Spin-2 Structure of the Gravitational Mediator: Phase-Stable Pair Emission and Quadrupole Radiation in the 0-Sphere Model
Year: 2026
Type: preprint
Abstract: [58 Structural — Resolution of Paper #49 Open Task (vi)] How can a spin-2 gravitational structure — the polarization content of a graviton — emerge from the internal motion of a single electron? In the 0-Sphere Model — an ongoing programme that describes the electron as two thermodynamic kernels joined by a circulating, light-like energy structure, with Zitterbewegung treated as a real internal oscillation — a single emission event carries only spin-1 polarization data and cannot, by itself, transform as a rank-2 tensor. The decisive observation of this paper is that the electron's internal oscillation produces not one but two emission events per Zitterbewegung cycle, half a cycle apart; together they form a pair whose relative phase is structurally fixed by the internal cycle, so that their combined polarization is a genuine rank-2 symmetric tensor carrying helicity \(\pm 2\). The spin-2 character of gravitational radiation is thus identified as a structural consequence of the \(\pi\)-periodicity of the internal motion, rather than being imposed as an external assumption. — Core Equations — Two ejection events per cycle (the central observation):\[ \Omega_T(\theta) \propto \sin(2\theta), \qquad E_{\text{ph}}(\theta) = \tfrac{1}{2}E_0\sin^2\theta \] The Thomas angular velocity \(\Omega_T\) carries a \(\sin(2\theta)\) factor of period \(\pi\), so its magnitude reaches an extremum at both \(\theta = \pi/2\) and \(\theta = 3\pi/2\); the photon-sphere kinetic energy \(E_{\text{ph}}\) reaches the ejection threshold \(E_0/2\) at the same two phases. Each Zitterbewegung cycle therefore produces two ejection events, separated by \(T_{\text{ZB}}/2\). The phase-stable pair as a rank-2 symmetric tensor:\[ h_{\mu\nu} \propto \varepsilon_{1\mu}\varepsilon_{2\nu} + \varepsilon_{1\nu}\varepsilon_{2\mu} \] The two emitted fragments carry polarization vectors \(\varepsilon_1\) and \(\varepsilon_2\); their combined two-index object transforms as a rank-2 symmetric tensor, whose trace-free part carries helicity \(\pm 2\) exclusively — the polarization content of the graviton. Quadrupole moment and radiated power:\[ Q_{ij}(t) = \frac{E_0\lambda_C^2}{c^2}\cos(\omega_{\text{ZB}}t)\left(\hat{e}_{x,i}\hat{e}_{x,j} - \tfrac{1}{3}\delta_{ij}\right), \qquad P_{\text{GW}} \sim 10^{-70}\,\text{W} \] The two-kernel mass-energy asymmetry generates a quadrupole moment oscillating at \(\omega_{\text{ZB}}\); its third time derivative drives gravitational radiation through the standard general-relativistic quadrupole formula, giving an order-of-magnitude power of \(10^{-70}\,\text{W}\) per electron. — Key Contributions — Two ejection events per ZB cycle: the \(\sin(2\theta)\) factor in the Thomas angular velocity has period \(\pi\), so the electron produces a second ejection event at \(\theta = 3\pi/2\) in addition to the first at \(\theta = \pi/2\), separated by half a Zitterbewegung period. A single-event picture captures only half of the per-cycle structure. The phase-stable pair as a rank-2 symmetric tensor: the combined polarization tensor \(h_{\mu\nu} \propto \varepsilon_{1\mu}\varepsilon_{2\nu} + \varepsilon_{1\nu}\varepsilon_{2\mu}\) carries the helicity \(\pm 2\) content of a graviton. A single spin-1 event cannot transform as a rank-2 tensor; the pair can. Phase stability from the \(4\pi\) boundary condition: the temporal separation between the two events cannot drift across cycles, because the internal energy returns to a definite single-kernel eigenstate at every half-period \(\theta = n\pi\) (Paper #1). This discrete phase renormalization replaces conventional quantum-optical coherence, which would conflict with the non-closing trajectory the model requires. Three structures unified by \(\sin(2\theta)\): the Thomas angular velocity, the timing of the two ejection events, and the oscillation of the quadrupole moment all trace to a single \(\sin(2\theta)\) factor. The \(\pi\)-periodicity is a structural necessity of any model deriving spin-2 from a Zitterbewegung-type oscillation. Quadrupole formula reproduction: the two-kernel mass-energy oscillation supplies the quadrupole source, and its third time derivative drives \(P_{\text{GW}}\) through the standard general-relativistic quadrupole formula — no dynamical postulate beyond the Hamiltonian identity of Paper #1 is required. Candidate \(\ell = 2\) identification: the quadrupole moment has the \(Y_2^0\) angular symmetry, making the phase-stable pair a candidate realization of the \(Y_\ell^m\) imprinting framework of Paper #57 at \(\ell = 2\). Full verification is recorded as an open task. — Series Position — This paper is the second in the five-paper sequence #57–#61 developing the microscopic gravitational mechanism of the 0-Sphere Model. Paper #57 identifies what is emitted — a \(Y_\ell^m\) excitation mode; the present paper identifies the spin-2 structure of that emission; Paper #59 establishes the bound-state condition under which the emission becomes directed and observable; Paper #60 situates these results within the Berry–Synge–Bonnet–Myers triangle. The paper resolves open task (vi) of Paper #49 — the rank-2 tensor proof for the ejected photon-sphere fragment — and builds on the \(\pi\)-periodic Thomas angular velocity of Paper #48 and the \(4\pi\) internal-cycle boundary conditions of Paper #1. The emission-axis assignment — the pair is ejected along \(\hat{e}_x\), the velocity direction — provides a first-principles basis for the gravity / electromagnetism directional decomposition proposed in Paper #52. The paper is presented as a Structural Proposal. Its principal open tasks are the full helicity-2 decomposition of the pair within the SU(2) holonomy framework of Paper #45, the derivation of Newton's constant \(G\) from the pair emission rate, and the verification of the \(\ell = 2\) identification with Paper #57. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 34 Geometrical Confinement: Emergent Gravitational Dynamics 10.5281/zenodo.18437010 45 Open-Path Spinorial Transport: Holonomy Sufficiency 10.5281/zenodo.18950974 46 Geometric Origin of the One-Half Factor: Thermal PE Floor 10.5281/zenodo.19010945 48 Geometric Origin of \(g=2\): U(1) Fiber Bundle 10.5281/zenodo.19227518 49 Photon-Sphere Fragmentation as a Gravitational Mediator 10.5281/zenodo.19393391 50 Rotation from Scalar Oscillation: Chirality and Helicity 10.5281/zenodo.19482145 52 Geometric Origin of the Weak Equivalence Principle: \(\lambda_C\) Cancellation Theorem 10.5281/zenodo.19583032 56 The Framework Boundary: Local-Field vs Directed-Path Descriptions 10.5281/zenodo.19900974 57 Spherical Harmonic Imprinting and the Physical Identity of Phase Information 10.5281/zenodo.19932176 59 Proton-Trapped Electron as the Generator of Spin-2 Gravitational Radiation 10.5281/zenodo.19932907 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi
Author: Hanamura, Satoshi
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7161414701