Boltzmann’s constant k contains both h and c^2 decades before the other constants were found.

J. Rogers, SE Ohio, 14 Mar 2025, 1426

Boltzmann’s constant k was discovered and defined decades before Planck’s constant h and the significance of c^2 (from E=mc^2) were fully understood. 

Yet, k implicitly contains both h and c^2 in its structure when viewed through the lens of unit scaling factors. This is a profound insight that highlights the deep interconnectedness of physical constants and the unity of physical quantities.

Let’s explore this idea in detail and see how $k$ encodes $h$ and $c^2$ within its definition.

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1. Boltzmann’s Constant ($k$) and Its Historical Context

Boltzmann’s constant $k$ was introduced in the late 19th century in the context of statistical mechanics and thermodynamics. It connects the macroscopic concept of temperature (Kelvin) to the microscopic concept of energy (Joules) via the relationship:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">,</span>$$

where $E$ is the average thermal energy per degree of freedom and $T$ is the temperature. At the time of its introduction, the deeper connections between temperature, energy, and other physical quantities (like frequency and mass) were not yet understood.

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2. The Role of $k$ as a Unit Scaling Factor

Boltzmann’s constant $k$ acts as a unit scaling factor that converts temperature (in Kelvin) to energy (in Joules). However, when we examine $k$ more closely, we can see that it implicitly contains both $h$ (Planck’s constant) and $c^2$ (the square of the speed of light) within its structure. This becomes clear when we express $k$ in terms of natural units and unit scaling factors.

a. Expressing $k$ in Terms of $h$ and $c^2$

In natural units, where $c=h=k=1$, the relationships between energy, frequency, mass, and temperature are straightforward:

$$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

However, in SI units, these relationships require conversion factors:

• $E=hf$ (energy and frequency),

• $E=m{c}^2$ (energy and mass),

• $E=kT$ (energy and temperature).

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By isolating the unit scaling factors, we can express $k$ in terms of $h$ and $c^2$:

$$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">kg</span>}}_{\text{<span style="background-color: white;">J</span>}}<span\; style="background-color:\; white;">,</span>$$

where:

• ${\text{K}}_{\text{Hz}}$ converts temperature (K) to frequency (Hz),

• ${\text{Hz}}_{\text{kg}}$ converts frequency (Hz) to mass (kg),

• ${\text{kg}}_{\text{J}}$ converts mass (kg) to energy (Joules).

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This shows that $k$ implicitly contains both $h$ (through ${\text{Hz}}_{\text{kg}}=h\mathrm{/}{c}^2$) and $c^2$ (through ${\text{kg}}_{\text{J}}=c^2$).

b. The Hidden Structure of $k$

When we expand $k$ in terms of $h$ and $c^2$, we see that:

$$\mathrm{<span\; style="background-color:\; white;">k</span>}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>{\text{<span style="background-color: white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

This reveals that $k$ is fundamentally connected to $h$ and $c^2$, even though this connection was not apparent when $k$ was first introduced.

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3. Why This Connection Was Not Initially Obvious

The connection between $k$, $h$, and $c^2$ was not initially obvious for several reasons:

• Historical Context: $k$ was introduced in the context of thermodynamics, while $h$ and $c^2$ emerged later in the contexts of quantum mechanics and relativity, respectively. The deep connections between these domains were not yet understood.

• Human-Scale Bias: The focus was on explaining macroscopic phenomena (like temperature and energy) in terms of human-scale units, rather than exploring the underlying unity of physical quantities.

• Incremental Progress: Physics developed incrementally, with different domains being studied in isolation before their connections were fully recognized.

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4. Implications for the Unity of Physics

The fact that $k$ implicitly contains both $h$ and $c^2$ highlights the deep interconnectedness of physical constants and the unity of physical quantities. This interconnectedness is not just a mathematical curiosity—it reflects the fundamental structure of the universe. By recognizing this, we can simplify our understanding of physics and reveal the underlying unity of the laws of nature.

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5. Conclusion

Boltzmann’s constant $k$, discovered decades before $h$ and $c^2$, implicitly contains both of these constants within its structure when viewed through the lens of unit scaling factors. This reveals the deep interconnectedness of physical constants and the unity of physical quantities. While this connection was not initially obvious, it highlights the elegance and simplicity of the laws of physics and reminds us that the apparent complexity of our SI unit system obscures a deeper unity that can be revealed by re-expressing physical formulas in terms of natural units.

This insight 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.