In this paper, the trajectory null scroll in $3$-dimensional Minkowski space-time $E_1^3$ is given by a firmly connected null oriented line moving with Cartan frame along null curve. Some theorems and results between curvatures of base curve and distribution parameter of this surface are obtained. Moreover, some theorems and results related to being developable and minimal of this surface are given. And also, some relationships among geodesic curvature, geodesic torsion and the curvatures of null base curve of trajectory null scroll are found.

Keywords: Null scroll, distribution parameter, mean curvature, Gaussian curvature