Geometry: from Isometries to Special Relativity

Description

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. 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