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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