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

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What is time? And space? How are these two apparently opposed elements indissolubly linked? This book starts from man's intrinsic need to ask himself questions and elaborate answers, and tries to analyze, through a simple and intuitive physical-mathematical language, some aspects of Einstein's Special and General Relativity, accompanied by equally clear and immediately understandable images. An interesting discussion of simultaneity and causality is presented together with the relativistic accelerated version of the original twin paradox. 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 Blank Page