A Geometric Approach to the Harmonicity of the Unit Frenet-Serret Vector Fields in a Minkowski Space E13

Abstract

In this paper, we firstly give worldline geometry of a moving spacelike particle in a 3-dimensional Minkowski space E-1(3) in terms of the Frenet-Serret frame elements. Then we compute energy on each Frenet-Serret vector fields in E-1(3). Here, we also show a necessary condition for the energy functional of the unit Frenet-Serret vector fields to have a critical point. We obtain the condition by using the natural technique of a variational problem. We see that this condition is expressed by only using geometrical description of the curvature and the torsion of the worldline belonging to the particle in E-1(3). Finally, we give physical understanding of the harmonicity for some special vector fields and we obtain such useful consequences.