On Angles and Pseudo-Angles in Minkowskian Planes

Leopold Verstraelen

doi:10.3390/math6040052

Abstract

The main purpose of the present paper is to well define Minkowskian angles and pseudo-angles between the two null directions and between a null direction and any non-null direction, respectively. Moreover, in a kind of way that will be tried to be made clear at the end of the paper, these new sorts of angles and pseudo-angles can similarly to the previously known angles be seen as (combinations of) Minkowskian lengths of arcs on a Minkowskian unit circle together with Minkowskian pseudo-lengths of parts of the straight null lines.

Highlights

In his 1908 lecture “Raum und Zeit"
At any given moment of time t, the angles between any two directions in the physical 3D Euclidean ( x, y, z) space at that moment are their standard original Euclidean angles; they are algebraically determined in terms of the group of the Euclidean rotations in a plane around a same point in this plane and they are geometrically measured by the Euclidean lengths of corresponding arcs on a Euclidean unit circle
From a natural scientific point of view there has been no immediate need to be occupied with looking for meaningful angles between two directions with arbitrary causal characters in planes of Minkowski

Summary

Introduction

In his 1908 lecture “Raum und Zeit" (cfr. Figure 1), Hermann Minkowski presented his indefinite geometry, which made possible the development of Lorentzian geometry and, more generally, of pseudo-Euclidean geometry and of pseudo-Riemannian geometry; (for references on these geometries, see e.g., [1,2,3,4,5,6,7,8,9] and the references in these books and chapters of books and articles). Like in several of his other recent papers, when it seems to the author of real importance for a better understanding of the text, he included a number of handmade figures In his experience, so much more than artificially made illustrations, such figures do essentially contribute to the readability of the paper. Much more than artificially made illustrations, such figures do essentially contribute to the readability of the paper This is very related to the real value of the drawings made on blackboards during proper lectures on mathematics and on the exact sciences. The author is very grateful for the editors of the journal Mathematics having been so kind to include the scans of ten handmade figures in the present paper

The Pseudo-Angles of Helzer

The Minkowskian Angles between Spacelike and Timelike Directions

The Minkowskian Angles between Null Directions

A Geometrical Meaning of the Minkowskian Angles and Pseudo-Angles

Conclusions