# A Mathematical Approach to Special Relativity

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A Mathematical Approach to Special Relativity introduces the mathematical formalisms of special and general relativity. Developed from the author’s experience teaching physics to students across all levels, the valuable resource introduces key concepts, building in complexity and using increasingly advanced mathematical tools as it progresses. Without assuming a background in calculus, the text begins with symmetry, before delving more deeply into Galilean relativity. Throughout, the book provides examples and useful "Guides to the Literature." This unique text emphasizes the experimental consequences and verifications of the underpinning theory in order to provide students with a solid foundation in this key area.

## Key Features

- Based on the professor’s 25+ years of experience teaching physics students at every level
- Covers key topics in special relativity, including some group theory, as well as an introduction to general relativity and basic differential geometry
- Contains numerous worked examples and "Guides to the Literature" throughout the text

## Readership

Students in undergraduate and graduate physics programs taking courses on Mathematical Methods for Physics, Special Relativity, Group Theory, General Relativity. Researchers and academics in physics and related STEM areas, who require a refresher to the subject

## Table of Contents

- Cover image
- Title page
- Table of Contents
- Copyright
- Dedication
- List of figures
- Biography
- Preface
- Acknowledgments
- Part 1: Physics
- Chapter 1: Galilean relativity
- 1.1. The homogeneity of space
- 1.2. The isotropy of space
- 1.3. Parity violation
- 1.4. Homogeneity and the origin of the coordinate system
- 1.5. Uniform rectilinear motion
- 1.6. Galilean principle of relativity
- 1.7. Inertial vs. noninertial frames
- 1.8. Time and clock
- 1.9. Newton's absolute time
- 1.10. Causal structure in Galilean relativity
- 1.11. Galilean boosts
- 1.12. Galilean addition of velocities
- 1.13. Transformation of kinetic energy and momentum
- 1.14. Preferred frame of reference
- 1.15. Classical Doppler effect from addition of velocities
- 1.16. Plane sound waves
- 1.17. Partial derivatives
- 1.18. Galilean transformations of some partial differential equations
- References
- Chapter 2: Failure of Newton's absolute time
- 2.1. Relativity and light
- 2.2. The Michelson–Morley experiment
- 2.3. Synchronization of clocks
- 2.4. Maximum velocity of propagation of interactions
- 2.5. No perfect rigid body
- 2.6. Relativity of simultaneity
- 2.7. Synchronization is a convention
- References
- Chapter 3: Lorentz boosts
- 3.1. Pure boost along the x-axis
- 3.2. World-lines
- 3.3. Causal structure in special relativity
- 3.4. Time dilation
- 3.5. Proper time
- 3.6. Length contraction
- 3.7. Boosts in other directions
- 3.8. Addition of velocities
- 3.9. Success: Fizeau experiment
- 3.10. Aberration of light
- 3.11. Thomas rotation and precession
- 3.12. Temperature of moving bodies
- 3.13. Transformation of accelerations
- 3.14. Proper acceleration
- 3.15. Hyperbolic motion
- References
- Chapter 4: Development of the formalism
- 4.1. 4-vectors
- 4.2. Index notation
- 4.3. Quadratic scalars
- 4.4. The 4-current
- 4.5. The 4-potential
- 4.6. Electromagnetic field tensor
- 4.7. 6-vectors
- References
- Chapter 5: Relativistic dynamics of particles
- 5.1. Momentum and energy of a particle
- 5.2. Lagrangian of a particle
- 5.3. Interaction of a particle with an electromagnetic field
- 5.4. Angular momentum 6-vector of a particle
- 5.5. Inertia
- 5.6. Inertia and the energy content
- 5.7. Kinematics of decays and collisions
- References
- Chapter 6: Electrodynamics in covariant form
- 6.1. Density and current density
- 6.2. Electric charge density and its current density
- 6.3. The Lorentz force
- 6.4. Energy-momentum-stress tensor of matter
- 6.5. The energy-momentum-stress tensor of the electromagnetic field
- 6.6. The conservation laws
- 6.7. Conservation of electric charge
- 6.8. Conservation of 4-momentum
- 6.9. Angular momentum and center of mass
- References
- Chapter 7: Noninertial frames
- 7.1. Curvilinear coordinates
- 7.2. Point observer vs. frame of reference
- 7.3. Time, distance, and simultaneity in general coordinates
- 7.4. Einstein synchronization in curvilinear coordinates
- 7.5. A Galilean change of coordinates on the Minkowski space-time
- 7.6. An expanding universe
- 7.7. A rotating frame
- 7.8. Nonrigid hyperbolic motion
- 7.9. The Rindler frame
- 7.10. Motion of free particles
- 7.11. Covariant derivative
- 7.12. Electrodynamics in curvilinear coordinates
- References
- Chapter 8: Gravity
- 8.1. Newtonian gravity
- 8.2. Newtonian gravitational field
- 8.3. Gravitational energy
- 8.4. Spherically symmetric fields
- 8.5. Newton's equivalence principle
- 8.6. Tidal force
- 8.7. Search for a relativistic theory of gravity
- 8.8. Einstein's equivalence principle
- 8.9. Gravitational Doppler effect
- 8.10. Gravity affects time
- 8.11. Proper and coordinate times
- 8.12. Strong gravitational field
- References
- Part 2: Mathematics
- Chapter 9: Mathematics of translations
- 9.1. Defining translations
- 9.2. The algebraic structure
- 9.3. Topological structure
- 9.4. Continuous and Lie groups
- 9.5. Generators of translations
- References
- Chapter 10: The rotation group
- 10.1. Rotations and inversions of the plane
- 10.2. Rotations and inversions of the space
- 10.3. Lie algebras
- 10.4. The Lie algebra so(3)
- 10.5. Topology of SO(3) and O(3)
- 10.6. Rotating the fields
- 10.7. The unitary groups U(2) and SU(2)
- 10.8. Topology of SU(2)
- 10.9. Exponentiating 2×2 anti-Hermitian matrices
- 10.10. Action of U(2) and SU(2) on R3
- 10.11. Representations of su(2)
- 10.12. Representations of so(3)
- 10.13. Tensors and index notation
- 10.14. Tensor product
- 10.15. The Euclidean groups
- References
- Chapter 11: The Lorentz group
- 11.1. The Minkowski space-time
- 11.2. The Lorentz group
- 11.3. The SO↑(1,3) matrices
- 11.4. Topology of the Lorentz group
- 11.5. Dependence on the velocity of light
- 11.6. Transforming the fields
- 11.7. Relation to SL(2,C)
- 11.8. The Poincaré group
- References
- Appendix A: Elementary analytic geometry
- A.1. Linear change of coordinates
- A.2. Rotations
- A.3. Hyperbolic rotations
- A.4. Hyperbolic quadratic forms
- Reference
- Appendix B: Active vs. passive rotations
- B.1. The rotation matrix
- B.2. Rotation of vectors
- B.3. Rotation of the coordinate system
- References
- Appendix C: Projective (Möbius) action
- References
- Appendix D: Metric, curvature, and geodesics of a surface
- D.1. Induced metric of a surface
- D.2. Gaussian curvature
- D.3. Motion of a free particle
- D.4. Geodesic motion
- D.5. Geodesics—shortest paths
- D.6. Generalization
- Reference
- Appendix E: Answers to selected problems
- Reference
- Index

## Product details

- No. of pages: 350
- Language: English
- Copyright: © Academic Press 2022
- Published: September 9, 2022
- Imprint: Academic Press
- Paperback ISBN: 9780323997089
- eBook ISBN: 9780323997096

## About the Author

### Ahmad Shariati

Professor Ahmad Shariati is an Associate Professor of Physics with Alzahra University, Iran, a public, women’s university. He has been teaching physics at a variety of levels for more than two decades, from high school students training for the International Physics Olympiad, to undergraduates, to graduate (MS and PhD) students. Active in research and professional groups such as APS and AAAS as well as teaching, Professor Shariati has also translated texts and articles to his native Persian

#### Affiliations and Expertise

Associate Professor, Department of Physics, Alzahra University, Tehran, Iran

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