The Impedance of Space II: Deriving Newtonian Gravity from Vacuum Hydrodynamics

Abstract We present a derivation of Newtonian gravity and the Poisson equation from the properties of the physical vacuum, characterized by a fundamental impedance Z0 ≈ 377 Ω. Building on the "Stiff" superfluid phase established in previous works (Papers 1–3), we demonstrate that in high-gradient stellar regimes, the vacuum undergoes a phase transition to a "Liquid" state. By modeling fundamental particles as topological defects (solitons) that act as sinks for vacuum energy, we derive the displacement field D directly from a conservation law analogous to Gauss's Law. We address the Hierarchy Problem by proposing that individual atoms act as "needles" that slip through the superfluid vacuum without engaging its full impedance, whereas macroscopic bodies engage the "Active Area" required to generate bulk gravity. We show that Newton's constant G is not fundamental, but is a composite parameter expressible in terms of the cosmological Hubble radius RH and the baryonic surface-density threshold Σψ. Finally, we derive the Schwarzschild metric's optical path predictions by treating the vacuum density variation as a dielectric response, ensuring full consistency with Solar System relativistic tests. Conceptual Overview: The Cosmic Fabric "Gravity is not a fundamental force. It is the pressure gradient of the vacuum reacting to the presence of matter." In this paper (Paper 4 in the series), we answer three fundamental questions: 1. Why is gravity so weak for atoms? (The "Needle" Analogy) We solve the Hierarchy Problem using hydrodynamics: The Atom (Needle): An atom is like a needle dropped into a superfluid ocean. It is too small to push the fluid effectively, so it slips through with almost no drag. This is why gravity is weak at quantum scales. The Star (Paddle): When atoms aggregate into a star, they act like a giant paddle. They engage the full "Impedance" (resistance) of the vacuum. This creates the massive pressure gradient we feel as weight. 2. Why does Newton's Law work here, but fail in Galaxies? Space has phase transitions, just like water turning to ice. We calculate that the Solar System (8.2 kpc from the Galactic Center) sits exactly at the transition boundary: Stiff Phase (Galactic Outer Regions): The vacuum is rigid. It resists bending, creating the extra force we mistake for "Dark Matter." Liquid Phase (Solar System): Near a star, the vacuum saturates and "melts." It behaves like a liquid obeying Pascal's Law, creating the inverse-square law of Newtonian Gravity. 3. Where does "Big G" come from? We show that Newton's Constant (G) is not a random number. It is determined by the size of the Universe and the stiffness of the vacuum: G ≈ c² / (4π² RH Σψ) Where: c: Speed of Light RH: Hubble Radius (Size of the Observable Universe) Σψ: Vacuum Surface Density (0.286 kg/m²) This formula predicts G ≈ 6.1×10⁻¹¹, which matches the measured value (6.67×10⁻¹¹) to within 10%. This implies that gravity is getting slightly weaker as the universe expands, which may explain Dark Energy. Series ContextThis is Paper 4 (Derivation) in the Impedance of Space series. Phase I: Galactic Dynamics Paper 1 (Discovery): Empirical Evidence of Baryonic Thresholds (DOI: 10.5281/zenodo.18035640) Paper 2 (Observation): Empirical Baryonic Surface-Density-Dependent Resonances (DOI: 10.5281/zenodo.18110723) Paper 3 (Mechanism): The Impedance of Space: Emergent Galactic Dynamics (DOI: 10.5281/zenodo.18319014) Phase II: Universal Gravitation Paper 4 (This Work): The Impedance of Space II: Deriving Newtonian Gravity from Vacuum Hydrodynamics. Paper 5 (Forthcoming): The Ultra-High-Density Frontier - Precision Tests across White Dwarfs and Neutron Stars. Paper 6 (Forthcoming): The Complete Vacuum Impedance Theory - Grand Unification of Galactic, Stellar, and Quantum Dynamics. Validation CodeThe Python scripts validating the derivation of G are available at: GitHub Repository