A Novel Approach to Relativistic Dynamics: Integrating Gravity, Electromagnetism and Optics

Description

This self-contained monograph provides a mathematically simple and physically meaningful model which unifies gravity, electromagnetism, optics and even some quantum behavior. The simplicity of the model is achieved by working in the frame of an inertial observer and by using a physically meaningful least action principle. The authors introduce an extension of the Principle of Inertia. This gives rise to a simple, physically meaningful action function. Visualizations of the geometryare obtained by plotting the action function. These visualizations may be used to compare the geometries of different types of fields. Moreover, a new understanding of the energy-momentum of a field emerges. The relativistic dynamics derived here properly describes motion of massive and massless objects under the influence of a gravitational and/or an electromagnetic field, and under the influence of isotropic media. The reader will learn how to compute the precession of Mercury, the deflection of light, and the Shapiro time delay. Also covered is the relativistic motion of binary stars, including the generation of gravitational waves, a derivation of Snell's Law and a relativistic description of spin. We derive a complex-valued prepotential of an electromagnetic field. The prepotential is similar to the wave function in quantum mechanics.  The mathematics is accessible to students after standard courses in multivariable calculus and linear algebra. For those unfamiliar with tensors and the calculus of variations, these topics are developed rigorously in the opening chapters. The unifying model presented here should prove useful to upper undergraduate and graduate students, as well as to seasoned researchers. Preface Contents Notation List of Figures List of Tables 1 Introduction 1.1 Physics via Geometry–A Historical Perspective 1.2 Unification and Simplicity 1.3 Overview of the Model 1.4 Outline of the Book 1.5 Alternative Theories 2 Classical Dynamics 2.1 Classical Fields 2.2 Motion in the Classical Fields 2.3 The Euler–Lagrange Equations 3 The Lorentz Transformations and Minkowski Space 3.1 Inertial Frames 3.2 Spacetime Transformations that Satisfy the Principle of Relativity 3.2.1 The Galilean Transformations 3.2.2 The Lorentz Transformations 3.3 Einstein Velocity Addition and Applications 3.3.1 Velocity Addition 3.3.2 Fiber Optic Gyroscopes and the Sagnac Effect 3.4 Minkowski Space 3.5 Four-Vectors, Four-Covectors, and Contraction 3.6 Relativistic Energy-Momentum 3.7 Relativistic Doppler Shift 3.8 Lorentz-Covariant Functions for a Single-Source Field 4 The Geometric Model of Relativistic Dynamics 4.1 The Relativity of Spacetime and the Extended Principle of Inertia 4.2 Geodesics on the Globe 4.3 The Geometric Action Function and Its Properties 4.4 Simple Action Function 4.5 Universal Relativistic Equation of Motion 4.5.1 The Equation of Motion Using Proper Time 4.5.2 The Equation of Motion Using tildeτ 5 The Electromagnetic Field in Vacuum 5.1 The Electromagnetic Field Tensor 5.2 The Four-Potential of a Single-Source Electric Field 5.3 The Electromagnetic Field of a Moving Source 5.4 The Electric and Magnetic Components of the Field … 5.5 The Energy-Momentum of an Electromagnetic Field 5.6 The Radiation Field 5.7 The Four-Potential of a General Electromagnetic Field 5.8 The Field of a Current in a Long Wire 5.9 Maxwell's Equations 5.10 Orbits of Charged Particles in a Static, Single-source Field 5.11 Circular Orbits 6 The Gravitational Field 6.1 The Gravitational Field of a Stationary, Static, Spherically … 6.2 Precession of Orbits in a Stationary, Static, Spherically Symmetric Gravitational Field 6.3 Periastron Advance of Binary Stars 6.4 Orbits in the Strong Field Regime 6.4.1 Circular Orbits 6.4.2 Elliptical Orbits 6.4.3 Hyperbolic-Like Orbits 6.5 Gravitational Lensing 6.6 Shapiro Time Delay 6.7 The Gravitational Field of Multiple Sources 6.8 The Gravitational Field of a Moving Source 6.9 Gravitational Waves 7 Motion of Light and Charges in Isotropic Media 7.1 The Photon Action Function of Rest Media 7.2 The Photon Action Function in Moving Media 7.3 Refraction of Light 7.4 Motion of a Charge in an Isotropic Medium at Rest 8 Spin and Complexified Minkowski Spacetime 8.1 History of the Spin of Particles 8.2 The State Space of an Extended Object Moving Uniformly 8.3 Complexified Minkowski Space as the State Space of an Extended Object 8.4 The Representation of the Spin of an Electron 8.5 Transition Probabilities of Spin States and Bell's Inequality 8.6 Motion of Particles with Spin in a Slow-varying Electromagnetic Field 9 The Prepotential 9.1 The Prepotential and the Four-Potential of a Field Generated by a Single Source 9.2 Representations of the Lorentz Group on Mc 9.3 Lorentz Invariance of the Prepotential and the Conjugation 9.4 The Four-Potential of a Moving Source 9.5 The Symmetry of the Complex Four-Potential 9.6 The Prepotential and the Wave Equation 9.7 The Electromagnetic Field Tensor of a Moving Source and its Self-Duality 9.8 The Prepotential of a General Electromagnetic Field Appendix References Index