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Celestial Trajectories
PDF Visuals Suite
OCTOBER 2026 • RESEARCH TREATISE • FLAT MINKOWSKI KINEMATICS

Recovery of Relativistic Celestial Trajectories as an Effective Field Limit of Internal Momentum Flux

A Flat-Space Kinematic Account of the Classical Tests of Relativistic Gravitation

Corley Kinnane • corley.kinnane@gmail.com

The Core Physical Ontological Shift

General Relativity attributes gravitational trajectory deflection to the pseudo-Riemannian curvature of a four-dimensional spacetime manifold, and so requires two kinds of entity: matter and geometry.

In this paper, the classical single-centre tests of relativistic gravitation are recovered on an uncurved flat Minkowski background (ℝ1,3, ημν) with only one kind of entity: momentum flux, formulated from three short axioms, two stated rules, and one calibration.

• No Coordinate Metric Curvature • Microscopic Sagitta Drift Generator • Empirical Equivalence Across All Current Tests
Download Research Paper (PDF • 50 Pages) Explore Interactive Visuals Suite Diagram Explanations ↓
EMPIRICAL VERIFICATION • RIGOROUS THEORETICAL RECOVERY

The Classical Tests of Relativistic Gravitation

Recovered from a single entity—momentum flux—on an uncurved Minkowski background without introducing metric field curvature.

Mercury Precession Planetary Orbits

43.0″ / Century Anomalous Perihelion Advance

Laplace's orbital aberration paradox is resolved through velocity-dependent retarded potential barrier cancellation. Solving the Binet orbit equation yields the exact post-Newtonian precession Δφ = 6πGM / [c²a(1 − e²)].

Target: 42.98″ ± 0.04″ 100% Match
Achromatic Light Deflection Starlight Bending

1.75″ Solar Limb Gravitational Starlight Bending

Light traverses null momentum flux streams. The dual-component arrival tensor integrates to the full relativistic factor of 2: θ = 4GM / (c²b), avoiding the historical failure modes of scalar flat-space models (γ = 1).

Eddington Benchmark θ = 4GM / c²b
Shapiro Delay Radar Echo

Logarithmic Signal Transit Delay

Radar propagation through the radial momentum debt field experiences an effective velocity retardation veff ≈ c(1 − 2GM / c²r), integrating precisely to the standard logarithmic Shapiro echo delay.

Cassini Experiment γ - 1 < 2.3 × 10−5
Gyroscopic Motion Gyroscope Precession

Geodetic Precession & Frame Dragging

Circulating momentum flux around rotating central bodies produces asymmetric lateral arrival flux, inducing de Sitter geodetic precession (6.60″/yr) and Lense-Thirring frame-dragging (0.039″/yr) measured by Gravity Probe B.

Gravity Probe B Exact Agreement
Clocks & Geoid Clock Cadence

Uniform Cadence on the Terrestrial Geoid

Atomic clock cadence is governed by the matter transit loop speed budget. Internal circulating speed adjusts under external flux streaming debt, reproducing gravitational redshift and equipotential clock synchronization.

Redshift & Clocks Δf / f = ΔΦ / c²
Empirical Demarcation Strong Fields

No Event Horizons or Physical Singularities

In ultra-strong fields, matter decelerates in internal circulation cadence and compresses asymptotically without freezing. Physical departures from GR are strictly demarcated to near-limb dynamic softening and non-singular collapse.

Strong Fields Non-Singular Limit
LIVE INTERACTIVE KINEMATIC SIMULATION SUITE

Circulating Momentum Flux & Relational Triad Suite

Experience the 5 dynamic models illustrating the flat-space derivation. Includes full real-time interactive controls, vector inspection, and an automated continuous tour.

Automated 60-Second Tour Mode Hands-Free

The suite includes an automatic playback feature that cycles through all 5 visual diagrams in 60-second intervals with reset synchronization, ideal for presentations or continuous self-guided study.

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Full Screen Suite
Diagram 1: Photon (m=0) locks to sector leading arc; massive particle (m>0) experiences angular lag corridor.
Interactive: Use sliders to modulate sector angles • Click anywhere on canvas to inspect vectors
PHYSICAL MECHANISMS • PAPER CONCORDANCE

Concordance: The Five Kinematic Diagrams

Each diagram in the visual suite directly maps to specific operational derivations in the treatise. Below is the technical description of each physical mechanism.

Diagram 1 Treatise Reference: Branch A §5.1, §5.4, §12 Photon Arc vs. Inertial Mass Deviation

Single Sector Momentum Flux • Arc Rotation & Curl

Considers a single isolated circulating momentum sector emitted from a source constituent. Seen from a target at distance L, the circulating flux traces a curved arc with radius R and angular opening Ω.

Null Particle (m = 0, Photon)

Propagates strictly at speed c and locks to the leading wavefront along the arc of the sector. It experiences pure tangential deflection along the arc without internal velocity lag.

Massive Particle (m > 0)

Sub-luminal speed (v < c) forces an angular lag inside the sector corridor. The particle continuously drifts away from the arc toward the tangent line, exposing the uncompensated rotational curl of an isolated sector.

Key Result: Sagitta h ≡ 0 in an isolated sector. Net inward gravity requires parity closure.
Live Simulation 1

Examine photon lock vs massive lag with variable sector width Ω.

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Diagram 2 Treatise Reference: §4, §6.1, Appendix A.11 Microscopic Parity Cancellation & Sagitta Accumulation

Paired Parity Cancellation & Monotonic Sagitta Accumulation

When a source consists of paired counter-circulating sectors (Clockwise and Counter-Clockwise), a remarkable geometric cancellation occurs. The tangential momentum components from the opposing sectors are equal in magnitude but opposite in direction:

Tangential Parity Cancellation:
+vtan − vtan = 0 (exact lateral parity cancellation)
Monotonic Inward Radial Residual:
haccum = R[1 − cos(Δθ / 2)] ≈ ½ R(Δθ)² > 0

While the transverse kick cancels to zero by symmetry, the geometric sagitta (the arc departure) does not cancel—both sectors deposit a strictly inward radial displacement. This singular sagitta is the microscopic seed of Newtonian attraction, derived as a necessary residual of delayed circular motion.

Live Simulation 2

Observe exact lateral cancellation leaving pure inward radial sagitta drift.

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Diagram 3 Treatise Reference: §6.2, §14.3, §14.4 Two Sources at Shared Target • Linear Vector Additivity

Dual Stationary Sources & Superposition of Relational Readout

General Relativity famously struggles with linear field superposition because the Einstein field equations are nonlinear (Gμν = 8πTμν). Diagram 3 demonstrates how multi-source fields interact in the internal momentum flux framework.

Vector Summation at Target T:
Δtotal = Δ⃗1 + Δ⃗2 = 2 haccum cos(α / 2) ûCM

Two separate sources (S1 and S2) each project counter-circulating sector pairs toward target T. Four crossing sectors interact at T. Lateral odd parity cancels symmetrically, while the inward sagitta displacements sum strictly as linear vectors toward the center-of-mass axis without requiring metric back-reaction or field non-linearities.

Live Simulation 3

Modulate source separation angle α and verify center-of-mass alignment.

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Diagram 4 Treatise Reference: §6.4, §10, Table 12 3D Relational Triad • Spherical Waves & Concurrent Median Inflow

3D Dynamical Relational Triad & Multi-Body Closure

Section 6.4 formulates the directed relational interaction ledger for an assembly of N constituents: Cdirected = ∑j=1N ∑k≠jN (j → k) = N(N − 1) directed channels. For N = 3 (a triad), there are 3 physical pairs yielding exactly 6 directed channels.

16 Alternating Sectors

8 Red CW and 8 Blue CCW sectors propagate outward as expanding spherical wavefronts across 3D space, demonstrating bilateral conservation without double counting.

Concurrent Median Inflow

Where counter-circulating fronts intersect, magenta interference tracks the emergent inward flow toward the triangle's geometric centroid, stabilizing multi-body orbital dynamics.

Live Simulation 4

Orbit the 3D camera to inspect spherical wave interference and triad centroid inflow.

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Diagram 5 Treatise Reference: §9, §11 (The Matter Interface), §11.6 Macroscopic Emergence • Crystalline Lattice & Cavendish Balance

Macroscopic Simulation & The Cavendish Emergence

While Diagrams 1–4 establish the microscopic kinematic cause, Diagram 5 demonstrates the bridge to macroscopic laboratory gravity (the Cavendish torsion balance). Particles carry the flux; atoms organize it into bound crystalline lattices.

Statistical Averaging of Chaotic Wobble: At the atomic scale, individual constituents undergo high-frequency thermal and quantum oscillation. When bound into a rigid lattice, the incoherent micro-wobbles cancel out via statistical phase averaging.

The Coherent Inward Attractor: What survives the ensemble average is a continuous, strictly directional 1/r² Newtonian acceleration between macroscopic test masses, demonstrating how macroscopic classical gravity emerges directly from delayed microscopic circulation.

Interactive Feature: Click and drag any crystal to disrupt the system and observe the restorative emergent gravitational pull.
Live Simulation 5

Interactive p5.js canvas with draggable masses and live torsion readout.

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THEORETICAL ARCHITECTURE • TREATISE SUMMARY

Architecture: Two Branches and One Kinematic Operator

The framework replaces metric curvature with an internal momentum flux ledger organized into two complementary branches.

Branch A §5 • The Green's Function

Projection & The Microscopic Generator

Describes how one source constituent projects momentum flux across the Minkowski substrate to a field point. Contains all of emission and the coupling half of reception.

  • • Source-Centred Arcs: Confined circulating flux carries non-vanishing internal curl.
  • • The Singular Sagitta: Finite propagation delay induces an uncancelled radial drift deficit.
  • • Inverse-Square Geometric Scaling: Conserved flux spreading in 3D Euclidean space automatically scales as 1/r².
Branch B §10 • Composition & Relaxation

Target Response & Kerr–Schild Recovery

Describes the internal rearrangement and relaxation of a target body responding to incoming external flux, manifesting what General Relativity models as coordinate curvature.

  • • Relaxation is Additivity: The target's internal circulation adjusts to restore local momentum conservation.
  • • Kerr–Schild Form Emergence: The summed Green's function generates the effective single-source metric gμν = ημν + 2H lμlν.
  • • Equivalence of Inertia and Weight: Persistent internal transit loops enforce gravitational mass equals inertial mass.

Ontological Comparison

Physical Domain General Relativity (Einstein 1915) Internal Momentum Flux (Kinnane 2026)
Spacetime Ontology Curved pseudo-Riemannian manifold (M, gμν) Uncurved flat Minkowski spacetime (ℝ1,3, ημν)
Entities Present Two kinds: Matter (energy-momentum) and Metric Geometry One kind: Conserved circulating momentum flux
Attractive Acceleration Geodesic motion through curved coordinates Geometric sagitta deficit from delayed circulation
Superposition Non-linear (metrics cannot be added) Strict linear vector additivity at relational readout
Strong Field Singularity Spacetime breakdown / infinite density singularities Finite asymptotic compression; no event horizons or infinities
FULL TREATISE DOWNLOAD

Download the Research Paper (October 2026)

The complete 50-page manuscript containing the axiomatic foundations, full mathematical derivations of all classical tests, PPN parameter evaluations, and domain demarcation.

Current Release: celestial_trajectories_v56.pdf • 50 Pages • Version 56 (October 2026)

Download Research Paper (PDF)
BibTeX Citation October 2026 
@article{kinnane2026celestial,
  title={Recovery of Relativistic Celestial Trajectories as an Effective Field Limit of Internal Momentum Flux: A Flat-Space Kinematic Account of the Classical Tests},
  author={Kinnane, Corley},
  year={2026},
  month={October}
}
HISTORICAL FOUNDATION • 2025 ARCHIVE

The Original Site: Conceptual Feasibility Study (2025)

The Corley Momentum Flux Theory • 17 Comprehensive Sections & Interactive p5.js Simulations

Explore 2025 Study

1. Feasibility Exploration

The original 2025 website served as the conceptual feasibility study for inverting the vacuum into a pressurized momentum plenum. It established the core physical argument that gravitational attraction is an emergent output of delayed geometry.

2. Preserved & Fully Functional

The old site has been shifted into its own dedicated subfolder (conceptual_study/) with all relative CSS styles, JavaScript simulations, and table of contents preserved and fully viewable.

3. Full AI Context Available

Researchers and AI analysis agents can access the preserved 17 sections, including the complete text archive (full_context.txt) directly from the subfolder.

Path: /conceptual_study/index.html (alias /old_site/)
Download full_context.txt Enter 2025 Archive →