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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2506.01146 |
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| _version_ | 1866914418140381184 |
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| author | Alexa, Anton |
| author_facet | Alexa, Anton |
| contents | We introduce the scalar function $C(v)=π(1-v^2/c^2)$ as a conformal factor associated, within the model, with longitudinal Lorentz contraction. Extending $C(v)$ to a one-parameter family $C(v,τ)$, we construct a variational scalar conformal flow that drives the factor toward the equilibrium $C=π$ without singularities. The main result is an explicit algebraic decay law for the energy functional: $E(τ)\sim τ^{-1/2}$ for generic initial data and $E(τ)\sim τ^{-5/2}$ for the physical initial condition $C(v,0)=π(1-v^2/c^2)$. More generally, if the initial deviation vanishes as $v^n$ near $v=0$, then $E(τ)\sim τ^{-(2n+1)/2}$. This behavior is explained by the gapless continuous spectrum of the relaxation operator, whose spectral measure satisfies $dμ(k)\sim k^{-1/2}dk$ 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 $C=π$ as the unique conformal representative whose curvature invariants agree with those of the unit $S^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Variational Scalar Conformal Flow for Lorentz-Contracted Geometry: Algebraic Decay and Canonical Normalization Alexa, Anton Mathematical Physics Differential Geometry 53E20, 53C20, 57K30, 83A05 We introduce the scalar function $C(v)=π(1-v^2/c^2)$ as a conformal factor associated, within the model, with longitudinal Lorentz contraction. Extending $C(v)$ to a one-parameter family $C(v,τ)$, we construct a variational scalar conformal flow that drives the factor toward the equilibrium $C=π$ without singularities. The main result is an explicit algebraic decay law for the energy functional: $E(τ)\sim τ^{-1/2}$ for generic initial data and $E(τ)\sim τ^{-5/2}$ for the physical initial condition $C(v,0)=π(1-v^2/c^2)$. More generally, if the initial deviation vanishes as $v^n$ near $v=0$, then $E(τ)\sim τ^{-(2n+1)/2}$. This behavior is explained by the gapless continuous spectrum of the relaxation operator, whose spectral measure satisfies $dμ(k)\sim k^{-1/2}dk$ 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 $C=π$ as the unique conformal representative whose curvature invariants agree with those of the unit $S^3$. |
| title | A Variational Scalar Conformal Flow for Lorentz-Contracted Geometry: Algebraic Decay and Canonical Normalization |
| topic | Mathematical Physics Differential Geometry 53E20, 53C20, 57K30, 83A05 |
| url | https://arxiv.org/abs/2506.01146 |