A New Angular Measurement in Minkowski 3-Space

Jinhua Qian,Young Ho Kim,Xueqian Tian,Jie Liu

doi:10.3390/math8010056

Jinhua Qian, Young Ho Kim + Show 2 more

Open Access

https://doi.org/10.3390/math8010056

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Abstract

In Lorentz–Minkowski space, the angles between any two non-null vectors have been defined in the sense of the angles in Euclidean space. In this work, the angles relating to lightlike vectors are characterized by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors in Minkowski 3-space. Meanwhile, the explicit measuring methods are demonstrated through several examples.

Highlights

In Einstein’s theory of relativity, time, together with the three dimension space, constitutes the four-dimension space–time
One of the remarkable things is that some classical research topics with Riemannian metric are generalized into Lorentz–Minkowski space with a pseudo-Riemannian metric [1,2]
Due to the causal character of vectors in this space, some simple problems become a little complicated and strange, especially the ones relating to null vectors, such as null curves, pseudo null curves, B-scrolls, marginally trapped surfaces and so on [3,4,5]

Summary

Introduction

In Einstein’s theory of relativity, time, together with the three dimension space, constitutes the four-dimension space–time. Due to the causal character of vectors in this space, some simple problems become a little complicated and strange, especially the ones relating to null vectors, such as null curves, pseudo null curves, B-scrolls, marginally trapped surfaces and so on [3,4,5]. One of the reasons is the angles relating to lightlike vectors that cannot be defined properly, which restrict some research, depending on angular measurement to some extent. Considering the relationship between any two independent lightlike vectors and the existing definitions of angles between any two non-null vectors in Minkowski space, an appropriate method is proposed to define the angles relating to lightlike vectors by the Frenet frame of a pseudo null curve and the angles between any two non-null vectors. The curves in this paper are regular and smooth unless otherwise stated

Preliminaries

The Angle between Any Two Lightlike Vectors

The Angle between a Lightlike Vector and a Spacelike Vector

The Angle between a Lightlike Vector and a Timelike Vector

Results