Wavefronts of traveling trajectories of geometric particles

Confining the traveling trajectory of a tachyon to the two-dimensional Lorentzian space forms, we describe the trajectory as a spacelike front in these Lorentzian space forms. Introducing the differential geometry of singular curves in Lorentzian space forms, that is, the hyperbolic space and de Sitter space, and applying the Legendrian duality theorems, we establish the moving frame along the front, whereby the definitions of the evolutes of spacelike fronts in Lorentzian space forms are presented and the geometric properties of these evolutes are investigated in detail. It is shown that these evolutes can be interpreted as wavefronts under the viewpoint of Legendrian singularity theory.