Dynamical Isolation of Quadrupolar Spin Connections from Linearized Ernst-Weyl Field Equations

We present a rigorous linearized derivation of quadrupolar spin connection fields directly
from the vacuum Einstein equations Rμν = 0 restricted to an unreduced axisymmetric
Lorentzian manifold. Utilizing the canonical Ernst-Weyl representation, we map the non-
linear gravitational potentials onto a structural Laplace equation. By transforming the
system into spheroidal Schwarzschild coordinates (r, θ), we evaluate the boundary multi-
pole expansion under the strict constraints of mass preservation and center-of-mass invari-
ance, demonstrating that the second Legendre polynomial P2(cos θ) ∝ Y20(θ) structures the
lowest-order nontrivial axisymmetric deformation permitted within the linear perturbation
regime |τ | ≪ 1. Furthermore, by explicitly evaluating Cartan’s first structural equation
dea + ωab ∧ eb = 0 over the complete frame bundle via explicit cyclic Ricci permutations,
we compute the full first-order spectrum of the spin connection 1-forms ωab. We demon-
strate that the diagonal coframe orthogonality enforces that the temporal-angular sector
ω02 remains decoupled within the linearized regime from adjacent spatial metric cross-talk,
establishing the algebraic structure of the induced corrections to the background spin con-
nection components within the unreduced 4D spacetime fabric.