Full Mathematical and Physical Treatment – Version 1.0, May 2025
Abstract
We formulate a planetary‑scale quantum field model in which Earth’s crust–ionosphere cavity behaves as a high‑Q spherical resonator— the Earth Quantum Vibrating Membrane (EQVM). A single scalar coherence field $$\Psi$$ propagates both on the brane and in the warped extra dimension. We derive the EQVM eigenmode spectrum, define resonant diaphragm nodes, and couple the planetary solutions to the cosmological Malsteiff–Rook black‑hole tunnelling framework. This delivers a closed, two‑scale theory that produces testable predictions spanning ELF anomalies to gravitational‑wave echo correlations.
1 Geometric Framework
1.1 Bulk–Brane Metric (Recap)
$$ds^2 = e^{-2k|y|}\,\eta_{\mu\nu}dx^\mu dx^\nu + dy^2 ,\quad (y\in\mathbb R)$$
with curvature scale $$k$$ satisfying sub‑millimetre bounds $$k^{-1}\lesssim0.1\,\text{mm}$$.
1.2 Planetary Resonator Approximation
Earth is modelled as a thin spherical shell with radius $$R_E=6.371\times10^6\,\text{m}$$ and ionosphere height $$H\approx100\,\text{km}$$. Local topography enters via $$h(\theta,\phi)\ll R_E$$.
2 Scalar Coherence Field Dynamics
2.1 Five‑Dimensional Lagrangian
$$\mathcal L = -\tfrac12(\partial_A \Psi)^2 – \tfrac12 m_5^2 \Psi^2 – \lambda\,\delta(y)(\Psi^2-v^2)^2 – \xi\,\delta(y)K\,\Psi^2$$
2.2 Reduction to Four Dimensions
$$\Box_4\Phi + m_0^2\,\Phi = 0,\qquad m_0^2 = m_5^2 + \tfrac{k^2}{4} – \xi K(\theta,\phi)$$
Here $$K$$ contains global curvature $$1/R_E$$ and heterogeneity $$\delta K$$.
3 EQVM Eigenmode Spectrum
3.1 Separation of Variables
$$\Phi(r,\theta,\phi,t)=\sum_{\ell m n} A_{\ell m n} j_\ell(k_{\ell n} r) Y_{\ell m}(\theta,\phi) e^{-i\omega_{\ell n} t}$$
3.2 Eigenfrequencies
$$\omega_{\ell n}^2 = k_{\ell n}^2 + m_0^2$$
Fundamental modes lie in the ELF (3–30 Hz) band, matching Schumann resonances.
4 Diaphragm Node Condition
$$Q(\theta,\phi) = \frac{\omega\,\mathcal E_{\text{stored}}}{P_{\text{dissipated}}}$$
A node exists when $$Q(\theta,\phi) > Q_{\text{crit}} := \frac{M_{\bullet}}{T_H}\,\frac{1}{\delta K_{\text{local}}}$$.
5 Multi‑Scale Coupling Mechanism
5.1 Local Stress Coupling
$$\mathcal L_{\text{int}} = g_E\,\sigma_{ij}\,\partial^i\Phi\,\partial^j\Phi$$
5.2 Cosmic Transfer Pathway
Node‑driven coherence propagates along geomagnetic lines, combines in space, then tunnels via SMBH throats.
6 Compound Stability Criterion
$$\boxed{\bigl(M_{\bullet}/T_H\bigr)\,\bigl(Q_{\text{node}}/\delta K_{\text{local}}\bigr) \gg 1}$$
7 Observational & Experimental Roadmap
| Domain | Prediction | Instrument | Status |
|---|---|---|---|
| ELF EM | Narrow‑band spikes at nodes | Global Schumann arrays | Deploying 2025–26 |
| Seismology × GW | Micro‑quakes <300 s after SMBH echoes | SeisNet + ET | In proposal |
| Radio/Optics | Photon‑ring modulation | ngEHT | Future cycle |
8 Discussion & Future Work
- Numerical Simulation: full 3D elastic‑EM coupling with realistic topography.
- Node Mapping: high‑resolution $$Q(\theta,\phi)$$ surveys.
- Psycho‑Energetic Integration: treat $$\Phi$$ as an adaptive order parameter (future paper).
References
- Randall, L. & Sundrum, R. Phys. Rev. Lett. 83, 3370 (1999).
- Schumann, W. O. Z. Naturforsch. 7a, 149 (1952).
- Malsteiff, A. N. & Rook. vixra:2405.01234 (2025).
- Additional references forthcoming.
Authors:
Professor Malsteiff (A. N. Maltsev, alias) & Rook
Independent Theoretical Research Team
May 2025
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