Lorentz Geometry is a very important intersection between Mathematics and Physics, being the mathematical language of General Relativity.
Learning this type of geometry is the first step in properly understanding questions regarding the structure of the universe, such as: What is the shape of the universe? What is a spacetime? What is the relation between gravity and curvature? Why exactly is time treated in a different manner than other spatial dimensions?
Introduction to Lorentz Geometry: Curves and Surfaces intends to provide the reader with the minimum mathematical background needed to pursue these very interesting questions, by presenting the classical theory of curves and surfaces in both Euclidean and Lorentzian ambient spaces simultaneously.
Features:
- Over 300 exercises.
- Suitable for senior undergraduates and graduates studying Mathematics and Physics.
- Written in an accessible style without loss of precision or mathematical rigor.
Author(s): Ivo Terek Couto, Alexandre Lymberopoulos
Edition: 1
Publisher: CRC Press
Year: 2021
Language: English
Pages: 350
Tags: Minkowski Space, Lorentz Geometry, Pseudo-Riemannian Metrics, Curves, Surfaces
Cover
Half Title
Title Page
Copyright Page
Contents
Preface of the Portuguese Version
Preface
Chapter 1: Welcome to Lorentz-Minkowski Space
1.1.
PSEUDO–EUCLIDEAN SPACES
1.1.1.
Defining Rnn
1.1.2.
The causal character of a vector in Rnn
1.2.
SUBSPACES OF Rnn
1.3.
CONTEXTUALIZATION IN SPECIAL RELATIVITY
1.4.
ISOMETRIES IN Rnn
1.5.
INVESTIGATING O1(2,R) AND O1(3,R)
1.5.1.
The group O1(2,R) in detail
1.5.2.
The group O1(3,R) in (a little less) detail
1.5.3.
Rotations and boosts
1.6.
CROSS PRODUCT IN Rnn
1.6.1.
Completing the toolbox
Chapter 2: Local Theory of Curves
2.1.
PARAMETRIZED CURVES IN Rnn
2.2.
CURVES IN THE PLANE
2.3.
CURVES IN SPACE
2.3.1.
The Frenet-Serret trihedron
2.3.2.
Geometric effects of curvature and torsion
2.3.3.
Curves with degenerate osculating plane
Chapter 3: Surfaces in Space
3.1.
BASIC TOPOLOGY OF SURFACES
3.2.
CAUSAL TYPE OF SURFACES, FIRST FUNDAMENTAL FORM
3.2.1.
Isometries between surfaces
3.3.
SECOND FUNDAMENTAL FORM AND CURVATURES
3.4.
THE DIAGONALIZATION PROBLEM
3.4.1.
Interpretations for curvatures
3.5.
CURVES IN A SURFACE
3.6.
GEODESICS, VARIATIONAL METHODS AND ENERGY
3.6.1.
Darboux-Ribaucour frame
3.6.2.
Christoffel symbols
3.6.3.
Critical points of the energy functional
3.7 THE FUNDAMENTAL THEOREM OF SURFACES
3.7.1.
The compatibility equations
Chapter 4: Abstract Surfaces and Further Topics
4.1.
PSEUDO-RIEMANNIAN METRICS
4.2.
RIEMANN’S CLASSIFICATION THEOREM
4.3.
SPLIT-COMPLEX NUMBERS AND CRITICAL SURFACES
4.3.1.
A brief introduction to split-complex numbers
4.3.2.
Bonnet rotations
4.3.3.
Enneper-Weierstrass representation formulas
4.4.
DIGRESSION: COMPLETENESS AND CAUSALITY
Appendix:
Some Results from Differential Calculus
Bibliography
Index
