book

Semi-Riemannian Geometry

by Stephen C. Newman

July 2019

Intermediate to advanced

625 pages

20h 7m

English

Wiley

Read now

Unlock full access

1.1 Vector Spaces1.2 Dual Spaces1.3 Pullback of Covectors1.4 Annihilators
2.1 Matrices2.2 Matrix Representations2.3 Rank of Matrices2.4 Determinant of Matrices2.5 Trace and Determinant of Linear Maps
3.1 Bilinear Functions3.2 Symmetric Bilinear Functions3.3 Flat Maps and Sharp Maps
4.1 Scalar Product Spaces4.2 Orthonormal Bases4.3 Adjoints4.4 Linear Isometries4.5 Dual Scalar Product Spaces4.6 Inner Product Spaces4.7 Eigenvalues and Eigenvectors4.8 Lorentz Vector Spaces4.9 Time Cones
5.1 Tensors5.2 Pullback of Covariant Tensors5.3 Representation of Tensors5.4 Contraction of Tensors
6.1 Contraction of Tensors6.2 Flat Maps6.3 Sharp Maps6.4 Representation of Tensors6.5 Metric Contraction of Tensors6.6 Symmetries of (0, 4)‐Tensors
7.1 Multicovectors7.2 Wedge Products7.3 Pullback of Multicovectors7.4 Interior Multiplication7.5 Multicovector Scalar Product Spaces

8.1 Orientation of ℝ m 8.2 Orientation of Vector Spaces8.3 Orientation of Scalar Product Spaces8.4 Vector Products8.5 Hodge Star
9.1 Topology9.2 Metric Spaces9.3 Normed Vector Spaces9.4 Euclidean Topology on ℝm
10.1 Derivatives10.2 Immersions and Diffeomorphisms10.3 Euclidean Derivative and Vector Fields10.4 Lie Bracket10.5 Integrals10.6 Vector Calculus
11.1 Curves inℝ3 11.2 Regular Surfaces inℝ3 11.3 Tangent Planes inℝ3 11.4 Types of Regular Surfaces in ℝ3 11.5 Functions on Regular Surfaces in ℝ3 11.6 Maps on Regular Surfaces inℝ3 11.7 Vector Fields Along Regular Surfaces in ℝ3
12.1 Curves in 12.2 Regular Surfaces in 12.3 Induced Euclidean Derivative in 12.4 Covariant Derivative on Regular Surfaces in 12.5 Covariant Derivative on Curves in 12.6 Lie Bracket in 12.7 Orientation in 12.8 Gauss Curvature in 12.9 Riemann Curvature Tensor in 12.10 Computations for Regular Surfaces in
13.1 Plane in 13.2 Cylinder in 13.3 Cone in 13.4 Sphere in 13.5 Tractoid in 13.6 Hyperboloid of One Sheet in 13.7 Hyperboloid of Two Sheets in 13.8 Torus in 13.9 Pseudosphere in 13.10 Hyperbolic Space in
14.1 Smooth Manifolds14.2 Functions and Maps14.3 Tangent Spaces14.4 Differential of Maps14.5 Differential of Functions14.6 Immersions and Diffeomorphisms14.7 Curves14.8 Submanifolds14.9 Parametrized Surfaces
15.1 Vector Fields15.2 Representation of Vector Fields15.3 Lie Bracket15.4 Covector Fields15.5 Representation of Covector Fields15.6 Tensor Fields15.7 Representation of Tensor Fields15.8 Differential Forms15.9 Pushforward and Pullback of Functions15.10 Pushforward and Pullback of Vector Fields15.11 Pullback of Covector Fields15.12 Pullback of Covariant Tensor Fields15.13 Pullback of Differential Forms15.14 Contraction of Tensor Fields
16.1 Exterior Derivatives16.2 Tensor Derivations16.3 Form Derivations16.4 Lie Derivative16.5 Interior Multiplication16.6 orientation16.7 Integration of Differential Forms16.8 Line Integrals16.9 Closed and Exact Covector Fields16.10 Flows
17.1 Smooth Manifolds with Boundary17.2 Inward‐Pointing and Outward‐Pointing Vectors17.3 orientation of Boundaries17.4 Stokes's Theorem
18.1 Covariant Derivatives18.2 Christoffel Symbols18.3 Covariant Derivative on Curves18.4 Total Covariant Derivatives18.5 Parallel Ranslation18.6 Torsion Tensors18.7 Curvature Tensors18.8 Geodesics18.9 Radial Geodesics and Exponential Maps18.10 Normal Coordinates18.11 Jacobi Fields
19.1 Semi‐Riemannian Manifolds19.2 Curves19.3 Fundamental Theorem of Semi‐Riemannian Manifolds19.4 Flat Maps and Sharp Maps19.5 Representation of Tensor Fields19.6 Contraction of Tensor Fields19.7 Isometries19.8 Riemann Curvature Tensor19.9 Geodesics19.10 Volume Forms19.11 orientation of Hypersurfaces19.12 Induced Connections
20.1 Hodge Star20.2 Codifferential20.3 Gradient20.4 Divergence of Vector Fields20.5 Curl20.6 Hesse Operator20.7 Laplace Operator20.8 Laplace–de Rham Operator20.9 Divergence of Symmetric 2‐Covariant Tensor Fields
21.1 Geodesics and Curvature on Riemannian Manifolds21.2 Classical Vector Calculus Theorems
22.1 Linear Isometries on Lorentz Vector Spaces22.2 Maxwell's Equations22.3 Einstein Tensor
B.1. GroupsB.2. Permutation GroupsB.3. RingsB.4. FieldsB.5. ModulesB.6. Vector SpacesB.7. Lie Algebras

Overview

An introduction to semi-Riemannian geometry as a foundation for general relativity

Semi-Riemannian Geometry: The Mathematical Language of General Relativity is an accessible exposition of the mathematics underlying general relativity. The book begins with background on linear and multilinear algebra, general topology, and real analysis. This is followed by material on the classical theory of curves and surfaces, expanded to include both the Lorentz and Euclidean signatures. The remainder of the book is devoted to a discussion of smooth manifolds, smooth manifolds with boundary, smooth manifolds with a connection, semi-Riemannian manifolds, and differential operators, culminating in applications to Maxwell’s equations and the Einstein tensor. Many worked examples and detailed diagrams are provided to aid understanding. This book will appeal especially to physics students wishing to learn more differential geometry than is usually provided in texts on general relativity.

STEPHEN C. NEWMAN is Professor Emeritus at the University of Alberta, Edmonton, Alberta, Canada. He is the author of Biostatistical Methods in Epidemiology and A Classical Introduction to Galois Theory, both published by Wiley.