Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface LHβ, a critical lightlike surface LSβ, and a degenerate curve LCβ, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant σ(t) that governs the emergence of specific singularity types, including C(2,3)×R2, SW×R, BF, C(BF), C(2,3,4)×R, and (2,3,4,5)-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane HSt0L and the curve β systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry.