Geometry, Symmetries, and Classical Physics: A Mosaic

Length: 480 pages
Edition: 1
Language: English
Publisher: CRC Press
Publication Date: 2021-12-29
ISBN-10: 0367535238
ISBN-13: 9780367535230
Sales Rank: #2138272 (See Top 100 Books)

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

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