On the Emergence of Kerr-Schild Geometry from Coherent Weyl Spinor Currents

Abstract

We analyse the conditions under which a Kerr–Schild metric structure [Formula: see text] emerges from a spacetime framework sourced by fermionic spin currents. The symmetric metric perturbation decomposes into an isotropic sector [Formula: see text] and an anisotropic sector [Formula: see text], where [Formula: see text] is the fermionic vector current. In the Weyl limit [Formula: see text], two conditions are simultaneously satisfied: the isotropic sector vanishes for coherent null configurations in which the integrated scalar contribution [Formula: see text] vanishes along the integral curves of the coherent null congruence, and the anisotropic sector collapses to a rank-one null structure via Fierz identities, which impose [Formula: see text]. Under the local coherence condition [Formula: see text], the non-local convolution with the Yukawa propagator reduces to the effective form [Formula: see text], with [Formula: see text] a scalar functional of the fermionic configuration. The metric then takes an effective Kerr–Schild form with [Formula: see text], formally expressed as a functional of the underlying spinor field. Within the same gradient expansion, the null congruence generated by [Formula: see text] is approximately geodesic and its transverse deformations are suppressed at leading order, consistently with the geometric properties of algebraically special Kerr–Schild spacetimes. The emergent geometry is exponentially screened beyond the characteristic scale [Formula: see text], where [Formula: see text] is the squared mass of the spin-induced sector, independent of the fermion mass. The construction establishes the algebraic rank-one null form of the effective metric response; it is not claimed that the resulting metric satisfies the vacuum or sourced Einstein equations, whose verification for specific profiles [Formula: see text] is left to future work.