This book systematically develops the mathematical foundations of the theory of relativity and links them to physical relations. For this purpose, differential geometry on manifolds is introduced first, including differentiation and integration, and special relativity is presented as tensor calculus on tangential spaces. Using Einstein's field equations relating curvature to matter, the relativistic effects in the solar system including black holes are discussed in detail.
The text is aimed at students of physics and mathematics and assumes only basic knowledge of classical differential and integral calculus and linear algebra.
Author(s): Rainer Oloff
Series: Graduate Texts in Physics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 274
City: Cham
Tags: General Relativity, Manifolds, Tensor Analysis, Black Holes, Cosmology
cover
Preface
Introduction
Contents
1 Differentiable Manifolds
1.1 Charts and Atlases
1.2 Topologisation
1.3 Submanifolds of mathbbRm
2 Tangent Vectors
2.1 The Tangent Space
2.2 Generation of Tangent Vectors
2.3 Vector Fields
2.4 The Lie Bracket
3 Tensors
3.1 Introduction
3.2 Multilinear Forms
3.3 Components
3.4 Operations with Tensors
3.5 Tensors on Euclidean Spaces
4 Semi-Riemannian Manifolds
4.1 Tensor Fields
4.2 Riemannian Manifolds
4.3 Bilinear Forms
4.4 Orientation
4.5 Spacetime
5 Theory of Special Relativity
5.1 Kinematics
5.2 Dynamics
5.3 Electrodynamics
6 Differential Forms
6.1 p-forms
6.2 The Wedge Product
6.3 The Hodge-Star Operator
6.4 Outer Derivative
6.5 The Maxwell Equations in Vacuum
7 The Covariant Differentiation of Vector Fields
7.1 The Directional Derivative in mathbbRn
7.2 The Levi-Civita Connection
7.3 Christoffel Symbols
7.4 The Covariant Derivative on Hypersurfaces
7.5 The Covariant Derivative in the Schwarzschild Spacetime
8 Curvature
8.1 The Curvature Tensor
8.2 The Weingarten Map
8.3 The Ricci Tensor
8.4 The Curvature of the Schwarzschild Spacetime
8.5 Connection Forms and Curvature Forms
9 Matter
9.1 Mass
9.2 Energy and Momentum of a Flow
9.3 The Energy-Momentum Tensor
9.4 Charge
9.5 Energy and Momentum in the Electromagnetic Field
9.6 The Einstein Field Equation
9.7 Spherically Symmetric Solutions
9.8 Outer and Inner Schwarzschild Metric
10 Geodesics
10.1 Time
10.2 The Euler-Lagrange Equation
10.3 The Geodesic Equation
10.4 The Geodesic Deviation
10.5 Perihelion Precession
10.6 Light Deflection
10.7 Red Shift
11 Covariant Differentiation of Tensor Fields
11.1 Parallel Transport of Vectors
11.2 Parallel Transport of Tensors
11.3 Calculation Rules and Component Representation
11.4 The Second Bianchi Identity
11.5 Divergence
12 The Lie Derivative
12.1 The Flow and Its Tangents
12.2 Pull-Back and Push-Forward
12.3 Axiomatic Set Up
12.4 The Derivative Formula
12.5 Component Representation
12.6 Killing Vectors
12.7 The Lie Derivative of Differential Forms
13 Integration on Manifolds
13.1 Introduction
13.2 Partition of Unity
13.3 Integrals
13.4 Manifolds with Boundary
13.5 Integral Theorems
13.6 Extremal Principles
14 Nonrotating Black Holes
14.1 The Schwarzschild Half Plane
14.2 Optics of Black Holes
14.3 The Kruskal Plane
15 Cosmology
15.1 Spaces of Constant Curvature
15.2 The Robertson-Walker Metric
15.3 Universe Models
16 Rotating Black Holes
16.1 The Kerr Metric
16.2 Other Representations of the Kerr Metric
16.3 Causal Structure
16.4 Covariant Derivative and Curvature
16.5 Conservation Theorems
17 A Glimpse of String Theory
17.1 Quantum Theory Versus Relativity Theory
17.2 Elementary Particles as Strings
17.3 The Extremal Principle
References
Index
