Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb{R}_2^3$ and $\mathbb{R}_1^2$

Abstract

The aim of this paper is to investigate the orthogonality of vectors to each other and the Gram-Schmidt method in the Minkowski space $\mathbb{R}_2^3$. Hyperbolic cosine formulas are given for all triangle types in the Minkowski plane $\mathbb{R}_1^2$. Moreover, the Pedoe inequality is explained for each type of triangle with the help of hyperbolic cosine formulas. Thus, the Pedoe inequality allowed us to establish a connection between two similar triangles in the Minkowski plane. In the continuation of the study, the rotation matrix that provides both point and axis rotation in the Minkowski plane is obtained by using the Lorentz matrix multiplication. Also, it is stated to be an orthogonal matrix. Moreover, the orthogonal projection formulas on the spacelike and timelike lines are given in the Minkowski plane. In addition, the distances of any point from the spacelike or timelike line are formulated.

Keywords

• gram-schmidt method
• lorentz triangle
• hyperbolic cosine formulas
• pedoe inequality
• lorentz matrix multiplication
• orthogonal projection