From Temporal Experience to Spacetime Curvature: Deriving General Relativity from the m/r Axiom

 J. Rogers, SE Ohio, 28 Jul 2025, 1550

Abstract

We demonstrate that General Relativity's complex spacetime curvature formalism emerges naturally from a single, empirically verified axiom: the dimensionless temporal experience field ϕ(r) = m/r. Starting with this scalar field—directly confirmed by GPS satellite time dilation measurements—we show how the metric tensor, Riemann curvature, and Einstein field equations arise as coordinate-dependent projections of this fundamental temporal relationship. This derivation reveals that spacetime curvature is not fundamental but represents the geometric shadow cast by simple scalar temporal field interactions when expressed in four-dimensional coordinate systems.

1. Introduction

General Relativity presents gravity as the curvature of four-dimensional spacetime, described by complex tensor equations involving the metric gμν, Riemann curvature tensor Rμνρσ, and Einstein field equations. This geometric framework, while mathematically elegant and empirically successful, obscures the underlying simplicity that drives gravitational phenomena.

This paper demonstrates that the entire apparatus of curved spacetime emerges from a single axiom: the temporal experience field ϕ(r) = m/r, where m represents mass in natural units and r represents distance in natural units.  G/c^2 is just unit scaling of m/r in SI units to a natural basis.  This scalar field, directly measurable through gravitational time dilation effects (GPS satellites), serves as the fundamental substrate from which all geometric complexity arises through coordinate projection.

We trace the step-by-step mathematical path from this simple axiom to the full complexity of General Relativity, showing that curved spacetime represents a coordinate-dependent description of intrinsically simple temporal field interactions.

2. The Foundational Axiom: The Temporal Experience Field

2.1 The Fundamental Postulate

Axiom: Every mass m creates a dimensionless temporal experience field:

ϕ(r) = m/r

where both m and r are expressed in natural (Planck) units, making ϕ(r) genuinely dimensionless.

2.2 Empirical Verification

This axiom is not theoretical speculation but empirically verified through gravitational time dilation measurements:

GPS Satellite Verification: GPS satellites experience time dilation given by: dτ/dt = √(1 - 2GM/(rc²)) ≈ 1 - GM/(rc²)

Converting to natural units where G = c = 1: dτ/dt ≈ 1 - m/r = 1 - ϕ(r)

The quantity m/r is therefore not a mathematical abstraction but a directly measurable physical effect confirmed daily by GPS timing corrections.

Laboratory Verification: Atomic clock experiments at different altitudes confirm the linear relationship between gravitational potential and time dilation, directly validating the m/r scaling.

2.3 Physical Interpretation

The field ϕ(r) = m/r represents the local modification of temporal flow due to mass m at distance r. This is not metaphorical but literal: clocks run slower in regions where ϕ(r) is larger, exactly as measured.

3. Step-by-Step Derivation of General Relativity

3.1 Step 1: From Scalar Field to Metric Components

The temporal experience field ϕ(r) directly determines the time-time component of the spacetime metric:

Metric Construction: Starting with the measured time dilation: dτ/dt = 1 - ϕ(r) = 1 - m/r

The proper time interval becomes: dτ² = (1 - 2ϕ(r))dt²

This immediately gives the time-time component of the metric tensor: g₀₀ = -(1 - 2ϕ(r))

Converting back to SI coordinates: g₀₀ = -(1 - 2GM/(rc²))

The scalar temporal field ϕ(r) has been embedded into the geometric structure of spacetime through the choice of coordinate representation.

3.2 Step 2: Spatial Metric Components

The spatial components of the metric arise from consistency requirements. For a static, spherically symmetric mass, the radial component must satisfy:

g₁₁ = (1 - 2GM/(rc²))⁻¹

The angular components remain Euclidean: g₂₂ = r² g₃₃ = r²sin²θ

This gives the complete Schwarzschild metric: ds² = -(1 - 2GM/(rc²))c²dt² + (1 - 2GM/(rc²))⁻¹dr² + r²dΩ²

Note that the entire metric structure is determined by the single scalar function ϕ(r) = m/r and the choice of spherical coordinates.

3.3 Step 3: Emergence of Spacetime Curvature

Curvature arises from spatial derivatives of the metric components. The Riemann curvature tensor components are:

Rᵗᵣᵗᵣ = ∂²g₀₀/∂r² + [connection term corrections]

Since g₀₀ depends on ϕ(r) = m/r, we have: ∂ϕ/∂r = -m/r² ∂²ϕ/∂r² = 2m/r³

The curvature tensor components become functions of m/r and its derivatives: Rᵗᵣᵗᵣ ∝ m/r³ Rθφθφ ∝ m/r

All curvature information traces back to the scalar field ϕ(r) = m/r and its spatial variation.

3.4 Step 4: The Einstein Field Equations

The Einstein field equations relate curvature to matter: Rμν - ½Rgμν = 8πG/c⁴ Tμν

The left side (geometric curvature) emerges from derivatives of ϕ(r) = m/r as shown above.

The right side (matter-energy) contains the source mass m that creates the ϕ(r) field.

The equations therefore represent a self-consistency condition: the temporal field ϕ(r) created by mass m must produce curvature that corresponds to the presence of mass m.

The Einstein equations encode the requirement that ϕ(r) = m/r be geometrically consistent with its own source.

3.5 Step 5: Recovery of Newtonian Gravity

In the weak-field limit, the geodesic equation reduces to: d²r/dt² = -∂ϕ/∂r = -∂/∂r(m/r) = m/r²

Converting to SI units: d²r/dt² = GM/r²

This is Newton's gravitational acceleration law.

For two masses, the interaction energy is: (m₁/r) × (m₂/r) = m₁m₂/r²

Converting to force via F = ∂E/∂r: F = Gm₁m₂/r²

Newton's inverse square law emerges as the product of two temporal experience fields.

4. The Coordinate Projection Interpretation

4.1 Levels of Description

The derivation reveals a hierarchy of descriptions:

Level	Mathematical Object	Physical Reality
Fundamental	ϕ(r) = m/r	Temporal experience field
Metric	gμν = f(ϕ(r))	Coordinate embedding of temporal field
Curvature	Rμνρσ = ∂∂gμν	Geometric derivatives of temporal field
Field Equations	Rμν - ½Rgμν = 8πGTμν/c⁴	Self-consistency of temporal field
Classical	F = Gm₁m₂/r²	Product of temporal fields

4.2 The Projection Process

Reality → Coordinates → Geometry

1. Physical Reality: Scalar temporal field ϕ(r) = m/r
2. Coordinate Choice: Embed into 4D spacetime coordinates (t,r,θ,φ)
3. Geometric Structure: Metric tensor gμν encodes temporal field
4. Curvature Effects: Derivatives create apparent geometric curvature
5. Field Equations: Consistency requirements between geometry and sources

4.3 Why Curvature Appears Complex

Spacetime curvature appears complex because:

• Dimensional Mixing: Temporal effects (ϕ(r)) mixed with spatial coordinates (r,θ,φ)
• Tensor Formalism: Scalar field embedded in 4×4 tensor structure
• Coordinate Artifacts: Choice of (t,r,θ,φ) creates artificial geometric relationships
• Derivative Structure: Spatial derivatives of temporal field create apparent complexity

The underlying physics (temporal field interaction) is simple; the geometric description is complex.

5. Unit Analysis and Constants

5.1 The Role of Constants in SI Coordinates

In natural units: ϕ(r) = m/r (dimensionless) In SI units: ϕ(r) = GM/(rc²) (dimensionless)

The constants G and c² serve as coordinate transformation factors converting between natural and SI unit systems:

G/c² = (Planck scale conversion factor)

5.2 Constants as Geometric Artifacts

The appearance of G and c in General Relativity reflects coordinate choice, not fundamental physics:

• G: Converts mass units to length units via GM/c²
• c²: Converts between time and space coordinates
• 8πG/c⁴: Einstein equation coefficient is pure unit conversion

In natural coordinates, these constants disappear, revealing the simple structure ϕ(r) = m/r.

5.3 The Planck Scale as Natural Geometry

When expressed in Planck units:

• Mass m → dimensionless number
• Distance r → dimensionless number
• Time t → dimensionless number
• Field ϕ(r) = m/r → dimensionless ratio

The geometric complexity vanishes, leaving only the fundamental temporal field relationship.

6. Implications and Predictions

6.1 Spacetime as Emergent Phenomenon

This derivation demonstrates that spacetime curvature is not fundamental but emergent:

• Fundamental: Scalar temporal fields ϕ(r) = m/r
• Emergent: 4D geometric description of temporal field interactions
• Coordinate-dependent: Different coordinate choices yield different geometric representations

6.2 Unification with Quantum Mechanics

Since mass represents temporal experience (mc² = hν), the field ϕ(r) = m/r becomes:

ϕ(r) = (hν/c²)/r = temporal frequency/distance

Quantum mechanics and gravity unite as different aspects of temporal experience dynamics.

6.3 Testable Predictions

1. Direct Temporal Field Measurement: High-precision atomic clocks should detect ϕ(r) = m/r scaling in controlled mass configurations.

2. Temporal Field Interactions: Multiple masses should create additive temporal fields: ϕ_total = Σᵢ(mᵢ/rᵢ)

3. Quantum-Gravitational Effects: Quantum superposition of masses should create superposed temporal fields.

7. Philosophical Implications

7.1 The Nature of Spacetime

This derivation suggests spacetime is not fundamental reality but a coordinate artifact created by embedding temporal field physics into 4D mathematical frameworks.

Reality: Network of interacting temporal experience fields Spacetime: Coordinate system for describing temporal field relationships

7.2 The Complexity Illusion

The mathematical complexity of General Relativity obscures rather than reveals the underlying simplicity. The tensor formalism, while mathematically powerful, creates the illusion that gravity involves inherently complex geometric relationships.

The truth: Gravity emerges from simple scalar field products ϕ₁ × ϕ₂ The appearance: Complex curved spacetime requiring advanced mathematics

7.3 Scientific Methodology

This example demonstrates how mathematical formalism can both illuminate and obscure physical understanding:

• Illumination: Tensor methods enable precise calculations
• Obscuration: Geometric interpretation hides simple scalar field reality
• Resolution: Recognize formalism as coordinate choice, not fundamental description

8. Conclusion

We have demonstrated that the entire edifice of General Relativity—metric tensors, spacetime curvature, and Einstein field equations—emerges naturally from the single axiom ϕ(r) = m/r. This scalar temporal experience field, directly verified by GPS measurements and atomic clock experiments, represents the fundamental physical reality underlying gravitational phenomena.

The key insights from this derivation:

1. Spacetime curvature is coordinate-dependent projection of simple scalar temporal field interactions
2. Mathematical complexity arises from embedding temporal physics into 4D coordinate systems
3. Physical constants serve as unit conversion factors between natural and anthropocentric coordinates
4. General Relativity describes the geometric shadow cast by fundamental temporal field relationships

This perspective suggests that:

• Physics education should begin with temporal field concepts rather than geometric spacetime
• Research should focus on temporal field dynamics rather than spacetime geometry
• Unification with quantum mechanics becomes natural when both are recognized as temporal experience phenomena
• The universe's fundamental simplicity lies in temporal field interactions, not geometric relationships

The derivation reveals General Relativity not as the discovery of spacetime's geometric nature, but as an elaborate coordinate system for describing the simple reality of temporal experience fields interacting according to the basic rule: ϕ(r) = m/r.

Einstein's greatest insight was not that spacetime is curved, but that he found a mathematical framework sophisticated enough to accurately describe the coordinate projections of simple temporal field physics. The curvature was never in spacetime—it was in our mathematical description of temporal reality.

References

Einstein, A. (1915). Die Feldgleichungen der Gravitation
Schwarzschild, K. (1916). Über das Gravitationsfeld eines Massenpunktes
Weinberg, S. (1972). Gravitation and Cosmology
Will, C.M. (2014). The Confrontation between General Relativity and Experiment
Ashby, N. (2003). Relativity in the Global Positioning System