Introduction to Relativity: In-Depth and Accessible

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface by Jean Iliopoulos, Dirac Prize 2007
Acknowledgments
Author
General Introduction
1 Relativity of Time and Distance
Introduction
1.1 Time and Space Are Not Absolute
1.1.1 Light Speed Invariance: An Affront to Common Sense
1.1.2 Time and Distance Relativity
1.1.2.1 Einstein’s Famous Train Scenario
1.1.2.2 Time and Distance Relativity
1.2 What Led Einstein to His Conclusion and the Principle of Relativity
1.2.1 The Principle of Relativity
1.2.1.1 Inertial Frame Definition
1.2.1.2 The Galilean Coordinate Transformation and the Classical Velocity Additive Law
1.2.1.3 The Principle of Relativity Was Supported by the Famous Newtonian Laws
1.2.2 The Issues Raised by Maxwell’s Equations
1.2.2.1 Case of an Electrical Charge Moving Beside a Magnet
1.2.2.2 The Galilean Coordinates Transformation Law is Contradicted
1.2.2.3 The Light Speed Invariance Issue
1.2.3 Einstein’s Solution to These Issues
1.3 Examination of the Notions of Distance and Time
1.3.1 Distance, Defined in an Inertial Frame
1.3.1.1 The Straight Line and the Distance
1.3.1.2 Distance Independence of the Location and the Time
1.3.1.3 Practical Limitations
1.3.2 Time, Defined in an Inertial Frame
1.3.2.1 Time in One Point of an Inertial Frame
1.3.2.2 Time in a Whole Inertial Frame
1.3.2.3 The Official Distance Unit Definition Based on the Time
1.4 Minkowski’s Powerful Concepts: The Event and the Proper Time
1.4.1 The Event
1.4.1.1 The Space-Time Separation Four-Vector
1.4.2 The Proper Time and the Proper Distance
1.4.2.1 The Proper Time
1.4.2.2 The Proper Distance
1.5 The Main Postulates of Special Relativity
1.5.1 The Universal Homogeneity and Isotropy
1.5.2 The Inertial Frames Equivalence Postulate
1.5.3 The Light Speed Invariance Postulate
1.6 Questions and Problems
Notes
2 The Lorentz Transformation
Introduction
2.1 Time Dilatation and Other Fundamental Laws
2.1.1 Distance Invariance along the Transverse Directions (y, z)
2.1.1.1 The Tennis Ball Scenario Showing the Invariance of Transverse Distances
2.1.2 The Fundamental Time Dilatation Law
2.1.2.1 Einstein’s Perfect Clock
2.1.2.2 The Famous Time Dilatation Scenario
2.1.2.3 Generalization and Consequences
2.1.2.4 The Time Dilatation Law Is Independent of the Direction of the Clock Motion
2.1.2.5 Beware of Abusive Usage of the Time Dilatation Law
2.1.3 The Length Contraction Law
2.2 Determination of the Lorentz Transformation
2.2.1 The Lorentz Transformation Must Be Linear
2.2.2 The Function Ф Has Only Four Parameters
2.2.3 The Two Scenarios to Obtain Four Equations Involving the Four Parameters
2.2.3.1 Scenario 1
2.2.3.2 Scenario 2
2.2.3.3 Conclusion: The Lorentz Transformation
2.2.3.4 The Reverse Lorentz Transformation
2.2.4 The Lorentz Transformation with the Matrix Format
2.2.5 If the Two Frames have Different Origin-Events, the Event Transformation Is an Affine Function
2.3 Validation of the Lorentz Transformation and First Applications
2.3.1 Experimental Aspects
2.3.1.1 The Rossi & Hall Experiment at Mount Washington
2.3.1.2 Theoretical Aspects
2.3.2 An Important Application: The Doppler Effect
2.3.2.1 Applying Classical Laws Contradicts the Principle of Relativity
2.3.2.2 The Doppler Effect Revisited in Light of Relativity
2.3.2.3 Complements and Remarks on the Doppler Effect
2.4 Minkowski Diagrams
2.4.1 Minkowski Diagrams with One Frame
2.4.1.1 Example of Minkowski Diagram with One Frame
2.4.2 Minkowski Diagrams with Two Combined Frames
2.4.2.1 The Time Dilatation and the Different Lengths of the Unit Frame Vectors in K and K’
2.5 Mathematical Supplements
2.5.1 Demonstration: The Invariance Under Translation Implies the Linearity of Ф
2.6 Questions and Problems
Notes
3 Other New Kinematics Laws, Causality and Accelerated Trajectories
Introduction
3.1 The New Velocity Composition Law and Consequences
3.1.1 The New Velocity Composition Law
3.1.1.1 The Transverse Components of the Object Velocity Are Not Invariant
3.1.2 Composition of Speeds Parallel to the Speed of the Frame K’ Relative to K
3.1.2.1 The Velocity Composition Curve
3.1.2.2 The Velocity’s Composition Law is Commutative
3.1.2.3 Case of Negative Speed v
3.1.2.4 The Reverse Velocity Transformation Law
3.1.3 The Speed c Is Both the Only Invariant Speed and the Maximum One
3.1.3.1 The Speed c Is the Only Invariant Speed
3.1.3.2 The Speed c cannot be Exceeded by Composing Speeds Below C
3.1.3.3 Further Arguments Showing That C Is the Maximum Speed
3.2 The Causality Principle and the Three Categories of Space-Time Intervals
3.2.1 The Causality Principle and the Relativistic Causality Condition
3.2.1.1 Scenario Showing That a Greater Speed than c Contradicts the Causality Principle
3.2.2 Causality Is an Intrinsic Notion, and the Three Types of Space-Time Intervals
3.2.3 Causality and the Space-Time Interval Invariance
3.2.3.1 The Lorentzian Norm Invariance: Mathematical Demonstration
3.2.4 Properties of Time-Like and Space-Like Intervals
3.2.4.1 Properties of Time-Like Intervals
3.2.4.2 Properties of Space-Like Intervals
3.3 The New Metric in the Space-Time Universe
3.3.1 The Lorentzian Norm Invariance: Physical Demonstration
3.3.2 The New Metric
3.4 The Time-Distance Equivalence
3.4.1 Time and Distance Can Be Naturally Expressed with a Single Unit
3.4.1.1 Equivalence Does Not Mean Identity
3.4.2 Physical Laws Are Simpler with Natural Units
3.4.2.1 The Conversion Method t → ct
3.4.2.2 The Lorentzian Norm Expressed with Natural Units, and its Invariance
3.4.2.3 The Lorentz Transformation Expressed with Natural Units
3.4.2.4 Some Reservations as to the General Use of Natural Units
3.5 Special Relativity in the Real World
3.5.1 Langevin’s Famous Paradox
3.5.1.1 Langevin’s Famous Twin Paradox
3.5.2 The “Clock Postulate” or the Proper Time Universality
3.5.3 The “Inertial Tangent Frame Law” (ITF Law)
3.5.3.1 The Inertial Tangent Frame (ITF): Definition and Main Properties
3.5.3.2 The “Inertial Tangent Frame Law” (ITF Law)
3.5.3.3 Langevin’s Travelers Paradox with a Differentiable Accelerated Trajectory
3.5.4 Implication on the Fundamental Law of Inertia
3.5.5 Langevin’s Travelers Paradox and the Relativistic Triangle Inequality
3.5.5.1 The Triangle Inequality of the Lorentzian Distance
3.5.6 Fundamental Implications on the Notion of Time
3.5.6.1 Implication on the Notion of Time: How to Synchronize Time?
3.6 Questions and Problems
Notes
4 The Next Revolution: Dynamics
Introduction
4.1 What’s Wrong with the Main Classical Laws?
4.1.1 Overview of the Main Classical Laws
4.1.1.1 The Newtonian Force
4.1.1.2 Case of Non-Inertial Frames: Inertial Forces
4.1.1.3 The Gravitational Force
4.1.1.4 The Momentum
4.1.1.5 The Concept of Energy
4.1.2 The Classical Momentum Definition Is Not Acceptable in Relativity
4.2 The Momentum in Relativity
4.2.1 Searching for the Relativistic Momentum Definition
4.2.1.1 The General Velocity Definition in Relativity
4.2.1.2 Seeking the Appropriate Time Source for the Relativistic Momentum
4.2.2 Momentum Validation and its Contravariance Property
4.2.2.1 Validity Conditions
4.2.2.2 The Momentum Is a Contravariant 4-Vector
4.2.2.3 The System Momentum Conservation Law Respects the Principle of Relativity
4.2.3 The Momentum Defined with Homogeneous Units
4.2.3.1 The Momentum with Homogeneous Units
4.2.3.2 The Momentum with Fully Homogeneous Units
4.3 Revolutionary Consequences of the Relativistic Momentum
4.3.1 The Inertia of an Object Increases with its Speed
4.3.2 The Meaning of the Mass
4.3.2.1 The Mass is Not Necessarily Constant
4.3.2.2 The Lorentzian Norm of the Momentum is Invariant and Represents the Mass
4.3.3 An Important Novelty: The System Mass
4.3.3.1 The Center of Momentum (CoM) Frame
4.4 Complements
4.4.1 The Total Mass of an Isolated System Remains Constant in Classical Physics
4.4.2 Relativistic Momentum Demonstration in a 3D Approach Using an Inelastic Collision
4.4.1.1 First Step: Demonstration That S[sup(→)] Must Be Extremely Close to V[sup(→)].γv
4.4.1.2 Second Step: Momentum Confirmation with an Elastic Collision
4.5 Questions and Problems
Notes
5 The Energy Revolution: E=mc²γ and Its Consequences
Introduction
5.1 The Relativistic Energy Expression
5.1.1 The Relativistic Energy and the Time Part of the Momentum
5.1.2 The Energy with Fully Natural Units and with the International Unit (Joule)
5.1.2.1 The Energy with Fully Natural Units
5.1.2.2 The Term c² Is there Because the International System of Unit Is Not the Natural One
5.1.3 First Remarks on the Relativistic Energy Expression
5.1.3.1 First Remarks on the Term mc²
5.1.3.2 First Remarks on the Kinetic Part
5.1.3.3 A Useful Relation: E² = m² c + p² c²
5.2 The Energy Inertia Equivalence and Other Revolutionary Consequences
5.2.1 The Energy Required to Accelerate an Object to the Speed C Is Infinite
5.2.2 The Energy Inertia Equivalence and Nuclear Reactions
5.2.2.1 Some Remarkable Cases of Exchanges Between Mass and Energy
5.2.2.2 Mass Changes During Inelastic Collisions
5.2.3 Nuclear Reactions: Fission and Fusion
5.2.4 Units Adapted to Particles and Typical Nucleons Masses
5.2.5 Naming and Physical Considerations
5.3 The Force in Relativity
5.4 Physical Considerations on the Photon
5.4.1 General Considerations on the Photon
5.4.1.1 The Photon Energy
5.4.1.2 The Photon Momentum
5.4.1.3 An Atom that Emits a Photon Loses Mass
5.4.2 The Famous Photoelectric Effect
5.5 Questions and Problems
Notes
6 Introduction to General Relativity
Introduction
6.1 The Principle of Equivalence
6.1.1 Statement of the Principle of Equivalence and first Consequences
6.1.2 Limitation of the Principle of Equivalence: The Tidal Effect
6.1.2.1 The Tidal Effect on Earth
6.2 Important Consequences of the Principle of Equivalence
6.2.1 The Bending of Light Near Very Massive Objects
6.2.2 Impacts of Gravity on Time
6.2.2.1 The Proper Time Universality
6.2.2.2 The Gravitational Time Dilatation
6.2.3 Revisit of the Fundamental Postulates and Concepts
6.3 Curved Surfaces: Distance and Curvature Characterization
6.3.1 Distance in 2D Surfaces
6.3.1.1 Distance in Simple 2D Plane Surfaces
6.3.1.2 Distance Calculation on a Curved Surface
6.3.1.3 Methodology for Distance Calculation on the Curved Surface
6.3.1.4 Distance Calculation Between Distant Points, and the Geodesic
6.3.2 Toward 4D: Case of 3D Deformable Volumes
6.3.2.1 Distorted 4D Space-Time Universe
6.3.2.2 The Lorentzian Distance
6.3.3 Relationship Between Distance and Chrono-Geometry
6.3.4 The Geodesic: Main Principles
6.4 The Equations of General Relativity
6.4.1 Main Principles Leading to the Equations of General Relativity
6.4.2 Confirmations of General Relativity: Mercury’s Perihelion and the Bending of Light
6.4.2.1 First Confirmation: Solution to the Mercury Perihelion Enigma
6.4.2.2 Schwarzschild’s Solution and the Bending of Light
6.5 Questions and Problems
Notes
7 Cosmological Consequences
Introduction
7.1 The Big Bang Theory
7.2 Black Holes: The Long and Chaotic Road That Led to Their Recognition and Discovery
7.3 Gravitational Waves: First Observations One Century After Their Prediction
7.3.1 Theoretical Aspects
7.3.2 First Gravitational Wave Detection on Earth in 2015
7.3.3 A New Powerful Tool for Cosmology
Notes
8 Epilogue: A Scientific and Epistemological Revolution
Introduction
8.1 How Relativity Was Accepted
8.2 An Epistemological Revolution
8.3 Who Was Albert Einstein
Notes
9 Solutions to Questions and Problems
9.1 Answers to Questions and Problems of Chapter 1
9.2 Answers to Questions and Problems of Chapter 2
9.3 Answers to Questions and Problems of Chapter 3
9.4 Answers to Questions and Problems of Chapter 4
9.5 Answers to Questions and Problems of Chapter 5
9.6 Answers to Questions and Problems of Chapter 6
Appendix 1: Summary of the Main Constants
Bibliography
Index