Helical Trajectory, Fixed-Endpoint Line Integrals, and the Emergence of the Spacetime Metric in the 0-Sphere Model
Year: 2026
Type: preprint
Abstract: [51 Essential] This paper presents a structural reconstruction of confinement and spacetime geometry within the 0-Sphere framework. The companion paper (#50) established that the orientation of the thermal geodesic in the internal phase space determines the electron/positron distinction as a Lorentz-invariant binary \( \chi = \operatorname{sign}\!\left(\frac{da}{dt}\right) \), confined to the transverse plane. The present paper extends that analysis to three dimensions, converts the planar rotation into a helix with two physically distinct velocity scales, establishes existence and uniqueness of the fixed-endpoint line integral via the Bonnet–Myers theorem, and proposes the spacetime metric as the second functional derivative of the resulting phase accumulation functional. — Core Results — Helical trajectory \[ x(\tau) = v_{\text{pitch}} \, \tau, \quad y(\tau) = \cos(\omega_{ZB} \tau), \quad z(\tau) = \sin(\omega_{ZB} \tau) \] Two velocity scales (no free parameters) \[ v_{ZB} \approx 0.04c \quad (\text{transverse, from } \gamma = 1 + a_e) \] \[ v_{\text{pitch}} \sim c \quad (\text{longitudinal, from } \Delta x \sim \lambda_C \text{ and } \omega_{ZB}) \] \[ \frac{v_{\text{pitch}}}{v_{ZB}} \approx 25 \] Bonnet–Myers confinement \[ \mathrm{Ric} \ge K \sim \lambda_C^{-2} \;\;\Rightarrow\;\; \mathrm{diam}(\text{photon sphere}) \le \pi \lambda_C \] Phase accumulation functional \[ \mathcal{S}(x_A, x_B) = \int_{\Gamma_{\mathrm{ext}}} \mathcal{M}_\mu \, dx^\mu \] Metric-emergence proposal \[ g_{\mu\nu}(x) = \left. \partial_{A\mu}\,\partial_{B\nu}\,\mathcal{S}_s(x_A, x_B) \right|_{x_A = x_B = x} \] (Synge reconstruction theorem applied to the symmetrized world function) — Key Contributions — Two-velocity-scale helix. The transverse Zitterbewegung speed \( v_{ZB} \approx 0.04c \) (from \( \gamma = 1 + a_e \)) and the longitudinal pitch speed \( v_{\text{pitch}} \sim c \) (from \( \Delta x \sim \lambda_C \)) are derived from the two-kernel architecture without free parameters. The ratio \( \approx 25 \) breaks the velocity uniformity of the Dirac–Hestenes helical model. Arcwise-connectivity prerequisite. \( S^0 = \{A, B\} \) is not arcwise connected; no line integral can be defined within it alone. The photon sphere \( S^n \) (\( n \ge 1 \)) supplies the minimal arcwise-connected embedding required. Confinement as a theorem. Applying the Bonnet–Myers theorem to the photon-sphere Ricci curvature \( \mathrm{Ric} \ge K \sim \lambda_C^{-2} \) yields \( \mathrm{diam}(S^n) \le \pi r \sim \lambda_C \), promoting the confinement domain \( \mathcal{D} \) of Paper #33 from an assumption to a geometric theorem. Kernel separation bounds. Bonnet–Myers provides the upper bound \( \pi \lambda_C \); the energy floor \( E_0/2 \) of Paper #46 provides the lower bound. Together: \[ 0 < \lvert x_A - x_B \rvert \le \pi \lambda_C \] without free parameters. Phase accumulation functional as Synge world function. \( \mathcal{S}(x_A, x_B) \) satisfies the structural axioms (coincidence vanishing, antisymmetry, correct mixed-derivative structure). Synge's reconstruction theorem then yields the metric directly. Metric-emergence proposal. \[ g_{\mu\nu} = \left. \partial_{A\mu}\partial_{B\nu}\mathcal{S}_s \right|_{\text{coincidence}} \] is verified in the flat-space limit (recovers \( \eta_{\mu\nu} \) to leading order). Derivation chain: Berry connection \( \rightarrow \) phase functional \( \rightarrow \) symmetrized world function \( \rightarrow \) Synge reconstruction. Internal vs. external observer. Co-rotating frame: \( g_{\text{internal}} = 2 \) (Dirac value, exact).Laboratory frame: \( g_{\text{external}} = 2(1 + a_e) = 2\gamma \). The anomalous magnetic moment is a special-relativistic observation effect, structurally identical to the muon lifetime extension. — Series Position — Paper #51 is the direct continuation of Paper #50 (companion) and builds on the line-integral programme of Papers #29–#31. It promotes a central assumption of Paper #33 to a theorem and descends one level below the derivative-order hierarchy of Paper #40, connecting the Berry connection to the spacetime metric via phase accumulation. The 0-Sphere Model series now spans 51 papers (2018–2026) deriving spin, anomalous magnetic moment, Zitterbewegung, confinement, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. — Key References (this paper) — # Title (abbreviated) DOI #50 Rotation from Scalar Oscillation [companion] 10.5281/zenodo.19482145 #31 Line Integrals as Fundamental Observables 10.5281/zenodo.18203433 #30 From Curvature to Connection 10.5281/zenodo.18135855 #29 Spinorial Phase Accumulation along Thermal Geodesics 10.5281/zenodo.18067760 #33 Geometrical Confinement: Rest Mass and Zitterbewegung 10.5281/zenodo.18356895 #40 On the Derivative-Order Mismatch 10.5281/zenodo.18736670 #24 Thermal Geodesics in the 0-Sphere Model 10.5281/zenodo.17765349 #46 Geometric Origin of the One-Half Factor 10.5281/zenodo.19010945 #10 Redefining Electron Spin and AMM 10.5281/zenodo.17764997 #1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 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: Hananamura, Satoshi
Language: English
Subfield: Astronomy and Astrophysics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7153823732