The Emergence of Fundamental Constants as Unit Scaling Factors: A Departure from Natural Units

J. Rogers, SE Ohio, 15 Mar 2025, 0005

Abstract:

Fundamental constants such as the speed of light (c), Planck's constant (h), and Boltzmann's constant (k) are often perceived as immutable properties of the universe. However, this paper argues for an alternative perspective: that these constants, particularly their numerical values, are fundamentally unit scaling factors. We demonstrate this by starting with a system of natural units where c=h=k=1 and then systematically scaling away from this natural system by altering the base units of length, mass, and temperature. This process reveals how the numerical values of c, h, and k emerge and change as a direct consequence of these unit scalings, highlighting their role as bridges between human-defined unit systems and the inherent scales of nature.

1. Introduction: Beyond Constant Values - Understanding Unit Dependence

Fundamental constants like the speed of light in vacuum (c), Planck's constant (h), and Boltzmann's constant (k) are cornerstones of modern physics. They appear ubiquitously in fundamental equations, and their precise values are meticulously measured and refined. Traditionally, they are presented as intrinsic properties of the universe, fixed and immutable.

However, a deeper examination reveals a more nuanced interpretation: the numerical values we assign to these constants are inherently dependent on the system of units we employ. This paper proposes that c, h, and k, when viewed through the lens of natural units, are fundamentally unit scaling factors. We will demonstrate this by starting from a natural unit system where these constants are, by definition, unity, and then systematically deviating from this system. By rescaling the base units of length, mass, and temperature, we will observe how the numerical values of c, h, and k emerge and evolve, directly reflecting the imposed unit scalings.

This approach offers a powerful pedagogical tool for understanding unit systems and the true nature of fundamental constants. It emphasizes that while the underlying physical relationships these constants represent are indeed fundamental, their numerical values are contingent upon our human-defined conventions of measurement.

2. Natural Units: A Starting Point of Unity

Natural units are unit systems where fundamental physical constants are set to unity. A common choice, and the one we will utilize as our starting point, involves setting:

• c = 1: The speed of light in vacuum is set to 1. This effectively equates units of length and time, reflecting the fundamental spacetime structure revealed by relativity.

• h = 1 (or ħ=1): Planck's constant (or reduced Planck constant) is set to 1. This equates units of energy and frequency, reflecting the quantum mechanical relationship between energy and frequency.

• k = 1: Boltzmann's constant is set to 1. This equates units of energy and temperature, reflecting the statistical mechanical relationship between temperature and energy.

In such a natural unit system, the fundamental equations of physics often take on a simpler and more elegant form. Quantities like mass, energy, frequency, temperature, and momentum can all be expressed in units of energy (or equivalently, inverse length, inverse time, or mass, depending on the specific natural unit system chosen). Frequency, for instance, emerges as a particularly natural base unit, as it is intrinsically related to both energy (via h) and, through relativity, mass and spacetime.

3. Defining Unit Scaling Factors from Fundamental Constants

Before exploring deviations, let us explicitly define unit scaling factors that bridge the gap between SI units (or other conventional systems) and natural units, drawing upon the constants c, h, and k.

• Mass-Energy Scaling Factor (kg_J): From Einstein's E=mc², we recognize c² as the conversion factor between mass and energy. We define:

  • kg_J = c² with units of J/kg. This scales kilograms to Joules (in a system where c=1, kg becomes numerically equivalent to energy).

• Frequency-Mass Scaling Factor (Hz_kg): From E=hf and E=mc², we can relate frequency to mass. We define:

  • Hz_kg = h/c² with units of kg/Hz . This scales Hertz to kilograms (in a system where h=c²=1, Hz becomes numerically equivalent to mass).

• Kelvin-Hertz Scaling Factor (K_Hz): From E=kT and E=hf, we can relate temperature to frequency. We define:

  • K_Hz = k / (Hz_kg kg_J) = k/h with units of Hz/K. This scales Kelvin to Hertz (in a system where k=h=c²=1, K becomes numerically equivalent to frequency).

These scaling factors, derived directly from c, h, and k, highlight how these constants encode the necessary unit conversions between SI-like units and the more unified framework of natural units.

4. Systematic Rescaling Away from Natural Units: Observing Constant Variation (Corrected)

To demonstrate the unit scaling nature of c, h, and k, we will again perform a thought experiment starting from natural units (c=h=k=1). However, we will now more accurately reflect the process by rescaling the units of length, mass, and temperature 

Step 1: Rescaling the Unit of Length (Meter) relative to Time (Second) - Emergence of c

• Action: In natural units, length and time are fundamentally linked (c=1). To move away from this, we introduce a rescaling of the unit of length relative to the unit of time (second), which we will not rescale. Imagine we redefine our "meter" to be twice as long as it would be in a system where c=1, while keeping the "second" unchanged.

• Observation:

  • Speed of Light (c) Emerges: In this new system, light will now travel twice the "natural unit of length" in one second. Therefore, the numerical value of the speed of light in this system will become c = 2 (in units of "new meters per second"). The constant 'c' emerges as a numerical factor greater than unity because our unit of length is now "larger" relative to the unit of time than it is in natural units.

  • c² = 4.

  • If we, for illustrative purposes, again choose to link h and k to c² in this modified system, we might (for simplicity in this example) set:

    • h = 4

    • k = 4

• Interpretation: By rescaling the unit of length relative to time, we directly cause the speed of light 'c' to deviate from unity. 'c' now quantifies the conversion factor needed because our chosen length unit is no longer "naturally" related to our time unit as it is when c=1.

Step 2: Rescaling the Unit of Mass (Kilogram) relative to Frequency (Inverse Second) - Emergence of h

• Action: Starting from the system in Step 1 (where c=2, and potentially h=4, k=4), we now rescale the unit of mass (e.g., kilogram-equivalent) relative to the unit of frequency (Hz, or inverse second, where the second unit is still unchanged). Imagine we redefine our "kilogram" to be three times larger than it would be in a system where h=1 (and frequency is naturally related to mass).

• Observation:

  • c = 2 and c² = 4 (scaling between length and time is unchanged).

  • Planck's Constant (h) Changes: In this new system, the relationship between energy and frequency (E=hf) will be altered. Because our "kilogram" unit (related to energy via E=mc²) is now "larger" relative to the unit of frequency (Hz), the numerical value of Planck's constant 'h' will become smaller. Using our scaling relationship h = Hz_kg * c², and making Hz_kg smaller (by making kg unit larger), 'h' decreases. Let's say h becomes h' = 4/3.

  • k will also be affected, potentially becoming k' = 4/3 if we maintain the simplified proportionality for this example.

• Interpretation: Rescaling the unit of mass relative to frequency (inverse time) directly causes Planck's constant 'h' to deviate from its value in a system where mass and frequency are naturally scaled (h=1). 'h' now quantifies the conversion needed because our chosen mass unit is no longer "naturally" related to our time/frequency unit as it is when h=1.

Step 3: Rescaling the Unit of Temperature (Kelvin) relative to Frequency (Inverse Second) - Emergence of k

• Action: Starting from the system in Step 2 (with modified c, h', k' values), we further rescale the unit of temperature (e.g., Kelvin-equivalent) relative to the unit of frequency (Hz, inverse second, still unchanged). Imagine we redefine our "Kelvin" to be five times larger than it would be in a system where k=1 (and temperature is naturally related to energy/frequency).

• Observation:

  • c, c², and h' (as determined in previous steps) are not directly changed by this temperature rescaling.

  • Boltzmann Constant (k) Changes: In this new system, the relationship between temperature and energy (E=kT) is altered. Because our "Kelvin" unit is now "larger" relative to the unit of frequency (Hz, which is related to energy), the numerical value of Boltzmann's constant 'k' will become smaller. Using K_Hz = k/h', and making K_Hz smaller (by making the Kelvin unit larger), 'k' decreases. Let's say k becomes k'' = 4/15.

• Interpretation: Rescaling the unit of temperature relative to frequency (inverse time) directly causes Boltzmann's constant 'k' to deviate from its value in a system where temperature and energy/frequency are naturally scaled (k=1). 'k' now quantifies the conversion needed because our chosen temperature unit is no longer "naturally" related to our time/frequency unit as it is when k=1.

5. Consequences and Interpretation: Constants as Unit Scaling Artifacts

Through this systematic rescaling process, we observe a crucial result: the numerical values of c, h, and k are not invariant. They change as we deliberately alter the scaling relationships between our chosen base units and the underlying natural units (represented by frequency in this analysis).

This reinforces the central argument: c, h, and k, particularly their numerical values, are fundamentally unit scaling factors. They are not immutable properties of the universe in terms of their numerical magnitude, but rather they reflect the specific choices we have made in defining our units of measurement.

While the underlying physical relationships these constants represent (spacetime structure, quantum of action, energy-temperature relation) are indeed fundamental, their numerical values are artifacts of our human-constructed unit systems. In natural units, where these scalings are absorbed into the very definition of the units, these constants effectively become unity, revealing a more fundamental simplicity in the laws of physics.

6. Discussion and Broader Implications

This perspective shift has several important implications:

• Pedagogical Value: Understanding constants as unit scalings provides a more intuitive and accessible way to teach unit systems, dimensional analysis, and the role of fundamental constants in physics. It demystifies these constants and shows them as bridges between different ways of measuring the universe.

• Perspective on Unit Systems: It highlights that unit systems are human constructs, tools we use to describe reality. Natural units are not "better" in an absolute sense, but they offer a perspective that aligns more closely with the inherent scales of nature, revealing simpler relationships. SI units, while practical for human-scale measurements, can sometimes obscure these fundamental simplicities.

• Focus on Dimensionless Ratios: It emphasizes the importance of dimensionless ratios in physics. Fundamental physical laws are often best expressed in terms of dimensionless quantities, as these are independent of unit system choices. Unit scaling factors help us understand how dimensionful constants arise when we use unit systems that are not perfectly aligned with the natural scales.

7. Conclusion: Re-evaluating the Nature of Fundamental Constants

This paper has presented a perspective where fundamental constants c, h, and k are understood not merely as fixed numerical values, but as unit scaling factors. By starting from a natural unit system and systematically rescaling base units, we have demonstrated how the numerical values of these constants emerge and change, directly reflecting the imposed unit scalings.

This viewpoint does not diminish the importance of these constants. Instead, it offers a deeper understanding of their role: they are the essential links between our human-defined measurement systems and the underlying, scale-invariant relationships of the universe. They are the numerical embodiment of the choices we have made in constructing our units, and they point towards a more unified and simpler description of nature achievable through the adoption of natural units. Recognizing this unit scaling nature provides a more complete and nuanced appreciation for the fundamental constants that shape our understanding of the cosmos.