This book provides advanced undergraduate physics and mathematics students with an accessible yet detailed understanding of the fundamentals of differential geometry and symmetries in classical physics. Readers, working through the book, will obtain a thorough understanding of symmetry principles and their application in mechanics, field theory, and general relativity, and in addition acquire the necessary calculational skills to tackle more sophisticated questions in theoretical physics.
Most of the topics covered in this book have previously only been scattered across many different sources of literature, therefore this is the first book to coherently present this treatment of topics in one comprehensive volume.
Key features:
- Contains a modern, streamlined presentation of classical topics, which are normally taught separately
- Includes several advanced topics, such as the Belinfante energy-momentum tensor, the Weyl-Schouten theorem, the derivation of Noether currents for diffeomorphisms, and the definition of conserved integrals in general relativity
- Focuses on the clear presentation of the mathematical notions and calculational technique
Author(s): Manousos Markoutsakis
Edition: 1
Publisher: CRC Press
Year: 2022
Language: English
Pages: 482
Tags: Manifolds, Tensors, Symmetry Groups, Fields, Riemannian Geometry, General Relativity
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Part I: Geometric Manifolds
1. Manifolds and Tensors
1.1. Differentiation in Several Dimensions
1.2. Differentiable Manifolds
1.3. Tangent Structure, Vectors and Covectors
1.4. Vector Fields and the Commutator
1.5. Tensor Fields on Manifolds
2. Geometry and Integration on Manifolds
2.1. Geometry and Metric
2.2. Isometry and Conformality
2.3. Examples of Geometries
2.4. Differential Forms and the Exterior Derivative
2.5. Integrals of Differential Forms
2.6. Theorem of Stokes
3. Symmetries of Manifolds
3.1. Transformations and the Lie Derivative
3.2. Symmetry Transformations of Manifolds
3.3. Isometric and Conformal Killing Vectors
3.4. Euclidean and Scale Transformations
Part II: Mechanics and Symmetry
4. Newtonian Mechanics
4.1. Galileian Spacetime
4.2. Newton's Laws of Mechanics
4.3. Systems of Particles and Conserved Quantities
4.4. Gravitation and the Shell Theorem
5. Lagrangian Methods and Symmetry
5.1. Applying the Principle of Stationary Action
5.2. Noether's Theorem in Mechanics
5.3. Galilei Symmetry and Conservation
6. Relativistic Mechanics
6.1. Lorentz Transformations
6.2. Minkowski Spacetime
6.3. Relativistic Particle Mechanics
6.4. Lagrangian Formulation
6.5. Relativistic Symmetry and Conservation
Part III: Symmetry Groups and Algebras
7. Lie Groups
7.1. Notion of a Group
7.2. Notion of a Group Representation
7.3. Lie Groups and Matrix Groups
8. Lie Algebras
8.1. Matrix Exponential and the BCH Formula
8.2. Lie Algebra of a Lie Group
8.3. Abstract Lie Algebras and Matrix Algebras
9. Representations
9.1. Representations of Groups and Algebras
9.2. Adjoint Representations
9.3. Tensor and Function Representations
9.4. Symmetry Transformations of Tensor Fields
9.5. Induced Representations
9.6. Lie Algebra of Killing Vector Fields
10. Rotations and Euclidean Symmetry
10.1. Rotation Group
10.2. Rotation Algebra
10.3. Translations and the Euclidean Group
10.4. Euclidean Algebra
11. Boosts and Galilei Symmetry
11.1. Group of Boosts
11.2. Group of Boosts and Rotations
11.3. Galilei Group
11.4. Galilei Algebra
12. Lorentz Symmetry
12.1. Lorentz Group
12.2. Spinor Representation of the Lorentz Group
12.3. Lorentz Algebra
12.4. Representation on Scalars, Vectors and Tensors
12.5. Representation on Weyl and Dirac Spinors
12.6. Representation on Fields
13. Poincare Symmetry
13.1. Meaning of Poincar e Transformations
13.2. Poincar e Group
13.3. Poincar e Algebra and Field Representations
13.4. Correspondence of Spacetime Symmetries
14. Conformal Symmetry
14.1. Conformal Group
14.2. Conformal Algebra
14.3. Field Transformations
14.4. Linearization of the Conformal Group
Part IV: Classical Fields
15. Lagrangians and Noether's Theorem
15.1. Introducing Fields
15.2. Action Principle for Fields
15.3. Scalar Fields
15.4. Spinor Fields
15.5. Maxwell Vector Field
15.6. Noether's Theorem in Field Theory
16. Spacetime Symmetries of Fields
16.1. Spacetime Symmetries and Currents
16.2. Versions of the Energy-Momentum Tensor
16.3. Conserved Integrals
16.4. Conditions for Conformal Symmetry
17. Gauge Symmetry
17.1. Internal Symmetries and Charge Conservation
17.2. Interactions and the Gauge Principle
17.3. Scalar Electrodynamics
17.4. Spinor Electrodynamics
Part V: Riemannian Geometry
18. Connection and Geodesics
18.1. Connection and the Covariant Derivative
18.2. Formulae for the Covariant Derivative
18.3. The Levi-Civita Connection
18.4. Parallel Transport and Geodesic Curves
19. Riemannian Curvature
19.1. Manifestation of Curvature
19.2. The Riemann Curvature Tensor
19.3. Algebraic Symmetries
19.4. Bianchi Identity and the Einstein Tensor
19.5. Ricci Decomposition and the Weyl Tensor
20. Symmetries of Riemannian Manifolds
20.1. Symmetric Spaces
20.2. Weyl Rescalings
20.3. The Weyl-Schouten Theorem
20.4. Group of Diffeomorphisms
Part VI: General Relativity and Symmetry
21. Einstein's Gravitation
21.1. Physics in Curved Spacetimes
21.2. The Einstein Equations
21.3. Schwarzschild Metric
21.4. Asymptotically Flat Spacetimes
22. Lagrangian Formulation
22.1. Action Principle in Curved Spacetimes
22.2. The Action for Matter Fields
22.3. The Action for the Gravitational Field
22.4. Diffeomorphisms and Noether Currents
23. Conservation Laws and Further Symmetries
23.1. Locally and Globally Conserved Quantities
23.2. On the Energy of Spacetime
23.3. Komar Integrals
23.4. Weyl Rescaling Symmetry
Part VII: Appendices
A. Notation and Conventions
A.1. Physical Units and Dimensions
A.2. Mathematical Conventions
A.3. Abbreviations
B. Mathematical Tools
B.1. Tensor Algebra
B.2. Matrix Exponential
B.3. Pauli and Dirac Matrices
B.4. Dirac Delta Distribution
B.5. Poisson and Wave Equation
B.6. Variational Calculus
B.7. Volume Element and Hyperspheres
B.8. Hypersurface Elements
C. Weyl Rescaling Formulae
D. Spaces and Symmetry Groups
Bibliography
Index
