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- Title
Topological structures of event horizons along framed null Cartan curves in Minkowski space.
- Authors
Yu, Xintong; Liu, Siyao; Wang, Zhigang
- 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 Δ 2 -dual to the tangent trajectory L (t) 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 ℒ and the curve in hyperbolic 3-space. A result indicates that surface ℒ is locally diffeomorphic to the swallowtail S W or cuspidal edge C E and is locally diffeomorphic to the (2 , 3 , 4) -cusp at certain a singular point. It is also shown that there exist deep relationships between the singularities of the surface ℒ and the curve and the order of contact between L (t) and elliptic quadric ℰ ( v 0 , − 1) or the order of contact between L (t) and spacelike hyperplane HP ( v 0 , − 1). Finally, we present several examples to describe the main results.
- Subjects
MINKOWSKI space; HYPERSURFACES; QUADRICS; SPACETIME; CURVES
- Publication
International Journal of Geometric Methods in Modern Physics, 2022, Vol 19, Issue 12, p1
- ISSN
0219-8878
- Publication type
Article
- DOI
10.1142/S0219887822501869