Abstract:
The prevailing defense for the fundamental nature of physical constants such as G, c, and h is that they are not definitions, but are instead empirically determined quantities derived from falsifiable experiments. This paper argues that this view, while factually correct, is based on a profound categorical error in interpretation. We demonstrate that the act of "measuring" a fundamental constant is, in fact, the act of calibrating an arbitrary, human-invented unit system (like the SI system) against the natural, invariant scaling of the universe (the Planck scale). Using a thought experiment involving the historical Mechanical Equivalent of Heat, we show that what an experiment measures depends entirely on the observer's chosen coordinate system of units. The empirical results for G, c, and ħ are therefore not revelations of mysterious cosmic numbers, but are the precise scaling factors—the Jacobians—required to translate between our local rulers and the fundamental geometry of reality. This re-contextualization dissolves the "fine-tuning" problem and reveals the constants not as the mystery, but as the Rosetta Stone that allows us to understand our own measurements.
1. Introduction: The Empiricist's Defense of Fundamental Constants
The most robust argument against the view that physical constants are mere definitions or coordinate artifacts is the appeal to experiment. A physicist will rightly state: "We do not invent the value of G; we measure it in a Cavendish experiment. We do not define c; we measure it with lasers and atomic clocks. These numbers are results, not axioms. They are properties of the world that we discover through empirical investigation."
This argument appears unassailable. It grounds the constants in the bedrock of the scientific method: observation and measurement. To challenge this seems to challenge the very foundation of physics as an empirical science. This paper will not challenge the validity of the experiments. Instead, we will challenge the interpretation of what these brilliant experiments are actually measuring. We will argue that they are not measuring a fundamental property of nature, but rather a fundamental relationship between our chosen system of units and an always unified nature.
2. The Parable of the Inch-Pound Alchemists: A Thought Experiment in Metrology
To clarify the core of our argument, let us imagine an isolated scientific community that developed its entire physics using the arbitrary units of the foot, the pound, and the second. In their study of thermodynamics, they define a unit of heat, the British Thermal Unit (BTU), as the heat required to raise one pound of water by one degree Fahrenheit. In their study of mechanics, they define a unit of energy, the foot-pound.
One day, they conduct a landmark experiment—Joule's paddlewheel experiment—to determine the relationship between mechanical work and heat. They meticulously measure how much mechanical energy (in foot-pounds) is required to generate a specific amount of heat (in BTUs). They discover a consistent, universal number, which they hail as a new fundamental constant of the universe, β (Beta), the "Mechanical Equivalent of Heat."
β ≈ 778.17 ft-lbf / BTU
A mystique develops around Beta. "Why 778.17?" they ask. "If this number were slightly different, the universe would be unrecognizable! Explaining the origin of this dimensionless constant is the primary goal of our 'Theory of Everything'." They are not wrong; they went into a lab and measured Beta. It is an empirical result.
Now, a visitor arrives who was trained in the SI system. The visitor examines their work and explains: "Your experiment is impeccable. However, you have not discovered a fundamental constant of the universe. You have discovered the conversion factor between your two different, arbitrary definitions of energy. You defined one unit of energy in terms of mechanics (the foot-pound) and another in terms of the thermal properties of water (the BTU). Beta is the scaling factor that connects them. In my system, we defined the Calorie in terms of water, and we found its mechanical equivalent is 4.186 Joules. The numbers are different because our arbitrary units are different. The underlying physics is the same."
The alchemists did not measure a property of the universe. They measured a property of their own rulers. Their "fundamental constant" was a coordinate artifact of their inconsistent and arbitrary choice of units.
3. The Real Experiment: Calibrating SI Units Against the Planck Scale
The parable of the alchemists is not an analogy; it is a direct parallel to the situation in modern physics. The "fundamental constants" G, c, h, and k_B are the "Betas" of our SI unit system.
Let us consider the Cavendish experiment to "measure G."
We take masses defined by the kilogram, an arbitrary cylinder of platinum-iridium.
We place them at distances defined by the meter, an arbitrary fraction of the Earth's circumference.
We measure the resulting force in Newtons, a unit derived from kilograms, meters, and the second, an arbitrary number of oscillations of a cesium atom.
The experiment is a masterpiece of precision. But what does the resulting number, G ≈ 6.674 x 10⁻¹¹ m³ kg⁻¹ s⁻², represent? It is not the "strength of gravity" in any absolute sense. It is the scaling factor required to make the simple, geometric law of universal gravitation (Force ~ mass₁ * mass₂ / distance²) yield the correct answer when we insist on using this peculiar, arbitrary, and anthropocentric set of units.
The experiment does not measure a mystery of the universe. It calibrates our local, human-made unit system against the invariant, natural structure of the universe. The same logic applies to the other constants:
Measuring c: This is the act of calibrating our meter stick against our stopwatch, revealing the universal geometric relationship between time and space.
Measuring h: This is the act of calibrating our unit of energy (the Joule) against our unit of frequency (the Hertz), revealing the fundamental granularity of action in the universe.
Measuring k_B: This is the act of calibrating our unit of energy (the Joule) against our unit of temperature (the Kelvin), revealing the statistical-mechanical nature of heat.
4. Two Interpretations of the Same Measurement
The empirical fact—that an experiment yields a number—is not in dispute. The interpretation of that fact is the entire point of contention.
| Premise: The Constants are fundamental. | Premise: The Planck Scale is fundamental. Our SI units are arbitrary. |
| Experiment: An experiment discovers the value of a mysterious constant (e.g., G). | Experiment: An experiment calibrates our arbitrary units against the Planck scale, revealing a scaling factor (which we name G). |
| Consequence: The constant is a property of nature. The Planck units are a curious mathematical construct derived from the constants. | Consequence: The constant is a property of our measurement system. It is the conversion factor that connects our arbitrary units to the unity that is always present in the fundamental Planck units. |
The Metrological Interpretation does not discard the experimental result; it gives it a more profound and rigorous meaning. The measurement is not the end of the mystery; it is the beginning of understanding our relationship to it.
5. The Special Case of Dimensionless Constants: Ratios, Not Rulers
A common and important objection to this metrological framework is the existence of dimensionless constants, most notably the fine-structure constant, α. The argument is as follows: "While constants with units like G or c may be artifacts of our unit system, α ≈ 1/137 is a pure number. Its value is independent of any choice of units. Therefore, α must be a truly fundamental, God-given number, and the mystery of its value remains."
This objection is insightful but ultimately mischaracterizes the nature of dimensionless quantities within our framework. The existence of α does not contradict the metrological interpretation; it powerfully reinforces it by highlighting the crucial distinction between scaling factors and structural ratios.
Let us re-examine the SI definition of α to reveal the profound clue hidden within its structure:
α = e² / (4πε₀ħc)
A rigorous dimensional analysis confirms that the units of the numerator (Charge squared) are identical to the combined units of the denominator (Permittivity × Action × Speed). The units cancel perfectly, leaving a pure, dimensionless number. This is not a coincidence. The formula is not a random soup of constants; it is a carefully balanced equation where the unit-scaling effects of ε₀, ħ, and c precisely conspire to vanish. This reveals that the SI equation is not creating a dimensionless number, but rather isolating a pre-existing one.
At the Planck scale, this becomes trivially obvious. By defining a natural unit system where the scaling factors ħ, c, and the electric permittivity factor ε₀ are all set to 1, the formula simplifies to its essential, geometric truth: α is the measure of the elementary squared charge e² itself, expressed as a pure ratio relative to the natural unit of action. No "constants" remain because no scaling is needed.
Therefore, the objection can be refuted as follows:
Constants with units exist because our human measurement axes (meter, kilogram, second) are misaligned with the natural axes of the universe. A dimensionless quantity like α has no units precisely because it is a ratio defined within the coherent, self-consistent system of the universe's natural axes.
The two categories are distinct:
Constants with Units (G, c, h, etc.): These are the Jacobians for the coordinate transformation from the Planck chart to our SI chart. They exist to correct for our choice of rulers.
Dimensionless Ratios (α): These are pure numbers that describe the intrinsic geometric or topological structure of reality itself, expressed in the natural Planck chart where no corrections are needed.
The existence of α is not a problem for this framework; it is its prime exhibit. It is a true "ratio of nature"—a pure structural number that requires no unit-scaling constants because it is already expressed in the native, dimensionless language of the universe. The mystery of "why 1/137" is a real, deep question of physics. But it is a question about the geometry of the universe, not about the arbitrary scale of our meter sticks.
6. Algebraic Equivalence vs. Ontological Priority: Deconstructing the Derivation of Planck Units
A subtle but pervasive error in the conventional pedagogy is the narrative that the Planck units are "derived from" the fundamental constants. This framing implies an ontological hierarchy: that the constants (G, c, h) are primary and the Planck units (l_P, m_P, t_P) are secondary, derivative constructs. This is a misreading of the basic algebra involved.
Consider the defining equation for the Planck length:
l_P = √(hG/c³)
This equation is a statement of algebraic equality. As with any such statement, it can be rearranged to solve for any of its variables without changing its truth value. For example, it is equally valid to write:
h = l_P²c³/G or G = hc³/l_P²
The choice of which variable to place on the left-hand side of the equation is a human convention; it does not establish a causal or foundational priority in nature. To say that l_P is "derived from" h, G, and c is no more correct than saying h is "derived from" l_P, G, and c. They are a self-consistent set of inter-definable quantities.
The relationship is not one of derivation but of algebraic equivalence. The set of dimensionful constants {h, G, c, k_B} and the set of Planck base units {l_P, m_P, t_P, T_P} are two different, but informationally equivalent, bases for describing the same physical reality.
This reveals the conventional practice for what it is: an arbitrary choice. The physics community has chosen to treat the constants as primary and the Planck units as derivative. We argue that this choice is historically contingent and conceptually backwards. A more logical and rigorous approach is to recognize the Planck units as the natural, orthogonal basis for reality and to define the dimensionful constants in terms of them:
c := l_P / t_P
h := m_P * l_P² / t_P
G := l_P³ / (m_P * t_P²)
k_B := m_P * l_P² / (t_P² * T_P)
This choice is not circular reasoning; it is a coordinate transformation. It is a declaration of intent to work in a more natural, less arbitrary coordinate system. Instead of measuring reality with human artifacts (the meter, the kilogram), we choose to measure it in terms of its own intrinsic, self-consistent scales. The so-called fundamental constants are then correctly revealed as the conversion factors—the components of the Jacobian matrix—that translate between our arbitrary local chart and nature's universal chart.
This procedure simply formalizes the common practice of "setting the constants to 1" in natural units. It is crucial to note, however, that a rigorous system must be built upon the physically discovered constant of quantization, h (the Planck constant), not the notational convenience ħ (the reduced Planck constant). The latter, ħ = h/2π, folds a pure geometric factor (2π) into a physical constant, creating a conceptual error that obscures the clean separation between physics and geometry. A truly natural system sets the fundamental scaling factors to unity, it does not arbitrarily rescale them for notational convenience.
By acknowledging the algebraic equivalence between the constants and the Planck units, we are free to choose the more logical and conceptually clear foundation. This choice demystifies the constants, clarifies their role as metrological scaling factors, and allows the simple, dimensionless structure of physical law to become manifest.
7. Conclusion: The Demotion of Constants and the Promotion of Structure
The argument that constants are fundamental because they are measured is a fallacy of interpretation. The experiments are real. The results are real. But the thing being measured is not a mysterious, irreducible property of the universe. The thing being measured is the conversion factor between the arbitrary coordinate system of our human-made units and the fundamental, always unified, invariant coordinate system of reality itself, which we call the Planck scale.
This perspective dissolves pseudo-problems like "fine-tuning" and correctly reframes our scientific quest. The goal is not to "explain" the numerical value of G in SI units, any more than the goal of the alchemists should have been to explain "778.17." The goal is to understand the simple, dimensionless, geometric, and topological relationships that govern the universe, for which the constants are merely the temporary, unit-dependent scaffolding we use to get there.
The experiments that measure our "fundamental constants" are the most important experiments in physics. Not because they reveal a mystery, but because they provide the calibration key that allows us to translate the language of our human rulers into the language of the cosmos. They are the Rosetta Stone, and it is time we stopped worshipping the stone itself and started paying attention to the translation.
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