J. Rogers, SE Ohio, 05 Apr 2025, 1900
The approach of using modular scaling factors derived from fundamental constants offers an elegant perspective on the transition between SI and natural units. This framework emphasizes how these scaling factors serve as bridges between different measurement systems, revealing the underlying unity in physical laws that is often obscured by our arbitrary choice of units.
Understanding Natural Units and Scaling Factors
Natural unit systems in physics are measurement systems where selected physical constants have been set to 1 through nondimensionalization of physical units6. For instance, setting the speed of light c=1 by rescaling the meter by current numerical value of c allows us to express mass and energy equivalently as E=m rather than E=mc². These systems simplify equations by eliminating conversion constants, but still require tracking dimensions to properly reinsert physical constants when needed6.
The International System of Units (SI) defines units through seven fundamental constants with exact numerical values, including the speed of light (c), Planck's constant (h), and Boltzmann's constant (k)2. These constants create intrinsic relationships between different physical quantities, which can be expressed through modular scaling factors.
Deriving Modular Scaling Factors from Fundamental Constants
The modular scaling approach identifies specific scaling factors that relate different SI units, particularly connecting mass, temperature, and energy to frequency:
Mass-Frequency Scaling Factor (Hz_kg)
From Einstein's mass-energy equivalence (E=mc²) and quantum theory (E=hf), we can derive:
mc² = hf
Therefore: m = (h/c²)f
This gives us the scaling factor Hz_kg = h/c², which has units of kg/Hz and represents the mass equivalent of 1 Hz1. Using the defined values of h and c from the SI system2:
Hz_kg ≈ 7.37249732×10⁻⁵¹ kg/Hz
We can see that Hz_kg = h/c^2 so we can define h = Hz_kg c^2
Temperature-Frequency Scaling Factor (K_Hz)
From thermal energy (E=kT) and quantum theory (E=hf), we can relate temperature to frequency:
kT = hf
Therefore: T = (h/k)f
The inverse gives us K_Hz = k/h, with units of Hz/K, representing the frequency equivalent of 1 K:
K_Hz ≈ 2.08366191×10¹⁰ Hz/K
We can see that K_Hz = k/h = k/( Hz_kg c^2 ) so we can define k = K_Hz Hz_kg c^2
Energy-Frequency Scaling
From quantum theory, E=hf provides a direct scaling between energy and frequency:
h ≈ 6.62607015×10⁻³⁴ J/Hz
This represents the energy equivalent of 1 Hz12.
But J is not an actual unit itself , you you can see that when we say Joule we really mean
kg m^2/s^2, So to rescale a J unit we have to first rescale all the base units, the kg and
meter, while keeping the second definition the same.
SI Unit Equivalencies Using These Scaling Factors
Using frequency (Hz) as our reference point, we can express equivalencies in the SI system based on the values of these scaling factors:
1 Hz corresponds to an energy of (Hz_kg × c²) Joules ≈ 6.62607015×10⁻³⁴ J
1 Hz corresponds to a mass of (Hz_kg) kilograms ≈ 7.37249732×10⁻⁵¹ kg
1 Hz corresponds to a temperature of (1/K_Hz) Kelvin ≈ 4.79924307×10⁻¹¹ K
These relationships highlight the numerical disparities in SI units for quantities that are fundamentally linked1. The factors (Hz_kg × c²), Hz_kg, and (1/K_Hz) are the SI scaling values that connect these different physical quantities.
Rescaling SI Units to Create Natural Units
To create a natural system where these equivalences are numerically 1:1, we can redefine our units by incorporating the scaling factors:
Natural Kilogram (kg')
We define the natural kilogram such that 1 kg' is numerically equivalent to 1 Hz':
1 kg' = (Hz_kg) kg_SI ≈ 7.37249732×10⁻⁵¹ kg_SI
This value was encoded into the Hz_kg ratio that defines h between Hz and kg unit scalings.
Natural Kelvin (K')
We define the natural Kelvin such that 1 K' is numerically equivalent to 1 Hz':
1 K' = (1/K_Hz) K_SI ≈ 4.79924307×10⁻¹¹ K_SI
This value was encoded into the K_Hz ratio that defines k between units of temperture
and frequency.
Natural Joule (J')
We define the natural Joule such that 1 J' is numerically equivalent to 1 Hz':
1 J' = (Hz_kg × c²) J_SI ≈ 6.62607015×10⁻³⁴ J_SI
Since h = (Hz_kg × c²) = 1 * 1^2 =1.
This approach is similar to the conversion methods described in the arXiv paper, where the correct factors of natural units are multiplied to the initial quantity to achieve the desired unit conversion1.
The Result in the Natural Unit System
By defining our units (kg', K', J') in this way—embedding the SI scaling factors Hz_kg, 1/K_Hz, and (Hz_kg × c²), respectively—we create a system where:
A mass measured as 1 kg' corresponds to the same fundamental quantity as a frequency measured as 1 Hz'
A temperature measured as 1 K' corresponds to the same fundamental quantity as a frequency measured as 1 Hz'
An energy measured as 1 J' corresponds to the same fundamental quantity as a frequency measured as 1 Hz'
Therefore, in this natural unit system: 1 K' = 1 Hz' = 1 kg' = 1 J'
This demonstrates that the specific SI values of the fundamental constants (c, h, k) function as numerical scaling factors required because SI units are arbitrarily defined relative to each other15.
In SI units:
4.799243073e-11 K = 1 Hz = 7.37249732e-51 kg = 6.62607015e-34 J
We rescale the units by these ratios and in natural units:
1 K = 1 Hz = 1 kg = 1 J
T <-- K_Hz --> Hz <-- Hz_kg --> kg <-- Hz_J --> J
As you can see the modular unit conversions allow you to step back and forth between all the equivalent properties of a particle. You don't have to do clear to the end of the line to energy town, you can get off at any station on the line. You can freely go between any stations using the modular scaling factors.
The Implications
While the result c=h=k=1 is a definitional consequence and this was not forced to happen or hand waved into existence, the process and the implication are far from unimportant:
Reveals True Role: It demonstrates conclusively that c, h, k are fundamentally unit conversion factors between concepts (space/time, energy/frequency, energy/temperature, mass/energy) that are intrinsically linked, but which our historical SI units treat as independent. Confirms the Links: The fact that we can successfully perform this rescaling using c, h, and k confirms that these constants correctly capture the fundamental relationships they represent. Shifts Focus: It moves the focus away from the arbitrary numerical values of constants in SI units (which depend entirely on the historical choice of meter, kg, second, Kelvin) and towards the underlying physical relationships and the inherent scales of nature. Simplification with Meaning: Yes, equations become simpler, but not just for algebraic convenience. They become simpler because the units themselves now directly reflect the unified physical reality that the constants mediate.
Applications in Computational Physics
The concept of scaling factors has practical applications in computational physics. For example:
In computational models of temperature-dependent systems, scaling factors can be used to account for temperature effects on various parameters4
In vibrational spectroscopy, scale factors are used to align quantum chemically computed harmonic frequencies with experimental data3
In relativistic calculations, natural units can simplify equations involving masses of stars or galactic scales5
By using the definitions of h and k it allows you to greatly simplify and reveal the exact unit scaling factors were really all that was being used all along. Examples follow:
Thermal de Broglie Wavelength:
Original: λ_th = h/(sqrt(T)*sqrt(m)*sqrt(2*pi*k))
Simplified: λ_th = c/(sqrt(2*pi*f_T*f_m))
Where f_T = T K_Hz and f_m = m * kg_Hz
Stephan-Boltzmann Formula:
Original: σ = 2*pi**5*k**4/(15*c**2*h**3)
Simplified: σ = 2*pi**5*K_Hz**4*Hz_kg/15
Planck law Formula:
Original: B(f T) = 2*f**3*h/(c**2*(e**(f*h/(T*k)) - 1))
Simplified: B(f T) = 2*f**3*Hz_kg/(e**(f/(T*K_Hz)) - 1)
This can be seen as a ratio of temperatures or frequencies.
Wien's Displacement Constant:
Original: x_peak = c*h/(T*k*λ_max)
Simplified: x_peak = c/(T*K_Hz*λ_max)
Debye Temperature:
Original: Θ_D = h*ν_D/k
Simplified: Θ_D = ν_D*Hz_K
Einstein Temperature:
Original: Θ_E = h*ν_E/k
Simplified: Θ_E = ν_E*Hz_K
Conclusion
The modular scaling approach reveals that fundamental constants like c, h, and k are conversion factors between arbitrarily defined units in our SI system. By identifying and incorporating these scaling factors into our unit definitions, we create natural unit systems where these constants become unity. This is a literal unity between properties that exists in SI unit system right now. This system can greatly simplify how formulas are presented, learned, and understood.
This approach demonstrates that the constants becoming 1 is not merely a mathematical convenience but reveals an underlying unity in physics that exists regardless of our unit choices. The physical significance lies in recognizing that what appear as separate quantities in our conventional unit system (mass, energy, frequency, temperature) are fundamentally interconnected aspects of the same physical reality. The scaling factors derived from c, h, and k reveal these intrinsic relationships, which become explicitly manifest when we redefine our units to make these relationships numerically transparent.
