A Variational Scalar Conformal Flow for Lorentz-Contracted Geometry: Algebraic Decay and Canonical Normalization

Abstract

We introduce the scalar function $C(v)=\pi(1-v^2/c^2)$ as a conformal factor associated, within the model, with longitudinal Lorentz contraction. Extending $C(v)$ to a one-parameter family $C(v,\tau)$, we construct a variational scalar conformal flow that drives the factor toward the equilibrium $C=\pi$ without singularities. The main result is an explicit algebraic decay law for the energy functional: $E(\tau)\sim \tau^{-1/2}$ for generic initial data and $E(\tau)\sim \tau^{-5/2}$ for the physical initial condition $C(v,0)=\pi(1-v^2/c^2)$. More generally, if the initial deviation vanishes as $v^n$ near $v=0$, then $E(\tau)\sim \tau^{-(2n+1)/2}$. This behavior is explained by the gapless continuous spectrum of the relaxation operator, whose spectral measure satisfies $d\mu(k)\sim k^{-1/2}dk$ near $k=0$. As an application, within the conformally homogeneous class of compact simply-connected $3$-manifolds with constant positive background curvature, the flow acts as a canonical normalization mechanism selecting $C=\pi$ as the unique conformal representative whose curvature invariants agree with those of the unit $S^3$.