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Geometry of Minkowski Space-Time

Part of the SpringerBriefs in Physics Book Series

DOI: 10.1007/978-3-642-17977-8

ISBN: 978-3-642-17976-1

Publisher: Springer Berlin Heidelberg - Springer Journals

Mar 4, 2011

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Chapter 1Introduction

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 1 –2

[It is largely known that the Theory of Special Relativity was born as a consequence of the demonstrated impossibility for the Maxwell’s electromagnetic (e.m.) theory of obeying Galilean transformations. The non-invariance of the e.m. theory under Galilean transformations induced the theoretical...
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Chapter 2Hyperbolic Numbers

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 3 –23

[Complex numbers can be considered as a two components quantity, as the plane vectors. Following Gauss complex numbers are also used for representing vectors in Euclidean plane. As a difference with vectors the multiplication of two complex numbers is yet a complex number. By means of this...
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Chapter 3Geometrical Representation of Hyperbolic Numbers

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 25 –31

[A relevant property of Euclidean geometry is the Pythagorean distance between two points. From this definition the properties of analytical geometry follow. In a similar way the analytical geometry in Minkowski plane is introduced, starting from the invariant quantities of Special Relativity. ]
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Chapter 4Trigonometry in the Hyperbolic (Minkowski) Plane

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 33 –55

[The correspondence between properties of complex numbers and Euclidean geometry allows to obtain an algebraic formalization of Euclidean geometry. Thanks to the equivalent properties between complex and hyperbolic numbers, the geometry of Minkowski space-time can be formalized in a similar...
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Chapter 5Equilateral Hyperbolas and Triangles in the Hyperbolic Plane

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 57 –72

[ The equilateral hyperbolas, represented in the Minkowski space-time, hold the same properties of circles in Euclidean plane and satisfy similar theorems. At the same time equivalent relations to the ones in Euclidean plane between circles and triangles are obtained in hyperbolic plane between...
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Chapter 6The Motions in Minkowski Space–Time (Twin Paradox)

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 73 –95

[All the curves in space-time plane, can be considered as a motion and the lengths of the time-like lines give the proper time. Therefore we can compare in a geometrical way, on different curves, the differences between proper times. Proper times are obtained by means of elementary mathematics...
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Chapter 7Some Final Considerations

Catoni, Francesco; Boccaletti, Dino; Cannata, Roberto; Catoni, Vincenzo; Zampetti, Paolo

Pages: 97 –99

[The modern view of geometry, that we recall in the Appendix, has allowed us to formalize the “geometry of space–time” so that we can work in this plane as we usually do for Euclidean plane geometry. Otherwise the obtained mathematical system, following Euclidean geometry, combine the logical...

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