Physical Constants as Unit Harmonization

 J. Rogers, SE Ohio

Introduction

The speed of light c, the gravitational constant G, and Planck's constant h are often described as the sacred pillars of modern physics—the fundamental constants that define the fabric of our universe. We ask why they have the values they do, imagining that these numbers hold the deepest secrets of cosmic design. This paper argues that this view is a profound misunderstanding.

We will demonstrate that these dimensional quantities are not fundamental properties of nature, but rather harmonization coefficients—artifacts generated by our system of measurement. They exist for one reason: to correct for our historically arbitrary choice to define independent scales for length, time, mass, and energy when, in reality, these quantities are intrinsically linked. The true, invariant physics lies not in these conversion factors, but in the dimensionless ratios and coordinate-free proportionalities they obscure. By separating the concerns of physical law from measurement convention, we will show how these "constants" emerge mechanically from our scaling choices and, in doing so, reveal the underlying unity they conceal.

The natural unit system in this paper is the original 1899 Planck units with h. This is because the simple proportion is E~f, not E~w. 

The Three Separate Concerns

Physical theories conflate three distinct concerns that should be separately defined:

1. Arbitrary measurement scales - Our conventional choices of meter, kilogram, second, etc. These have no physical content.

2. Invariant physical ratios - The actual relationships between physical quantities that don't depend on measurement choices. This is the real physics.

3. Dimensional constants - Numbers like c, G, h, k_B whose function is to harmonize our arbitrarily scaled measurement axes.

Dimensional vs. Dimensionless Quantities

Before proceeding, we must distinguish two fundamentally different types of quantities:

Dimensional constants like c, G, h, and k_B have units (m/s, N·m²/kg², J·s, J/K). These serve to harmonize our measurement systems and are the subject of this analysis.

Dimensionless ratios like the fine structure constant α ≈ 1/137, mass ratios between particles, and coefficients like 8π are pure numbers. These are coordinate-free, unit-independent quantities that represent the actual physics - the "fingerprints" of reality that remain regardless of how we choose to measure. These are not "constants" in the sense we critique here; they are the invariant relationships that constitute physical law.

The Core Insight

We incorrectly assume our measurement axes are independent. We believe we can choose meters and seconds independently, kilograms and joules independently, and so on. But the underlying physics connects these quantities through fixed ratios. Dimensional constants exist to convert between our independent conventional choices and the coupled reality.

Natural Units Reveal Unified Physical Reality

Natural units were never about "setting constants to 1" for convenience. They reveal something deeper: the quantities we measure as mass, length, time, temperature, energy, momentum, and force are not independent entities but are different facets of a single, unified physical reality. They are fully interconvertible, and the laws of physics are the rules of this conversion.

In natural units:

T_nat = f_nat = m_nat = 1/length_nat = E_nat = p_nat = F_nat

These equivalences show that we are measuring different aspects of the same underlying physical reality. The physics is this equivalence chain - the coordinate-free relationships between quantities.

The Tautology of Measurement

To scale from natural units to SI units, we perform what can be called a "tautology of measurement" - multiplying each quantity by 1, expressed as its Planck unit ratio:

T_nat × (T_P/T_P) 

= f_nat × (t_P/t_P) 

= m_nat × (m_P/m_P) 

= (1/length_nat) × (l_P/l_P) 

= E_nat × (E_P/E_P) 

= p_nat × (p_P/p_P) 

= F_nat × (F_P/F_P)

This is perfectly legal mathematically and adds zero information about physics. We're just encoding our choice of SI scaling.  X_si = X_nat * X_P is the pattern to watch.

Now we can express each quantity in SI units:

    T_si/T_P = f_si × t_P = m_si/m_P = l_P/length_si = E_si/E_P = p_si/p_P = F_si/F_P

The dimensionless ratios remain identical to the natural unit chain.

How Constants Emerge from Scaling Choices

Pick any two quantities from this chain and scale them to SI. For example, energy and frequency:

First we start with the physics:

E_nat ~ f_nat

Then we add the human scaling as a meaningless tautology:

E_nat * 1 = f_nat * 1

E_nat × E_P/E_P = f_nat × t_P/t_P 

Since E_si = E_nat * m_P and f_si = f_nat/t_P 

E_si/E_P = f_si × t_P

Solve for E_si:

E_si = f_si × (E_P × t_P) = h × f_si

The constant h is simply the product E_P × t_P - a unit conversion factor that appears because we independently scaled our energy axis and our time axis, when they're actually measuring coupled aspects of the same physical reality.

The physical law is the proportionality E ∝ f. The constant h is not part of the physical law itself; it is a conversion factor that emerges solely from our choice to measure energy in Joules and frequency in Hertz. The physics is the relationship, not the scaling number.

Why Some Equations Have No Constants

Consider Newton's second law: F = ma

When traced back through Planck units:

F_si/F_P = (m_si/m_P) × (l_si/l_P)/(t_si/t_P)²

The Planck unit ratios cancel completely on both sides, leaving:

F_si = m_si × a_si

No constants appear because the dimensional structure is balanced. The scaling factors for force, mass, and acceleration cancel through their interdependence.

Constants appear exactly where our independent axis scaling choices create an imbalance that doesn't self-cancel.

Generating Constants from Dimensional Imbalance

Constants can be mechanically generated from the dimensional imbalance in any equation. Consider Einstein's field equations.

In natural units:

G_μν_nat = 8π T_μν_nat

The actual physics is the dimensionless ratio 8π - a pure number independent of any measurement convention.

When scaling to SI:

• Left side: G_μν has dimensions 1/L²
• Right side: T_μν has dimensions Force/L²

The dimensional imbalance requires multiplication by 1/Force:

Required scaling factor = 1/F_P

And indeed: G/c⁴ = 1/F_P

The constant G/c⁴ is deterministically generated by our unit scaling choices, not derived from nature.

What Constants Really Are

Dimensional constants are not fundamental properties of nature. They are harmonization coefficients:

• c - Harmonizes our independent time axis to our independent length axis
• G - Part of harmonizing our mass/energy axes to our spatial curvature axis
• h - Harmonizes our energy axis to our frequency axis
• k_B - Harmonizes our temperature axis to our energy axis

The actual physics consists of coordinate-free, dimensionless relationships:

• The fine structure constant α ≈ 1/137 (a charge ratio with human convention removed)
• The coefficient 8π in Einstein's equations
• Ratios between particle masses
• All proportionalities in natural units (E_nat ∝ f_nat, E_nat ∝ m_nat)

These dimensionless quantities tell us something real about nature. But c, G, h, k_B are artifacts of pretending our measurement axes are independent when they're not.

The Physics Lives in the Proportionalities

When we write E = hf or E = mc², the dimensional constants (h and c²) are unit-dependent artifacts. The physics - the actual physical law - is:

• E ∝ f (energy is proportional to frequency)
• E ∝ m (energy is proportional to mass)

These proportionalities are what Einstein and Planck discovered. The constants of proportionality depend entirely on how we chose to scale our measurement axes. In natural units, these constants vanish, revealing the underlying unity: E_nat = f_nat = m_nat.

Historical Context

Einstein understood this about mass-energy equivalence. He insisted the physics was E ∝ m, and that c² simply converts between how we measure mass versus how we measure energy. The proportionality was the discovery, not the constant.

Newton designed his laws to be unit-invariant. F = ma works the same regardless of whether you use feet, meters, or furlongs. He deliberately structured his physics so arbitrary human measurement choices would cancel out.

Modern physics has reified unit conversion factors into "fundamental constants of nature" and built ontology around them, asking questions like "why does c have this value?" - which is like asking "why are there 2.54 centimeters in an inch?"

The Software Engineering Perspective

Physics equations are code that mixes concerns:

• Business logic (invariant ratios)
• Presentation layer (measurement conventions)

Constants are magic numbers hardcoded throughout instead of being properly isolated as configuration parameters.

A properly refactored physics would separate:

1. Natural ratio layer - The actual physics (coordinate-free proportionalities and dimensionless ratios)
2. Measurement interface layer - The Jacobian mappings (Planck units)
3. SI presentation layer - Human-convenient units

Constants exist because we're calling across layers without proper abstraction. This isn't philosophy - it's software engineering applied to physics notation, revealing that our theories are more obfuscated than necessary.

Conclusion

The dimensional constants we treat as fundamental are conversion factors generated by our choice to scale measurement axes independently. The real physics consists of:

• Coordinate-free proportionalities (E ∝ f, E ∝ m)
• Dimensionless ratios (α, mass ratios, geometric coefficients)

These are the unit-independent fingerprints of our universe. Everything else is bookkeeping required to translate between our arbitrary measurement conventions and physical reality.