The Equivalence of SI Units with Constants and Planck Scale Physics: A Dimensional Analysis

J. Rogers, SE Ohio, 28 Jul 2025, 1453

Abstract

We demonstrate that the SI unit system with physical constants is mathematically identical to the Planck unit system, differing only by coordinate transformation. What physics interprets as "fundamental constants" are revealed to be Jacobian coefficients for basis rotation between measurement coordinates. This equivalence dissolves the artificial distinction between "natural units" and "physical constants," showing that both describe the same underlying dimensional relationships in different coordinate representations.

1. Introduction

Modern physics treats the Planck scale as a convenient mathematical construction while regarding physical constants (ℏ, c, G, k_B) as fundamental properties of nature. This paper demonstrates that these perspectives describe identical mathematical structures. The SI system with constants is the Planck system expressed in non-orthogonal coordinates, with constants serving as coordinate transformation coefficients.

2. The Dimensional Equivalence

2.1 Planck Units as Natural Coordinates

In Planck units, fundamental relationships become dimensionless:

• Energy: E_P = 1 (dimensionless)
• Length: l_P = 1 (dimensionless)
• Time: t_P = 1 (dimensionless)
• Mass: m_P = 1 (dimensionless)

Physical laws in these coordinates reduce to simple proportionalities:

• Gravity: F ∝ M₁M₂/r²
• Energy-frequency: E ∝ f
• Uncertainty: Δx × Δp ∝ 1

2.2 SI Units as Coordinate Projections

The same relationships in SI coordinates require transformation coefficients:

• Gravity: F = GM₁M₂/r²
• Energy-frequency: E = hf
• Uncertainty: Δx × Δp ≥ h/2

The boldface terms are not fundamental constants but coordinate transformation coefficients (Jacobians) relating SI basis vectors to Planck basis vectors.

3. The Jacobian Structure

3.1 Dimensional Jacobian Matrix

The transformation between SI and Planck coordinates is governed by:

[SI quantities] = [Jacobian Matrix] × [Planck quantities]

Where the Jacobian elements are combinations of c, h, G, k_B.

3.2 Specific Transformations

Gravitational Constant:

• G = l_P³/(m_P × t_P²) = (coordinate scaling factor)
• Not a "strength of gravity" but a basis rotation coefficient

Planck's Constant:

• h = m_P × l_P²/t_P = (coordinate scaling factor)
• Not "quantum of action" but a dimensional conversion factor

Speed of Light:

• c = l_P/t_P = (coordinate scaling factor)
• Not "cosmic speed limit" but a unit ratio

4. The Proof: Computational Verification

4.1 The Dimensional Algorithm

Physical laws can be derived algorithmically:

1. Substrate Relation: Express relationship in dimensionless Planck coordinates
2. Coordinate Transform: Apply Jacobian to convert to SI coordinates
3. Result: Standard physical law with "constants"

4.2 Examples

Einstein's Mass-Energy Equivalence:

• Planck scale: E/E_P = M/m_P (dimensionless)
• Jacobian transform: E = (E_P/m_P) × M = c² × M
• Result: E = Mc²

Newton's Gravity:

• Planck scale: F/F_P = (M₁M₂/m_P²) × (l_P²/r²)
• Jacobian transform: F = (F_P × l_P³)/(m_P × t_P² × m_P²) × M₁M₂/r²
• Result: F = GM₁M₂/r²

5. The Constants as Epicycles

5.1 Historical Parallel

Just as Ptolemaic epicycles were mathematical constructs preserving geocentric coordinates, physical constants are mathematical constructs preserving anthropocentric measurement coordinates.

Ptolemaic System:

• Complex epicycles → Simple ellipses (heliocentric coordinates)

Modern Physics:

• Complex constants → Simple proportionalities (Planck coordinates)

5.2 The Dimensional Fraud

Physics has mislabeled coordinate artifacts as fundamental properties:

h is not "action" - it has dimensions [M⋅L²⋅T⁻¹] = mass × area/time = spatial flux coefficient

G is not "gravitational strength" - it's the coordinate conversion factor between mass-space-time units

c is not "speed limit" - it's the metric ratio between space and time coordinates

6. Implications

6.1 Unified Physics

There are no separate quantum mechanics, relativity, and thermodynamics. There is only temporal substrate physics viewed through different coordinate projections:

• Quantum mechanics: Temporal dynamics in small-scale coordinates
• General relativity: Temporal dynamics in large-scale coordinates
• Thermodynamics: Collective temporal behavior

6.2 The End of Constants

In natural coordinates:

• No gravitational constant (gravity is geometric)
• No Planck's constant (quantum effects are temporal)
• No Boltzmann constant (temperature is frequency)
• No speed of light (space and time have natural ratio)

6.3 Simplified Universe

Physical reality consists of:

• Single substrate: Temporal experience (mc² = hν)
• Single interaction: Temporal field products ((m₁/r) × (m₂/r))
• Single scaling: Planck coordinates are reality's natural units

7. Conclusion

The SI unit system with physical constants and the Planck unit system are mathematically identical - they are the same dimensional structure expressed in different coordinates. What physics calls "fundamental constants" are coordinate transformation artifacts with no more fundamental reality than the epicycles of Ptolemaic astronomy.

This equivalence reveals that:

1. The universe is already unified at the Planck scale
2. Constants are coordinate-dependent illusions
3. Physical complexity emerges from measurement coordinate misalignment
4. Unity is found by working in nature's natural coordinates

The deepest truth of physics is not that reality is complex, but that reality is simple and our coordinate systems are complex. The Planck scale is not an exotic frontier - it is the natural coordinate system where the universe's fundamental simplicity becomes apparent.

References

References

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   • Foundational development of Planck units and Planck’s constant.

2. Planck, M. (1899). Natürliche Maßeinheiten. Übers. von J. Rapior, A. Weller, W. Wöß. European Physical Journal H, 39(2), 205–207 (2014).

   • Planck's original proposal of "natural units" and dimensional arguments.

3. Bridgman, P. W. (1922). Dimensional Analysis. New Haven: Yale University Press.

   • The classic treatise on defining and relating physical units via dimensional analysis.

4. Duff, M. J., Okun, L. B., & Veneziano, G. (2002). Trialogue on the number of fundamental constants. Journal of High Energy Physics, 2002(03), 023. arXiv:physics/0110060

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For a synthetic overview and critique (including the novel interpretation of constants as purely Jacobian coefficients):

• See the discussion in Duff et al., 2002.

• [Misner, Thorne, Wheeler, 1973] analyze how constants vanish in natural coordinates, reflecting the same insight as described in this work.