Why Einstein, Planck, and Boltzmann's Constants Could Be Different Than You Think

J. Rogers, SE Ohio, 02 Apr 2025, 2222

 The Hidden Truth About Physics' Most Famous Numbers

We've all seen them - those fundamental numbers etched into the bedrock of physics:

• The speed of light, c: 299,792,458 m/s

• Planck's constant, h: 6.626 × 10⁻³⁴ J·s

• Boltzmann's constant, k: 1.381 × 10⁻²³ J/K
  
  

Textbooks present these as universal truths, constants woven into the fabric of reality itself. Historically these have been empirically measured. But they were measured within a system that defined units of measure for length, mass, time, and temperature.  And each of these units are scaled by a different amount.  Like Morpheus' revelation to Neo in The Matrix, we've been living in a constructed reality:

"You've been living in a dream world, Neo. This... is the real world."

Our measurement systems built that dream world - where light's speed seems mysteriously fixed at 299,792,458, where energy and mass appear separated by a squared constant, where temperature and energy need conversion factors. But peel back the layers, and you find these constants are just the rules of our self-imposed simulation - the exchange rates between units we created independently before understanding their fundamental unity.

 

The Babylonian Code: How Ancient Timekeeping Defined the Modern Second

Before we can understand why the speed of light is 299,792,458 m/s, we need to talk about something far more ancient - the second. It is so ubiquitous that we don't even think about it.  Even natural unit systems just accept that the second is still the second.  That humble unit we take for granted was first carved into history by Babylonian astronomers nearly 4,000 years ago.

The Original Timekeepers

• Divided Earth's rotation into 24 hours (base-60 system)

• Split hours into 60 minutes

• Further divided minutes into 60 seconds

This sexagesimal system became our global standard through centuries of astronomical observations and mechanical clocks. But here's the critical point few realize:

Every modern constant traces back to this arbitrary choice.

When we defined the second in 1967 as "the duration of 9,192,631,770 periods of cesium-133 radiation," we weren't discovering nature's clock - we were anchoring our measurements to an upgraded version of the millennia old Babylonian system.

The Domino Effect

1. Time: Second defined by atomic vibrations

2. Length: Meter defined by how far light travels in 1/299,792,458 seconds

3. Mass: Kilogram now defined via Planck's constant (which includes seconds)

4. Temperature: Kelvin defined via Boltzmann's constant (also needing seconds)
   
   

Like Morpheus revealing the Matrix's code, we see our "fundamental" constants are actually conversion layers built atop this ancient timekeeping foundation. The universe doesn't care about base-60 math - but our entire physics edifice rests on this accident of history.

Even the very definition of what 1Hz is depends on the length of the second. If the second were 10 times as long, our current 1Hz would just be 0.1 Hz.  But the physics would be the same, just be different numbers defining the relationships. 

The Meter That Changed Everything

Let's start with everyone's favorite constant - the speed of light. That big number (299,792,458) seems so precise, so fundamental. But here's the secret: it's exact because we made it exact.

In 1983, scientists did something radical. They fixed the speed of light at exactly 299,792,458 m/s and redefined the meter to match. Before this, we measured light's speed. Afterward, we defined it.

This means the number isn't special - we chose it. If aliens used "light-years" as their base distance unit, their version of c would simply be 1. The physics hasn't changed - just how we write it down.

But c is just a unit conversion.  It is just the meter, our unit of length, defined against how far light travels in a second, so when you say 5 meters what you are really saying is simply 5/c seconds of light travel.  Those are two equivalent ways of saying the identical thing in different units of measure, so time is length in our universe. 

Einstein's Unit Converter

That famous equation E=mc²? It's often presented as revealing mass can become energy. But Einstein himself saw deeper, writing in 1946:

"Mass and energy are both but different manifestations of the same thing."

The c² is just doing unit conversion between equivalent units of measurement:

• Energy (joules) = kg·m²/s²

• Mass (kg) needs m²/s² to match
  
  We are not converting mass to energy.  We are saying that all things have both mass and energy, two different distinct properties measured in different ways, and these these two properties are always scaled from each other by c^2 in particles in our unit system of measurement.  We can see that in the context of relativity that kg_K = c^2 with units of J/kg so we are really just saying  E = m kg_J, a simple unit scaling from the unit definition of kg to the unit definition of energy in SI unit system of measurement. 

When advanced physicists set c=1 in natural units, the equation simplifies to E=m - revealing the pure equivalence Einstein described. The constant didn't disappear - we just stopped letting unit conversions obscure the physics.  The equivalence between mass and energy is the true physics.

Planck's Quantum Translator

But if time and length and energy are just equivalences between each other, and frequency is based on time, then why would converting a frequency to an energy be any less an equivalence than mass and energy were?  We are already equating mass to time through length, so equating mass to frequency is in no way different, just two properties that are equivalent measures of the same thing seen two different ways in our units of measure. 

Planck's constant h connects energy to frequency (E=hν). But look at the units:

• Energy: kg·m²/s²

• Frequency: 1/s
  
  

h is just translating between these measurement systems. Since 2019, we've locked h to an exact value to define the kilogram - further proof it's about our units, not nature's laws.

But what exactly is h scaling?   We can simply see that Hz_kg = h/c^2  with units of kg/Hz.So we can simply define h = Hz_kg c^2.  This means that in our current units of measure we have defined the kg against time such that we have Hz_kg number of Hz per kilogram.   This means that all along h was a composite of two simple unit scaling factors.  One related Hz to mass and another related mass to energy. 

This means that h is not a quantum of action with units of J s, it is a unit conversion between frequency and  energy, between Hz and J units as we have defined them in our human SI units of measure system.  h has the value it has because we defined the kg away from 1kg = 1Hz, because we defined c away from 1m = 1 light second.

Boltzmann's Temperature Bridge

Similarly, Boltzmann's k links temperature to energy (E=kT). But temperature already is energy - we just invented Kelvin as a separate scale. k is just the conversion factor between these human inventions.

This means that we can simple see that temperature is another equivalence.   K_Hz = k/h= k/(Hz_kg c^2), this has units of Hz/K and is how many Hz there are per degree of Kelvin are equivalent measures.   We can simply see that k = K_Hz Hz_kg c^2.  k has the value it does because of how far away from 1K = 1 Hz we scaled the Kelvin unit of measurement. 

What Really Matters are Equivalences.

If these famous numbers are just unit conversions, what's left? The truly fundamental physics is that everything is equivalent in a particle.  If you know the rest mass and any other property, you can figure out what every other property of that particle is. 

What natural unit systems were trying to do all along was to align the units of measure together, so that 1K = 1Hz = 1kg = 1J, but the confusions that constants were some sort of fundamental truth confused them into thinking it was the constants that you somehow had to force to unity with extra dimensions or extra particles.  But that unity we have always seeked was already present if we could just get our units of measure out of the way. 

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A New Perspective

This isn't to say c, h, and k aren't important. They're crucial for doing actual physics in our human-scale world and SI units of measurement. But understanding them as conversion factors rather than cosmic truths gives us a clearer picture of reality.

As Einstein might say, we're seeing through the illusion of units to the deeper unity beneath. The constants haven't changed - but how we understand them has fundamentally shifted.

"The laws of physics must be so constructed that they apply to systems of reference in any kind of motion. Only thus can we attain the goal of generalizing the theory of relativity." 

-Albert Einstein, (From "The Foundation of the General Theory of Relativity")

The Takeaway

Next time you see these famous numbers, remember: they're more like the "exchange rate" between different measurement systems than fundamental laws. The real physics is what remains when you strip away all human conventions - the pure relationships that exist regardless of how we choose to measure them.

After all, the universe doesn't care what units we use. It just is.

Constants as Unit-System Artifacts

The perspective presented offers a profound reinterpretation of physical constants, positioning them not as fundamental quantities but as explicit conversion factors inherent in our measurement systems. This contrasts with traditional natural unit systems (like Planck or Stoney units) where constants are artificially set to 1. Instead, the constants' numerical values directly encode the scaling ratios required to reconcile disparate unit definitions in the SI system. Here's a breakdown of this framework and its implications:

Core Insight: Constants as Unit-System Artifacts

Physical constants like $h$, $c$, and $k_B$ arise from the mismatch between how we define units (e.g., kg, Hz, K) and the deeper physical relationships between quantities. For example:

• $h\mathrm{/}{c}^2$ defines Hz_kg (kg/Hz), the ratio between mass and frequency units.

• $k_B\mathrm{/}h$ defines K_Hz (Hz/K), the ratio between temperature and frequency units.

These ratios reflect how far SI units deviate from a hypothetical "natural" system where:

$$\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">kg</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">Hz</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">K</span>}<span\; style="background-color:\; white;">\equiv </span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}\text{<span style="background-color: white;">J</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

In such a system, constants like $h$ and $k_B$ would naturally equal 1, eliminating the need for explicit conversion factors.

Key Relationships and Simplifications

By expressing constants as products of these ratios, physical laws simplify dramatically:

1. Planck's Constant:
   $h=\text{Hz\_kg}\cdot c^2$
   This redefines $h$ as a bridge between frequency and mass via $c^2$, aligning with $E=m{c}^2$.

2. Boltzmann's Constant:
   $k_B=\text{K\_Hz}\cdot \text{Hz\_kg}\cdot c^2$
   Here, $k_B$ becomes a chain of conversions: temperature → frequency → mass → energy.

Practical Implications

1. Simplified Formulas:
   Constants can be replaced by unit-conversion ratios derived from SI definitions:

   • Hz_kg = $h\mathrm{/}{c}^2\approx 7.372\times {10}^{-51}\text{kg/Hz}$
     The mass equivalent of 1 Hz.

   • K_Hz = $k_B\mathrm{/}h\approx 2.084\times {10}^{10}\text{Hz/K}$
     The frequency equivalent of 1 K.

   This reframes equations like the Stefan-Boltzmann law:

   • Original: $\sigma =\frac{2{\pi }^5{k}_B^4}{15c^2{h}^3}$

   • Simplified: $\sigma =\frac{2{\pi }^5}{15}\text{Hz\_kg}\cdot {\text{K\_Hz}}^4$
     Constants replaced by unit ratios inherent in SI.

2. Dimensional Transparency:
   Constants reveal how SI units could be redefined for coherence:

   • Redefining the kilogram via frequency ($1\text{kg}\equiv c^2\mathrm{/}h\cdot \text{Hz}$) would collapse $h$ and $c$ into unit definitions, rendering them numerically 1.

3. Anthropocentric Critique:
   The SI system’s historical development prioritized human-scale measurements (e.g., kilogram prototype, atomic clocks), creating artificial divides between energy, mass, and temperature. Constants like $h$ and $k_B$ exist to reconcile these divides.

Why This Isn’t “Natural Units”

Traditional natural units discard dimensions entirely (e.g., length = time), relying on “silent” constants for real-world calculations24. This framework instead:

• Retains SI dimensions while exposing their arbitrary scaling.

• Explains constant magnitudes as artifacts of unit definitions (e.g., $c$’s large value reflects the meter’s impracticality for relativity).

• Avoids dimensionless confusion by keeping constants’ dimensional roles explicit2.
  
  
  

The reinterpretation of energy formulas through the lens of E = mc² as a foundational pattern reveals intriguing nested relationships between physical constants, suggesting that constants like $h$, $c^2$, and $k_B$ encode scaling factors rather than fundamental truths. Here's how this nesting structure emerges:

Nested Relationships in Energy Formulas

1. Starting with $E=m{c}^2$:

   • The equation expresses the equivalence between mass ($m$) and energy ($E$) through the speed of light squared ($c^2$). This serves as the foundation for all energy-related formulas.

2. Mass as Frequency Equivalent:

   • Substituting $m=f\cdot {\text{Hz}}_{\text{kg}}$ (where ${\text{Hz}}_{\text{kg}}=h\mathrm{/}{c}^2$), we get:

     $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">f</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

   • Here, Planck's constant ($h$) emerges as a conversion factor between frequency ($f$) and energy ($E$).

3. Frequency as Temperature Equivalent:

   • Recognizing that $f=T\cdot K_{\text{Hz}}$ (where $K_{\text{Hz}}=k_B\mathrm{/}h$), we substitute into the equation:

     $$\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">m</span>}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">\cdot </span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">k</span>}}_{\mathrm{<span\; style="background-color:\; white;">B</span>}}\mathrm{<span\; style="background-color:\; white;">T</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

   • Boltzmann's constant ($k_B$) now emerges as a composite factor encoding temperature-frequency-energy equivalence.

Russian Nesting Doll Structure of Constants

The constants are not independent but are nested within one another:

• $c^2$ is embedded in $h$:
  $h=c^2\cdot {\text{Hz}}_{\text{kg}}$, where $H{z}_{\text{kg}}=h\mathrm{/}{c}^2\mathrm{.}$

• $h$ is embedded in $k_B$:
  $k_B=h\cdot K_{\text{Hz}},$ where $K_{\text{Hz}}=k_B\mathrm{/}h\mathrm{.}$

This nesting implies that these constants are scaling factors arising from how SI units are defined rather than fundamental properties of nature. They encode relationships between mass, frequency, temperature, and energy.

Unified Chain of Equivalences

From this perspective, all energy-related formulas follow a consistent chain:

$$\mathrm{<span\; style="background-color:\; white;">K</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">temperature</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>{\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">frequency-temperature scaling</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">f</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">frequency</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">mass-frequency scaling</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;">(</span>\text{<span style="background-color: white;">mass</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</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;">mass-energy scaling</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">⟷</span>\mathrm{<span\; style="background-color:\; white;">E</span>}<span\; style="background-color:\; white;">(</span>\text{<span style="background-color: white;">energy</span>}<span\; style="background-color:\; white;">)</span>$$

Each constant acts as a bridge between adjacent quantities:

• K_Hz   = k_B/h:  Converts temperature to frequency.

• Hz_kg = h/c^2:  Converts frequency to mass.

• c^2                   :  Converts mass to energy.

Conclusion

The constants $h$, $c$, and k_B are not fundamental truths but byproducts of SI’s fragmented unit definitions. Recognizing them as scaling factors reveals a path toward a more coherent measurement system—one where energy, mass, frequency, and temperature share a unified foundation. This view doesn’t invalidate SI but highlights how its architecture shapes our perception of physical laws. By redesigning units around physical relationships (e.g., defining mass via frequency), we could eliminate these constants entirely, achieving true natural units organically