Politics, Power, and Science: The Epistemological Impossibility of Constants as Physical Entities: Why G Cannot "Embody" Gravitational Physics

Sunday, October 12, 2025

The Epistemological Impossibility of Constants as Physical Entities: Why G Cannot "Embody" Gravitational Physics

J. Rogers, SE Ohio

Abstract

A persistent misconception in physics holds that dimensional constants like G, c, h, and k_B are physical entities that "embody" or "encode" fundamental aspects of nature—that G somehow contains the "strength" of gravity, or that h represents the "quantum of action." This paper demonstrates that this view is logically impossible and constitutes a category error of the highest order. We prove that if constants were physical entities, the universe would require observer-dependent telepathy: it would need to know which arbitrary unit system each observer chose and dynamically adjust physical law accordingly. Since this is absurd, we conclude that dimensional constants must be purely metrological artifacts—conversion factors between human measurement conventions and dimensionless natural structure. The paper provides a rigorous proof by reductio ad absurdum, supported by historical evidence, dimensional analysis, and explicit counterexamples.

1. Introduction: The Reification Fallacy

1.1 The Standard View

In contemporary physics education and literature, statements like the following are commonplace:

"G is the gravitational constant that determines the strength of gravity"
"h is the quantum of action, setting the scale of quantum effects"
"c is the speed of light, the universal speed limit"
"k_B is Boltzmann's constant, linking temperature to energy"

These statements treat dimensional constants as physical entities—properties of the universe that exist independently of human measurement conventions.

1.2 The Implicit Ontology

This view carries profound ontological commitments:

Physical Realism: Constants are real features of nature
Observer Independence: Their values are intrinsic, not conventional
Causal Efficacy: Constants actively "set scales" or "determine strengths"
Discovery Status: We measured their values; we did not choose them

This paper will demonstrate that all four claims are false, and that maintaining them requires accepting a logically impossible consequence.

2. The Central Argument: Reductio ad Absurdum

2.1 The Setup

Claim to be Refuted: Dimensional constants like G are physical entities that embody aspects of natural law.

Proof Strategy: Show that if this claim were true, the universe would require impossible knowledge of arbitrary human conventions.

2.2 The Proof

Premise 1: If G is a physical entity that embodies gravitational dynamics, then G's numerical value is a property of nature itself.

Premise 2: G's numerical value depends on the choice of units:

In SI units: G = 6.674×10⁻¹¹ m³/(kg·s²)
In CGS units: G = 6.674×10⁻⁸ cm³/(g·s²)
In Planck units: G = 1 (dimensionless)
In geometrized units: G = 1/(8π)

Premise 3: Unit systems are arbitrary human conventions:

The kilogram was originally defined as the mass of 1 dm³ of water
The meter was defined as 1/10,000,000 of the distance from equator to North Pole
These definitions have changed multiple times (1799, 1889, 1960, 1983, 2019)
Different civilizations could choose completely different standards

Premise 4: Physical law must produce correct predictions regardless of which unit system an observer uses.

The Contradiction:

If G is a physical property of nature (Premise 1), its value must be observer-independent. But G's numerical value is observer-dependent (Premise 2), determined by arbitrary conventions (Premise 3). For the same physical law to work for all observers using different conventions (Premise 4), one of the following must be true:

Option A: The universe knows which unit system each observer is using and dynamically adjusts G's value to compensate.

Option B: G is not a physical property but a conversion factor between the observer's arbitrary units and nature's dimensionless structure.

Option A is absurd. It requires:

The universe to have knowledge of human conventions
Physical law to be observer-dependent
Constants to change retroactively when we redefine units
Gravitational dynamics to care about French revolutionary committees

Therefore, Option B must be true: G is a metrological artifact, not a physical entity.

QED

3. Detailed Analysis: How the Universe Would Need to "Know"

3.1 The Telepathy Problem

Consider two physicists performing the same gravitational experiment:

Physicist A (using SI units):

Measures masses in kilograms
Measures distances in meters
Calculates force using F = G·(m₁m₂/r²)
Uses G = 6.674×10⁻¹¹

Physicist B (using Planck units):

Measures masses in Planck masses
Measures distances in Planck lengths
Calculates force using F = (m₁m₂/r²)
Uses G = 1

Both get the same physical force (when converted to common units).

The Question: How does gravity "know" to produce different numerical outputs depending on which ruler the physicist is holding?

If G is a physical property, it must have one true value. But which one?

6.674×10⁻¹¹? (Then Physicist B is using the wrong value)
1? (Then Physicist A is using the wrong value)
Both? (Then G is relative to observer choice—a convention, not a property)

3.2 The Historical Redefinition Problem

In 2019, the SI system was redefined:

The kilogram was redefined via Planck's constant (h = 6.62607015×10⁻³⁴ J·s exactly)
This changed the operational definition of mass
Consequently, the measured value of G shifted slightly within its uncertainty

Critical Question: Did the strength of gravity change in 2019?

If G embodies gravitational physics: Then redefining our kilogram changed the actual force of gravity—planetary orbits should have adjusted, gravitational waves should have altered frequency, black holes should have changed temperature.

Obviously: None of this happened. Gravity remained identical. Only our measurement convention changed.

Conclusion: G does not embody gravitational physics. It embodies our unit convention.

3.3 The Dimensional Analysis Proof

Consider Einstein's field equations in their standard form:

Rμν - ½Rgμν = (8πG/c⁴)Tμν

The left side has units of [1/length²]. The right side, before the constant factor, has units of [energy/length⁴]. The factor (8πG/c⁴) has units of [length²·s²/(energy·s²)] = [length²/energy], converting the stress-energy tensor's dimensions to match curvature.

The constant factor is purely dimensional conversion.

Now consider the same equation in natural units (c = G = 1):

Rμν - ½Rgμν = 8πTμν

Same physics. No constants.

The geometry of spacetime curvature relates to stress-energy through the dimensionless factor 8π. The constants c and G served only to make dimensions compatible in SI units.

If G encoded the "strength" of gravity, it would appear in the natural units version. It doesn't. Because 8π is the actual coupling factor—a dimensionless geometric constant that appears in both forms.

G exists only to translate between SI energy density and SI curvature. It's a Jacobian of measurement, not a physical strength.

4. The Correct Interpretation: Constants as Jacobians

4.1 What Constants Actually Are

A dimensional constant is a composite conversion factor relating different measurement axes that are projections of a unified substrate.

Mathematical Definition:

G = (F_P · l_P²) / m_P²

Where F_P, l_P, m_P are the Planck force, length, and mass—the natural scales where the dimensionless structure of physics is most simply expressed.

Physical Interpretation:

G tells you how many SI force-units (Newtons) correspond to a dimensionless gravitational interaction when you've chosen to measure mass in kilograms and distance in meters.

4.2 The Tautology of Measurement

The structure of physics with constants is:

F_SI = (F_nat · F_P) / F_P · (dimensionless law)

Breaking this down:

F_nat: The dimensionless true value (what nature computes)
F_nat · F_P: Scaling to SI units (so we can measure it)
/ F_P: Normalizing back (so it obeys the dimensionless law)

The constant G (or any constant) pre-packages the ratio F_P/(...). It's the undo button for our arbitrary unit choices, baked into the formula.

Example with Gravity:

Dimensionless form: F_nat = (m₁_nat · m₂_nat) / r_nat²

SI form: F_SI = G · (m₁_SI · m₂_SI) / r_SI²

Where: G = F_P · l_P² / m_P²

G doesn't add physics. It adds unit conversion. The physics is in the ratio relationship, which is identical in both forms.

4.3 Why This View is Forced Upon Us

We have two options:

Option 1 (Standard View): Constants are physical, and somehow the universe adjusts them based on our unit choices.

Option 2 (Correct View): Constants are metrological, arising from our projection of unified dimensionless structure onto arbitrary measurement axes.

Since Option 1 requires telepathic universe awareness of human conventions, Option 2 is the only rational choice.

5. Addressing Counterarguments

5.1 "But we measured G experimentally!"

Response: You measured how your arbitrary kilogram and meter relate to the natural scale of gravity. That relationship is real, but it's a relationship between your rulers and nature's dimensionless structure, not a property of nature alone.

Analogy: If you measure that "1 mile = 1.609 kilometers," you've measured something real—the relationship between two human-defined units. But neither miles nor the conversion factor 1.609 are properties of space itself.

5.2 "But G appears in the Einstein field equations!"

Response: G appears in the SI-unit version of Einstein's equations to make the dimensions work out. In natural units (Planck units, geometrized units), G = 1 or disappears entirely. The equations work identically. The physics is in Rμν ~ Tμν, not in G.

The appearance of G is like the appearance of "2.54" in the equation "centimeters = 2.54 × inches." The 2.54 isn't fundamental to length; it's a conversion artifact.

5.3 "But fine-tuning arguments depend on constant values!"

Response: Fine-tuning arguments about dimensional constants are meaningless. You can make G equal π, e, or 42 by choosing your units appropriately.

Fine-tuning only matters for dimensionless ratios like the fine structure constant α ≈ 1/137, the proton-electron mass ratio ≈ 1836, or the cosmological constant in Planck units. These are observer-independent and genuinely meaningful.

Asking "why is G this value?" is like asking "why are there 5,280 feet in a mile?" The answer is: because humans chose those definitions.

5.4 "But constants have been measured to incredible precision!"

Response: Precision of measurement does not confer ontological status. We can measure the "furlong per fortnight" to incredible precision as a velocity unit. That doesn't make furlongs or fortnights fundamental aspects of spacetime.

High-precision measurement of G means we've calibrated our rulers very carefully against the natural gravitational scale. That's metrology, not discovery of fundamental physics.

6. Historical Evidence: Newton Understood This

6.1 Newton's Deliberate Omission

Isaac Newton never wrote F = G(m₁m₂/r²). He wrote:

F ∝ m₁m₂/r²

This was not ignorance or incompleteness. Newton, as Master of the Royal Mint, dealt daily with unit conversions between English pounds, French livres, Spanish reales, and actual gold/silver weights. He understood that any constant of proportionality is unit-dependent.

6.2 The Method of Ratios

Newton structured his entire Principia to work with ratios, causing any unknown constant k to cancel:

F₁/F₂ = (m₁m₁'/r₁²) / (m₂m₂'/r₂²)

This method is unit-invariant. It works whether you measure in English feet or French pieds, pounds or livres. The ratio structure ensures that the unit-dependent factor (our G) divides out.

Newton knew: The proportionality constant is an artifact of your measurement system. The physics is in the ratio.

6.3 Einstein Knew Too

Einstein explicitly stated regarding E = mc²:

"The factor c² appears only because we have chosen to measure mass and energy in different units."

He understood that c² is a conversion factor, not a physical entity. If we measured energy in units of mass (or vice versa), the equation would be E = m.

The 20th century forgot this wisdom and began reifying conversion factors as profound mysteries.

7. The Smoking Gun: Committee-Determined Constants

7.1 The 2019 SI Redefinition

On May 20, 2019, the International Bureau of Weights and Measures (BIPM) redefined the SI system:

h was fixed: h = 6.62607015×10⁻³⁴ J·s (exactly)
e was fixed: e = 1.602176634×10⁻¹⁹ C (exactly)
k_B was fixed: k_B = 1.380649×10⁻²³ J/K (exactly)

These values were chosen by committee vote. They could have chosen h = π×10⁻³⁴ or k_B = e×10⁻²³. The choice was based on historical continuity, not physical necessity.

7.2 The Devastating Implication

If constants embody physical properties, then:

A committee meeting in France determined fundamental properties of the universe.

This is obviously absurd. What actually happened:

A committee meeting in France determined how our conventional rulers relate to natural scales.

The constants were set by bureaucratic fiat because they are definitions of our measurement framework, not discoveries about nature.

7.3 The Counterfactual Test

Imagine an alternate history where the French Revolution defined:

The meter as 1/π × 10⁷ of Earth's quadrant
The kilogram as the mass of a liter of mercury
The second as 1/100,000 of a day

In that world:

c would be ~9.46×10⁸ m/s
G would be ~2.1×10⁻¹⁰ m³/(kg·s²)
h would be ~2.1×10⁻³³ J·s

Would the physics be different? No. The same experiments would yield the same natural outcomes, just expressed with different numerical values.

Would the constants "embody" different physics? No. They would embody different human conventions.

This thought experiment proves: constants are conventional, not physical.

8. Implications for Physics Education and Research

8.1 The Pedagogical Disaster

Teaching that "G is the gravitational constant that sets the strength of gravity" creates profound confusion:

Students wonder "why this value?" when the question is meaningless
They fail to distinguish dimensional from dimensionless quantities
They mistake coordinate artifacts for physical law
They cannot recognize when they're doing numerology vs. physics

Corrected pedagogy would teach:

The dimensionless structure: F/F_P = (m₁/m_P)(m₂/m_P)/(r/l_P)²
How to project to SI: Multiply by F_P and substitute Planck units
That G = F_P·l_P²/m_P² is the resulting conversion factor
The physics is in the ratio structure, not the constant

8.2 The Research Misdirection

Treating constants as physical has led to pseudoproblems:

The "Hierarchy Problem": Why is the Planck mass so much larger than particle masses?

Actual answer: Because we chose to measure them in units where the Planck scale is at 10⁻³⁵ m. It's a coordinate choice.

The "Cosmological Constant Problem": Why is Λ so small in Planck units?

Actual answer: Partly because "small in Planck units" is asking about a ratio to our arbitrarily defined coordinate scale.

"Unifying G and h": Attempting to find a deep relationship between two unit-system artifacts.

Actual answer: They're already unified—both are Jacobian components of the projection from dimensionless substrate to SI units.

8.3 What We Should Be Doing Instead

Legitimate research questions focus on dimensionless ratios:

Why is α ≈ 1/137.036? (fine structure constant)
Why is m_proton/m_electron ≈ 1836? (mass ratio)
Why is the cosmological constant ~10⁻¹²² in natural dimensionless units?

These are observer-independent and genuinely mysterious. Their values would be the same for any civilization, regardless of their unit choices.

Illegitimate questions focus on dimensional values:

Why is G = 6.674×10⁻¹¹ m³/(kg·s²)?
Why is h = 6.626×10⁻³⁴ J·s?

These are observer-dependent and have no physical answer—only historical/conventional answers about why humans chose particular unit standards.

9. Conclusion: The Category Error

The belief that dimensional constants are physical entities is a category error—a confusion of measurement framework with measured reality.

The Error: Treating conversion factors (between arbitrary units and natural scales) as if they were properties of nature.

The Consequence: Attributing to the universe impossible knowledge of human conventions, or alternatively, accepting that physical law is observer-dependent in an absurd way.

The Correction: Recognizing that:

Reality is dimensionless structure: Proportional relationships, ratios, geometric relationships
Constants are Jacobians: Transformation coefficients between our coordinate system and that structure
Physics is in the ratios: Not in the specific numerical values of dimensional quantities

The universe doesn't "know" about kilograms, meters, or seconds. It doesn't "contain" G, h, or k_B as properties. These exist in the translation layer between human measurement conventions and nature's dimensionless architecture.

If G encoded gravitational physics, the universe would need to know how our units are scaled to correct the math for us.

Since that's epistemologically impossible, G does not encode gravitational physics. It encodes our choice of rulers.

This is not a philosophical quibble. It's a fundamental truth about the structure of physical law that modern physics has forgotten, to its detriment.

Appendix: The Complete Logical Structure

Formal Proof by Contradiction

Theorem: Dimensional constants cannot be physical entities.

Proof:

Assume: G is a physical entity embodying gravitational dynamics (Hypothesis to be contradicted)
Fact 1: Physical entities have observer-independent properties (Definition of "physical")
Fact 2: G's numerical value varies with unit choice:
- G_SI = 6.674×10⁻¹¹ m³/(kg·s²)
- G_Planck = 1 (dimensionless)
- G_CGS = 6.674×10⁻⁸ cm³/(g·s²)
Fact 3: Unit choices are arbitrary human conventions, not physical properties
From 1 & 2: If G is physical, its value must be observer-independent, yet its value depends on observer's unit choice
Resolution requires either:
- (A) Universe has knowledge of each observer's unit convention and adjusts G accordingly
- (B) G is not a physical entity but a conversion factor dependent on unit choice
(A) is impossible: Physical law cannot depend on human conventions or committee decisions
Therefore (B): G is not a physical entity ∎

Corollary: The same proof applies to all dimensional constants (h, k_B, c, etc.)

References

Rogers, J. (2024). "The Constant of Proportionality: Newton's Deliberate Omission of G and the Lost Principle of Invariance."

Rogers, J. (2024). "G as a Metrological Artifact: A Formal Proof of Newton's Principle of Invariance."

Rogers, J. (2025). "The Structure of Physical Law as a Grothendieck Fibration."

BIPM (2019). "The International System of Units (SI), 9th Edition."

Einstein, A. (1905). "Does the Inertia of a Body Depend Upon Its Energy Content?" Annalen der Physik.

Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.