Spin Induced Geometry: Emergence of Metric and Torsional Sectors from Spinor Source

Abstract: We present a geometric framework in which both metric and torsional degrees of freedom emerge dynamically from spinor currents, without being postulated as fundamental properties of the affine connection. The fundamental dynamical variable is a rank-three field carrying local Lorentz indices, governed by a massive Klein--Gordon equation sourced by fermionic spin currents. Its projection onto spacetime indices yields a rank-two tensor with no definite symmetry; the symmetric and antisymmetric sectors define, respectively, an effective spin-induced metric and the torsional degrees of freedom. Both sectors are massive and Yukawa-suppressed, ensuring decoupling from long-range gravitational dynamics. Unlike Einstein--Cartan theory, torsion here is propagating rather than algebraically constrained. A key consequence is that spinless test particles follow geodesics of the effective metric and are therefore indirectly sensitive to spin currents through the emergent geometric structure~ -- ~a mechanism absent in both standard General Relativity and Einstein--Cartan theory. The spinorial structure of the source is analyzed across three regimes: general Dirac, Weyl, and Majorana fermions, each giving rise to a distinct geometric phase. In the Majorana limit, the geometry becomes purely axial-torsional, admitting topologically non-trivial configurations such as vortices and Skyrmion-like structures, which emerge dynamically from the spinorial source.