Null Cartan Normal Helices in Minkowski Space-Time

Abstract

A complete theory of null Cartan normal helices in Minkowski space-time $\mathbb{E}^4_1$ is developed. Two algebraic conditions, obtained by successive differentiation of the helix invariant along a unit $C$-constant normal field, fully characterize null Cartan helices; the quadratic condition yields two mutually orthogonal helix axes in the Lorentzian metric. Special field types are analyzed and null Cartan cubics are shown to be normal helices. On a timelike hypersurface, a Darboux frame with six curvature functions is constructed from first principles, the normal isophotic condition is shown to reduce to a linear first-order ODE, and the existence of normal silhouettes in $\mathbb{E}^4_1$ is established.