The Unity of Physical Quantities Revealed by Unit Scaling Factors

 J. Rogers, SE Ohio, 14 Mar 2025, 1305

---

Abstract

This report explores the profound unity of physical quantities—such as temperature (T), mass (m), frequency (f), and energy (E)—through the lens of unit scaling factors encoded in the fundamental constants $c$ (speed of light), $h$ (Planck’s constant), and $k$ (Boltzmann’s constant). By isolating and canceling these scaling factors, we demonstrate that these quantities are fundamentally equivalent, and this equivalence extends to other physical concepts like curved spacetime, inertia, momentum, gravity, and worldline scaling. This framework reveals that the apparent complexity of our SI unit system obscures a deeper unity in the laws of physics, which can be made explicit by re-expressing physical formulas in terms of natural units.

---

1. Introduction

The fundamental constants $c$, $h$, and $k$ are traditionally seen as independent physical quantities with specific values in the SI unit system. However, a closer examination reveals that these constants encode unit scaling factors that bridge the gap between natural units (arising from the fundamental laws of physics) and our human-scale SI units. By isolating these scaling factors, we can simplify physical formulas and reveal the underlying unity of physical quantities. This report demonstrates how this framework applies to key physical concepts and shows that the equivalence of temperature, mass, frequency, and energy extends to a broader unification of physics.

2. Unit Scaling Factors Encoded in Fundamental Constants

The fundamental constants $c$ (speed of light), $h$ (Planck’s constant), and $k$ (Boltzmann’s constant) encode the precise unit scaling factors needed to convert between natural units and our SI unit system. By isolating these scaling factors, we can reveal the deep unity of physical quantities and simplify the relationships between them.

2.1. Speed of Light ($c$) and the Meter

The speed of light, $c$, defines the scaling between space (meters) and time (seconds). In natural units, setting $c=1$ establishes the equivalence:

$$\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>meter</span>}<span>=</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>light-second</span>}\mathrm{<span>.</span>}$$

This means that $c$ acts as a conversion factor between space and time. Furthermore, $c^2$ serves as the unit conversion from mass ($m$) to energy ($E$) via the relationship $E=m{c}^2$. This equivalence shows that mass and energy are interchangeable, with $c^2$ acting as the scaling factor. We can define:

$${\text{<span>kg</span>}}_{\text{<span>J</span>}}<span>=</span>{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}<span>,</span>$$

which has units of $\text{J/kg}$.

2.2. Planck’s Constant ($h$) and the Kilogram

Planck’s constant, $h$, relates energy (Joules) to frequency (Hz). By isolating the unit scaling factors within $h$, we can extract the conversion factor ${\text{Hz}}_{\text{kg}}$, which converts frequency (Hz) in natural units to mass (kg) in SI units:

$${\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>=</span>\frac{\mathrm{<span>h</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>,</span>$$

which has units of $\text{kg/Hz}$. This allows us to express $h$ as:

$$\mathrm{<span>h</span>}<span>=</span>{\text{<span>Hz</span>}}_{\text{<span>J</span>}}<span>=</span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>{\text{<span>kg</span>}}_{\text{<span>J</span>}}<span>,</span>$$

with units of $\text{J/Hz}$. This relationship demonstrates the frequency-mass equivalence, where mass ($m$) can be expressed in terms of frequency ($f$):

$$\mathrm{<span>m</span>}<span>=</span>\mathrm{<span>f</span>}<span>\cdot </span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}\mathrm{<span>.</span>}$$

If we were to rescale the kilogram such that ${\text{Hz}}_{\text{kg}}=1$, then:

$$\mathrm{<span>h</span>}<span>=</span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}<span>=</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>kg/Hz</span>}<span>\cdot </span><span>(</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>m/s</span>}{<span>)</span>}^{\mathrm{<span>2</span>}}<span>=</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>J/Hz</span>}\mathrm{<span>.</span>}$$

This rescaling effectively converts the meter and kilogram to natural units, where, for example, $5\text{kg}\sim 5\text{Hz}$.

2.3. Boltzmann’s Constant ($k$) and the Kelvin

Boltzmann’s constant, $k$, relates energy (Joules) to temperature (Kelvin). By isolating the unit scaling factors within $k$, we can extract the conversion factor ${\text{K}}_{\text{Hz}}$, which converts temperature (K) in SI units to frequency (Hz) in natural units:

$${\text{<span>K</span>}}_{\text{<span>Hz</span>}}<span>=</span>\frac{\mathrm{<span>k</span>}}{{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>{\text{<span>kg</span>}}_{\text{<span>J</span>}}}<span>,</span>$$

which has units of $\text{Hz/K}$. This allows us to express $k$ as:

$$\mathrm{<span>k</span>}<span>=</span>{\text{<span>K</span>}}_{\text{<span>J</span>}}<span>=</span>{\text{<span>K</span>}}_{\text{<span>Hz</span>}}<span>\cdot </span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>{\text{<span>kg</span>}}_{\text{<span>J</span>}}<span>,</span>$$

with units of $\text{J/K}$. This relationship demonstrates the temperature-frequency equivalence, where frequency ($f$) can be expressed in terms of temperature ($T$):

$$\mathrm{<span>f</span>}<span>=</span>\mathrm{<span>T</span>}<span>\cdot </span>{\text{<span>K</span>}}_{\text{<span>Hz</span>}}\mathrm{<span>.</span>}$$

If we were to rescale the Kelvin such that ${\text{K}}_{\text{Hz}}=1$, then:

$$\mathrm{<span>k</span>}<span>=</span>{\text{<span>K</span>}}_{\text{<span>Hz</span>}}<span>\cdot </span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>{\text{<span>kg</span>}}_{\text{<span>J</span>}}<span>=</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>Hz/K</span>}<span>\cdot </span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>kg/Hz</span>}<span>\cdot </span><span>(</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>m/s</span>}{<span>)</span>}^{\mathrm{<span>2</span>}}<span>=</span>\mathrm{<span>1</span>}\text{<span>\hspace{0.17em}</span>}\text{<span>J/K</span>}\mathrm{<span>.</span>}$$

2.4. Implications for Unit Scaling

The values of $c$, $h$, and $k$ encode the exact unit scaling factors needed to convert between natural units and our SI unit system. Specifically:

• $c$ defines the scaling between space and time.

• $h$ defines the scaling between frequency and mass.

• $k$ defines the scaling between temperature and frequency.

By isolating these scaling factors, we can see that our SI unit system differs from natural units only by the scaling encoded in these constants. While rescaling our unit system to make ${\text{Hz}}_{\text{kg}}=1$ and ${\text{K}}_{\text{Hz}}=1$ would be impractical for everyday use, it reveals the underlying unity of physical quantities and simplifies the relationships between them. This framework demonstrates that the apparent complexity of our SI unit system obscures a deeper simplicity in the laws of physics, which can be made explicit by re-expressing physical formulas in terms of natural units.

---

3. The Role of Fundamental Constants as Unit Scaling Factors

3.1 Speed of Light ($c$)

The speed of light, $c$, relates space (meters) and time (seconds). In natural units, setting $c=1$ defines the unit of space in terms of the unit of time (e.g., 1 meter = 1 light-second). This equivalence is the foundation of relativity and shows that space and time are interconnected.

3.2 Planck’s Constant ($h$)

Planck’s constant, $h$, relates energy (Joules) and frequency (Hz). In natural units, setting $h=1$ defines the unit of energy in terms of the unit of frequency. This equivalence is the foundation of quantum mechanics and shows that energy and frequency are interconnected.

3.3 Boltzmann’s Constant ($k$)

Boltzmann’s constant, $k$, relates energy (Joules) and temperature (Kelvin). In natural units, setting $k=1$ defines the unit of energy in terms of the unit of temperature. This equivalence is the foundation of statistical mechanics and shows that energy and temperature are interconnected.

---

4. Equivalence of Temperature, Mass, Frequency, and Energy

The relationships encoded in $c$, $h$, and $k$ reveal the equivalence of temperature, mass, frequency, and energy:

• Energy and Mass: $E=m{c}^2$ shows that mass and energy are equivalent, with $c^2$ acting as the conversion factor.

• 
  

• Energy and Frequency: $E=hf$ shows that energy and frequency are equivalent, with $h$ acting as the conversion factor.

• 
  

• Energy and Temperature: $E=kT$ shows that energy and temperature are equivalent, with $k$ acting as the conversion factor.

These relationships mean that $T$, $m$, $f$, and $E$ are all different manifestations of the same underlying physical quantity, just measured in different units.

---

5. Extending the Equivalence to Other Physical Quantities

The equivalence of $T$, $m$, $f$, and $E$ extends further to other physical quantities because they are all interconnected through the fundamental constants and the structure of physics. For example:

5.1 Curved Spacetime

In general relativity, the curvature of spacetime is related to the distribution of energy and mass via the Einstein field equations:

$${\mathrm{<span>G</span>}}_{\mathrm{<span>\mu </span>}\mathrm{<span>\nu </span>}}<span>=</span>\frac{\mathrm{<span>8</span>}\mathrm{<span>\pi </span>}\mathrm{<span>G</span>}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>4</span>}}}{\mathrm{<span>T</span>}}_{\mathrm{<span>\mu </span>}\mathrm{<span>\nu </span>}}<span>,</span>$$

where $G_{\mu \nu }$ describes the curvature of spacetime and $T_{\mu \nu }$ describes the distribution of energy and mass. Since energy and mass are equivalent ($E=m{c}^2$), this means that curved spacetime is also equivalent to energy, mass, frequency, and temperature.

5.2 Inertia and Momentum

Inertia is a property of mass, and momentum ($p$) is related to mass and velocity ($p=mv$). Since mass is equivalent to energy ($E=m{c}^2$), inertia and momentum are also equivalent to energy, frequency, and temperature.

5.3 Gravity

Gravity is the result of curved spacetime, which is itself equivalent to energy and mass. This means that gravity is also equivalent to frequency and temperature.

5.4 Worldline Scaling

In relativity, the worldline of a particle is a path through spacetime, and its scaling is related to the proper time ($\tau$), which depends on the energy and momentum of the particle. Since energy and momentum are equivalent to mass, frequency, and temperature, worldline scaling is also equivalent to these quantities.

---

6. Simplifying Physical Formulas

By isolating and canceling the unit scaling factors encoded in $c$, $h$, and $k$, we can simplify physical formulas and reveal their underlying structure. For example:

6.1 Planck’s Law

The traditional form of Planck’s Law for spectral radiance is:

$$\mathrm{<span>B</span>}<span>(</span>\mathrm{<span>f</span>}<span>,</span>\mathrm{<span>T</span>}<span>)</span><span>=</span>\frac{\mathrm{<span>2</span>}\mathrm{<span>h</span>}{\mathrm{<span>f</span>}}^{\mathrm{<span>3</span>}}}{{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}<span>\cdot </span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>e</span>}}^{\mathrm{<span>h</span>}\mathrm{<span>f</span>}\mathrm{<span>/</span>}<span>(</span>\mathrm{<span>k</span>}\mathrm{<span>T</span>}<span>)</span>}<span>-</span>\mathrm{<span>1</span>}}\mathrm{<span>.</span>}$$

By expanding the constants and canceling redundant factors, we can rewrite this as:

$$\mathrm{<span>B</span>}<span>(</span>\mathrm{<span>f</span>}<span>,</span>\mathrm{<span>T</span>}<span>)</span><span>=</span>\mathrm{<span>2</span>}{\mathrm{<span>f</span>}}^{\mathrm{<span>3</span>}}{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>\cdot </span>\frac{\mathrm{<span>1</span>}}{{\mathrm{<span>e</span>}}^{\mathrm{<span>f</span>}\mathrm{<span>/</span>}<span>(</span>\mathrm{<span>T</span>}{\text{<span>K</span>}}_{\text{<span>Hz</span>}}<span>)</span>}<span>-</span>\mathrm{<span>1</span>}}<span>,</span>$$

which makes it clear that the formula describes the relationship between frequency and temperature without the clutter of unit conversions.

6.2 Stefan-Boltzmann Constant

The traditional form of the Stefan-Boltzmann constant is:

$$\mathrm{<span>\sigma </span>}<span>=</span>\frac{\mathrm{<span>2</span>}{\mathrm{<span>\pi </span>}}^{\mathrm{<span>5</span>}}{\mathrm{<span>k</span>}}^{\mathrm{<span>4</span>}}}{\mathrm{<span>15</span>}{\mathrm{<span>h</span>}}^{\mathrm{<span>3</span>}}{\mathrm{<span>c</span>}}^{\mathrm{<span>2</span>}}}\mathrm{<span>.</span>}$$

By expanding the constants and canceling redundant factors, we can rewrite this as:

$$\mathrm{<span>\sigma </span>}<span>=</span>\frac{\mathrm{<span>2</span>}{\mathrm{<span>\pi </span>}}^{\mathrm{<span>5</span>}}}{\mathrm{<span>15</span>}}<span>\cdot </span>{\text{<span>K</span>}}_{\text{<span>Hz</span>}}^{\mathrm{<span>4</span>}}<span>\cdot </span>{\text{<span>Hz</span>}}_{\text{<span>kg</span>}}<span>,</span>$$

which reveals the underlying simplicity of the formula.

6.3 Thermal de Broglie Wavelength

The traditional form of the thermal de Broglie wavelength is:

$${\mathrm{<span>\lambda </span>}}_{\text{<span>th</span>}}<span>=</span>\frac{\mathrm{<span>h</span>}}{\sqrt{\mathrm{<span>2</span>}\mathrm{<span>\pi </span>}\mathrm{<span>m</span>}\mathrm{<span>k</span>}\mathrm{<span>T</span>}}}\mathrm{<span>.</span>}$$

By expanding the constants and canceling redundant factors, we can rewrite this as:

$${\mathrm{<span>\lambda </span>}}_{\text{<span>th</span>}}<span>=</span>\frac{\mathrm{<span>c</span>}}{\sqrt{\mathrm{<span>2</span>}\mathrm{<span>\pi </span>}{\mathrm{<span>f</span>}}_{\mathrm{<span>m</span>}}{\mathrm{<span>f</span>}}_{\mathrm{<span>T</span>}}}}<span>,</span>$$

where $f_m=m\cdot {\text{kg}}_{\text{Hz}}$ and $f_T=T\cdot {\text{K}}_{\text{Hz}}$. This makes it clear that the formula describes the relationship between mass, temperature, and wavelength without the distraction of unit conversions.

7. Why This Unity Is Not Obvious

The unity of physical quantities is not immediately obvious in our SI unit system for several reasons:

• Historical Development: Physics developed incrementally, with different domains (e.g., mechanics, thermodynamics, electromagnetism) being studied in isolation before their connections were fully understood.
  
  

• Human-Scale Bias: Our intuition is rooted in human-scale experiences, where units like meters, kilograms, and seconds feel natural. The idea of redefining these units in terms of fundamental constants (e.g., $c$, $h$, $k$) was counterintuitive and required a shift in perspective.
  
  

• Complexity of Constants: The constants $c$, $h$, and $k$ encode complex relationships between different physical quantities, and it took time to recognize their role as unit scaling factors.
  
  

It’s entirely possible—and even likely—that we’ve overcomplicated things by not fully recognizing the role of fundamental constants like $c$, $h$, and $k$ as unit scaling factors rather than independent physical quantities. This overcomplication arises from historical development, human-scale biases, and the incremental way physics has evolved over centuries. Let’s break this down and explore why this might be the case.

---

7.1. Historical Development and Incremental Progress

Physics has developed incrementally, with different domains (e.g., mechanics, thermodynamics, electromagnetism, quantum mechanics) being studied in isolation before their connections were fully understood. Each domain introduced its own set of units and constants, which were later found to be interconnected. For example:

• $c$ was discovered in the context of electromagnetism and later became central to relativity.

• $h$ was introduced in quantum mechanics to explain phenomena like the photoelectric effect.

• $k$ emerged in statistical mechanics to connect microscopic particle behavior to macroscopic thermodynamics.

Because these constants were discovered in different contexts, their role as unit scaling factors was not immediately obvious. Instead, they were treated as independent physical quantities with specific values, which added complexity to the equations and obscured their unifying role.

---

7.2. Human-Scale Bias and Practicality

Our intuition is rooted in human-scale experiences, where units like meters, kilograms, and seconds feel natural. This bias led us to develop a unit system (SI units) that is convenient for everyday use but not necessarily aligned with the natural scales of the universe. For example:

• The meter is based on human-scale lengths, not the natural scale of spacetime.

• The kilogram is based on human-scale masses, not the natural scale of energy or frequency.

• The Kelvin is based on human-scale temperatures, not the natural scale of energy or frequency.

This misalignment between our unit system and the natural scales of the universe introduced unnecessary complexity, as we had to constantly use constants like $c$, $h$, and $k$ to convert between these scales.

---

7.3. Overcomplication in Notation and Interpretation

The traditional notation and interpretation of physical formulas often obscure the underlying unity of physical quantities. For example:

• The appearance of $h$ in quantum mechanics is often interpreted as a fundamental feature of quantum theory, rather than a unit scaling factor that converts between frequency and energy.

• The appearance of $k$ in thermodynamics is often interpreted as a fundamental feature of statistical mechanics, rather than a unit scaling factor that converts between temperature and energy.

• The appearance of $c$ in relativity is often interpreted as a fundamental limit, rather than a unit scaling factor that converts between space and time.

By treating these constants as independent physical quantities, we’ve added layers of complexity to our understanding of physics that could be eliminated by recognizing their role as unit scaling factors.

---

7.4. The Simplicity of Natural Units

In natural units, where $c=h=k=1$, the equations of physics become much simpler and more transparent. For example:

• The equivalence of mass and energy ($E=m{c}^2$) becomes $E=m$.

• The equivalence of energy and frequency ($E=hf$) becomes $E=f$.

• The equivalence of energy and temperature ($E=kT$) becomes $E=T$.

This simplification reveals the underlying unity of physical quantities and eliminates the need for arbitrary unit conversions. It also highlights the fact that the complexity of our SI unit system is not intrinsic to the laws of physics but is instead an artifact of our choice of units.

---

7.5. Overcomplication in Theories of Everything

Many "theories of everything" (e.g., string theory, loop quantum gravity) are implicitly or explicitly attempting to describe a universe where the natural units $c=h=k=1$ are inherently true. These theories seek to unify the fundamental forces and constants of nature by revealing deeper symmetries or structures that make the apparent complexity of our universe emerge from a simpler, more fundamental framework.

However, the complexity of these theories often arises from the need to reconcile our human-scale unit system with the natural scales of the universe. By recognizing the role of $c$, $h$, and $k$ as unit scaling factors, we can simplify these theories and reveal their underlying unity.

---

7.6. Conclusion: Yes, We Overcomplicated Things

It’s entirely possible—and even likely—that we’ve overcomplicated things by not fully recognizing the role of fundamental constants as unit scaling factors. This overcomplication arises from historical development, human-scale biases, and the incremental way physics has evolved. By isolating and canceling these scaling factors, we can simplify physical formulas, reveal their underlying structure, and gain deeper insights into the laws of physics.

This doesn’t mean that the work done so far is wrong or unnecessary—it simply means that we can now see the bigger picture and recognize the unity of physical quantities that was always there, hidden beneath the complexity of our unit system. This perspective is a powerful reminder that the laws of physics are fundamentally simple and elegant, and that our task is to uncover this simplicity beneath the apparent complexity.

---

8. Implications for Physics

The unity of physical quantities has profound implications for our understanding of physics:

• Simplification: By recognizing the equivalence of $T$, $m$, $f$, and $E$, and extending this equivalence to other quantities, we can simplify physical formulas and reveal their underlying structure.
  
  

• Universality: The unity of physical quantities highlights the universality of the laws of physics. The same fundamental relationships apply at all scales, from the quantum realm to the cosmological scale.
  
  

• Deeper Insights: By seeing the unity of physical quantities, we can gain deeper insights into the structure of the universe and the relationships between different physical phenomena.
  
  

---

9. Conclusion

The framework presented in this report reveals the deep unity of physical quantities—such as temperature, mass, frequency, and energy—through the lens of unit scaling factors encoded in the fundamental constants $c$, $h$, and $k$. This equivalence extends further to other physical concepts like curved spacetime, inertia, momentum, gravity, and worldline scaling, highlighting the interconnectedness of physics. By isolating and canceling these scaling factors, we can simplify physical formulas, reveal their underlying structure, and gain deeper insights into the laws of physics. This framework demonstrates that the apparent complexity of our SI unit system obscures a deeper unity in the laws of physics, which can be made explicit by re-expressing physical formulas in terms of natural units.