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Chapter 28: ψ-Hollow Core and Gravitational Drift

The Empty Heart of Matter​

Common sense suggests planetary cores must be dense, compressed by overlying mass. But Ψhē Cosmology reveals a startling possibility: collapse dynamics can create hollow cores—regions where the collapse field reverses, creating gravitational voids at the hearts of worlds. These hollows drift and evolve, making planetary interiors far stranger than imagined.

28.1 Hollow Core Formation​

Definition 28.1 (Core Reversal): A hollow core exists where: ∇2ψ<0\nabla^2\psi < 0

within radius r < R_hollow. Negative Laplacian indicates collapse divergence rather than convergence.

28.2 The Hollowing Mechanism​

Theorem 28.1 (Central Cavity): For sufficient rotation Ω, a cavity forms when: Ω2>4πGρcentral3\Omega^2 > \frac{4\pi G\rho_{central}}{3}

Centrifugal effects overcome central collapse.

Proof: Rotating collapse fields develop negative pressure at the center. When rotation exceeds critical value, this creates an expanding cavity until pressure balances at R_hollow. ∎

28.3 Gravitational Anomalies​

Hollows create unexpected fields:

Definition 28.2 (Interior Field): Inside a hollow core: g(r)=4πG3ρshell(r−Rinner)g(r) = \frac{4\pi G}{3}\rho_{shell}(r - R_{inner})

Gravity increases linearly from zero at the cavity center.

28.4 Core Drift Dynamics​

Hollows don't stay centered:

Theorem 28.2 (Drift Velocity): The hollow migrates at: v⃗drift=∇ψ×Ω⃗∣ψ∣2\vec{v}_{drift} = \frac{\nabla\psi \times \vec{\Omega}}{|\psi|^2}

Coriolis forces drive systematic displacement.

28.5 Oscillation Modes​

Hollows ring like bells:

Definition 28.3 (Cavity Modes): Normal frequencies: ωn=n(n+1)cs2Rhollow2\omega_n = \sqrt{\frac{n(n+1)c_s^2}{R_{hollow}^2}}

where c_s is sound speed. Seismic waves reveal hollow dimensions.

28.6 Magnetic Consequences​

Hollows affect dynamos:

Theorem 28.3 (Field Disruption): Magnetic field vanishes where: r<Rhollow−δskinr < R_{hollow} - \delta_{skin}

The cavity cannot support currents, creating a magnetic void.

28.7 Density Inversions​

Matter redistributes around hollows:

Definition 28.4 (Shell Densification): Density peaks at: ρmax=ρ0exp⁡(GMcs2Rhollow)\rho_{max} = \rho_0 \exp\left(\frac{GM}{c_s^2 R_{hollow}}\right)

Compressed shells surround low-density cavities.

28.8 Thermal Signatures​

Hollows alter heat flow:

Theorem 28.4 (Temperature Profile): Inside the cavity: T(r)=Twall(rRhollow)2/5T(r) = T_{wall}\left(\frac{r}{R_{hollow}}\right)^{2/5}

Radiative equilibrium replaces conduction.

28.9 Stability Conditions​

Not all hollows persist:

Definition 28.5 (Collapse Threshold): Cavities collapse when: Pexternal>2γRhollowP_{external} > \frac{2\gamma}{R_{hollow}}

where γ is surface tension. Small hollows are unstable.

28.10 Multi-Hollow Systems​

Planets can host multiple voids:

Theorem 28.5 (Void Interaction): Two hollows merge when separation: d<R1+R2+2λψd < R_1 + R_2 + 2\lambda_\psi

Creating complex interior geometries.

28.11 Observable Predictions​

Hollow cores leave signatures:

  1. Seismic Shadows: Waves cannot traverse cavities
  2. Moment of Inertia: Lower than solid sphere
  3. Magnetic Poles: Offset from rotation axis
  4. Free Oscillations: Anomalous frequencies
  5. Gravitational Harmonics: Non-standard J₂, J₄ coefficients

Each hints at interior voids.

28.12 The Hollow Earth Realized​

Science fiction imagined hollow worlds; Ψhē theory makes them possible. Not through fantasy but through rigorous collapse dynamics—rotation and field interaction naturally creating vast interior spaces. Planets become shells, their hearts empty yet structured, drifting voids shaping surface geology and magnetic fields.

The cosmos builds its worlds not solid but hollow—cathedral spaces hidden within.

Next: Chapter 29: ψ-Auroras as Collapse Rebound Fields