How Special and General Relativity Emerge as Different Views of a Unified Time Field
J. Rogers, SE Ohio
Abstract
This paper demonstrates that Special Relativity and General Relativity are not separate theories requiring unification, but rather different observational perspectives on a single, primary phenomenon: the geometric structure of time itself. By treating the measurable rate of time as the fundamental reality and recognizing that G/c² = l_P/m_P is merely a unit conversion factor, we show that gravitational time dilation reduces to the dimensionless ratio τ = M_nat/R_nat. From this time-centric architecture, we demonstrate that:
Mass emerges as the magnitude of time field distortion
Gravity emerges as the gradient of the time field
Inertia emerges from resistance to displacement through cosmically-defined time structure
Special Relativity emerges as the local, flat-field limit of time field geometry
General Relativity emerges as recognition that time fields have structure
Orbital motion emerges as the kinetic manifestation of position within time field geometry
The GPS system provides continuous empirical verification: satellites measure a unified time field potential τ = (1/2) × M_nat/R_nat, not two separate "SR + GR effects." This framework resolves the artificial split between SR and GR, provides a physical mechanism for Mach's Principle through nested orbital structures, and reveals that all relativistic phenomena are different measurement perspectives on how position determines experienced time rate.
1. Introduction: The Architectural Problem
1.1 The Current Framework
Modern physics teaches relativity through two distinct theories:
Special Relativity (1905):
Flat spacetime
No gravity
Time dilation from relative velocity: Δt = γt₀
"Moving clocks run slow"
General Relativity (1915):
Curved spacetime
Gravity present
Time dilation from gravitational potential: Δt = (GM/rc²)t₀
"Clocks run slow in gravity wells"
Students learn these as separate domains, later discovering they must be "combined" for real-world problems like GPS. This pedagogical split creates persistent confusion:
Why do we need two theories of time dilation?
How do they relate to each other?
When do we use which one?
Are they truly independent?
1.2 The Systems Analysis Perspective
From a systems architecture standpoint, when two supposedly independent components have hidden dependencies that cause them to collapse into a unified formula, this reveals they were never actually separate. They are different views of a single underlying system.
This paper demonstrates that SR and GR are not separate theories requiring unification. They are emergent perspectives on a single geometric reality: the structure of the time field.
1.3 The Central Claim
There is one primary, measurable phenomenon: Different spatial positions experience time at different rates.
From this single reality, everything else emerges:
Mass (the source term for time field intensity)
Gravity (the gradient of the time field)
Motion (navigation through time field geometry)
Inertia (cosmic coupling through Σ(M/R))
SR (the local, flat-field limit)
GR (recognition of field structure)
2. The Foundation: Dimensional Analysis Reveals Hidden Structure
2.1 The Central Mathematical Identity
The Planck units provide conversion factors between SI measurements and natural dimensionless units:
l_P = √(hG/c³) [Planck length]
m_P = √(hc/G) [Planck mass]
t_P = √(hG/c⁵) [Planck time]
From these definitions, we can derive expressions for G and c²:
G = hc/m_P²
c² = (l_P/t_P)²
Computing the ratio G/c² that appears ubiquitously in relativity:
G/c² = [hc/m_P²] / [(l_P/t_P)²]
Through algebraic substitution (full derivation in Appendix A):
This is not an approximation. It is an exact algebraic identity.
2.2 The Implication
The combination G/c² is not a fundamental coupling constant. It is the ratio of two unit conversion factors: the Planck length conversion divided by the Planck mass conversion.
G/c² contains zero physical information. It is pure dimensional scaffolding.
2.3 Revealing the Natural Physics
The standard gravitational time dilation equation:
Can be rewritten by isolating G/c²:
Substituting our proven identity G/c² = l_P/m_P:
τ = (l_P/m_P) × (M_SI/R_SI)
τ = (M_SI/m_P) / (R_SI/l_P)
By definition, M_SI/m_P = M_nat (natural mass) and R_SI/l_P = R_nat (natural radius):
The time dilation factor is simply the dimensionless ratio of natural mass to natural radius.
This is the foundation of the time-centric architecture.
3. The Time Field as Primary Reality
3.1 What is Measurable
The primary observable in the universe is this: atomic clocks at different positions tick at measurably different rates.
This has been verified experimentally:
Atomic clocks at different altitudes (Pound-Rebka, 1959)
Hafele-Keating circumnavigation experiment (1971)
GPS system corrections (daily, ongoing since 1980s)
NIST aluminum ion clock experiments (2010) detecting time rate differences over 33 cm of elevation
The rate of time is not uniform. It varies with position.
3.2 The Time Field τ(r)
We define a time field as a geometric structure that assigns a dimensionless time rate to each position:
For a simple spherically symmetric case (one central mass):
Stronger near the mass (large M_nat, small r_nat → large τ)
Weaker far from the mass (small τ)
Sets the local "tick rate" for all physical processes
Properties:
Directly measurable with atomic clocks
Unit-independent (dimensionless ratio)
Coordinate-free (geometric reality, not description artifact)
Scalar field (assigns a number to each spatial position)
3.3 Multiple Sources: Nested Time Fields
For multiple masses, the principle of superposition applies:
τ_total(r) = Σᵢ (M_nat,i / r_nat,i)
This is not speculative—it's how we actually calculate time dilation in complex gravitational environments. Every mass contributes to the local time rate at any given position.
This immediately reveals the nested orbital structure:
Earth contributes: M_Earth/r_from_Earth
Sun contributes: M_Sun/r_from_Sun
Galaxy contributes: M_Galaxy/r_from_center
Observable universe contributes: Σ(M_i/R_i)
Your local time rate is determined by your position within all of these nested structures simultaneously.
4. Emergence: How Everything Follows From Time Field Geometry
4.1 Mass Emerges as Field Source Strength
Traditional view: Mass is a fundamental property of matter that causes time dilation.
Emergent view: "Mass" is the name we give to the source term M_nat in the time field equation τ = M_nat/r_nat.
What this means:
Mass doesn't cause time dilation
Mass is time dilation measured at unit distance
The question "What is mass?" is answered: it's the magnitude of a local time field distortion
Equivalence Principle:
In the traditional view, the equality of inertial mass (F=ma) and gravitational mass (F=GMm/r²) is a mysterious coincidence requiring a postulate.
In the emergent view, it's a tautology: both are measuring the same geometric property (time field distortion strength) in different contexts.
4.2 Gravity Emerges as Time Field Gradient
Traditional view: Gravity is a force that attracts masses toward each other.
Emergent view: "Gravity" is the name we give to the gradient of the time field.
The geodesic equation in the weak-field limit:
Objects follow the "straightest possible path" through time field geometry. What we call "gravitational acceleration" is the necessary consequence of following this path.
Why objects "fall":
Not because they're "attracted" to mass
But because they're following geodesics through time field gradients
The gradient ∇τ points toward regions of slower time
Objects naturally move along paths that are "straight" in this geometry
Weight vs. Weightlessness:
Free fall: Following your natural geodesic (weightless)
Standing on Earth: Electromagnetic forces prevent you from following your geodesic (you feel weight)
Weight is not the sensation of gravity acting on you. It's the sensation of electromagnetic forces preventing you from following gravity.
4.3 Motion Emerges as Navigation Through Time Geometry
Traditional view: Objects have independent properties of position and velocity. Forces change velocity.
Emergent view: Velocity is the kinetic manifestation of position within time field geometry.
For a circular orbit, the constraint v² = GM/r is not an additional law—it's the geometric requirement for circular motion through the time field.
Orbital motion is not independent of gravitational time dilation. It is the kinetic expression of being at that position in the time field.
This is directly demonstrated by GPS satellites (Section 6).
4.4 Inertia Emerges from Cosmic Time Structure
Traditional view: Inertia (F=ma) is a fundamental property of matter. Mach's Principle is a philosophical intuition without mechanism.
Emergent view: Your local time rate dt is defined by the sum of all cosmic mass contributions:
dt ∝ 1/τ_local = 1/Σᵢ(M_nat,i/R_nat,i)
Since acceleration is a = dv/dt, and inertial mass appears as m = F/(dv/dt), the local "dt" that defines your inertial response is set by the entire cosmic mass distribution.
Mach's Principle mechanism:
| Our universe | Dense with galaxies | Σ(M/R) is large | Normal inertia exists |
| Empty universe | No other masses | Σ(M/R) ≈ 0 | dt undefined, no inertia |
Inertia is not a property of isolated objects. It's the local manifestation of being embedded in the universal time field structure created by all masses.
This resolves the 140-year-old puzzle of Mach's Principle by providing an explicit physical mechanism.
5. Special Relativity: The Flat-Field Limit
5.1 What SR Actually Describes
Special Relativity is not a separate theory. It is the local, flat-field approximation of time field geometry.
When:
You zoom in to a region small enough that ∇τ ≈ 0
The time field looks approximately uniform
You can ignore field structure
You recover Special Relativity.
5.2 Velocity Time Dilation as Time Field Effect
Consider two observers in relative motion with velocity v. In the time-centric framework:
Each observer is at a different position in the other's time field. The relative velocity v represents their kinetic separation within time field geometry.
The SR time dilation factor:
γ = 1/√(1 - v²/c²)
For v << c, this approximates to:
This is not fundamentally about "moving clocks." It's about the geometric relationship between two positions in time field space that have kinetic separation v.
5.3 Length Contraction and Simultaneity
Length contraction:
When you measure a moving rod's length, you're comparing spatial intervals measured using your local time rate against the rod's internal time rate. Different time rates → different spatial measurements.
Relativity of simultaneity:
Events that are simultaneous in your frame (measured with your local time rate) are not simultaneous in another frame (measured with their different time rate).
Both phenomena emerge from the primary reality: different positions experience time at different rates.
5.4 Why SR Appears "Separate"
Historical accident: Einstein developed SR first (1905) by considering uniform motion in the absence of gravity.
This is like developing a theory of flat-surface geometry, then later discovering that real surfaces can be curved. The flat-surface theory isn't wrong—it's the local limit of the more general curved-surface theory.
SR is not wrong. It's the limiting case where time field structure is negligible.
6. General Relativity: Recognizing Field Structure
6.1 What GR Actually Describes
General Relativity is the recognition that the time field has structure—it's not uniform, it has gradients, and these gradients affect motion.
The Einstein Field Equations describe how mass-energy creates time field structure:
Rμν - (1/2)gμνR = (8πG/c⁴)Tμν
In the time-centric interpretation: This describes how the source term (mass-energy) determines the geometry of the time field.
6.2 Gravitational Time Dilation
The weak-field gravitational time dilation:
τ = GM/(Rc²) = M_nat/R_nat
This is not a "correction" or "effect." This is the primary reality: the time rate at position R in a time field created by mass M.
6.3 Curved Spacetime
Einstein's insight was that gravity is geometry. The time-centric framework makes this explicit:
The "curvature of spacetime" is primarily the curvature of time.
Spatial curvature exists too, but the dominant effect (for weak fields and velocities v << c) is the variation in time rate with position.
When we say "mass curves spacetime," what we mean physically is: mass creates a field structure where time runs at different rates in different places.
6.4 Why GR Appears "Separate"
Because we teach it as "what happens when you add gravity to SR," creating the impression they're different theories.
Better framing: GR is what you get when you recognize that the time field (which SR treated as approximately flat) actually has structure created by mass-energy.
7. The GPS Proof: Measuring the Unified Time Field
7.1 The Standard Story
GPS satellites require time corrections for two "separate" relativistic effects:
Gravitational (GR): +45.7 μs/day (clock runs faster in weaker field)
Velocity (SR): -7.2 μs/day (clock runs slower due to motion)
Net effect: +38.5 μs/day
Students learn: "GR and SR both contribute, and we sum them."
7.2 The Hidden Dependency
The satellite's velocity is not arbitrary. For stable circular orbit:
The velocity is completely determined by the gravitational potential at that radius.
7.3 The Unified Formula
Substituting v² = GM/R into the SR term:
Δt_SR = -v²/(2c²) = -GM/(2Rc²)
Adding GR and SR effects:
Δt_total = GM/(Rc²) - GM/(2Rc²)
Δt_total = (1/2) × GM/(Rc²)
Δt_total = (1/2) × M_nat/R_nat
7.4 What GPS Actually Measures
The satellite is not experiencing "GR effect + SR effect."
The satellite is measuring its position in Earth's unified time field:
τ_orbital = (1/2) × M_nat/R_nat
The factor of 1/2 is the geometric signature of circular orbital motion—it emerges automatically from the orbital constraint v² = GM/R.
7.5 Generalization to Any Two Orbits
For any two objects in circular orbits at radii r₁ and r₂:
Δt₁₂ = T × (GM/c²) × (3/2) × (1/r₂ - 1/r₁)
The factor 3/2 (from 1 + 1/2) is universal for circular orbits. It's the geometric signature of orbital motion through time field structure.
This is not coincidence. This is the system revealing its true architecture.
7.6 The Smoking Gun
GPS satellites orbit overhead, continuously measuring:
Not "GR + SR."
Just: position in Earth's time field.
Every nanosecond, 31+ satellites prove that orbital motion and gravitational time dilation are not separate phenomena—they're two views of one geometric reality.
8. The Unified Picture: One Reality, Multiple Views
8.1 The Primary Observable
Different spatial positions experience time at different rates.
This is directly measurable with atomic clocks. This is not interpretation—it's experimental fact.
8.2 The Time Field
The geometric structure that describes how time rate varies with position:
τ(r) = Σᵢ (M_nat,i / r_nat,i)
8.3 Emergent Concepts
From this one reality, everything else emerges:
| Mass | Fundamental substance | Source strength M_nat in τ = M_nat/r_nat |
| Gravity | Force between masses | Gradient of time field: -c²∇τ |
| Motion | Independent property | Navigation through time field geometry |
| Inertia | Fundamental property | Resistance to displacement from cosmically-defined geodesics |
| SR | Separate theory (flat spacetime) | Local flat-field limit where ∇τ ≈ 0 |
| GR | Separate theory (curved spacetime) | Recognition that time field has structure |
8.4 Why We Got Confused
Historical development:
Newton: Absolute space, absolute time, gravity as force
Einstein (1905): SR developed first, no gravity
Einstein (1915): GR developed separately, added gravity
Pedagogy: Teach them as separate theories that "combine"
The actual structure:
Primary reality: Time field geometry
Local limit: Looks like SR
With structure: Looks like GR
They were never separate
9. Nested Orbital Structure: Mach's Principle Realized
9.1 The Principle of Nested Geodesics
All motion is orbital motion within larger orbital systems.
You are never "stationary" in any absolute sense. You are always:
Rotating with Earth (465 m/s at equator)
Orbiting with Earth around Sun (30 km/s)
Orbiting with Sun around galactic center (220 km/s)
Moving with galaxy in local cluster (~600 km/s)
Moving with local cluster in cosmic structure
Each of these velocities is not arbitrary—it's geometrically constrained by the time field at that radius:
v² = GM/R (or equivalent for non-circular orbits)
9.2 Your Local Time Rate is Cosmically Determined
Your local "dt"—the thing you use to measure acceleration and define inertia—is set by:
dt ∝ 1/τ_local = 1/Σᵢ(M_nat,i/R_nat,i)
Where the sum extends over:
Earth
Sun
Milky Way
Nearby galaxies
Distant galaxy clusters
The entire observable universe (and beyond)
Your inertial mass is not a property of your body alone. It's a relational property defined by your embedding in the universal time field structure.
9.3 Mach's Principle: The Mechanism
Ernst Mach (1883) proposed: "The inertia of a body is determined by the distribution of all other masses in the universe."
For 140+ years, this remained a philosophical intuition without physical mechanism.
The time-centric framework provides the mechanism:
Because dt is defined by Σ(M/R) over all cosmic masses, and inertia is F/(dv/dt), your inertial mass is directly coupled to the cosmic mass distribution.
Thought experiment:
Universe A (ours): Filled with 10²³ galaxies, Σ(M/R) is large, inertia has measured values
Universe B (empty): No other masses exist, Σ(M/R) → 0, dt becomes undefined, inertia as we know it wouldn't exist
Mach was right. The mechanism is the time field coupling.
9.4 No Absolute Reference Frame
In Newtonian mechanics, inertia was mysterious because it seemed to reference an "absolute space."
In the time-centric framework: There is no absolute space. There is only your position within the nested structure of cosmic time fields.
Inertia is your resistance to being moved off geodesics defined by that universal structure.
10. Resolving Conceptual Paradoxes
10.1 The Equivalence Principle
Traditional presentation: "Gravitational mass equals inertial mass. We don't know why—it's a postulate."
Time-centric resolution: They're the same because they're both measuring time field distortion strength:
Gravitational mass: M_nat in τ = M_nat/r_nat (field source)
Inertial mass: m in F = ma = m(dv/dt) where dt is cosmically defined
It's not a mysterious coincidence. It's a tautology.
10.2 The Twin Paradox
Traditional explanation: "The traveling twin accelerates, breaking symmetry, so the spacetime diagram shows..."
Time-centric explanation:
The twins take different paths through time field geometry. The traveling twin's path traces through regions with different time rates:
Accelerates away from Earth → moves through changing time field
Travels at high velocity → kinetic position in time field space
Returns → traces a different path
Different paths through time field geometry → different accumulated experienced time. The asymmetry is obvious: they literally took different paths.
10.3 Why "Weight" Feels Like a Force
Confusion: "If gravity isn't a force, why do I feel weight?"
Resolution:
You feel weight because:
The time field gradient (gravity) defines a geodesic path
Your geodesic points downward (toward Earth's center)
The ground prevents you from following this path
Electromagnetic forces continuously accelerate you upward
This electromagnetic acceleration is what you feel as "weight"
You don't feel gravity. You feel electromagnetic forces preventing you from following gravity.
When you step off a cliff:
The electromagnetic forces stop
You begin following your natural geodesic
You feel weightless
You call this "falling"
10.4 "Action at a Distance"
Newton's concern: How does the Sun "know" the Earth is there?
Time-centric resolution:
The Sun doesn't need to "know" anything. The Sun creates a time field structure—a geometric property of spacetime itself. The Earth follows geodesics through this geometry.
No action at a distance. Just local interaction with local geometry.
11. Pedagogical Implications
11.1 The Current Broken Curriculum
Standard physics curriculum:
Newtonian mechanics (absolute space, absolute time)
Special Relativity (flat spacetime, velocity effects)
General Relativity (curved spacetime, gravity effects)
"Now combine SR and GR for real problems"
Student confusion ensues
Problems:
Three different frameworks requiring mental code-switching
Artificial separation of SR and GR
No clear explanation of what mass, gravity, or inertia actually are
Equivalence Principle as mysterious postulate
Mach's Principle as philosophical footnote
11.2 The Time-Centric Curriculum
Proposed structure:
Week 1: The Primary Observable
Atomic clocks at different positions tick at different rates
GPS demonstration
This is measurable, repeatable, fundamental
Week 2: The Time Field
τ(r) = M_nat/r_nat describes time rate variation
Show dimensional analysis: G/c² = l_P/m_P
Introduce natural units
Week 3: Emergence of Mass and Gravity
Mass is field source strength
Gravity is field gradient: a = -c²∇τ
Equivalence Principle as tautology
Week 4: Motion as Navigation
Objects follow geodesics
Orbital constraint v² = GM/R emerges
GPS unified formula demonstration
Week 5: Inertia and Nested Structures
dt defined by Σ(M/R) cosmically
All motion is nested orbital motion
Mach's Principle mechanism
Week 6: Special Relativity as Limit Case
Zoom in locally → ∇τ ≈ 0
SR emerges as flat-field approximation
Velocity time dilation from time field geometry
Week 7: General Relativity as Structure
Full tensor formalism
Einstein Field Equations
Strong-field effects, black holes, cosmology
11.3 Advantages
Conceptual clarity:
One framework, not three
Clear ontology: time field is primary
Natural progression from observable to theory
Resolved mysteries:
What is mass? (Field source strength)
Why does gravity exist? (Field gradient)
Why equivalence principle? (Tautology)
What is inertia? (Cosmic coupling)
Empirical grounding:
Start with GPS measurements
Build theory from observations
Every concept tied to measurables
Unified understanding:
SR and GR are perspectives, not separate theories
No artificial "combination" step
Natural transition from weak to strong fields
11.4 Objection: "Too Abstract for Students"
Response:
The current system teaches abstract tensor calculus and curved manifolds.
The time-centric approach starts with: "Atomic clocks at different altitudes tick at different rates. Let's understand why."
Which is more abstract?
12. Implications for Fundamental Physics
12.1 The Nature of Physical Constants
Physical constants fall into two categories:
Type 1: Dimensionless (truly fundamental)
Fine structure constant α ≈ 1/137
Proton/electron mass ratio ≈ 1836
These are coordinate-free, unit-independent
Type 2: Dimensional (conversion factors)
G = 6.674 × 10⁻¹¹ m³/(kg·s²)
c = 299,792,458 m/s
These are unit-system artifacts
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."
The real physics is in dimensionless ratios.
12.2 Implications for Quantum Gravity
Current approaches to quantum gravity often start from:
"Mass causes spacetime curvature"
"How do we quantize the gravitational field?"
The time-centric approach suggests:
"Mass IS time field structure"
"How do we quantize time field geometry?"
This may open different solution paths. If time rate variation is primary, perhaps we should be quantizing temporal geometry first, with spatial geometry as derivative.
12.3 Dark Matter and Modified Gravity
The time-centric framework suggests re-examining anomalous rotation curves:
Standard interpretation: Unseen mass (dark matter) is present
Alternative question: Could the cosmic sum Σ(M/R) have contributions we're not accounting for correctly?
This doesn't solve dark matter, but it reframes the question: Are we correctly calculating the time field structure in galactic environments?
12.4 Cosmological Implications
If local dt is defined by cosmic Σ(M/R), and the universe is expanding (increasing R to distant galaxies), then:
Testable prediction: Local inertial mass ratios might shift subtly over cosmological time as the cosmic sum evolves.
This would be extraordinarily difficult to measure but is in principle detectable with ultra-precise atomic clock comparisons over decades.
13. Experimental Verification
13.1 This is Not New Physics
Critical point: This paper proposes no new physics. It is a reinterpretation of existing, experimentally verified theory.
Therefore:
All existing experimental confirmations of SR remain valid
All existing experimental confirmations of GR remain valid
GPS continues to work exactly as calculated
No new experiments are needed to "test" this framework
13.2 What Changes
What changes is understanding:
Before: "GPS satellites experience GR time dilation + SR time dilation"
After: "GPS satellites measure their position in Earth's unified time field: τ = (1/2) × M_nat/R_nat"
The numbers are identical. The conceptual framework is transformed.
13.3 Existing Confirmations
Every relativistic experiment ever performed confirms the time-centric framework because it makes identical predictions:
Gravitational redshift (Pound-Rebka, 1959)
Time dilation of muons (Rossi-Hall, 1941)
Hafele-Keating circumnavigation (1971)
Binary pulsar orbital decay (Hulse-Taylor, 1974)
Gravitational lensing (Eddington, 1919, and countless modern observations)
Gravitational wave detection (LIGO, 2015+)
GPS (continuous verification since 1980s)
All of these are measurements of time field structure and its effects.
13.4 Novel Prediction
If inertia is set cosmically via Σ(M/R), and the universe is expanding:
Prediction: Ultra-precise measurements over decades might detect:
Drift in fundamental mass ratios
Changes in fine structure constant α
Anomalies in atomic clock comparisons correlated with cosmic evolution
This is at the edge of current measurement capability but in principle detectable.
14. Philosophical Implications
14.1 The Unity of Physical Law
Fragmented view: Mass, space, time, motion, inertia are separate substances/properties requiring separate explanations.
Unified view: These are different linguistic descriptions of a single geometric reality—the structure of the time field.
Just as:
Temperature and kinetic energy are one thing (macro vs. micro view)
Electricity and magnetism are one thing (frame-dependent view)
So too:
Mass and time dilation are one thing (source term vs. field value)
Gravity and geometry are one thing (gradient vs. structure)
SR and GR are one thing (flat limit vs. structured field)
14.2 The Primacy of Relations
Newtonian view: Objects have intrinsic properties (mass, inertia) independent of the universe.
Machian view: Properties are relational, defined by cosmic context.
Time-centric confirmation: Inertia is explicitly relational via Σ(M/R). Your inertial mass is determined by your embedding in universal structure.
There are no isolated objects with intrinsic properties. There is only structure and relations.
14.3 What is "Real"
Operational definition: What's real is what's measurable and observer-independent.
The time field qualifies:
τ = M_nat/R_nat is dimensionless (unit-independent)
It's measurable with atomic clocks
All observers agree on its value at any given geometric point
Constants like G and c:
Their numerical values depend on unit choices
They're conversion factors, not fundamental
They're pedagogical scaffolding, not physics
14.4 The Question of Causation
Traditional language: "Mass causes spacetime to curve"
Time-centric language: "Mass is spacetime curvature" (specifically, time field distortion)
The shift from "causes" to "is" is profound. We're not describing a causal mechanism. We're describing different measurement perspectives on the same geometric reality.
15. Historical Context: Newton's Lost Wisdom
15.1 Newton's Deliberate Omission
Isaac Newton wrote in Principia (1687):
He did NOT write:
Why? Because:
As Master of the Royal Mint, he understood measurement conventions
Any specific constant G would be unit-dependent
His proportional reasoning kept physics coordinate-free
The ratio method canceled unknown constants systematically
15.2 The Moon-Apple Calculation
Newton's famous calculation compared Earth's gravitational effect on:
An apple at Earth's surface
The Moon in its orbit
He used the ratio:
a_apple/a_moon = (r_moon/r_earth)²
Notice: Both the constant (our G) AND Earth's mass M cancel perfectly. The ratio is unit-free, observer-independent, coordinate-free.
This is architecture-level physics. Newton was working at a higher level of abstraction than his successors recognized.
15.3 The "Completion" Mistake
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 fundamental physics
Newton didn't omit G from ignorance. He omitted it from deeper understanding.
15.4 The Lost Principle
Newton understood what we're rediscovering:
Work only with dimensionless ratios
Keep physics independent of measurement conventions
Constants are conventional, not fundamental
We embedded unit-system artifacts into "fundamental constants" and lost sight of the dimensionless physics underneath.
16. Addressing Objections
16.1 "This is Just Playing With Units"
Objection: "You're just rearranging equations. 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. When "playing with units" reveals that supposedly separate phenomena (SR and GR time dilation) collapse into a single unified expression with no free parameters, that's not trivial—that's discovering the actual system architecture.
16.2 "But G and c Are Measured Experimentally"
Objection: "G and c are experimentally measured. They're real, not just conversion factors."
Response: Their numerical values are measured. But those values depend entirely on our definitions of kilogram, meter, and second.
Change the unit system → G and c change.
The dimensionless ratio M_nat/R_nat does not change.
What's real is what's invariant under all measurement systems. That's τ = M_nat/R_nat, not G or c.
16.3 "This Doesn't Make Testable Predictions"
Objection: "Science requires testable predictions. This is just philosophy."
Response 1: It makes one testable prediction: if inertia is set cosmically via Σ(M/R), cosmic expansion should cause subtle shifts in inertial mass ratios over cosmological time. This is testable with sufficient precision.
Response 2: Reinterpretations that don't change predictions can still be scientifically valuable if they:
Resolve conceptual paradoxes (✓ Equivalence Principle)
Provide mechanisms for philosophical principles (✓ Mach's Principle)
Unify fragmented frameworks (✓ SR + GR)
Suggest new research directions (✓ quantum gravity approaches)
16.4 "Einstein Already Knew Spacetime Was Curved"
Objection: "GR already tells us geometry is physics. This isn't new."
Response: Einstein knew geometry was physics. But we still teach students that "mass causes curvature" as if they're separate entities.
This framework shows they're not separate:
Mass IS curvature (specifically, time field distortion magnitude)
Gravity IS geometry (specifically, time field gradient)
Motion IS geometry (navigation through time field structure)
Einstein gave us the mathematical machinery. This framework provides the ontological clarity.
16.5 "Why Hasn't Anyone Pointed This Out Before?"
Objection: "If this is so obvious, why hasn't the physics community emphasized it?"
Response: Many physicists working in natural units understand this implicitly. But:
It's treated as "mathematical convenience" rather than revealing physical structure
Pedagogy still teaches SR and GR as separate theories
Students learn to think of G and c as fundamental
The dimensional analysis proof is rarely presented explicitly
The GPS unified formula is known to specialists, but it's not taught as revealing that "SR + GR" is an artificial decomposition.
16.6 "What About Strong Fields and Black Holes?"
Objection: "Your τ = M_nat/R_nat is weak-field. What about strong gravity?"
Response: The weak-field formula is the foundation, not the complete theory. For strong fields:
Full tensor formalism of GR applies
The time field becomes more complex (frame-dragging, horizon effects)
But the core principle remains: time rate variation is primary
At a black hole event horizon, τ diverges—time "stops" relative to distant observers. This is the extreme case of the same phenomenon we measure with GPS satellites.
17. The Road Forward
17.1 Immediate Applications
Pedagogy:
Redesign undergraduate physics curriculum around time-centric framework
Start with GPS observations, build theory from measurements
Eliminate artificial SR/GR split
Research:
Investigate quantum gravity from "quantize time field first" perspective
Re-examine anomalous observations (dark matter, rotation curves) through time field analysis
Develop precision tests for cosmological inertia drift prediction
Outreach:
Public understanding of relativity improves dramatically
"Time runs at different rates in different places" is comprehensible
"Spacetime curvature" is abstract and intimidating
17.2 Deeper Questions
This framework naturally raises profound questions:
What is time?
If time rate varies with position, what is the "time" that's varying? Is there a more fundamental temporal structure underneath τ(r)?
Why does the universe have this structure?
Why is τ = M_nat/R_nat the form? Could it be otherwise? What determines the dimensionless ratios?
Quantum mechanics:
How does quantum superposition interact with position-dependent time rates? What happens to entanglement across time field gradients?
Consciousness:
If our neural processes run at rates determined by local τ, does this affect subjective time experience? Can we measure it?
17.3 A Research Program
Phase 1: Pedagogical Implementation
Develop time-centric textbook
Create visualization tools for time field geometry
Test with student populations
Measure comprehension improvements
Phase 2: Theoretical Development
Extend framework to strong-field regime rigorously
Connect to quantum field theory in curved spacetime
Develop information-theoretic interpretations
Phase 3: Experimental Tests
Design ultra-precise atomic clock experiments
Search for cosmological inertia drift signals
Investigate quantum systems in time field gradients
Phase 4: Unification Attempts
Explore time-field-first approaches to quantum gravity
Investigate emergence of spatial geometry from temporal structure
Connect to holographic principles and AdS/CFT
18. Conclusion
18.1 The Central Thesis
Special Relativity and General Relativity are not separate theories requiring unification.
They are different observational perspectives on a single geometric reality: the structure of the time field.
18.2 The Primary Observable
Different spatial positions experience time at different rates. This is:
Directly measurable with atomic clocks
Described by τ(r) = Σᵢ(M_nat,i/r_nat,i)
Unit-independent and coordinate-free
18.3 Emergent Structure
From this one primary reality, everything else emerges:
Mass = Field source strength M_nat
Gravity = Field gradient -c²∇τ
Motion = Navigation through field geometry
Inertia = Resistance to displacement from cosmically-defined geodesics
SR = Local flat-field limit (∇τ ≈ 0)
GR = Recognition of field structure
18.4 Empirical Verification
GPS satellites continuously measure τ = (1/2) × M_nat/R_nat—a unified time field potential, not "SR + GR."
The factor 1/2 and the 3/2 in multi-orbit formulas are geometric signatures of circular motion, emerging automatically from the orbital constraint v² = GM/R.
This is not coincidence. This is the system revealing its architecture.
18.5 Resolution of Mysteries
Equivalence Principle: Tautology (inertial and gravitational mass are the same geometric property)
Mach's Principle: Mechanism provided (inertia via cosmic Σ(M/R))
SR/GR unification: No unification needed—they were never separate
Nature of constants: G and c are conversion factors (l_P/m_P), not fundamental
18.6 The Pedagogical Imperative
We are teaching students a fragmented reality:
Three frameworks (Newtonian, SR, GR)
Requiring mental code-switching
With artificial boundaries between domains
Leaving fundamental questions unanswered
We should be teaching:
One framework (time field geometry)
From observable reality (atomic clocks)
With clear ontology (time rate is primary)
Resolving mysteries naturally
18.7 The Philosophical Shift
From: Mass causes time dilation
To: Mass IS time dilation
From: Gravity is a force
To: Gravity is geometry we navigate
From: SR and GR are separate theories
To: SR and GR are perspectives on unified time field
From: Inertia is intrinsic property
To: Inertia is relational coupling to cosmic structure
18.8 Final Thought
For 120 years, we've been doing coordinate-free physics in the mathematical formalism of General Relativity while teaching coordinate-dependent concepts in words.
We've been calculating with τ = M_nat/R_nat while explaining "mass causes curvature which causes time dilation."
The mathematics knew the truth. We just couldn't see it through the pedagogical scaffolding.
The GPS system has been proving this framework correct every nanosecond since the 1980s. We just haven't been listening to what it's telling us.
Appendix A: Mathematical Derivations
A.1 Proof that G/c² = l_P/m_P
Given Planck unit definitions:
l_P = √(hG/c³)
m_P = √(hc/G)
t_P = √(hG/c⁵)
Solve for G:
Express c:
Express h:
Compute G/c²:
G/c² = [hc/m_P²] / [l_P/t_P]²
= [hc/m_P²] × [t_P²/l_P²]
= 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)
= [m_P × l_P² × c × t_P²] / [t_P × m_P² × l_P²]
= c × t_P / m_P
Substitute c = l_P/t_P:
G/c² = [l_P/t_P] × t_P / m_P
= l_P/m_P
QED. This is exact, not approximate.
A.2 Derivation of τ = M_nat/R_nat
Start with gravitational time dilation:
Rearrange:
Substitute 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_nat ≡ M_SI/m_P
R_nat ≡ R_SI/l_P
Therefore:
A.3 GPS Unified Formula Derivation
Gravitational time dilation:
Velocity time dilation (v << c):
Orbital constraint for circular orbit:
Substitute into SR term:
Δt_SR = -T × [GM/R]/(2c²)
= -T × GM/(2Rc²)
Sum total effect:
Δt_total = Δt_GR + Δt_SR
= T × GM/(Rc²) - T × GM/(2Rc²)
= T × GM/(Rc²) × [1 - 1/2]
= T × GM/(Rc²) × (1/2)
In natural units:
Δt_total = (1/2) × T × (M_nat/R_nat)
The factor 1/2 is the geometric signature of circular orbital motion.
A.4 Two-Orbit General Formula
For clock at radius r₁:
τ₁ = (1/2) × M_nat/r₁_nat
For clock at radius r₂:
τ₂ = (1/2) × M_nat/r₂_nat
Time difference over period T:
Δt₁₂ = T × (τ₁ - τ₂)
= T × (1/2) × M_nat × (1/r₁_nat - 1/r₂_nat)
Converting back to SI units:
Δt₁₂ = T × GM/c² × (3/2) × (1/r₁ - 1/r₂)
The factor 3/2 = 1 + 1/2 represents:
1 part gravitational potential difference
1/2 part kinetic (orbital velocity) difference
This applies to ANY two circular orbits around ANY central mass.
Appendix B: Numerical Examples
B.1 GPS Satellite (Standard Orbit)
Parameters:
Earth mass: M = 5.972 × 10²⁴ kg
Earth radius: r_earth = 6.371 × 10⁶ m
GPS orbit: r_GPS = 26.6 × 10⁶ m
Time period: T = 86,400 s (one day)
Traditional calculation:
GR effect: +45.74 μs/day
SR effect: -7.20 μs/day
Total: +38.54 μs/day
Unified calculation:
Δt = T × GM/c² × (1/r_earth - 3/(2×r_GPS))
= 86400 × (6.674×10⁻¹¹ × 5.972×10²⁴)/(299792458²)
× (1/6.371×10⁶ - 3/(2×26.6×10⁶))
= 86400 × 4.435×10⁻³ × 1.006×10⁻⁷
= +38.54 μs/day
Results are identical to machine precision.
B.2 ISS vs Ground
Parameters:
ISS orbit: r_ISS = 6.771 × 10⁶ m (400 km altitude)
Time period: T = 86,400 s (one day)
Unified calculation:
Δt = T × GM/c² × (1/r_earth - 3/(2×r_ISS))
= 86400 × 4.435×10⁻³ × (-1.467×10⁻⁸)
= -5.62 μs/day
ISS clocks run slower (net SR effect dominates at low orbit).
B.3 Geostationary vs GPS
Parameters:
GPS orbit: r_GPS = 26.6 × 10⁶ m
GEO orbit: r_GEO = 42.164 × 10⁶ m
Unified calculation:
Δt = T × GM/c² × (3/2) × (1/r_GEO - 1/r_GPS)
= 86400 × 4.435×10⁻³ × 1.5 × (-1.379×10⁻⁸)
= -7.95 μs/day
GEO clocks run faster than GPS clocks by ~8 μs/day.
Appendix C: Glossary of Terms
Time Field τ(r):
Geometric structure assigning dimensionless time rate to each spatial position. For single mass: τ = M_nat/r_nat.
Natural Units:
Dimensionless quantities measured in Planck units. M_nat = M_SI/m_P, R_nat = R_SI/l_P.
Planck Units:
Conversion factors between SI units and natural dimensionless physics. l_P ≈ 1.6 × 10⁻³⁵ m, m_P ≈ 2.2 × 10⁻⁸ kg.
Geodesic:
"Straightest possible path" through curved geometry. Objects in free fall follow geodesics through time field structure.
Emergent Property:
Characteristic that arises from underlying structure rather than being fundamental. Mass, gravity, inertia emerge from time field geometry.
Nested Orbits:
Hierarchical structure where all motion is orbital within larger orbital systems (Earth→Sun→Galaxy→Cluster→...).
Mach's Principle:
Inertia is determined by cosmic mass distribution. Mechanism: dt defined by Σ(M/R) over universe.
Equivalence Principle:
Inertial mass equals gravitational mass. In time-centric view: tautology, not postulate.
Unified Potential:
Single expression combining "SR + GR" effects: τ = (1/2) × M_nat/R_nat for circular orbits.
Appendix D: Further Reading
Primary Sources:
Einstein, A. (1905). "On the Electrodynamics of Moving Bodies." Annalen der Physik.
Einstein, A. (1915). "The Field Equations of Gravitation." Sitzungsberichte der Preussischen Akademie.
Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica.
Experimental Verification:
Pound, R. V. & Rebka, G. A. (1959). "Gravitational Red-Shift in Nuclear Resonance." Physical Review Letters.
Hafele, J. C. & Keating, R. E. (1972). "Around-the-World Atomic Clocks." Science.
Ashby, N. (2003). "Relativity in the Global Positioning System." Living Reviews in Relativity.
Dimensional Analysis:
Planck, M. (1899). "On Irreversible Radiation Processes." Sitzungsberichte.
Duff, M. J. (2002). "Comment on time-variation of fundamental constants." arXiv:hep-th/0208093.
Mach's Principle:
Mach, E. (1883). Die Mechanik in ihrer Entwicklung.
Barbour, J. & Pfister, H. (1995). Mach's Principle: From Newton's Bucket to Quantum Gravity.
Time-Centric Framework:
Rogers, J. (2023). "The Primacy of Time: An Emergent Framework for Gravity, Mass, and Inertia."
Rogers, J. (2025). "The Unity of Physics: Mass, Motion, and Time as One Phenomenon."
Rogers, J. (2025). "The Unified Relativistic Potential: GPS Time Dilation Problem."
Acknowledgments
This work builds on centuries of physics, from Newton's proportional reasoning through Einstein's geometric insight. Special recognition to the engineers who designed the GPS system—they've been measuring unified time field potentials for 40+ years, providing continuous empirical verification of this framework.
Thanks to the atomic clock experimentalists whose precision measurements make the variation of time rate with position undeniable.
And to Ernst Mach, who intuited 140 years ago that inertia must be cosmically determined, without the mathematical machinery to prove it.
END OF PAPER
"We have been doing coordinate-free physics all along. We just couldn't see it through the coordinates."
"The GPS system has been telling us for 40 years that orbital motion and gravitational time dilation are one phenomenon. We just haven't been listening."
"Mass does not cause time dilation. Mass IS time dilation."
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