Topological structures of event horizons along framed null Cartan curves in Minkowski space

Abstract

In this work, we model two event horizons associated with a null Cartan curve as two lightlike hypersurfaces, respectively. We define a lightlike surface and a spacelike surface whose images coincide with the sets of critical values of two event horizons, and meanwhile we present two curves whose images coincide with the sets of critical values of these two surfaces, respectively. Using the singularity theory, we characterize the local topological structures of two event horizons, two surfaces and two curves at their singularities by means of two new invariants. Moreover, we also present a spacelike braneworld model along the particle as a spacelike surface in hyperbolic 3-space. An important fact shows that from the viewpoint of Legendrian dualities, this surface is [Formula: see text]-dual to the tangent trajectory [Formula: see text] of the null Cartan curve in Lorentz–Minkowski space-time. Meanwhile, we also consider a curve whose image is the set of critical values of this surface in hyperbolic 3-space. The third invariant of the null Cartan curve characterizes the singularities of the surface [Formula: see text] and the curve [Formula: see text] in hyperbolic 3-space. A result indicates that surface [Formula: see text] is locally diffeomorphic to the swallowtail [Formula: see text] or cuspidal edge [Formula: see text] and [Formula: see text] is locally diffeomorphic to the [Formula: see text]-cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface [Formula: see text] and the curve [Formula: see text] and the order of contact between [Formula: see text] and elliptic quadric [Formula: see text] or the order of contact between [Formula: see text] and spacelike hyperplane [Formula: see text]. Finally, we present several examples to describe the main results.