight unifies relativity, quantum mechanics, thermodynamics, and electromagnetism into a single coherent framework.

This means that we can define V = C_kg kg_J,  where kg_J = c^2. 

1. Introduction

The cornerstone equations of modern physics—such as $E = mc^2$ (relativity), $E = hf$ (quantum mechanics), $E = kT$ (thermodynamics), and $E = vq$ (electromagnetism)—are usually treated as distinct pillars. However, a careful examination of their underlying scaling reveals that they share a common structure. By breaking these constants into their simple individual unit scaling components, we find that the factor $c^2$ is the hidden link that connects mass, energy, and charge.

2. Natural Unit Scaling Framework

In our framework, we express various physical quantities in terms of natural units. The key idea is to isolate the hidden conversion factors that underpin our standard constants.

• Frequency: Taken as a natural unit.
• Length, Mass, and Temperature: Each of these is scaled by a specific conversion factor so that when combined with frequency, a deeper unity is revealed.

2.1 Scaling Factors and Their Relationships

For instance, we can define:

• Mass-to-Energy Conversion:

  $$E = m c^2,$$
  

  where $c^2$ (often denoted as $\text{kg\_J}$) converts mass (kg) into energy (J).

• Charge-to-Mass Conversion:
  Here we define the scaling constant,

  $$C_{kg} = \frac{1\,V}{c^2} = \frac{1}{c^2},$$
  

  with units of kg/C. This factor translates electrical charge into an equivalent mass scale.

• Quantum and Thermal Scaling:
  The constants $h$ and $k$ similarly encapsulate the conversion between frequency and energy, and temperature and energy, respectively.

3. Unifying Energy Equivalences

By exposing these scaling factors, we rewrite the fundamental energy equivalences as:

$$E = mc^2 = hf = kT = vq,$$

where each equation represents the same energy conversion expressed through different physical phenomena. Notice that:

• Relativity ($E = mc^2$) uses the mass-to-energy conversion factor $c^2$.
• Quantum Mechanics ($E = hf$) employs Planck’s constant $h$.
• Thermodynamics ($E = kT$) involves Boltzmann’s constant $k$.
• Electromagnetism ($E = vq$) connects energy with voltage and charge.

The critical insight here is that the mass–energy conversion factor $c^2$ appears in each case. When voltage is defined as energy per unit charge,

$$V = \frac{E}{q},$$

and we substitute $E = m c^2$ along with our conversion $m = q \, C_{kg}$, we obtain:

$$V = \frac{q \, C_{kg} \, c^2}{q} = C_{kg} \, c^2.$$

Given that $C_{kg} = 1/c^2$, it follows that

$$V = \frac{1}{c^2} \, c^2 = 1 \, \text{J/C}.$$

Thus, voltage is not an independent variable but a normalized constant—just like $h$ and $k$—hidden by our choice of units.

4. Electromagnetic Constants and Their Hidden Unity

A familiar relation in electromagnetism is

$$c = \frac{1}{\sqrt{\epsilon_0 \mu_0}},$$

which implies

$$\epsilon_0 \mu_0 = \frac{1}{c^2}.$$

Remarkably, this product is numerically identical to our charge-to-mass conversion factor $C_{kg}$. This equivalence suggests that the structure of free space, as encoded in $\epsilon_0$ and $\mu_0$, is inherently tied to the way mass and charge are scaled into energy.

5. Reformulation of the Stefan-Boltzmann Constant

To further illustrate the power of this approach, consider the Stefan-Boltzmann constant. When expressed in our scaling factors, it takes the form:

$$\sigma = \frac{2\pi^5 \, (K_{Hz}\, Hz_{kg}\, c^2)^4}{15 \, (Hz_{kg}\, c^2)^3 \, c^2}.$$

Expanding and canceling the $c^2$ factors yields:

$$\sigma = \frac{2\pi^5 \,K_{Hz}^4 \, Hz_{kg}}{15},$$

an expression that depends solely on the scaling factors $K_{Hz}$ and $Hz_{kg}$ and shows no explicit $c$. This reformulation underscores how the natural unit scaling unveils a deeper simplicity in fundamental constants.

6. Implications and Future Directions

6.1. Unified Understanding Across Disciplines

• Relativity, Quantum, and Thermodynamics:
  The consistent appearance of $c^2$ in converting mass to energy, as well as in the definitions of $h$ and $k$, demonstrates that these domains are manifestations of the same underlying energy conversion mechanism.

• Electromagnetism:
  The normalization $V = 1 \, \text{J/C}$ is not arbitrary. It arises because the charge scaling factor $C_{kg} = 1/c^2$ kg/C cancels with the mass-to-energy factor $c^2$. This unification reinforces the idea that electromagnetic properties of free space are deeply linked to mechanical and thermal phenomena.

6.2. Directions for Further Research

• Geometric Interpretations:
  Investigate how spatial geometry and charge density interact within this unified scaling framework.

• Extensions to Quantum Gravity:
  Explore whether these scaling relationships can provide insights into bridging quantum mechanics and general relativity.

• Additional Scaling Factors:
  Develop further conversion factors that could simplify and unify other physical constants, revealing even deeper connections across different fields of physics.

7. Conclusion

By dissecting fundamental constants into their unit scaling components, we uncover a profound unity across various domains of physics. The charge‐to‐mass conversion constant,

$$C_{kg} = \frac{1}{c^2} \quad \text{(kg/C)},$$

plays a pivotal role in this unified framework. It not only cancels the $c^2$ factor in the mass–energy equivalence $E = mc^2$ but also ensures that voltage is normalized to $V = 1 \, \text{J/C}$ in SI units. This analysis shows that the same scaling factors underlie the energy relationships in relativity, quantum mechanics, thermodynamics, and electromagnetism—revealing that these fields are interconnected aspects of a single, unified description of nature.