c, h, and k simply relate SI unit scaling of meter, kg, and K units of measure to natural units.

The constants c, h, and k are not mysterious or deeply fundamental in the way we often imagine; instead, their role is purely to relate the scaling of our human-defined units (like meters, kilograms, and Kelvin) to the definition of time as 1 second. This insight demystifies their purpose and reveals that achieving natural units is far simpler than previously thought — it’s a matter of understanding how our unit system works and adjusting it accordingly.

Why This Wasn't Seen Before

Historically, we misunderstood the relationship between SI units and natural units due to several reasons:

1. Misinterpretation of Constants

• Constants like $c$, $h$, and $k$ were often treated as "deeply fundamental" values intrinsic to nature itself.

• The large or small numerical values of these constants in SI units obscured their true purpose: unit scaling.

• We failed to recognize that these constants are simply conversion factors arising from how we defined units like meters, kilograms, and Kelvin relative to seconds.

2. Complexity Bias

• Physicists often assumed that achieving natural units required a "theory of everything" or some deeper understanding of the universe.

• Natural units were seen as something inherently tied to quantum gravity or unification theories, rather than something already encoded in the constants themselves.

• This led to unnecessary complexity, overlooking the simple fact that natural units emerge directly from redefining our unit scales.

3. Lack of Focus on Unit Systems

• Unit systems were treated as arbitrary human conventions rather than tools for understanding physical relationships.

• We didn’t examine how constants like $c$, $h$, and $k$ encode the relationships between different unit scales (length, mass, temperature) and time.

• Instead of asking "What are these constants really doing?", we focused on their numerical values without realizing they reflect our choice of units.

The Physics Behind Constants: Scaling Time

The breakthrough insight is that the physics encoded in $c$, $h$, and $k$ is entirely about scaling temperature, frequency, mass, and length relative to time (1 second). Here’s how:

1. Speed of Light ($c$): Relating Length to Time

• The definition of $c=299,792,458\text{ m/s}$ reflects how meters are scaled relative to seconds.

• By redefining the meter such that 1 meter = distance light travels in 1 second, $c$ naturally becomes 1 m/s in natural units.

2. Planck Constant ($h$): Relating Mass and Frequency

• Key Relationship: Energy is linked to frequency via $E=hf$, where $h=6.626\times {10}^{-34}\text{J}\text{cdotp}\text{s}$ in SI units.

• Scaling Factor: The term ${\text{Hz}}_{\text{kg}}=h\mathrm{/}{c}^2$ (units: kg/Hz) tells us how many kilograms correspond to one hertz in SI units:

  $${\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">h</span>}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">7.372</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">51</span>}}\text{<span style="background-color: white;"> kg/Hz</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  This very small number reflects how SI defines kilograms (macroscopic mass) relative to hertz (quantum-scale frequency).

• Natural Redefinition: By redefining the kilogram such that $1\text{ kg}=1\text{ Hz}$, we set:

  $${\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;"> kg/Hz</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  This naturally leads to $h=1\text{ J/Hz}$, collapsing the distinction between mass and frequency.

3. Boltzmann Constant ($k$): Relating Temperature and Frequency

• Key Relationship: Thermal energy is linked to temperature via $E=kT$, where $k=1.38\times {10}^{-23}\text{ J/K}$ in SI units.

• Scaling Factor: The term $K_{\text{Hz}}=k\mathrm{/}h$ (units: Hz/K) tells us how many hertz correspond to one kelvin in SI units:

  $${\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">k</span>}}{\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">2.083</span>}<span\; style="background-color:\; white;">\times </span>{\mathrm{<span\; style="background-color:\; white;">10</span>}}^{\mathrm{<span\; style="background-color:\; white;">10</span>}}\text{<span style="background-color: white;"> Hz/K</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  This large number reflects how SI defines Kelvin (macroscopic temperature scales) relative to hertz.

• Natural Redefinition: By redefining Kelvin such that $1\text{ K}=1\text{ Hz}$, we set:

  $${\mathrm{<span\; style="background-color:\; white;">K</span>}}_{\text{<span style="background-color: white;">Hz</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">1</span>}\text{<span style="background-color: white;"> Hz/K</span>}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

  This naturally leads to $k=1\text{ J/K}$, making temperature directly proportional to energy.

Key Insight

These scaling factors (${\text{Hz}}_{\text{kg}}$ and $K_{\text{Hz}}$) reveal that:

1. $h$ does not directly map frequency to mass; instead, it encodes the relationship through the intermediate scaling factor, $c^2$.

2. $k$ does not directly map temperature to frequency; instead, it encodes this relationship through its ratio with $h$.

By redefining kilograms and Kelvin relative to hertz, we naturally achieve a system where:

• $h=1$, reflecting energy-frequency equivalence ($E=f$).

• $k=1$, reflecting energy-temperature equivalence ($E=T$).

Natural Units Are Already Encoded in SI Constants

The constants $c$, $h$, and $k$ are not external impositions — they are intrinsic parts of our unit system:

• They exist because SI defines length (meters), mass (kilograms), and temperature (Kelvin) independently from time (seconds).

• Their numerical values reflect the scaling gaps between these definitions and natural relationships.

By redefining length (meters), mass (kilograms), and temperature (Kelvin) relative to time (seconds), these constants naturally collapse to 1:

• $c=1\text{ m/s}$ because length is defined by light travel time.

• $h=1\text{ J/Hz}$ because mass is defined by frequency equivalence.

• $k=1\text{ J/K}$ because temperature is defined by energy equivalence.

We Assumed Complexity Where Simplicity Exists

Historically, we thought achieving natural units required breakthroughs in theoretical physics or a "theory of everything." In reality:

• Natural units don’t require new physics — they emerge directly from redefining our  base unit scales of meter, kg, and Kelvin units of measure by relationships already encoded in $c$, $h$, and $k$.

• The complexity was an illusion created by misunderstanding what these constants represent: unit scaling artifacts.

Implications

This realization has profound implications for physics:

1. Natural Units Are Accessible Right Now

We don’t need a theory of everything or new discoveries — natural units are already embedded in the constants we use daily.

2. Simplified Understanding

By recognizing that constants like $c$, $h$, and $k$ are scaling factors tied to unit definitions:

• We can simplify equations (e.g., $E=m{c}^2\to E=m,E=hf\to E=f,E=kT\to E=T$).

• We gain deeper insight into how physical quantities relate across different domains (energy ↔ mass ↔ frequency ↔ temperature).

3. A New Perspective on Unit Systems

Unit systems are not arbitrary conventions — they are tools for exploring relationships between physical quantities. By understanding their structure:

• We can move seamlessly between SI and natural units.

• We can better appreciate how constants like $c$, $h$, and $k$ unify diverse phenomena.

Conclusion

The constants $c$, $h$, and $k$ reveal a hidden simplicity in physics: they encode natural unit relationships directly into our current SI system by scaling length, mass, temperature, and frequency relative to time (seconds). This insight shows that achieving natural units doesn’t require complexity or new theories — it’s simply a matter of redefining base units based on relationships already encoded in these constants.