Geometric approaches to establish the fundamentals of Lorentz spaces ℝ 2 3 and ℝ 1 2

Sevilay Çoruh Şenocak; Salim Yüce

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 4, page 549-567
  • ISSN: 0862-7959

Abstract

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The aim of this paper is to investigate the orthogonality of vectors to each other and the Gram-Schmidt method in the Minkowski space ℝ 2 3 . Hyperbolic cosine formulas are given for all triangle types in the Minkowski plane 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.

How to cite

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Çoruh Şenocak, Sevilay, and Yüce, Salim. "Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb {R}_2^3$ and $\mathbb {R}_1^2$." Mathematica Bohemica 149.4 (2024): 549-567. <http://eudml.org/doc/299621>.

@article{ÇoruhŞenocak2024,
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.},
author = {Çoruh Şenocak, Sevilay, Yüce, Salim},
journal = {Mathematica Bohemica},
keywords = {Gram-Schmidt method; Lorentz triangle; hyperbolic cosine formulas; Pedoe inequality; Lorentz matrix multiplication; orthogonal projection},
language = {eng},
number = {4},
pages = {549-567},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb \{R\}_2^3$ and $\mathbb \{R\}_1^2$},
url = {http://eudml.org/doc/299621},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Çoruh Şenocak, Sevilay
AU - Yüce, Salim
TI - Geometric approaches to establish the fundamentals of Lorentz spaces $\mathbb {R}_2^3$ and $\mathbb {R}_1^2$
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 4
SP - 549
EP - 567
AB - 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.
LA - eng
KW - Gram-Schmidt method; Lorentz triangle; hyperbolic cosine formulas; Pedoe inequality; Lorentz matrix multiplication; orthogonal projection
UR - http://eudml.org/doc/299621
ER -

References

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