Today, Relativity is becoming an integrated aspect of engineering fields. Its application to the Global Positioning System (GPS), extends in usage from smart watches to the navigation of cars, airplanes (drones) and even autonomous tractors. In rather expensive particle accelerators, physicists are everyday "playing" with Relativistic Billiards, common to the betatrons of cancer therapy using electrons. Computer programs, such as "ray tracing" methods, are enhanced to simulate objects in relativistic motion, which now offer us relativistic visualizations of accretion disks around compact, astrophysical objects like Black Holes. Against the backdrop of the applications explained throughout the chapters, this book takes on a practical and intuitive approach in introducing the Lorentz invariance of light propagation and space-time concepts. The book begins with simple mathematics, like the classical Pythagoras formula for energy-momentum "triangles". Later, readers will find the intuitive vector calculus reemerging in the expansion of full relativistic expressions. Prepared with instructive diagrams of recent experiments, even the layperson can grasp the essential study of Relativity and marvel at its applications within this book.
Author(s): Eckehard W Mielke
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 207
City: Singapore
Contents
Preface
1. Introduction: “Dawn of Time”
1.1 The Concept of Time in Ancient Cultures
1.2 The Clock Story
1.3 The Definition of the Second
1.3.1 Precursors of the second
1.3.2 First standards
1.4 What is the Most Accurate Clock in the Universe?
1.5 A Brief Biography of Albert Einstein
1.5.1 1905: A special year for physics
1.6 Einstein and Mileva Maric: A Failed Collaboration?
1.6.1 Ludek Zakel: A son “in theory”
References
2. Invariant Light Velocity
2.1 Concepts of Light Propagation
2.2 Rømer and the Moons of Jupiter
2.3 Terrestrial Measurements of Fizeau
2.4 The Michelson–Morley Interferometer
2.5 Decisive Lorentz Contraction
2.6 Isotropy of Light
2.6.1 Comparison with the velocity of neutrinos and
gravitational waves
2.7 Superluminal Velocities?
2.7.1 Tunnel effect
2.8 Homework
References
3. Time Dilation
3.1 Idealized Einstein Clock
3.1.1 Comparison of light clocks in different
inertial frames
3.2 Time Dilation of Muons
3.3 Gravitational Time Dilation
3.3.1 Hafele and Keating experiment
3.4 The Apparent Twin Paradox
3.5 Time Dilation Measured by Optical Clocks
3.6 The GPS Navigation System
3.7 Homework
References
4. Relativistic Shape of Mechanics
4.1 Relativistic Momentum and Force
4.2 Relativistic Energy
4.2.1 Unitary transformation in QED
4.2.2 Examples
4.2.2.1 Rest energy of an electron
4.2.2.2 Fast moving electron
4.3 Homework
Reference
5. Spacetime Diagrams
5.1 Spacetime: Minkowski Diagrams
5.1.1 Observer at rest
5.1.2 Observer in relative motion: Lorentz transformations
5.2 Spacetime Interval
5.2.1 The calibration of the axes
5.2.2 Example: Space rocket
5.2.3 Hyperbolic intersections
5.2.4 Lorentz contraction
5.3 Simultaneity
5.4 Temporal Order: Past, Present, Future, and Causality
5.4.1 Time machines?
5.5 Homework
References
6. Rapidity
6.1 Galilean Relativity
6.2 Lorentz Transformations
6.3 Rapidity in 2D
6.4 Generalized Lorentz Transformations
6.5 Velocity: The General Lorentz Transformations
6.5.1 Aberration
6.5.1.1 Searchlight effect
6.6 Fizeau Interferometer
6.6.1 “Aether drag” experiment in the undergraduate lab
6.6.2 Drag coefficient from Einstein’s addition of velocities
6.7 Homework
References
7. Relativistic Kinematics
7.1 Four-Dimensional Cartesian Coordinates
7.1.1 Invariant distance
7.1.2 Proper time
7.1.3 Time dilation
7.2 Four-D Velocities
7.2.1 Quadratic invariant
7.3 Acceleration in 4D
7.4 Relativistic Momentum
7.5 Quadratic Invariant for the Momentum
7.6 Relativistic Energy Corrections
7.7 Relativistic Shape of Mechanics
7.8 Spinning Top
7.9 Group Velocity of a Particle
7.10 Homework
References
8. Relativistic Billiards
8.1 Collision of Identical Particles
8.1.1 Elastic collisions
8.2 Mandelstam Variables
8.2.1 Center of mass
8.3 Inelastic Collision of Particles in the CM Frame
8.3.1 Scattering angle
8.4 Compton Effect
8.5 Homework
Reference
9. Electrodynamics in 4D
9.1 The Poincaré
Transformations
9.2 Electric Currents in 4D
9.3 Unifying Vector Potential
9.3.1 Aharonov–Bohm effect
9.3.2 Electromagnetic fields derived from potentials
9.3.3 Magnetic moments
9.4 Maxwell’s Equations in Matter
9.5 Electric and Magnetic Fields of a Charge in Uniform Motion
9.5.1 Nanoscopically thin nickel needle as “fake” magnetic monopole
9.6 Lorentz Force
9.6.1 Synchrotron
9.7 Homework
References
10. Relativistic Ray Tracing
10.1 Relativistic Doppler Effect
10.2 Aberration
10.2.1 Bradley discovered the aberration of light
10.2.2 Superluminal velocities in twin jets?
10.3 Invisibility of the Lorentz Contraction
10.3.1 Lampa–Terrell–Penrose rotation
10.4 Visualization: Angular Compression and
Increased Intensity
10.5 Ray Tracing in General Relativity
10.5.1 Relativistic jet emanating from the galaxy M87
10.6 Homework
References
Appendix A: Spacetime Groups
A.1 Group Axioms
A.2 Rotations and Euler Angles
A.3 Homework
A.4 Inhomogeneous Galilean Group
A.5 Homework
A.6 Poincaré Transformations
A.7 Lorentz Group
A.7.1 Lorentz group parameters
A.7.2 Algebraic conditions
A.7.3 Lorentz factor
A.8 Homogeneous and Isotropic Spacetime
Transformations
Reference
Appendix B: Units of the International System
B.1 Homework
References
Bibliography
Index
