Looking at Time from a Physics Perspective

The subject of time continues to be a subject of extensive research in the development of new theories of physics. This new volume is addressed to students who are starting a graduate program in physics or electrical engineering interested in complementing their studies of relativity theory and quantum physics, applying the knowledge they have acquired about these themes to the analysis of situations where the issue of time measurement is relevant. This is the case, for example, of clock synchronization, transit times of optical signals through dielectric and absorbing media, lifetimes of excited atomic states, among others. These topics, in addition to being of great importance to theoretical physicists, are the basis of many technological developments. For example, global positional systems (GPS) are based on the predictions of relativity theory about time and the effect of gravity over time measurement. Divided into six chapters, the volume discusses how the concept of time is present in the main fields of physics, such as classical mechanics, electrodynamics, quantum mechanics and theory of relativity. Illustrative examples and case studies are included in each chapter. The volume includes an analysis of themes related to time such as causality and the arrows of time, spooky action at distance and Einstein-Podolsky-Rosen Paradox, quantum mechanics and entangled states, apparent superluminal velocity, and time reversal. This book, Looking at Time from a Physics Perspective , will contribute to the understanding of concepts learned in courses on classical mechanics, electrodynamics, quantum mechanics and relativity, reviewing the implications of the time variable for the description of the different physical phenomena at the microscopic and macroscopic level. Cover Half Title Title Page Copyright Page Dedication About the Author Table of Contents Abbreviations Preface Introduction 1. Time in Classical Mechanics Introduction Time-Reversal Symmetry Time Reversal for a Simple Case: The Dynamics of 1 Projectile Reversible Laws of Motion in Classical Mechanics Dissipation and Time-Reversal Asymmetry Time Reversal of a Conservative System Time-Reversal Symmetry of the Dynamics of a Relativistic Particle Time Reversal of a Mechanical Wave Time Translational Symmetry Conclusions of Chapter 1 2. About Time in Electromagnetic Theory Introduction Time-Reversal Invariance of Maxwell Equations in Vacuum Time Reversal of Electromagnetic Waves in Vacuum Time Reversal of Material Relations Effect of Dispersion in Time-Reversal Symmetry Time Reversal of Processes with Transfer of Energy Doppler Effect and Time Reversal Causality in Electrodynamics An Arrow of Time for Electromagnetic Radiation Optical Dirac-Like Equation and Time-Reversal Symmetry Electromagnetic Nonreciprocity and Time Reversal Application of Time Reversal for Locating Lightning Discharges Conclusions of Chapter 2 3. Time in Non-Relativistic Quantum Mechanics Introduction Time-Reversal Symmetry Breaking of Time-Reversal Symmetry by an External Magnetic Field Time Reversal of the Photoelectric Effect Time-Reversal Operator for Systems with Spin Time Reversal and Geometric Phases Effect of Dissipation in a Quantum System Entanglement and Causality What About Time Symmetry in Quantum Optics? PT—Symmetric Quantum Mechanics Conclusions of Chapter 3 4. Time in Special Relativity Theory Introduction Fundamental Postulates of Relativity Theory Time and Length Measurements The Space Time and Lorentz Transformations About the Simultaneity of Events Space Time and the Light Cone Some Reflections About Causality and Time Causality and Superluminal Signals Relativistic Transformation of Velocities Time Dilation of the Lifetime of a Particle Time and Relativistic Doppler Effect Lorentz Transformations of Other Physical Quantities A Note About Superluminal Particles Conclusions of Chapter 4 5. Time in General Relativity Theory Introduction Equivalence Principle Gravity and Light Geometry of Space Time Plausibility Arguments for Schwarzschild Metric Gravity and Time Effects of Gravitational Field Due to Earth on Time Measurements Application of General Relativity Theory to Global Positioning System An Overview of Einstein’s Field Equations General Relativity and Blackholes Time Reversal Symmetry in an Expanding Universe Model Time Reversal Symmetry in Cosmology Time Traveling Through Wormholes: Is it Possible? Quantum Entanglement According to General Relativity Measurement of Intervals Between Events in the Space Time General Relativity Theory and Quantum Mechanics Conclusions of Chapter 5 6. A Glance at Some Recent Theories About Time in Physics Introduction Time in the Cellular Automaton Theory The Thermal Time Hypothesis Time as a Discrete Dynamical Variable Quantum Operator for Time in Non-Relativistic Quantum Mechanics Quantum Operator for Time in Relativistic Quantum Mechanics Time in Quantum Gravity Exploring the Physics of Time Travel Conclusions and Epilogue of Chapter 6 Index