The Handwaving of Constants to 1: Understanding the Real Unit Scaling Behind Natural Units

J. Rogers, 09 Mar 2025, 1927

Introduction

In theoretical physics, it is common practice to set fundamental constants like the speed of light , Planck’s constant , and Boltzmann’s constant to 1 in so-called “natural units.” This is often done with a casual hand wave, justified by the idea that natural units make equations simpler and reveal deeper relationships between physical quantities. However, this approach typically lacks a concrete explanation of how these constants are actually being rescaled. Thus, it cannot be used as a proof of any theory.

The reality is that setting these constants to 1 is not just a conceptual trick—it requires precise unit re-scalings. These re-scalings are always overlooked, leaving the transformation from SI units to natural units incomplete and unexplained. In this article, we will explore the exact transformations needed to achieve a real, mathematically consistent system of natural units, and what that means for familiar quantities like mass, length, and temperature.

---

The Traditional Hand Wave: How Constants Are “Set to 1”

In standard physics discussions, setting constants to 1 is typically justified as a matter of convenience. We define a new system where:

• eliminates conversion factors between space and time.

• eliminates the distinction between energy and frequency.

• eliminates the distinction between temperature and energy.

However, in most presentations, the exact unit scalings needed to make this work are never explicitly addressed. Instead, it is assumed that this transformation is self-evident and that natural units arise naturally (ironically).

But if c, h, and k all have specific numerical values in SI units, then to make them numerically 1, we must rescale the base units of mass, length, and temperature accordingly.

The key question is: what are those exact re-scalings?

---

The Real Unit Rescaling: Deriving the Necessary Transformations

The Real Unit Rescaling: Deriving the Necessary Transformations

To transform SI units into a real, usable system where fundamental constants become 1, we must properly rescale the base units of kilograms, meters, and kelvins. The correct transformations follow these precise scaling factors:

• Speed of light ($c$): Converts meters into light-seconds.

  • The new unit of length is defined by scaling meters by $c$, so that:
    
    $$1 \text{ new meter} = c \times 1 \text{ meter} = 1 \text{ light-second}$$
    

• Hz_kg: Converts mass into a natural frequency scale.

  • The correct mass unit scaling follows from the relationship between energy and frequency, given by:
    
    $$1 \text{ new kg} = \frac{1}{\text{Hz}_{\text{kg}}} \times 1 \text{ kg}$$
    
  • This means that the new natural unit of mass is $\frac{c^2}{h}$ times bigger (about 10^50 times bigger) than the current kilogram. 

• K_Hz: Converts temperature into a natural frequency scale.

  • The correct temperature unit scaling follows from the relationship between temperature and energy:
    
    $$1 \text{ new K} = K_{\text{Hz}} \times 1 \text{ K}$$
    
  • This means that temperature is measured in terms of equivalent frequency, aligning it directly with energy.

This rescaling process ensures that $c = 1$, $h = 1$, and $k = 1$ without handwaving, resulting in a fully consistent natural unit system.

---

The Impact on Mass, Length, and Temperature

If we actually apply these scalings to define a real natural unit system, the consequences are striking:

• Mass: The kilogram is absurdly large in natural units. The natural unit of mass would be 10^50 times larger  than the current definition of the kg, showing that SI mass is wildly disproportionate compared to fundamental scales.

• Length: The meter transforms into a natural unit equivalent to a light-second, meaning it is about 3 × 10^8 times larger.

• Temperature: The Kelvin unit transforms based on , which is slightly larger than , meaning that natural temperature units are significantly larger than SI Kelvin values.

This perspective completely removes the mystery surrounding the values of and . Their values are exactly what is needed to convert SI units into a consistent frequency-based system. The constants are not “fundamental” in the sense of being arbitrary physical properties; rather, they are just unit conversion factors reflecting the historical choices made in defining SI units.

---

Conclusion

The common practice of “setting constants to 1” is often treated as a conceptual trick, but in reality, it requires precise rescalings of our base units. The real transformation is not arbitrary—it follows naturally from the structure of our measurement system.

The key takeaways are:

• The values of , , and are not mysterious—they are exactly what is needed to bridge the gap between SI and natural units.

• To truly set these constants to 1, we must explicitly rescale kg, meters, and Kelvin by the factors , , and , respectively.

• This transformation makes mass incredibly small, length incredibly large, and temperature larger than expected.

By understanding these rescalings, we move beyond the handwaving and gain a deeper appreciation for the role of fundamental constants. Rather than being arbitrary, they are simply the natural conversion factors that emerge from our choice of measurement system.

Natural units aren’t just a conceptual tool—they are a real mathematical framework, and now we know how to actually achieve them.