This textbook offers a geometric perspective on special relativity, bridging Euclidean space, hyperbolic space, and Einstein’s spacetime in one accessible, self-contained volume. Using tools tailored to undergraduates, the author explores Euclidean and non-Euclidean geometries, gradually building from intuitive to abstract spaces. By the end, readers will have encountered a range of topics, from isometries to the Lorentz–Minkowski plane, building an understanding of how geometry can be used to model special relativity.
Beginning with intuitive spaces, such as the Euclidean plane and the sphere, a structure theorem for isometries is introduced that serves as a foundation for increasingly sophisticated topics, such as the hyperbolic plane and the Lorentz–Minkowski plane. By gradually introducing tools throughout, the author offers readers an accessible pathway to visualizing increasingly abstract geometric concepts. Numerous exercises are also included with selected solutions provided.
Geometry: from Isometries to Special Relativity offers a unique approach to non-Euclidean geometries, culminating in a mathematical model for special relativity. The focus on isometries offers undergraduates an accessible progression from the intuitive to abstract; instructors will appreciate the complete instructor solutions manual available online. A background in elementary calculus is assumed.
Author(s): Nam-Hoon Lee
Series: Undergraduate Texts in Mathematics
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 264
Tags: Euclidean Plane, Spere, Stereographic Projection, Hyperbolic Plane, Lorentz-Minkowski Plane, Special Relativity
Preface
Contents
Dependence Chart
1 Euclidean Plane
1.1 Isometries
Exercises
1.2 Three Reflections Theorem
Exercises
1.3 Rotations and Translations
Exercises
1.4 Glide Reflections and Orientation
Exercises
2 Sphere
2.1 The Sphere S2 in R3
Exercises
2.2 Isometries of the Sphere S2
Exercises
2.3 Area of a Spherical Triangle
Exercises
2.4 Orthogonal Transformations of Euclidean Spaces
3 Stereographic Projection and Inversions
3.1 Stereographic Projection
Exercises
3.2 Inversions on the Extended Plane
Exercises
3.3 Inversions on the Sphere S2
Exercises
3.4 Representation of the Sphere in the Extended Plane
Exercises
4 Hyperbolic Plane
4.1 Poincaré Upper Half-Plane H2
Exercises
4.2 H2-Shortest Paths and H2-Lines
Exercises
4.3 Isometries of the Hyperbolic Plane
Exercises
4.4 Hyperbolic Triangle and Hyperbolic Area
Exercises
4.5 Poincaré Disk
Exercises
4.6 Klein Disk
Exercises
4.7 Euclid's Fifth Postulate: The Parallel Postulate
Exercises
5 Lorentz–Minkowski Plane
5.1 Lorentz–Minkowski Distance
Exercises
5.2 Relativistic Reflections
Exercises
5.3 Hyperbolic Angle
Exercises
5.4 Relativistic Rotations
Exercises
5.5 Matrix and Isometry
Exercises
5.6 Relativistic Lengths of Curves
Exercises
5.7 Hyperboloid in R2,1
Exercises
5.8 Isometries of R2,1
Exercises
6 Geometry of Special Relativity
6.1 R3,1 and the Special Relativity of Einstein
Exercises
6.2 Causality
Exercises
6.3 Causal Isometry
Exercises
6.4 Worldline
Exercises
6.5 Kinetics in R3,1
Exercises
Answers to Selected Exercises
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Bibliography
Index
Symbol Index
