Beyond Special Relativity: Looking for the Intrinsic Properties of Space-Time

Physics Research and Technology
Beyond Special Relativity
Contents
Acknowledgments
Chapter 1Au Lecteur: APhilosophical-LiteraryJourney through Time andSpace
1.1. A Crack in the Crystal Palace
1.2. A Geopolitical Crisis
1.3. A Short Excursus on Relativity in Literature
1.4. The Enigma of Time in Art
1.5. An Open Conclusion
1.6. Book’s Aim
Part ISpecial Relativity
Subpart ATowards Special Relativity
Chapter 2 MathPrerequisites
2.1. Something about Tensors
Vector and their Invariant Character
Vector as a Matrix
Tensors
2.2. Hyperbolic Functions
Chapter 3The Crisis of ClassicalPhysics: A Falling House ofCards
3.1. Aether Historical Theories in Brief
3.2. The Michelson-Morley Experiment
Arm Parallel to the AetherWind
Arm Perpendicular to the AetherWind
Conclusion
Chapter 4Definitions, Postulates andPrinciples
4.1. Basic Definitions
4.2. Galilean Transformation
4.3. Newton’s Fundamental Laws of Dynamics
4.4. Basic Postulates and Principles
4.5. Frame of References and Frame of Coordinates
Chapter 5Where It All Began: TheLight Clock
5.1. Perpendicular Distances Do Not Change
5.2. The Physical Apparatus
5.3. Two Systems of References
Who Is Really Moving with Respect To?
5.4. Train’s Time
5.5. Rail’s Time
How Is It Possible to Measure It?
Straightness of Trajectory
5.6. The Role Played by the Pythagorean Theorem
5.7. Time Dilation?
5.8. Distance Contractions
5.9. Indirect Relativistic Measures
5.10. Criticism to the Contraction of Distances
5.11. Light’s Climb Rate
5.12. Aberration Angle
Chapter 6The Lorentz Transformations
6.1. Lorentz Space-Equation and Its Inverse
6.2. Lorentz Time-Equation and Its Inverse
Finite Difference Lorentz Equations
6.3. Properties of Lorentz Equations
Deriving Time Dilation and Space Contractions
Length Contraction
Time Dilation
Deriving the Composition of Velocities
6.4. Invariant Interval
Chapter 7Simultaneity and Causality
7.1. Introduction
7.2. Einstein’s Train Paradox
Train Track’s Reference Frame
Train’s Reference Frame
Time Interval Measured by the Train Track
Time Interval Measured by the Train
7.3. Train Paradox - Light Sensors’ Variation
Train Track’s Reference Frame
Train’s Reference Frame
Time Interval Measured by the Train Track
Time Interval Measured by the Train
7.4. Car and Garage Paradox
The Nature of the Paradox
Solving the Paradox
Time Interval Measured by the Car
7.5. Chronological Order
7.6. Causality
7.7. Conclusion
Subpart BSpecial Relativity: Kinematics
Chapter 8Lorentz-Minkowsky’sSpacetime
8.1. Four-Position
8.2. Four-Velocity
8.3. Four-Acceleration
8.4. Lorentz-Minkowsky’s Metric
8.5. Deriving with Respect to Proper Time
Quoting Two Proper Quantities
Quoting Two Non-Proper Quantities: the Celerity
Deriving the Four-Position with Respect to Proper Time
8.6. Universe Lines
8.7. A Look towards Dynamics
Energy
Total Energy
Heat and Work
Four-Momentum
Four-Force
8.8. Some Interesting Solved Exercises
Chapter 9The Accelerated Motion
9.1. A Common Misconception about SR
9.2. Definition of Uniformly Accelerated Motion
9.3. Defining Four-Acceleration
The Derivative of g with respect to t
The Derivative of g~v with respect to t
9.4. The Laws of Uniformly Accelerated Motion
The Square Norm of 4−Acceleration
The 4−Acceleration in MITCF, namely w.r.t. t,s
Attempts to Find out the 4−Acceleration w.r.t. t, s
Law of Motion and ItsWorldline
An Interesting Link with the SEP
Relationships between Proper and Non-Proper Spaces
9.5. Equation Summary
9.6. Boundary Analysis
When Speed Is Much Smaller than c
When Speed Tends to c
9.7. Still Parabolic Motion? No, Hyperbolic!
9.8. Speed versus Time Graphs
9.9. Rindler’s Metric for L-M Spacetime
Defining Rindler’s Coordinates and Metric
Metric Interpretation
Time Dilation in Rindler’s Metric
9.10. Horizons
Horizon’s Analysis in Rindler’s Coordinates
Relative Relativistic Uniformly Accelerated Motions
9.11. Moving Away in Opposite Directions
9.12. A Little Excursion in Cosmology
Universe’s Shape
The Singularity - Single Point Misconception
Homogeneous and Isotropic
Cosmological Principle
Universe Is Homogeneous so It Does Not Have an Edge or a Centre
Ball-Like - Positive-Curved Universe
Flat-Infinite Universe
Flat-Finite Universe
And so, What’s the Universe Like?
9.13. Round Trip to the Edge of the Universe
Photon Moving on a Rubber Carpet
The “Hubble’s Law” and the Expanding Universe
Reaching the Edge of a Finite-Flat Universe
SkyWill Be Forever Black
Chapter 10The Accelerated TwinParadox
10.1. The Original Twin Paradox
Explaining the Paradox
An Everyday Paradox
10.2. Context and Data
10.3. The Paradox No Longer Exists
First Phase: Acceleration
Second Phase: Uniform Rectilinear Motion
Third Phase: Deceleration
Solution
Part IIA Glimpse at GeneralRelativity
Chapter 11Gravitational Lensing andProofs of General Relativity
11.1. Geodesics
11.2. Tangent Spaces
Christoffel Symbols Geometric Definition
Christoffel Symbols Metric Expression
11.3. Schwarzschild Metric
An Historical Note
11.4. Maximal Aging
The Role of Acceleration
11.5. The Eddington Experiment
11.6. The Briatore-Leschiutta Experiment
Solving the Problem: A Prediction of the Result
This Paper-Sheet Is TooWide!
Part IIIConclusion
Chapter 12Relativity in a Nutshell
Part IVAppendices
Appendix AHyperbolic Functions
A.1. Preamble
Trigonometric Functions
Hyperbolic Functions
A.2. Definitions
Inverse Hyperbolic Functions
A.3. Defining the Hyperbolic Functions
A.4. The Hyperbolic Tangent
A.5. Full Geometric Interpretation
Bibliography
About the Authors
Index
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