Politics, Power, and Science: The Unity of Physics: Mass, Motion, and Time as One Phenomenon

Monday, October 27, 2025

The Unity of Physics: Mass, Motion, and Time as One Phenomenon

A Dimensional Analysis Revealing the Hidden Structure of General Relativity

J. Rogers, SE Ohio

Abstract

This paper demonstrates through pure dimensional analysis that the gravitational constant G and speed of light c appearing in General Relativity are not fundamental physical constants but unit conversion factors between SI units and natural dimensionless ratios. By performing algebraic substitutions using the Planck unit definitions, we prove the mathematical identity G/c² = l_P/m_P, revealing that gravitational time dilation is simply the dimensionless ratio τ = M_nat/R_nat. This identity, already embedded in existing physics equations, shows that mass, inertia, gravity, motion, and time dilation are not separate phenomena but different measurement perspectives on a single geometric reality: the structure of time itself. Using the GPS system as an empirical case study, we demonstrate that orbital motion is not independent of gravitational time dilation but is the kinetic manifestation of position within a time field. This analysis provides a physical mechanism for Mach's Principle and reveals that Newton's deliberate omission of a gravitational constant reflected a deeper understanding of coordinate-free physics than is commonly recognized.

1. Introduction: Reading What Physics Already Says

The standard equation for weak-field gravitational time dilation is:

τ = GM/(Rc²)

where τ is the dimensionless time dilation factor, G is the gravitational constant, M is mass, R is radius, and c is the speed of light.

This paper makes no modification to this equation. Instead, we perform a rigorous dimensional analysis to reveal what it has been saying all along. We will prove that G/c² is not a fundamental coupling between mass and geometry, but rather a compound conversion factor between SI units and natural dimensionless units. This is not speculation—it is a mathematical theorem derivable from the definitions of Planck units.

The implications are profound: mass is not a substance that causes time dilation; mass IS time dilation. Gravity is not action-at-a-distance; it is local interaction between time field gradients. Motion is not primary; it is the consequence of navigating through non-uniform time geometry. These are not new theories—they are the necessary logical consequences of removing the unit-system artifacts from our existing equations.

2. The Planck Units as Conversion Factors

2.1 The Standard Definition

Planck units are derived by setting fundamental constants equal to 1:

h = c = G = k_B = 1

From these, we derive the Planck length, mass, and time:

l_P = √(hG/c³) ≈ 4.051 × 10⁻³⁵ m
m_P = √(hc/G) ≈ 5.456 × 10⁻⁸ kg
t_P = √(hG/c⁵) ≈ 1.351 × 10⁻⁴³ s

Note: We use Planck's constant h, not the reduced constant ℏ = h/(2π). Using ℏ implicitly sets π = 1, which is a category error—treating a dimensionless mathematical constant as if it were a physical unit.

2.2 The Reinterpretation: Planck Units as Jacobians

However, we can equivalently view these as conversion factors (Jacobians) between SI units and natural dimensionless units:

M_SI = M_natural × m_P (converts dimensionless mass to kilograms)
R_SI = R_natural × l_P (converts dimensionless length to meters)
T_SI = T_natural × t_P (converts dimensionless time to seconds)

This is not a new interpretation—it is simply making explicit what has always been true. When we measure "70 kilograms," we are implicitly stating that the object's dimensionless mass is M_natural = 70 kg / m_P ≈ 3.2 × 10⁹ in natural units.

3. The Central Proof: G/c² is a Unit Conversion Artifact

3.1 Deriving the Identity

The Planck definitions provide us with expressions for G and c in terms of the Planck units:

From m_P = √(hc/G), we can solve for G:

m_P² = hc/G
G = hc/m_P²

We also know that:

c = l_P/t_P

And from the Planck definitions:

h = m_P × l_P²/t_P

Now we compute G/c²:

G/c² = [hc/m_P²] / [l_P/t_P]²

G/c² = [hc/m_P²] × [t_P²/l_P²]

G/c² = hc × t_P² / (m_P² × l_P²)

Substitute h = m_P × l_P²/t_P:

G/c² = [m_P × l_P²/t_P] × c × t_P² / (m_P² × l_P²)

G/c² = [m_P × l_P² × c × t_P²] / [t_P × m_P² × l_P²]

G/c² = c × t_P / m_P

Since c = l_P/t_P:

G/c² = [l_P/t_P] × t_P / m_P

G/c² = l_P / m_P

3.2 The Theorem

G/c² = l_P/m_P

This is not an approximation. It is an exact algebraic identity derived from the definitions of Planck units. The combination G/c² that appears ubiquitously in General Relativity is nothing more than the ratio of the Planck length conversion factor to the Planck mass conversion factor.

G/c² contains zero new physical information. It is a pure unit-scaling artifact.

4. Revealing the Natural Physics: τ = M_nat/R_nat

4.1 The Substitution

Return to the standard time dilation equation:

τ = GM/(Rc²)

Rearrange to isolate G/c²:

τ = (G/c²) × (M/R)

Substitute our proven identity G/c² = l_P/m_P:

τ = (l_P/m_P) × (M_SI/R_SI)

Rearrange:

τ = (M_SI/m_P) / (R_SI/l_P)

By definition, M_SI/m_P = M_natural and R_SI/l_P = R_natural:

τ = M_natural / R_natural

4.2 The Physical Meaning

The time dilation factor—the thing we measure with atomic clocks, the thing that makes GPS work—is simply the dimensionless ratio of natural mass to natural length.

This ratio is:

Unit-independent: Any observer using any measurement system will calculate the same τ
Coordinate-free: It describes the geometry of spacetime, not our description of it
Already in our equations: We just couldn't see it through the G and c² obscuring it

The physics is breathtakingly simple: gravitational time dilation is a pure geometric ratio.

5. Mass as Time Field Geometry

5.1 The Ontological Shift

The equation τ = M_nat/R_nat forces a radical reinterpretation:

Old View:

Mass is a substance
Mass causes a gravitational field
The gravitational field causes time dilation
Mass → Field → Time Effect

New View:

Time dilation field is fundamental (τ as a function of position)
"Mass" is the name we give to the source term M_nat in this field
M_nat/R_nat directly gives the local time rate
Mass IS the time field structure

Mass does not cause time dilation. Mass IS time dilation measured at unit distance.

5.2 The Time Field

We can define a time field τ(r) = M_nat/r_nat that describes how time rate varies with position around a massive object. This is not a new field—it is simply recognizing what the metric tensor has been describing all along.

Properties of the time field:

Stronger near massive objects (large M_nat, small r_nat)
Weaker far from masses (small M_nat/r_nat)
Sets the local "tick rate" of all clocks
Objects follow geodesics through this field

6. Gravity as Local Time Field Interaction

6.1 Newton's Law in Natural Units

The standard gravitational force equation:

F = G × m₁m₂/r²

But all G does is remove the unit scaling for si units

F/F_P = (l_P/m_P)² × m₁m₂/r²

Solve for F in si units

F = F_P × (l_P/m_P)² × m₁m₂/r²

G = F_P × (l_P/m_P)²

In natural units (setting conversion factors aside):

F_nat = m₁_nat × m₂_nat / r_nat²

But we now understand that m_nat is actually the time field strength at unit distance. Therefore:

F_nat = (m₁_nat / r_nat) × (m₂_nat / r_nat)

F_nat = τ₁ × τ₂, separated by length r_nat

where τ₁ and τ₂ are the time field values associated with each mass.

6.2 The Physical Interpretation

Gravity is not action-at-a-distance between masses. It is the local interaction of two time field distortions separated by distance r.

Each "mass" creates a time field distortion
These distortions interact at their respective locations
The "force" describes how strongly coupled they are through the geometry
Distance r defines the geometric separation in the time field

Gravity is not mysterious action-at-a-distance. It is local time field gradient interaction.

6.3 Why Objects Fall

Objects don't fall because they are "attracted" to mass. They fall because:

Mass creates a gradient in the time field (∂τ/∂r ≠ 0)
The natural path (geodesic) through this gradient curves toward regions of slower time
"Falling" is following this natural path
"Weight" is the force required to prevent following this path

Objects move toward slower time, following the geometry of the time field.

7. The GPS Proof: Orbital Motion as Time Field Expression

7.1 The Standard GPS Calculation

GPS satellites require time corrections for two "separate" relativistic effects:

Gravitational time dilation (GR):

Δt_grav = GM/(Rc²)

Velocity time dilation (SR):

Δt_vel = -v²/(2c²)

Engineers sum these effects and apply the total correction. The system works with extraordinary precision.

7.2 The Hidden Constraint

However, the satellite's velocity is not independent. For a stable circular orbit:

v² = GM/R

This is not optional—it is the geometric requirement for orbital motion. The velocity is completely determined by the gravitational potential at radius R.

7.3 The Unification

Substitute the orbital constraint into the velocity term:

Δt_vel = -(GM/R)/(2c²)

Now sum the two "separate" effects:

Δt_total = GM/(Rc²) - GM/(2Rc²)
Δt_total = GM/(2Rc²)

The two effects collapse into a single unified potential.

7.4 In Natural Units

Convert to natural units using G/c² = l_P/m_P:

Δt_total = (1/2) × (l_P/m_P) × (M/R)
Δt_total = (1/2) × (M_nat/R_nat)

The GPS satellite is directly measuring the dimensionless geometric ratio M_nat/R_nat at its orbital radius. The factor of 1/2 emerges from the orbital constraint—it is the geometric expression of circular motion within the time field.

7.5 The Profound Implication

The satellite's velocity is not causing "special relativistic time dilation" that adds to "general relativistic time dilation."

The velocity is the kinetic manifestation of being at that position in the gravitational time field.

The two "effects" were never separate. They are two partial views of a single geometric reality: the unified time field value at radius R.

8. Motion as Navigation Through Time Geometry

8.1 Redefining Motion

If time field geometry is fundamental, then motion must be reinterpreted:

Old View:

Objects have position and velocity
Forces change velocity
Time dilation is a side effect of motion

New View:

Time field geometry exists (primary)
Objects follow geodesics through this geometry
Motion is the spatial manifestation of navigating time field contours
Velocity is not independent—it reflects position in the field

8.2 Free Fall as Natural Motion

When you are in free fall:

You follow a geodesic through the time field
No forces act on you
You experience weightlessness
This is the natural state

8.3 Standing Still as Forced Acceleration

When you stand on Earth's surface:

The ground prevents you from following your geodesic
Electromagnetic forces continuously accelerate you upward
You experience weight (proper acceleration ≈ 9.8 m/s²)
This is a forced, non-natural state

You are not "at rest." You are being continuously accelerated away from your natural geodesic by the ground beneath your feet.

8.4 Weight is Not Gravity

The sensation of weight is not the sensation of gravity pulling you down. It is the sensation of the ground pushing you up, preventing you from following the time field gradient.

Weight is gravity being denied, not gravity being felt.

9. Inertia and Mach's Principle

9.1 The Historical Problem

Mach's Principle (1883) states: "The inertia of a body is determined by the distribution of all other masses in the universe."

This remained a philosophical intuition without a physical mechanism for over 140 years.

9.2 The Mechanism: Cosmically-Defined Time

If all motion is orbital (nested hierarchy from planets to galaxies), then our local time rate is determined by:

τ_local = Σ (M_nat,i / R_nat,i)

where the sum is over all masses in the observable universe. And probably from places we cannot currently observe depending on how old the universe really is.

Our local "dt"—the thing we use to define acceleration and inertia—is a relational property set by the cosmic mass distribution.

9.3 Inertia Emerges

Inertia is defined as:

F = m × a
a = dv/dt

Therefore m = F/(dv/dt).

But if dt itself is set by Σ(M_i/R_i) from all cosmic masses, then:

The inertial mass we measure is a direct consequence of the total mass distribution of the universe.

9.4 The Thought Experiment

Universe A (ours):

Filled with galaxies
Σ(M/R) is large
Local dt is set by this cosmic sum
Inertia exists with measured values

Universe B (empty):

No other masses exist
Σ(M/R) ≈ 0
Local dt is undefined by cosmic structure
Inertia as we know it would not exist

Mach's Principle is not a separate postulate. It is a necessary consequence of time being a relational, globally-coupled property.

10. The Unity of Five Concepts

We can now see that five apparently distinct phenomena are one:

10.1 Mass

What it is: The source term M_nat in the time field equation τ = M_nat/R_nat

Physical meaning: The magnitude of time field distortion at unit distance

10.2 Time Dilation

What it is: The local value of the time field τ(r) = M_nat/r_nat

Physical meaning: The actual, measurable rate of clock ticking at position r

10.3 Gravity

What it is: The gradient of the time field ∇τ = -M_nat/r_nat²

Physical meaning: The geometric curvature that defines geodesic paths

10.4 Inertia

What it is: Resistance to displacement through the cosmically-defined time field

Physical meaning: m = F/(dv/dt) where dt is set by Σ(M_i/R_i)

10.5 Motion

What it is: Navigation through time field geometry

Physical meaning: Spatial manifestation of position within non-uniform time structure

11. Newton's Wisdom Recovered

11.1 The Deliberate Omission

Isaac Newton wrote:

F ∝ m₁m₂/r²

He did not write F = k × m₁m₂/r² with a specific constant k (our G) because:

As Master of the Royal Mint, he understood that conversion constants are conventional, not fundamental
He knew that any specific numerical k would be unit-dependent
His ratio method canceled the constant in every calculation
He was working at a higher level of abstraction than his successors

11.2 The Lost Principle

Newton's proportional reasoning was architecturally rigorous:

Work only with dimensionless ratios
Cancel unknown constants systematically
Keep physics independent of measurement conventions

His moon-apple calculation:

a_apple/a_moon = (r_moon/R_earth)²

This is coordinate-free physics. The ratio is the same for any observer using any units. Both k (our G) and M_earth cancel perfectly.

11.3 The Modern Error

When Cavendish measured G in 1798, physics celebrated it as "completing" Newton's theory. But:

Newton's theory was already complete (proportionally)
Cavendish calibrated it for a specific unit system (SI)
We mistook the calibration for the physics
We embedded unit-system artifacts into "fundamental constants"

Newton didn't omit G from ignorance. He omitted it from deeper understanding.

12. Why This Was Hidden

12.1 The Pedagogical Trap

Physics education presents:

F = G × m₁m₂/r²

with G ≈ 6.674 × 10⁻¹¹ m³/(kg·s²)

Students learn:

G is a "fundamental constant of nature"
It has the value it does for "deep reasons"
Finding that value was a triumph of experimental physics

They don't learn:

G's value is entirely determined by our choice of kilogram, meter, and second
Change the unit system → change G
The physics is in the dimensionless ratio, not in G

12.2 The Conceptual Barrier

We teach that:

Mass is fundamental (measured in kg)
Length is fundamental (measured in m)
Time is fundamental (measured in s)
G connects them

This obscures that:

M_nat, R_nat, T_nat are dimensionless (the actual physics that are invariant under all measurement)
m_P, l_P, t_P are conversion factors (unit system artifacts)
G is built from these conversions

We've been teaching unit conversions as if they were physics.

12.3 The Mathematical Camouflage

The equation τ = GM/(Rc²) looks like it contains fundamental constants G and c. But:

τ = (l_P/m_P) × (M_SI/R_SI)

reveals that G and c are merely the machinery for converting SI measurements to natural dimensionless ratios.

The constants were hiding the physics, not revealing it.

13. Experimental Verification

13.1 This is Not a New Theory

Critically, this paper proposes no new physics. It is a dimensional analysis of existing theory. Therefore:

All existing experimental confirmations of GR remain valid
GPS continues to work precisely as calculated
No new experiments are needed to "test" this framework

13.2 What Changes

What changes is interpretation:

Before: "GPS satellites experience GR time dilation + SR time dilation"

After: "GPS satellites measure their position in Earth's unified time field"

The numbers are identical. The understanding is transformed.

13.3 A Testable Prediction

If inertia is set by Σ(M_i/R_i) over the cosmos, and the universe is expanding, then:

R_i to distant galaxies increases over time
Σ(M_i/R_i) slowly decreases
Local inertial mass ratios might shift subtly

Prediction: Ultra-precise measurements over decades might detect:

Drift in fundamental mass ratios
Changes in fine structure constant
Anomalies in atomic clock comparisons

This is at the edge of current measurement capability but in principle detectable.

14. Philosophical Implications

14.1 The Nature of Constants

"Fundamental constants" fall into two categories:

Type 1: Pure numbers (dimensionless)

Fine structure constant α ≈ 1/137
Proton/electron mass ratio ≈ 1836
These are coordinate-free physics

Type 2: Unit-dependent values

G = 6.674 × 10⁻¹¹ m³/(kg·s²)
c = 299,792,458 m/s
These are conversion factors

We've been treating Type 2 as if they were Type 1, asking "why does G have this value?" when the answer is "because we defined the kilogram, meter, and second this way."

14.2 The Structure of Physical Law

Physical laws should be:

Coordinate-free (same for all observers)
Dimensionless (pure geometric ratios)
Unit-independent (not tied to human conventions)

Newton understood this. General Relativity embodies it (in the tensor formulation). But our pedagogical presentation obscures it.

This paper is a call to teach the dimensionless physics first, and the unit conversions second.

14.3 The Unity of Reality

The deepest implication is ontological:

Mass, time, space, motion, and inertia are not separate substances or properties. They are different linguistic descriptions of a single geometric reality.

Just as:

Temperature and kinetic energy are one thing (viewed macroscopically vs microscopically)
Electricity and magnetism are one thing (viewed in different reference frames)

So too:

Mass and time dilation are one thing (viewed as source term vs field value)
Gravity and geodesic curvature are one thing (viewed as force vs geometry)
Inertia and cosmic time coupling are one thing (viewed locally vs globally)

Physics is unified not because we found connections between separate phenomena, but because we recognized they were never separate.

15. Conclusion

This paper has proven through dimensional analysis that:

G/c² = l_P/m_P (exact algebraic identity)
τ = M_nat/R_nat (gravitational time dilation is a pure dimensionless ratio)
Mass is time field geometry (not substance causing time dilation)
Gravity is local time field interaction (not action-at-a-distance)
Motion is navigation through time geometry (not independent phenomenon)
Inertia emerges from cosmic time coupling (Mach's Principle mechanism)

None of this required new physics. It required only:

Recognizing Planck units as conversion factors
Performing algebraic substitution
Reading what the equations already say

GPS satellites orbiting overhead prove this framework every nanosecond: their orbital motion is not independent of their gravitational time dilation—it is the kinetic expression of their position within Earth's time field.

Newton's deliberate omission of a gravitational constant reflected his understanding that constants are conventional, not fundamental. His proportional reasoning kept the physics coordinate-free. We lost this wisdom when we reified G as fundamental.

The physics is not in G, c, or any unit-dependent constant. The physics is in the invariant, dimensionless ratios of the universe—ratios that are true no matter how they are measured.

Mass does not cause time change. Mass IS time change.

Gravity is not a force. Gravity is the geometry we navigate.

Motion is not primary. Motion is the manifestation of existing in non-uniform time.

These are not speculations. They are the necessary logical consequences of removing unit-system artifacts from equations that have been validated for over a century.

We have been doing coordinate-free physics all along. We just couldn't see it through the coordinates.

References

Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica
Einstein, A. (1915). "Die Feldgleichungen der Gravitation." Sitzungsberichte der Preussischen Akademie der Wissenschaften
Cavendish, H. (1798). "Experiments to Determine the Density of the Earth." Philosophical Transactions of the Royal Society
Mach, E. (1883). Die Mechanik in ihrer Entwicklung
Rogers, J. (2025). "The Constant of Proportionality: Newton's Deliberate Omission of G and the Lost Principle of Invariance"
Rogers, J. (2025). "From GPS to Mach's Principle: On the Unity of Motion, Time, and Inertia"
Ashby, N. (2003). "Relativity in the Global Positioning System." Living Reviews in Relativity, 6(1)
Planck, M. (1899). "Über irreversible Strahlungsvorgänge." Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften

Appendix A: Mathematical Summary

Planck Definitions (using h, not ℏ):

l_P = √(hG/c³)
m_P = √(hc/G)
t_P = √(hG/c⁵)

Unit Conversions:

M_SI = M_nat × m_P
R_SI = R_nat × l_P
T_SI = T_nat × t_P

The Central Identity:

G/c² = l_P/m_P

Time Dilation (Standard Form):

τ = GM/(Rc²)

Time Dilation (Natural Form):

τ = M_nat/R_nat

Gravity (Standard Form):

F = Gm₁m₂/r²

Gravity (Natural Form):

F_nat = τ₁ × τ₂ / r_nat²

GPS Unified Potential:

Δt_total = (1/2) × GM/(Rc²) = (1/2) × M_nat/R_nat

Mach's Principle:

τ_local = Σᵢ (M_nat,i / R_nat,i)
m_inertial ∝ 1/τ_local ∝ Σᵢ (M_nat,i / R_nat,i)

Appendix B: For the Skeptical Reader

Objection 1: "This is just playing with units. You haven't changed the physics."

Response: Exactly. That's the point. The physics already says this. We've been obscuring it with unit-dependent notation.

Objection 2: "But G and c are measured experimentally. They're real."

Response: Their numerical values are measured. But those values depend entirely on our definitions of kg, m, and s. Change the unit system, and G and c change. The dimensionless ratio M_nat/R_nat does not change.

Objection 3: "This doesn't make testable predictions."

Response: It makes one: if inertia is set cosmically via Σ(M/R), then cosmic expansion should cause subtle shifts in inertial mass ratios over cosmological time. This is testable with sufficient precision.

Objection 4: "Einstein already knew spacetime was curved. This isn't new."

Response: Einstein knew geometry was physics. But we still teach that "mass causes curvature" as if they were separate. This framework shows they are not separate—mass IS the curvature (specifically, of time).

Objection 5: "Why hasn't anyone pointed this out before?"

Response: They have. Physicists working in natural units understand this. But it's been treated as a "mathematical convenience" rather than revealing the actual structure of physical law. And it's not taught this way to students, who learn to think of G and c as fundamental.

The GPS system proves this interpretation daily. Every satellite measures τ = M_nat/R_nat. We just call it "GM/(Rc²)" because we insist on working in SI units.