The Special Theory of Relativity - A Mathematical Approach

Description

This textbook expounds the major topics in the special theory of relativity. It provides a detailed examination of the mathematical foundation of the special theory of relativity, relativistic mass, relativistic mechanics, and relativistic electrodynamics. As well as covariant formulation of relativistic mechanics and electrodynamics, the text discusses the relativistic effect on photons. A new chapter on electromagnetic waves as well as several new problems and examples have been included in the second edition of the book. Using the mathematical approach, the text offers graduate students a clear, concise view of the special theory of relativity. Organized into 15 chapters and two appendices, the content is presented in a logical order, and every topic has been dealt with in a simple and lucid manner. To aid understanding of the subject, the text provides numerous relevant worked-out examples in every chapter. The mathematical approach of the text helps students in their independent study and motivates them to research the topic further. Preface to the Second Edition Preface to the First Edition Contents About the Author 1 Pre-relativity and Galilean Transformation 1.1 Failure of Newtonian Mechanics 1.2 Galilean Transformations 1.3 Galilean Transformations in Vector Form 1.4 Non-inertial Frames 1.5 Galilean Transformation and Laws of Electrodynamics 1.6 Attempts to Locate the Absolute Frame 2 Michelson–Morley Experiment and Velocity of Light 2.1 Attempts to Locate Special Privileged Frame 2.2 The Michelson–Morley Experiment (M–M) 2.3 Phenomena of Aberration: Bradley's Observation 2.4 Fizeau's Experiment 2.5 The Relativistic Concept of Space and Time 3 Lorentz Transformations 3.1 Postulates of Special Theory of Relativity 3.2 Lorentz Transformations 3.2.1 Lorentz Transformation Between Two Inertial Frames of Reference (Non-axiomatic Approach) 3.2.2 Axiomatic Derivation of Lorentz Transformation 3.2.3 Lorentz Transformation Based on the Postulates of Special Theory of Relativity 3.3 The General Lorentz Transformations 3.4 Thomas Precession 4 Mathematical Properties of Lorentz Transformations 4.1 Length Contraction (Lorentz–Fitzgerald Contraction) 4.2 Time Dilation 4.3 Relativity of Simultaneity 4.4 Twin Paradox in Special Theory of Relativity 4.5 Car–Garage Paradox in Special Theory of Relativity 4.6 Real Example of Time Dilation 4.7 Terrell Effects 5 More Mathematical Properties of Lorentz Transformations 5.1 Interval 5.2 The Interval Between Two Events Is Invariant Under Lorentz Transformation 6 Geometric Interpretation of Space–Time 6.1 Space–Time Diagrams 6.2 Some Possible and Impossible World Lines 6.3 Importance of Light Cone 6.4 Relationship Between Space–Time Diagrams in S and S1 Frames 6.5 Geometrical Representation of Simultaneity, Space Contraction and Time Dilation 6.5.1 Simultaneity 6.5.2 Space Contraction 6.5.3 Time Dilation 7 Relativistic Velocity and Acceleration 7.1 Relativistic Velocity Addition 7.2 Relativistic Velocity Transformations 7.3 Relativistic Acceleration Transformations 7.4 Uniform Acceleration 7.5 Relativistic Transformations of the Direction Cosines 7.6 Application of Relativistic Velocity and Velocity Addition Law 7.6.1 The Fizeau Effect: The Fresnel's Coefficient of Drag 7.6.2 Aberration of Light 7.6.3 Relativistic Doppler Effect 8 Four-Dimensional World 8.1 Four-Dimensional Space–Time 8.2 Proper Time 8.3 World Velocity or Four Velocities 8.4 Lorentz Transformation of Space and Time in Four-Vector Form 9 Mass in Relativity 9.1 Relativistic Mass 9.1.1 First Method Based on Hypothetical Experiment of Tolman and Lews 9.1.2 Second Method Based on a Thought Experiment 9.1.3 Third Method 9.2 Experimental Verification of Relativistic Mass 9.3 Lorentz Transformation of Relativistic Mass 10 Relativistic Dynamics 10.1 Four Forces or Minkowski Force 10.2 Four Momenta 10.3 Relativistic Kinetic Energy 10.4 Mass–Energy Relation 10.5 Relation Between Momentum and Energy 10.6 Evidence in Support of Mass–Energy Relation 10.7 Force in Special Theory of Relativity 10.8 Covariant Formulation of Newton's Law 10.9 Examples of Longitudinal Mass and Transverse Mass 10.10 The Lorentz Transformation of Momentum 10.11 The Expression p2 - E2c2 Is Invariant Under Lorentz Transformation 11 Photon in Relativity 11.1 Photon 11.2 Compton Effect 11.3 The Lorentz Transformation of Momentum of Photon 11.4 Minkowski Force for Photon 12 Relativistic Lagrangian and Hamiltonian 12.1 Relativistic Lagrangian 12.2 Relativistic Hamiltonian Function 12.3 Covariant Lagrangian and Hamiltonian Formulation 12.4 Lorentz Transformation of Force 12.5 Relativistic Transformation Formula for Density 13 Electrodynamics in Relativity 13.1 Relativistic Electrodynamics 13.2 Equation of Continuity 13.3 Maxwell's Equations 13.4 Derivation of Equation of Continuity from Maxwell's Equations 13.5 Displacement Current 13.6 Transformation for Charge Density 13.7 Four Current Vector 13.8 Equation of Continuity in Covariant Form 13.9 Transformation of Four Current Vector 13.10 Maxwell's Equations in Covariant Form 13.10.1 The d'Alembertian Operator is Invariant Under Lorentz Transformation 13.10.2 Lorentz-Gauge Condition in Covariant Form 13.10.3 Gauge Transformations 13.11 Transformation of Four Potential Vector 13.12 The Electromagnetic Field Tensor 13.13 Lorentz Transformation of Electromagnetic Fields 13.14 Maxwell's Equations Are Invariant Under Lorentz Transformations 13.15 Lorentz Force on a Charged Particle 13.16 Electromagnetic Field Produced by a Moving Charge 13.17 Relativistic Lagrangian and Hamiltonian Functions … 14 Electromagnetic Waves 14.1 Introduction 14.2 Wave Equation for Magnetic Intensity "0245H 14.3 Wave Equation for Electric Field Strength "0245E 14.4 Electromagnetic Waves in a Non-conducting Dielectric Medium 14.5 Poynting's Theorem (Energy Conservation) 14.6 Boundary Conditions 14.7 Plane Electromagnetic Waves in a Non-conducting Isotropic Medium 14.8 Plane Electromagnetic Waves in a Conducting Medium 14.9 Skin Depth 14.10 Wave Guides 14.11 Coulomb Gauge 14.12 Hertz Vector 14.13 A Brief Introduction of Relativistic Wave Equation 15 Relativistic Mechanics of Continua 15.1 Relativistic Mechanics of Continuous Medium (Continua) Appendix Appendix A Appendix Appendix B Appendix References Index