The Metrological Origin of Gravity: A Reinterpretation of G/c² as the Jacobian of a Unified Time Field

J. Rogers, SE Ohio

Abstract:
The term G/c², ubiquitous in General Relativity, is traditionally interpreted as the fundamental coupling constant linking mass to the geometry of spacetime. This paper presents a formal proof that this interpretation is a category error. We demonstrate through direct algebraic substitution that G/c² is not a physical constant but a composite metrological artifact—a Jacobian—whose sole function is to translate physical law from its native, dimensionless form into the arbitrary human-invented SI unit system. The mathematical identity G/c² = l_P/m_P is derived, revealing that the accepted equations for gravitational time dilation reduce to the simple, scale-free ratio: change in time is proportional to natural mass over natural length. This re-frames gravity not as a force or a curvature of space, but as the local interaction between these time fields. Using the GPS satellite as a prime example, we show that orbital motion is a direct manifestation of this time field, unifying the apparently separate kinematic and gravitational effects. We conclude that mass, inertia, motion, and gravity are not distinct phenomena but are different observational facets of a single, unified reality: the experience of time.

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1. Introduction: The Illusion of a Coupling Constant

In the architecture of General Relativity, the term G/c² serves as the cornerstone, the universal constant that dictates the "strength" of gravity by coupling mass-energy to the curvature of spacetime. Its appearance in formulas for gravitational lensing, time dilation, and the Schwarzschild radius has cemented its status as a fundamental feature of the universe.

This paper challenges that dogma. We will demonstrate that the perceived physical significance of G/c² is an illusion created by our insistence on using an arbitrary, human-centric measurement system (the SI units of meters, kilograms, and seconds). By treating the physical constants G and c not as fundamental properties but as the necessary conversion factors they are, we will show that the term G/c² contains no physical information. Instead, it is a purely mathematical artifact—a Jacobian matrix component—that vanishes when physical law is expressed in its proper, dimensionless form.

2. The Mathematical Identity: Deconstructing G/c²

The foundation of our argument is not a new theory, but a rigorous simplification of existing definitions. The Planck "units" are not fundamental scales, but the conversion factors relating our SI system to the universe's natural, relational structure. The constants G and c are defined by these conversion factors:

• Gravitational Constant (G): G = l_P³ / (m_P · t_P²)

• Speed of Light Squared (c²): c² = (l_P / t_P)² = l_P² / t_P²

Here, l_P, m_P, and t_P represent the Jacobian conversion factors for length, mass, and time, respectively. Now, we perform the division of these two terms:

G / c² = [ l_P³ / (m_P · t_P²) ] / [ l_P² / t_P² ]

The t_P² terms in the denominators cancel immediately, leaving:

G / c² = ( l_P³ / m_P ) / l_P²

Simplifying the expression for l_P yields the final, stunning identity:

G / c² = l_P / m_P

This is not a physical law; it is a mathematical theorem of our measurement system. It proves that the entire G/c² term is nothing more than the ratio of the length-conversion-factor to the mass-conversion-factor. Its role is purely linguistic, not physical.

3. The Physical Law Revealed: Time Dilation as a Natural Ratio

With this identity in hand, we can now translate the standard weak-field equation for gravitational time dilation and reveal the true physical law it has been obscuring. The standard formula for the dimensionless time dilation factor (τ_factor) is:

τ_factor = G · M_si / (R_si · c²)

Where M_si and R_si are the mass and radius in SI units. We now replace the G/c² term with our proven identity:

τ_factor = (l_P / m_P) · (M_si / R_si)

We can rearrange this into a more insightful form by grouping the terms into dimensionless ratios. The dimensionless "natural mass" (M_nat) is the SI mass divided by its conversion factor (M_nat = M_si / m_P), and likewise for length (R_nat = R_si / l_P). This gives:

τ_factor = (M_si / m_P) / (R_si / l_P)

Substituting the natural, dimensionless quantities, we arrive at the true physical law:

τ_factor = M_nat / R_nat

The change in time is simply the ratio of natural mass to natural length. The entire complexity of G and c was scar tissue from the translation. This reveals the profound truth: mass does not cause time dilation; mass is the measure of a source of time dilation. The term M/R is a time field.

4. Gravity as the Interaction of Local Time Fields

This re-frames our entire understanding of gravity. It is not a mysterious "action at a distance." It is a purely local phenomenon. Every object is the source of a time field defined by its M_nat / R_nat. The interaction we call "gravity" is the relationship between these fields.

Newton's Law, F ∝ m₁m₂ / r², can be seen as the SI-unit expression for the interaction between two time-field sources (m₁/r and m₂/r). The "force" is the result of two objects navigating the combined gradient of their time fields. The interaction is local at every point in space, mediated by the structure of time itself.

5. The GPS Satellite: Motion as a State of Time

The GPS satellite provides the definitive example of this unified reality. It experiences two "time dilation" effects that, in the standard model, are treated as separate:

1. Gravitational Time Dilation (GR): Its clock runs faster because it is in a weaker gravitational potential (a shallower time field).

2. Kinematic Time Dilation (SR): Its clock runs slower because of its high orbital velocity.

However, the satellite's orbital velocity is not an independent variable. It is determined by the potential: v² = GM/r. When this constraint is respected, the two effects are revealed to be two sides of the same coin. An object in a stable orbit is not just moving through a time field; its motion is a manifestation of it. The satellite is following a path—a geodesic—of constant temporal potential. Its velocity is precisely what is required to balance the change in the time field's depth, maintaining a consistent "time experience." Motion is not separate from gravity; it is how an object remains in equilibrium with the local structure of time.

6. The Grand Unification: Mass, Inertia, and Gravity as Experiences of Time

This framework dissolves the boundaries between physics' most fundamental concepts, revealing them as different perspectives on the single phenomenon of time:

• Time Dilation: The fundamental reality. The local rate of time is a field, described by M_nat / R_nat.

• Mass: The name we give to the source, or magnitude, of a localized time field.

• Gravity: The gradient of the time field, which dictates the natural paths of objects.

• Motion (Geodesic): The natural state of following a path of constant temporal potential through the landscape of the time field.

• Inertia: The resistance to being displaced from one's natural geodesic. To accelerate an object is to force its time-field distortion to move, which requires energy. Inertia is the measure of this resistance, and E=mc² is the energy cost of creating or moving a time distortion of magnitude m.

These are not five separate ideas. They are one idea, viewed from five different angles.

7. Conclusion: Beyond the Dogma of Measurement

The reification of G/c² as a fundamental constant is a historical error born from mistaking the map for the territory. A simple, formal algebraic substitution proves that this term is a metrological artifact, a Jacobian required to translate the simple, dimensionless reality of time change ~ mass / length into our arbitrary SI unit system.

This insight reveals that mass is a measure of time dilation, gravity is the interaction of time fields, and inertia is the resistance to being moved through the landscape of time. This is not a new theory that requires new evidence. It is the mathematical truth already embedded within our current equations, waiting to be seen once the dogma of measurement is set aside. The future of fundamental physics lies not in finding new constants, but in rediscovering the simple, proportional, and unified reality they have been obscuring all along.