The Eddington Redemption: A Unit-Invariant Framework for 21st Century Physics

 

Abstract
A century ago, Arthur Eddington proposed that dimensional "fundamental constants" (c, h, G, k) are not intrinsic properties of nature but conversion factors between human-defined units. He argued that only dimensionless ratios (like the fine-structure constant, α) are physically meaningful. Despite his mathematical rigor, his ideas were dismissed as "numerology" and suppressed by the emerging quantum field theory establishment.

This paper presents a computational framework that realizes Eddington’s vision, demonstrating that:

1. All dimensional constants reduce to unity under proper unit scaling.

2. The apparent "fundamentality" of c, h, and k is an artifact of radically different unit definitions between each unit of measure.

3. Modern physics can be reformulated in a unit-scaling aware way, dissolving artificial problems like the "hierarchy problem."

We provide open-source tools for automated unit rescaling, proving that Eddington was correct—and that his exclusion from mainstream physics delayed progress for a century.

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1. Introduction: Eddington’s Lost Revolution

In the 1920s, Arthur Eddington made a radical claim:

"The so-called fundamental constants are merely conversion factors between human-defined units. The true laws of physics are dimensionless."

He attempted to derive physics purely from unit-invariant ratios, arguing that:

• c, h, and G are scaffolding, not fundamental.

• The fine-structure constant (α) is the real physics.

Yet his work was dismissed as numerology, and physics instead embraced:

• Dimensional quantum field theory (QFT).

• The romanticization of c and h as "deep constants."

• Artificial problems like "Why is gravity weak?" (a unit mismatch).

This paper corrects history by formalizing Eddington’s insight into a computational framework.

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2. The Eddington Continuation: A Unit-Invariant Framework

2.1 Core Principles

The framework rests on three realizations:

1. Units are arbitrary coordinate choices (like Cartesian vs. polar coordinates) that are simply a scaled version of natural units.

2. Physical laws are simple equivalences—the radically different unit scaling of our units of measure obscure this simple unity.

3. "Fundamental constants c, h and k " are conversion factors, e.g.:

   • c converts time to space. It is the defintion of the meter.

   • h converts frequency to energy.  m/f * c^2 defines h, the m/f ratio defines the kg unit scale now.

   • k converts temperature to energy. f/T m/f c^2 defines k, the f/T ratio defines the K unit scale right now.

2.2 Mathematical Foundation

Let natural units (c = h = k = 1) define the "true" scaling. Then:

• SI units are a rescaled representation, introducing artificial constants.

• Any dimensional equation can be rewritten in unit scaling aware form.

Example: Planck’s Law

• SI form:

  $${\mathrm{<span\; style="background-color:\; white;">B</span>}}_{\mathrm{<span\; style="background-color:\; white;">\nu </span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">\nu </span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">T</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\frac{\mathrm{<span\; style="background-color:\; white;">2</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}{\mathrm{<span\; style="background-color:\; white;">\nu </span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">h</span>}\mathrm{<span\; style="background-color:\; white;">\nu </span>}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">T</span>}}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">1</span>}}$$

• Unit-Scaling Aware form:

  $$B(f,T)=\frac{2f^3\cdot \text{Hz\_kg}}{e^{f\mathrm{/}(T\cdot \text{K\_Hz})}-1}$$

  (All h, k, and c^2 terms are now unit scaling ratios.)

2.3 Computational Implementation

We provide Python modules that:

1. Automatically rescale equations from SI to natural units.
   

2. Expose unit-dependence as a computational artifact, not physics.

3. Verify that all dimensional constants reduce to 1 under unit of measurement scaling.
   

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3. Implications: Dissolving Pseudoproblems

3.1 The "Hierarchy Problem" is a Unit Mismatch

• Claim: Gravity seems "weak" because G is small compared to h and c.

• Reality: In natural units (G = 10^-86), there is no hierarchy—g is a measured force between two masses of 1 units of mass at 1 unit of distance.  G has nothing to do with c or h because G is not about unit scaling.

3.2 Quantum Gravity’s "Unification" is current wrong.

• In natural units, GR and QFT already use the same scales (c = h = G = 1).
  This is actually wrong.  G is not unit scaling, if you scale the meter and kg such that c=h=1 then it is physically impossible for G to also be 1. 

• The problem is assuming that "E=" constants which are unit scaling between equivalent properties are the same thing as "F=" force constants that are measured force values measured in newtons with extra units added to make the property squared divided by radius squared  proportion dimensionless. Where that dimensionless proportion is 1, then the force law formulas become a simple  F = G newtons where ( meter^2/kg^2  * m1m2/r^2) =1.  

• What this means is that there is a curve of space time, and G is a measurement of the force along that curve under specific unit defined conditions.  Currently two 1 kg masses a meter apart is when that proportion is equal to 1. 

• Because this is not understood, quantum gravity is asking the wrong question.

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4. Why Eddington Was Suppressed—And Why Now is Different

4.1 The 20th Century Suppression

Eddington’s ideas threatened:

1. The rise of quantum field theory, which embedded ħ and c into its mathematical formalism, making them appear indispensable.

2. The cultural and scientific impact of relativity, which elevated c and G as cornerstones of modern physics, inadvertently reinforcing their perception as fundamental rather than scale-dependent.

3. The institutionalization of metrology, which relied on SI units for standardization, making unit-invariant approaches seem abstract or impractical

4.2 The 21st Century Opportunity

Now, we have:

1. The 2019 SI redefinition, which fixed h, e, and k as exact (admitting they’re conventions).

2. Computational physics, which can automate unit rescaling and prove equivalence.

3. A crisis in theoretical physics, forcing a re-examination of foundations.

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5. Conclusion: Finishing Eddington’s Revolution

Eddington was right. Dimensional constants are human conventions. The true laws of physics are unit-invariant ratios.  But these constants are useful artifacts in human convenient unit systems of measurement. But only if we break these out as individual unit scaling factors so they are clear what they are doing in formulas.

Call to Action

1. Adopt unit-invariant teaching (natural units first in the earliest education of children, then teach SI later as a practical approximation, a coordinate scaling of natural units into a unit of measure convenient for humans).

2. Reformulate theories using modular unit scaling factors (e.g., express h as Hz_kg kg_J).

3. Use computational tools to automate rescaling and expose unit artifacts.

The era of "fundamental constant" mysticism is over. The future is seeing all valid unit systems as coordinate scalings of each other and that constants are the unit mapping between equivalent measures.

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Supplementary Material

• Modular Unit Scaling Framework
  
• Physics Unit Coordinate System

Acknowledgments
To Arthur Eddington—who saw the truth a century ago, and to the open-source community, which makes its realization possible today.