Phase Cycles and Lorentz-Dilation: An Action-Groupoid Extension of a Lightlike Functor
Fuente:
Zenodo
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Recurso digital |
| Publié: |
Zenodo
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866902324033617920 |
|---|---|
| author | Meisner, Edward |
| author_facet | Meisner, Edward |
| contents | <p>This paper presents the Lorentzian completion of the phase‑cycle/lightlike functor introduced in <em>Paper #1</em>. The earlier construction defined a category of phase cycles Pθ, a category Sc of lightlike intervals along a single null ray, and a functor Fθ:Pθ→Sc that is an isomorphism of thin groupoids. That functor is inherently limited to one null direction and pure rescaling morphisms.</p> <p>The present work introduces the minimal categorical enrichment needed to extend that construction to <strong>all nonzero Lorentzian interval vectors</strong>. Let (V,q) be a real 1+1-dimensional Lorentzian vector space and let</p> <div> <div>G=R>0×SO+(1,1)</div> </div> <p>act by positive dilation and Lorentz transformation. The source is enriched by adding a Lorentzian direction variable u∈V∖{0} and quotienting the redundant scale in the pair (Δθ,u). This yields the quotient space</p> <div> <div>Xθ=((0,∞)×(V∖{0}))/ ∼,(Δθ,u)∼(rΔθ,u/r),</div> </div> <p>and the enriched phase‑cycle category Pθ=Xθ//G. The target is the Lorentz‑dilation action groupoid</p> <div> <div>SLor=(V∖{0})//G.</div> </div> <p>The main result is that the functor</p> <div> <div>Fθ([Δθ,u])=ΔθΔθPl u,Fθ(λ,Λ)=(λ,Λ),</div> </div> <p>is an <strong>isomorphism of action groupoids</strong></p> <div> <div>Pθ ≅ SLor.</div> </div> <p>The lightlike functor of Paper #1 is recovered as the <strong>pure‑rescaling null‑ray slice</strong> of this enriched construction. The result is purely structural: no physical constants, dynamical laws, or curvature data are derived.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20316224 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Phase Cycles and Lorentz-Dilation: An Action-Groupoid Extension of a Lightlike Functor Meisner, Edward category theory action groupoids Lorentzian geometry Lorentz-dilation phase cycles thin groupoids null rays lightlike intervals timelike and spacelike intervals equivariant functors Lorentz transformations 1+1 dimensional spacetime structural mathematics <p>This paper presents the Lorentzian completion of the phase‑cycle/lightlike functor introduced in <em>Paper #1</em>. The earlier construction defined a category of phase cycles Pθ, a category Sc of lightlike intervals along a single null ray, and a functor Fθ:Pθ→Sc that is an isomorphism of thin groupoids. That functor is inherently limited to one null direction and pure rescaling morphisms.</p> <p>The present work introduces the minimal categorical enrichment needed to extend that construction to <strong>all nonzero Lorentzian interval vectors</strong>. Let (V,q) be a real 1+1-dimensional Lorentzian vector space and let</p> <div> <div>G=R>0×SO+(1,1)</div> </div> <p>act by positive dilation and Lorentz transformation. The source is enriched by adding a Lorentzian direction variable u∈V∖{0} and quotienting the redundant scale in the pair (Δθ,u). This yields the quotient space</p> <div> <div>Xθ=((0,∞)×(V∖{0}))/ ∼,(Δθ,u)∼(rΔθ,u/r),</div> </div> <p>and the enriched phase‑cycle category Pθ=Xθ//G. The target is the Lorentz‑dilation action groupoid</p> <div> <div>SLor=(V∖{0})//G.</div> </div> <p>The main result is that the functor</p> <div> <div>Fθ([Δθ,u])=ΔθΔθPl u,Fθ(λ,Λ)=(λ,Λ),</div> </div> <p>is an <strong>isomorphism of action groupoids</strong></p> <div> <div>Pθ ≅ SLor.</div> </div> <p>The lightlike functor of Paper #1 is recovered as the <strong>pure‑rescaling null‑ray slice</strong> of this enriched construction. The result is purely structural: no physical constants, dynamical laws, or curvature data are derived.</p> |
| title | Phase Cycles and Lorentz-Dilation: An Action-Groupoid Extension of a Lightlike Functor |
| topic | category theory action groupoids Lorentzian geometry Lorentz-dilation phase cycles thin groupoids null rays lightlike intervals timelike and spacelike intervals equivariant functors Lorentz transformations 1+1 dimensional spacetime structural mathematics |
| url | https://doi.org/10.5281/zenodo.20316224 |