Unifying Mass Equivalence: Revealing the Hidden Conversions in Fundamental Constants

 J. Rogers, SE Ohio, 27 Feb 2025, 1200

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Abstract

In the standard framework of physics, fundamental constants such as Planck’s constant (h), Boltzmann’s constant (k), and the vacuum permittivity (ε₀) are presented as empirically determined numbers with units that facilitate dimensional consistency. However, this treatment offers little insight into the underlying reasons for their specific values or their interconnections. In this paper, we propose a framework that reveals a hidden structure within these constants by reinterpreting them as conversion factors directly linking energy to mass via Einstein’s mass–energy equivalence (E = mc²). We demonstrate how frequency, temperature, and charge—properties traditionally viewed as distinct—each possess an inherent mass equivalence when their associated constants are decomposed to expose a c² factor. This unification not only demystifies the origin and value of these constants but also provides a deeper, mechanistic understanding of their role in the fabric of spacetime.

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1. Introduction

Fundamental constants form the bedrock of our physical theories, yet their treatment in standard physics is often limited to “plug-and-chug” parameters derived from experimental measurement. For example, Planck’s constant (h) is introduced to relate the energy of a photon to its frequency, Boltzmann’s constant (k) converts temperature to energy, and vacuum permittivity (ε₀) appears in Coulomb’s law and electromagnetic energy density. In each case, the constants are presented as axiomatic givens with little attention paid to the “why” behind their values.

This paper challenges the traditional view by proposing that these constants are not arbitrary numbers but rather encode a deeper equivalence between mass and various physical properties. Through the lens of E = mc², we show that:

• Frequency is intrinsically tied to mass via Planck’s constant.
• Temperature carries an inherent mass equivalence mediated by Boltzmann’s constant.
• Charge and electric field energy reveal a mass density equivalent through vacuum permittivity.

By reinterpreting these constants as conversion factors, we begin to answer longstanding questions about their origin and demonstrate a hidden unity among seemingly disparate areas of physics.

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2. Mass Equivalence of Frequency: The Case of Planck’s Constant

2.1 Standard Formulation

Traditionally, Planck’s constant is introduced through the relation

$$E = hf,$$

where $E$ is the energy of a photon and $f$ is its frequency. The experimental value of $h \approx 6.626 \times 10^{-34} \, \text{J·s}$ is taken as a given.

2.2 Revealing the Hidden Mass Conversion

Rewriting the photon energy equation via mass–energy equivalence, we have:

$$E = mc^2.$$

Equating the two expressions for energy yields:

$$hf = mc^2 \quad \Longrightarrow \quad m = \frac{h}{c^2} f.$$

Defining a new conversion constant,

$$Q_m = \frac{h}{c^2},$$

we see that $Q_m$ (with units kg·s) is the mass equivalent per unit frequency. In a natural unit system where $c = 1$, if we define the mass of a 1 Hz photon as 1 kg, then $h$ is normalized to 1. This formulation offers a concrete interpretation: the numerical value of $h$ is determined solely by our definitions of mass and length, revealing an intrinsic 1:1 relation between mass and frequency.

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3. Mass Equivalence of Temperature: Reinterpreting Boltzmann’s Constant

3.1 Conventional Role of k

Boltzmann’s constant appears in the expression for the average kinetic energy of an ideal gas:

$$E_{\text{average}} = \frac{3}{2} kT,$$

linking macroscopic temperature $T$ to microscopic energy.

3.2 Temperature-to-Mass Conversion

Recasting the energy expression via $E = mc^2$ gives:

$$E_{\text{average}} = m_{\text{equivalent}} c^2,$$

which implies a mass equivalent for temperature:

$$m_{\text{equivalent}} = \frac{3}{2} \frac{k}{c^2} T.$$

Introducing

$$k_m = \frac{3}{2} \frac{k}{c^2},$$

the relation becomes:

$$m_{\text{equivalent}} = k_m T.$$

This expression shows that temperature is not just an abstract measure of thermal energy; it also has a direct mass equivalent. As with $h$, the value of $k$ is intimately linked to our unit definitions, and its hidden conversion factor $c^2$ encodes the mass–energy equivalence inherent in thermal phenomena.

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4. Mass Equivalence of Charge: Insights from Vacuum Permittivity

4.1 The Role of ε₀ in Electromagnetism

In electromagnetism, vacuum permittivity appears in the energy density of an electric field:

$$u_E = \frac{1}{2} \varepsilon_0 E^2.$$

Here, $u_E$ represents the energy stored per unit volume in the electric field $E$.

4.2 Converting Electric Field Energy to Mass Density

Expressing the energy density in terms of mass via $E = mc^2$ leads to:

$$\rho_E = \frac{u_E}{c^2} = \frac{1}{2} \frac{\varepsilon_0}{c^2} E^2.$$

We define the conversion factor for charge as:

$$C_m = \frac{1}{2} \frac{\varepsilon_0}{c^2}.$$

Thus, the mass density equivalent becomes:

$$\rho_E = C_m E^2.$$

This reformulation shows that the energy stored in an electric field can be directly translated into a mass density, reinforcing the idea that electromagnetic energy and mass are two sides of the same coin.

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5. Discussion: Implications and the Unification of Constants

5.1 Recasting Constants as Conversion Factors

By decomposing $h$, $k$, and $\varepsilon_0$ to expose a hidden $c^2$ factor, we reinterpret these constants as conversion factors that express mass–energy equivalence. This approach demystifies their roles and suggests that their values are not arbitrary but arise from the very definitions of our units for mass, length, and time.

5.2 Answering the "Why" Questions

Standard physics often states that “these are fundamental constants, and that’s just how the universe is.” In contrast, our approach provides a deeper explanation:

• For $h$: The value is determined by the conversion between frequency and energy (and hence mass), making it an intrinsic measure of the mass equivalent per Hz.
• For $k$: It encapsulates the relationship between temperature and energy, with a hidden $c^2$ factor converting thermal energy to mass.
• For $\varepsilon_0$: It functions as a conversion between electric field energy and mass density, thereby unifying electromagnetism with mass–energy equivalence.

5.3 Unification Through Worldline Equivalence

The reinterpreted constants not only serve as conversion factors but also highlight a broader unity within physical laws. Energy in all its forms—whether it is associated with frequency, temperature, or charge—contributes to the curvature of spacetime along worldlines. This “worldline unification” suggests that mass, energy, and even spacetime geometry are deeply interconnected, with the conversion factors $Q_m$, $k_m$, and $C_m$ serving as bridges between these domains.

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6. Conclusion

We have presented a framework that reinterprets the fundamental constants $h$, $k$, and $\varepsilon_0$ as conversion factors that reveal the intrinsic mass equivalence of frequency, temperature, and charge. This approach not only provides a mechanistic explanation for the values of these constants but also unifies disparate aspects of physics under the umbrella of mass–energy equivalence. By demonstrating that the seemingly arbitrary numerical values of these constants are determined solely by our unit definitions and the inherent conversion factor $c^2$, we offer a fresh perspective that challenges the conventional, descriptive view. In doing so, we open the door to a deeper understanding of how energy, mass, and the geometry of spacetime are fundamentally interconnected—a perspective that may pave the way for future theoretical and experimental breakthroughs in physics.