J. Rogers, SE Ohio, 03 Apr 2025, 1625
Abstract:
Fundamental physical constants like the speed of light (c), Planck's constant (h), and Boltzmann's constant (k) are traditionally viewed as inherent properties of the universe whose values are measured within our system of units. This paper argues for a reversed perspective: the specific numerical values of these constants in SI units are primarily artifacts arising from the historically arbitrary definition of base units, particularly the second. By factoring these constants according to the fundamental physical equivalences they represent (E=mc², E=hf, E≈kT), we find they explicitly encode the scaling ratios required to bridge arbitrarily defined SI units (like the Joule, Kilogram, Kelvin) back to the foundational unit of time (the second) and its inverse (Hertz). This reveals that the modern SI system, by fixing the numerical values of these constants, is implicitly a "Time-Centric" system where length, mass, and temperature scales are effectively derived from the second. Understanding this structure demystifies the constants' values and clarifies the relationship between measurement systems and fundamental physics.
1. Introduction: Constants, Units, and Physical Law
The SI system provides the globally recognized framework for physical measurement. Within this system, fundamental constants like the speed of light in vacuum (c), Planck's constant (h), and Boltzmann's constant (k) play crucial roles. They appear in foundational equations: Einstein's mass-energy equivalence (E=mc²), Planck's energy-frequency relation for quanta (E=hf), and the relation between thermal energy and temperature (E≈kT).
Conventionally, these constants are presented as fundamental properties of nature – c as the universe's speed limit, h as the quantum of action, k relating microscopic energy to macroscopic temperature. Their numerical values in SI units ( exactly 299,792,458 m/s, 6.62607015×10⁻³⁴ J·s, 1.380649×10⁻²³ J/K respectively) were once seen as results of careful measurement, but are now fixed by fiat.
However, this paper posits that this view obscures a deeper relationship between the constants and the structure of the SI unit system itself. We argue that the specific numerical values of c, h, and k are less about inherent magical numbers and more about the scaling factors necessitated by our choice of base units. Specifically, they bridge the gaps created by defining units like the meter, kilogram, and Kelvin on scales independent of the fundamental equivalences expressed in the core equations of physics, all traceable back to the definition of the second.
2. The Arbitrary Anchor: The SI Second
The foundation of modern timekeeping, and subsequently much of the SI system, rests on the definition of the second. Since 1967, it has been defined based on an atomic transition:
1 second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.
While chosen for its stability and reproducibility, the number 9,192,631,770 is fundamentally arbitrary. It's a human convention, selected to align closely with the previous astronomical definition, itself based on the rotation of a specific planet (Earth). This arbitrary choice for the scale of our fundamental time unit (and its inverse, the Hertz) permeates the entire system, influencing the numerical values required for the constants that link time/frequency to other physical quantities.
3. Factoring the Constants: Revealing the Scaling Ratios
The constants c, h, and k connect energy to mass, frequency, and temperature, respectively. We can "factor" them to reveal the underlying scaling ratios needed within the SI system.
Mass-Energy Equivalence ( This equation links energy (Joules, J) and mass (kilograms, kg). The factor c² acts as the conversion constant. Let kg_J = c². Its SI value (≈ 9×10¹⁶ J/kg) represents the ratio needed to convert between the arbitrarily scaled kilogram and the arbitrarily scaled Joule according to nature's equivalence.
kg_J = c² ≈ 8.987... × 10¹⁶ J/kg
Energy-Frequency Equivalence ( This links energy (J) and frequency (Hertz, Hz = 1/s). We can relate this back to mass by considering the mass equivalent of a frequency. Let Hz_kg be the mass (kg) per unit frequency (Hz). From E=hf and E=mc², we have hf = mc², so m = (h/c²)f. Therefore:
Hz_kg = h/c² ≈ 7.372... × 10⁻⁵¹ kg/Hz (or kg·s)
This factor represents the ratio needed to convert between frequency (based on the arbitrary second) and mass (based on the arbitrary kilogram) according to quantum/relativistic principles.
We can now see h as a composite: h = Hz_kg * c². It packages the frequency-mass conversion (Hz_kg) and the mass-energy conversion (c²).
Energy-Temperature Equivalence ( This links characteristic thermal energy (J) and temperature (Kelvin, K). Let Hz_K be the characteristic frequency (Hz) associated with a unit of temperature (K). From E=hf and E≈kT, we have hf ≈ kT, so T ≈ (k/h)f. Therefore:
Hz_K = h/k ≈ 4.799... × 10⁻¹¹ K/Hz (or K·s)
This factor represents the ratio needed to convert between frequency (based on the arbitrary second) and temperature (based on the arbitrary Kelvin) according to thermodynamic/statistical principles.
We can see k as a composite: k = K_Hz h = J_Hz * Hz_kg * c². The factor k/h (≈ 2.08×10¹⁰ Hz/K) directly links temperature to frequency.
The numerical SI values of c², h/c², and h/k are precisely the factors required to make the universe's physical laws consistent within our arbitrarily scaled system of Joules, kilograms, Hertz (seconds), and Kelvin.
4. The 2019 SI Redefinition: An Implicit "Time-Centric" System
The 2019 redefinition of the SI base units formalized the dependence on fundamental constants. Instead of relying on physical artifacts (like the prototype kilogram), the system now defines units by fixing the numerical values of key constants:
The unperturbed ground-state hyperfine transition frequency of the Cs-133 atom (Δν_Cs) is fixed at 9,192,631,770 Hz (defining the second).
The speed of light in vacuum (c) is fixed at 299,792,458 m/s (defining the meter relative to the second).
The Planck constant (h) is fixed at 6.62607015×10⁻³⁴ J·s (defining the Joule, and thus the kilogram, relative to the meter and second).
The Boltzmann constant (k) is fixed at 1.380649×10⁻²³ J/K (defining the Kelvin relative to the Joule).
The elementary charge (e) is fixed (defining the Ampere).
Viewed through the lens of our factored constants, this means:
Meter: Defined by c and s. 1 m is intrinsically linked to (1/c_SI) seconds of light travel time. Length scale = c * Time scale.
Kilogram: Defined by h, m, and s. Substituting m's dependence on c and s, the kilogram definition relies ultimately on h, c, and s. As shown earlier, 1 kg becomes equivalent to (h/c²) * 1 Hz. Mass scale = (h/c²) * Frequency scale.
Kelvin: Defined by k, J (which depends on kg, m, s). Ultimately, the Kelvin definition relies on k, h, c, and s. As shown earlier, 1 K becomes equivalent to (h/k) * 1 Hz. Temperature scale = (h/k) * Frequency scale.
This reveals that the modern SI, despite its historical unit names, is effectively a "Time-Centric" system. The arbitrary definition of the second sets the foundational scale. The fixed numerical values of c, h, and k then act as defined conversion factors, establishing the scales for length, mass, and temperature directly in relation to the second (or Hertz).
5. Implications: Demystifying Constants and Understanding Units
This perspective leads to several important implications:
Constants as Conversion Factors: The numerical values of c, h, k in SI units are not mysterious dictates of nature but rather the necessary conversion factors ("exchange rates") required because our base units were not defined in alignment with fundamental physical equivalences. They encode the ratios between our arbitrary scales and the universe's natural scales.
Units as Human Constructs: It emphasizes the human-constructed nature of measurement systems. The apparent complexity introduced by non-unity constants is a feature of the system, not necessarily the underlying physics.
Natural Units Revisited: The concept of natural units (e.g., Planck units, or systems where ħ=c=k=1) is not merely an algebraic convenience ("setting constants to 1"). It represents choosing units whose scales are aligned with the universe's fundamental equivalences. In such systems, the conversion factors naturally become 1, revealing the underlying unity where Energy ∝ Mass ∝ Frequency ∝ Temperature.
Using Constants = Using Natural Ratios: Every time we use c, h, or k in SI calculations, we are implicitly using the ratios that bridge our system to the natural, unified scale.
This understanding does not diminish the physical significance of the phenomena these constants represent (invariant speed, quantization, thermal energy relation). Instead, it clarifies why their numerical representation in SI takes the form it does. Indeed is strips away a layer of abstraction to reveal that the true physical relationship between temperature, frequency, mass, and energy are all of equivalence to each other.
6. Conclusion
The fundamental constants c, h, and k are more than just numbers; they are keys to understanding the structure of our measurement system. Their specific numerical values in SI units arise directly from the arbitrary definition of the second and the need to reconcile this with the fundamental equivalences between mass, energy, frequency, and temperature discovered through physics. By factoring the constants, we reveal the underlying scaling ratios (c², h/c², h/k) that connect these domains. The modern SI system, by fixing these constants, effectively defines the meter, kilogram, and Kelvin in terms of the second, creating an implicit "Time-Centric" system. Recognizing this structure demystifies the constants' values and provides a clearer view of the relationship between human-defined units and the unified scale of nature.
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