How Modern Metrology Implicitly Adopted the Planck Reference Frame
J. Rogers, SE Ohio
Abstract
The 2019 redefinition of SI base units is widely described as "fixing fundamental constants for measurement stability." This paper demonstrates that the redefinition actually accomplished something more profound: it fixed the Jacobian transformation between the arbitrary SI coordinate system and the natural Planck coordinate system. By defining units through exact values of c, h, and Δν_Cs, the SI now explicitly defines each measurement axis as a scaled rotation of basis from the original, h-based Planck units. This transforms the SI from an artifact-based system to a Jacobian-fixed projection of natural units—though this structural reality remains unacknowledged in official documentation.
1. Introduction: What Really Changed in 2019
The Official Story
The Bureau International des Poids et Mesures (BIPM) describes the 2019 redefinition as fixing numerical values of fundamental constants to eliminate dependence on physical artifacts, improve measurement stability, and enable better precision across scales.
The Mathematical Reality
What actually happened was more fundamental: The SI explicitly became a Jacobian-projected coordinate system. By fixing the values of c, h, k_B, e, and Δν_Cs, the 2019 redefinition fixed the transformation matrix from SI to Planck units, making each SI unit a defined multiple of its Planck counterpart. The Planck units are now implicitly embedded in the definition of every SI measurement.
2. The Pre-2019 System: Artifacts and Measurements
How Units Were Defined (Pre-2019)
Kilogram: Mass of the International Prototype Kilogram (IPK).
Meter: Fixed via c.
Second: Fixed via Δν_Cs.
The Measurement Process
The logical flow was to use these artifacts to measure constants, from which the Planck units were then derived as uncertain quantities. This paper uses Max Planck's original 1899 formulation based on the Planck constant h, not the modern reduced constant ħ (h-bar).
Step 1: Start with physical artifacts.
Step 2: Use these to measure h and G (with uncertainty).
Step 3: Calculate the h-based Planck units from measured constants:
m_P = √(hc/G) ≈ 5.456 × 10⁻⁸ kg (with uncertainty from G)
l_P = √(hG/c³) ≈ 4.051 × 10⁻³⁵ m (with uncertainty from G)
t_P = √(hG/c⁵) ≈ 1.351 × 10⁻⁴³ s (with uncertainty from G)
The Planck units were derived quantities with measurement uncertainty, primarily from G.
3. The 2019 System: Fixed Jacobians
How Units Are Now Defined (Post-2019)
The 2019 redefinition reversed the logical flow: Fixed Constants → Planck units (implicitly fixed) → SI units (defined by Jacobian).
The Seven Exact Definitions
The SI is now defined by fixing these exact values:
Δν_Cs = 9,192,631,770 Hz
c = 299,792,458 m/s
h = 6.62607015 × 10⁻³⁴ J·s
e = 1.602176634 × 10⁻¹⁹ C
k_B = 1.380649 × 10⁻²³ J/K
N_A = 6.02214076 × 10²³ mol⁻¹
K_cd = 683 lm/W
By fixing c, h, and Δν_Cs, you have completely determined the relationships for the mechanical Planck units. The entire uncertainty of the absolute Planck scale is now isolated in the ongoing measurement of G.
4. The Jacobian Structure
The SI system is now a coordinate chart defined by its fixed Jacobian transformation from Planck space. The components of this transformation are the physical constants.
The Transformation Equations (h-based)
Meter ↔ Planck length:
1 m = (1 / l_P) * l_P ≈ 2.4684 × 10³⁴ l_P
Kilogram ↔ Planck mass:
1 kg = (1 / m_P) * m_P ≈ 1.8330 × 10⁷ m_P
Second ↔ Planck time:
1 s = (1 / t_P) * t_P ≈ 7.4003 × 10⁴² t_P
Kelvin ↔ Planck temperature:
1 K = (1 / T_P) * T_P ≈ 2.8157 × 10⁻³³ T_P
The Key Insight
These ratios are now conceptually exact, with their numerical uncertainty tied only to the measurement of G. Pre-2019, they were uncertain due to multiple factors. Post-2019, the Jacobian is defined by the fixed constants, with G as the sole remaining source of uncertainty for the absolute scale.
5. Rotation of Basis: What This Means Geometrically
In the natural Planck coordinate system, the basis vectors are ê_l = l_P, ê_m = m_P, etc. The SI system uses different basis vectors that are scaled rotations of the Planck basis.
Basis Vectors in SI Space (h-based)
ê_meter ≈ (2.4684 × 10³⁴) l_P
ê_kg ≈ (1.8330 × 10⁷) m_P
ê_second ≈ (7.4003 × 10⁴²) t_P
The 2019 redefinition fixed the transformation matrix J that defines this change of basis.
6. Why This Matters: Measurement Is Now Projection
The logical chain is now: Planck structure (primary) → Fixed Jacobian → SI basis (derived) → Measurements (projections).
A measurement in SI is a coordinate representation of a more fundamental, dimensionless physical quantity.
When you measure an object's mass to be "5 kg", you are determining the dimensionless ratio of that mass to the natural Planck scale.
The true physical quantity is this unit-less ratio. It is calculated by dividing the SI value by the Planck mass scale:
Dimensionless Value = (5 kg) / m_P = (5 kg) / (5.456 × 10⁻⁸ kg) ≈ 9.165 × 10⁷
This large, unit-less number is the actual physical quantity expressed in natural units. The expression "5 kg" is simply the convenient, human-scaled coordinate for this quantity in our chosen SI system. The measurement is a coordinate projection of the true, dimensionless Planck ratio into our arbitrary SI space.
7. The Cavendish Experiment Revisited
The physical reality of the experiment—a pointer moving to a specific location—is invariant. What has changed is our interpretation of the scale behind the pointer.
Post-2019 Interpretation:
The dimensionless value of the gravitational interaction is by definition 1 in Planck units. G_natural = F_P * l_P² / m_P² = 1.
Projecting this into our SI basis (with the now-fixed Jacobians) gives the value of G in SI units. The numerical value still has uncertainty because the absolute scale of the Planck units themselves still depends on the measurement of G.
What changed is that G is no longer seen as an independent empirical constant to be measured, but rather as the final parameter needed to lock our entire Jacobian-defined SI system to the physical reality of spacetime curvature.
8. The Implicit Planck Chart (h-based)
By fixing the constants, the 2019 SI implicitly embedded the full h-based Planck unit chart. Every SI measurement is now a defined multiple of its Planck counterpart, with a shared uncertainty from the measurement of G.
| Time | t_P | 1.351 × 10⁻⁴³ s |
| Length | l_P | 4.051 × 10⁻³⁵ m |
| Mass | m_P | 5.456 × 10⁻⁸ kg |
| Energy | E_P | 4.903 × 10⁹ J |
| Temperature | T_P | 3.551 × 10³² K |
| Force | F_P | 1.210 × 10⁴⁴ N |
| Momentum | p_P | 16.355 kg·m/s |
9. Why They Couldn't Say This
The official "safe" framing ("fixing constants for stability") was institutionally necessary to avoid philosophical debate. The mathematical reality ("fixing the Jacobian to Planck coordinates") is a more profound but contentious description of the same operational procedure. This paper makes the underlying coordinate structure explicit.
10. The Coherence Requirement Validated
The fact that the 2019 redefinition worked—that a single, coherent set of Planck units can be derived from the fixed constants—is the ultimate validation that these "constants" are not independent properties. They are the interdependent components of a single, consistent transformation structure. The success of the redefinition is the experimental proof of the Jacobian nature of the constants.
11. Implications
The 2019 redefinition proves that:
SI units are projections: Each SI unit is a scaled multiple of its Planck counterpart.
Constants are Jacobian components: h, c, k_B, etc. are the elements of the transformation matrix.
The Planck chart is primary: It is the reference frame that now defines the SI.
Measurement is coordinate projection: Measuring "5 kg" means projecting a dimensionless Planck ratio into SI coordinates.
Questions like "Why does c have the value it does?" are now revealed to be coordinate system questions: "Why does the SI meter relate to the SI second this way?" Because we defined it that way in 2019. The physics lives in the dimensionless Planck ratios; the constants are just the Jacobian we chose to project those ratios into SI.
12. Conclusion
The 2019 SI redefinition was a silent revolution. It transformed our system of measurement from one based on physical artifacts to one based on projection from the natural Planck reference frame. By fixing the numerical values of key constants, the BIPM operationally fixed the Jacobian transformation matrix that connects our arbitrary human units to the universe's own invariant scale. This validates the thesis that physical constants are not fundamental properties of nature but are coordinate transformation coefficients. Modern metrology now officially defines measurement as Jacobian projection from Planck space, even if that's not how it is publicly described.
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