On Null Curves in Minkowski 3-Space and Its Fractal Folding

A.E El-Ahmady,A.T M-Zidan,Malak E Raslan

doi:10.5539/mas.v14n4p90

Abstract

In this paper, a form for Frenet equations of all null curves in Minkowski 3-space has been presented. New types of foldings of curves are obtained. The connection between folding, deformation and Frenet equations of curves are also deduced.

Highlights

The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by
A curve is said to be regular if ( ) ≠ 0 for all ∈, ∈ is space-like if its velocity vectors are space-like for all ∈, for time-like and null
For the unit speed curve ( ) with non-null frame vectors, we distinguish three cases depending on the causal character of ( ) and its Frenet equations are as follows

Summary

Introduction

The Minkowski 3-space is the Euclidean 3-space provided with the standard flat metric given by. Since is an indefinite metric, recall that a vector ∈ is said space-like if ( , ) > 0 = 0, time-like if ( , ) < 0 and null (light-like) if ( , )=0. An arbitrary curve = ( ) in can locally be space-like, time- like or null(light-like), if all of its velocity vectors ( ) are respectively, space-like, time-like or null (light-like) respectively. A curve is said to be regular if ( ) ≠ 0 for all ∈ , ∈ is space-like if its velocity vectors are space-like for all ∈ , for time-like and null. If is a null curve, we can re-parameterize it such that, 〈 ( ), ( )〉 = 0 and ( ) ≠ 0, recall the norm of a vector is given by ‖ ‖ = | ( , )|.

Preliminary Notes

Form of Frenet Equations of Null Curves in Minkowski 3-Space

Folding of Null Curves