Gravity: The Sagitta of Delayed Momentum

A Mechanical Re-Interpretation of Vacuum Energy and Gravity

Constructed and Copyright © 2025 by Corley Kinnane.

Abstract: Standard General Relativity describes gravity as the curvature of non-Euclidean spacetime. This unified work presents the Vacuum Momentum Transfer (VMT) Theory, which proposes that this curvature is an emergent, statistical phenomenon resulting from the conservation of momentum propagating through Einstein’s Vacuum Energy.

By adopting a perspective where all internal energy is treated as Momentum Flux, we reveal that the finite speed of propagation creates a geometric degradation—a Sagitta (or "arrow") of inward acceleration. This inversion of the causal chain resolves the major anomalies of modern cosmology (Dark Matter, Dark Energy, and the Hubble Tension) without postulating new particles.

1. The Perspective Shift: Interpreting Energy as Momentum

In relativity, energy and momentum are not distinct entities, but components of the same tensor. Einstein's field equations sum these components to describe curvature.

• The Euclidean Base: To understand the shape of this curvature mechanistically, I am permitted to choose a specific perspective: one where I interpret all the energy within an atom as Momentum Flux. This allows me to "flatten" the map onto a Euclidean grid.
• Not an Ontology, but a Map: I am not claiming that mass "does not exist." I am stating that for the purpose of mapping the interaction, I can treat the energy-momentum tensor entirely as flux.

2. The Seed of Motion: Rotational Flux & Symmetry

I do not challenge the Standard Model's description of the atom. Instead, I focus on the external flux an atom generates. I model this interaction as a balanced rotational exchange.

• Symmetry and Position: The symmetric, rotating flux gives the atom a definitive "sense of position" relative to its neighbors.
• Newton's Third Law: I strictly enforce Newton's Third Law. Every rotational influence is balanced by a counter-influence.

3. Local Conservation: The Spherical Distribution

The magnitude of the rotational nudge (α) is determined by strictly enforcing Local Conservation of Energy. To conserve total energy, flux density must decrease proportional to the surface area of a sphere (4πR²), necessitating the inverse square law.

4. The Mechanism: The Geometric Sagitta

Why does balanced rotation lead to attraction? The answer lies in the geometry of the interaction combined with the speed of light.

• The Arc vs. The Chord: When you rotate a vector by +α and then by -α, you do not return to the origin via a straight line. You trace an arc.
• The Sagitta: The vector sum leaves a tiny residual component pointing directly back to the center. This residue is the Sagitta. It represents the difference between the "inertial tangent" and the "relativistically delayed position."

5. Simulation vs. Reality: The Validity of the Discrete Proxy

This simulation uses discrete time steps. I acknowledge that the universe is not discrete in this specific way. However, using discrete time here is a deliberate and valid mapping technique.

• Statistical Equivalence: In a continuous universe with delayed momentum transfer, interactions suffer from phase accumulation. In a discrete simulation, interactions suffer from integration error. I argue that on average, across a large ensemble of atoms, the "error" profile of discrete time acts as a faithful proxy for the "phase accumulation" profile of delayed momentum.
• The "Fixed" Error: Standard simulations use symplectic integrators to erase this error, effectively simulating a universe where c = ∞. By allowing the discrete error to persist, I capture the physical consequences of finite signal speed without needing to simulate infinite continuous history.

6. Simulation 1: Mapping the Influence

In this first simulation, I visualize the Principle of Influence. I allow the atoms to update their positions directly based on the accumulated Sagitta of their neighbors. This "clumping" visual confirms that the geometric mechanism naturally produces a convergent, gravity-like attraction.

7. Mach's Principle and the Ensemble Effect

Gravity in this model is an emergent statistical property of the Ensemble. A distant object forms interaction triangles with every atom in a dense clump. While individual phases average out, the geometric Sagittas accumulate.

8. Foundational Axioms of VMT

The "Sagitta" mechanism relies on a physical medium to transfer the momentum. The VMT theory rests on five inescapable physical principles:

• Momentum is Fundamental: The conservation of linear and angular momentum (P^μ) is the primary law of the universe. Energy is treated as the time component of momentum.
• The Medium is Vacuum Energy: Space is not empty. It contains a physical medium—Vacuum Energy (Λ)—which is required to sustain momentum transfer. We do not postulate a "new" field; we describe the mechanical action of the existing Vacuum Energy.
• Locality & The c-Limit: Momentum transfer is not instantaneous. It propagates through the Vacuum Energy at the speed of light (c).
• Inertia is Emergent: Inertial mass (m) is not intrinsic. It is the resistance to momentum change, derived solely from the confinement of internal angular momentum (L_internal) within a bound particle (e.g., quarks/electrons).
• Gravity is a Statistical Residual: Gravity is the non-zero vector sum of momentum transfers from a remote source. It is "unblockable" because one cannot block the conservation of momentum.

9. The Mathematical Framework & The Photon

The theory is formally described by the Momentum Tensor (P^μν) within the Vacuum Energy.

(Momentum must be conserved within the Vacuum Energy)

The Photon and the Null Geodesic (The Factor of Two)

The specific behavior of light (bending twice as much as Newtonian predictions) is derived mechanically:

Massive Particles (Bound):

• Possess Binding Energy and Confinement.
• This confinement generates the "geometric error" that radiates Gravity (The Residual).
• This confinement generates Inertia, which resists the Vacuum Stress.
• Result: Minimal bending (Factor of 1).

Photons (Unbound):

• Possess Intrinsic Spin (s=1) but No Binding Energy and No Confinement.
• Because they are unbound, they do not radiate a momentum residual. They are pure travelers through the medium.
• Because they lack confinement, they possess No Inertia to resist the Vacuum Stress.
• Result: They are completely subject to both the density (P⁰⁰) and the full, unmitigated tension of the vacuum (P^ij). This forces them to follow the path of maximum distortion (Factor of 2).

10. The Three Pillars of Resolution

The VMT Theory resolves the "Dark" anomalies by attributing them to mechanical properties of the atom and the vacuum medium.

Dark Matter: The Variable G (η)

The Concept: The Gravitational Constant (G) is not universal. It is a local coupling factor (η) determined by the Nuclear Confinement (B_avg) and Atomic Asymmetry of the source matter.

The Mechanism: Atoms with low binding energy and high internal asymmetry (wobble) radiate the momentum residual more effectively into the Vacuum Energy.

The Chemical Signature:

• Neutral Hydrogen (H): Minimal confinement (B_avg ≈ 0) + High Proton Wobble (Unbalanced Electron) = Anomalously High G (η ≫ 1).
• Helium/Iron: High confinement (B_avg > 0) + Symmetric Core/Paired Electrons = Standard G (η ≈ 1).

The Resolution: The "missing mass" in galactic halos is an illusion caused by using the wrong G for the Hydrogen-rich halo gas.

Ionization Prediction: Neutral Hydrogen (H) (wobbling proton) gravitates stronger than Ionized Hydrogen (H⁺) (lone, symmetric proton). This matches observations where cold neutral halos show the anomaly, while hot ionized intergalactic plasma does not.

Dark Energy: Vacuum Tension

The Concept: Dark Energy is not a new substance. It is the intrinsic stress/tension of the Vacuum Energy itself.

The Mechanism: In the absence of matter, the Vacuum Energy maintains a base level of tension (negative pressure) to satisfy momentum conservation laws, driving cosmic expansion.

Cosmological Redshift: Momentum Dissipation

The Concept: Redshift is the "friction" of light.

The Mechanism: As photons travel through the Vacuum Energy, they continuously transfer an infinitesimal amount of momentum to the medium's stress field.

The Unification: The energy lost by photons (Redshift) is exactly the energy required to sustain the Vacuum Tension (Dark Energy).

11. Expanding the Concept: From Mapping to Physics

The first simulation successfully demonstrates that the geometry of rotational flux leads to attraction. However, simply moving atoms based on position updates is a kinematic map, not a dynamic physics simulation. To prove this theory is compatible with macroscopic reality, I must extend the concept to handle Momentum Transfer.

The Requirement for Momentum

In the real universe, energy is conserved. An atom cannot simply "move" because a geometric rule says so; it must be accelerated by a force, and that force must come from an exchange of momentum.

• Inertia: I must grant my atoms true inertia (linear momentum). They drift in straight lines unless acted upon.
• Impulse, not Displacement: The "Sagitta" calculated in Part 1 is not a shift in position, but an Impulse Vector added to the atom's momentum.

12. The Virtual Cavendish Experiment

To demonstrate this visually, I have constructed a virtual version of the Cavendish Experiment. Instead of a cloud of gas, I simulate macroscopic solid objects ("sugar cubes") formed by crystalline lattices.

• Solidity: The atoms are bound by internal electromagnetic-like springs. This allows the object to maintain its shape and disperse momentum internally as heat (vibration).
• No Fake Gravity: I do not program a gravitational force between the cubes. I only program the Rotational Flux between individual atoms.
• The Result: The accumulation of thousands of microscopic rotational impulses creates a net macroscopic attraction. The solids drift together, colliding and interacting according to Newton's laws.

13. Key Predictions and Tests

This theory moves beyond speculation by offering falsifiable predictions that distinguish it from the Standard Model.

The Solar System Evolution Test

Prediction: The Sun's effective gravity (G_Sun) is weakening over time as it converts high-G Hydrogen into low-G Helium.
Observable: Planetary orbits (and Earth's distance) should expand slightly faster than can be accounted for by mass loss alone.

The Exoplanet Comparative Test

Prediction: Comparing two stars of equal mass, the Younger Star (Hydrogen-rich) will exert a stronger gravitational pull than the Older Star (Helium-rich).
Observable: Exoplanets orbiting young stars should have tighter orbital periods than those around old stars of the same mass.

The Galactic Halo Test

Prediction: The Dark Matter anomaly must correlate with the chemical state of the gas.
Observable: Strong anomalies in cold, neutral Hydrogen (H) regions; weak anomalies in hot, ionized Hydrogen (H⁺) regions.

14. Conclusion: The Power of Inversion

The Vacuum Momentum Transfer Theory succeeds by accepting the reality of the vacuum. By acknowledging that Vacuum Energy is the active medium for momentum, we eliminate the need for:

• Gravitons (Gravity is emergent).
• Dark Matter Particles (Gravity is chemically variable).
• Dark Energy Fields (Vacuum stress is intrinsic).
• String Theory (Gravity is not a quantum force).

This framework restores a mechanical, causal logic to the universe, unifying the very small (atomic confinement) with the very large (galactic rotation) through the single, unblockable law of Momentum Conservation.

Simulation Source Code

This single-file application contains the code for both the Atomic Mapping and the Cavendish Flux simulations.