The Mathematics of Minkowski Space-Time

With an Introduction to Commutative Hypercomplex Numbers

verfasst von: Francesco Catoni, Dino Boccaletti, Roberto Cannata, Vincenzo Catoni, Enrico Nichelatti, Paolo Zampetti

Verlag: Birkhäuser Basel

Buchreihe : Frontiers in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Hyperbolic numbers are proposed for a rigorous geometric formalization of the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers as a simple extension of the field of complex numbers is extensively studied in the book. In particular, an exhaustive solution of the "twin paradox" is given, followed by a detailed exposition of space-time geometry and trigonometry. Finally, an appendix on general properties of commutative hypercomplex systems with four unities is presented.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Complex numbers represent one of the most intriguing and emblematic discoveries in the history of science. Even if they were introduced for an important but restricted mathematical purpose, they came into prominence in many branches of mathematics and applied sciences. This association with applied sciences generated a synergistic effect: applied sciences gave relevance to complex numbers and complex numbers allowed formalizing practical problems. A similar effect can be found today in the “system of hyperbolic numbers”, which has acquired meaning and importance as the Mathematics of Special Relativity, as shown in this book. Let us proceed step by step and begin with the history of complex numbers and their generalization.

Chapter 2. N-Dimensional Commutative Hypercomplex Numbers

Abstract

As summarized in the preface, hypercomplex numbers were introduced before the linear algebra of matrices and vectors. In this chapter, in which we follow a classical approach, their theory is developed mainly by means of elementary algebra, and their reference to a representation with matrices, vectors or tensors is just for the practical convenience of referring to a widely known language. In particular, the down or up position of the indexes, which in tensor calculus are named covariance and contravariance, respectively, indicates if the corresponding quantities (vectors) are transformed by a direct or inverse matrix (see Section 2.1.4). We also use Einstein’s convention for tensor calculus and omit the sum symbol on the same covariant and contravariant indexes; in particular, we indicate with Roman letters the indexes running from 1 to N − 1, and with Greek letters the indexes running from 0 to N − 1.

Chapter 3. The Geometries Generated by Hypercomplex Numbers

Abstract

The groups of transformations were introduced and formalized by S. Lie in the second half of the XIXth century. The concept of group is well known; here we only recall the definition of transformations or Lie groups that is used in this section.

Chapter 4. Trigonometry in the Minkowski Plane

Abstract

We have seen in Section 3.2 how commutative hypercomplex numbers can be associated with a geometry, in particular the two-dimensional numbers can represent the Euclidean plane geometry and the space-time (Minkowski) plane geometry. In this chapter, by means of algebraic properties of hyperbolic numbers, we formalize the space-time geometry and trigonometry. This formalization allows us to work in Minkowski space-time as we usually do in the Euclidean plane, i.e., to give a Euclidean description that can be considered similar to Euclidean representations of non-Euclidean geometries obtained in the XIXth century by E. Beltrami [2] on constant curvature surfaces, as we recall in Chapter 9.

Chapter 5. Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox)

Abstract

In this chapter we show how the formalization of trigonometry in the pseudo-Euclidean plane allows us to treat exhaustively all kinds of motions and to give a complete formalization to what is called today the “twin paradox”. After a century this problem continues to be the subject of many papers, not only relative to experimental tests [1] but also regarding physical and epistemological considerations [51]. We begin by recalling how this “name” originates.

Chapter 6. General Two-Dimensional Hypercomplex Numbers

Abstract

In this chapter we study the Euclidean and pseudo-Euclidean geometries associated with the general two-dimensional hypercomplex variable, i.e., the algebraic ring (see Section 2.2)

$$ \{ z = x + uy; u^2 = \alpha + u\beta ; x,y,\alpha ,\beta \in R; u \notin R\} , $$

(6.0.1)

and we show that in geometries generated by these numbers, ellipses and general hyperbolas play the role which circles and equilateral hyperbolas play in Euclidean and in pseudo-Euclidean planes, respectively.

Chapter 7. Functions of a Hyperbolic Variable

Abstract

For real variables, the definition of polynomials (linear combinations of powers) stems from the definitions of elementary algebraic operations. Since for complex variables the same algebraic rules hold, also for them the polynomial can be defined and, grouping together the terms with and without the coefficient i, we can always express them as P(z) = u (x, y)+i v (x, y), where u, v are real functions of the real variables x, y.

Chapter 8. Hyperbolic Variables on Lorentz Surfaces

Abstract

In this chapter we start from two fundamental memoirs of Gauss [39] and Beltrami [3] on complex variables on surfaces described by a definite quadratic form. By using the functions of a hyperbolic variable, we extend the results of the classic authors to surfaces in space-times described by non-definite quadratic forms.

Chapter 9. Constant Curvature Lorentz Surfaces

Abstract

It is known that, in the XVIIIth century, the growth of a new physics also drove the mathematics into new ways with respect to Euclidean geometry which, following Plato and Galileo, had been considered the “measure and interpretation” of the world. From then on new subjects emerge, such as differential calculus, complex numbers, analytic, differential and non-Euclidean geometries, functions of a complex variable, partial differential equations and, at the end of the XIXth century, group theory.

Chapter 10. Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle)

Abstract

In Chapter 5 we have seen how hyperbolic trigonometry, introduced in the flat pseudo-Euclidean plane, has allowed us a complete treatment of accelerated motions and a consequent formalization of the twin paradox. In this second chapter concerning physical applications we shall see how the expansion from algebraic properties to the introduction of functions of hyperbolic variable allows an intriguing extension to general relativity of the symmetry of hyperbolic numbers, just introduced through special relativity.

Backmatter

Titel: The Mathematics of Minkowski Space-Time
verfasst von: Francesco Catoni
Dino Boccaletti
Roberto Cannata
Vincenzo Catoni
Enrico Nichelatti
Paolo Zampetti
Verlag: Birkhäuser Basel
Electronic ISBN: 978-3-7643-8614-6
Print ISBN: 978-3-7643-8613-9
DOI: https://doi.org/10.1007/978-3-7643-8614-6