E=mc² is known as the most famous but least understood equation in physics. This two-volume textbook illuminates this equation and much more through clear and detailed explanations, new demonstrations, a more physical approach, and a deep analysis of the concepts and postulates of Relativity.
Volume II contains, notably:
- In Special Relativity: complementary explanations, alternative demonstrations relying on more advanced means and revealing other aspects. Further topics: accelerated objects and the Relativistic force, nuclear reactions, the use of hyperbolic trigonometry, the Lagrangian approach, the Relativistic Maxwell’s equations.
- In General Relativity: tensors, the affine connection, the covariant derivative, the geodesic equation, the Schwarzschild solution with two of its great consequences: black holes and the bending of light; further axiomatic considerations on time, space, matter, energy and light speed.
- In Cosmology: the FLRW Metric, the Friedman equation, the cosmological constant, the four ideal cosmological Models.
These subjects are presented in a concrete and incremental manner, and illustrated by many case studies. The emphasis is placed on the theoretical aspects, with rigorous demonstrations based on a minimum set of postulates. The mathematical tools dedicated to Relativity are carefully explained for those without an advanced mathematical background.
Both volumes place an emphasis on the physical aspects of Relativity to aid the reader’s understanding and contain numerous questions and problems (147 in total). Solutions are given in a highly detailed manner to provide the maximum benefit to students.
This textbook fills a gap in the literature by drawing out the physical aspects and consequences of Relativity, which are otherwise often second place to the mathematical aspects. Its concrete focus on physics allows students to gain a full understanding of the underlying concepts and cornerstones of Relativity.
Author(s): Paul Bruma
Edition: 1
Publisher: CRC Press
Year: 2022
Language: English
Commentary: Publisher PDF
Pages: 272
City: Boca Raton, FL
Tags: Lorentz Transformation; Relativistic Laws of Dynamics; General Relativity; Cosmological Models
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface by Jean Iliopoulos, Dirac Prize 2007
1 Steps before the Lorentz Transformation
Introduction
1.1 Observations and Experiments That Helped Characterize the Ether
1.1.1 Stellar Aberrations
1.1.1.1 Calculations and Consequences
1.1.1.2 Classical and Relativistic Calculations in the General Case
1.1.2 The Michelson–Morley Experiment
1.1.2.1 Presentation of the Michelson–Morley Experiment
1.1.2.2 Calculations of the Fringe Shift
1.2 Complements to the Principle of Relativity
1.2.1 The Existence of a Permanent Center of Symmetry between Two Inertial Frames
1.2.1.1 The Existence of a Speed of K″ Such That at
t″ = 1, an Observer in O″ States: O″O = O″O′
1.2.2 Fundamental Implications of the Principle of Relativity
1.2.2.1 The Proper Time and the Proper Distance Are the Same in All Inertial Frames
1.2.2.2 The Speed of Light Is the Same in K and in K′
1.2.2.3 The Reciprocity of Velocities between Inertial Frames
1.2.2.4 Any Physical Experiment Gives the Same
Results in K as the Same Experiment in K′
1.2.2.5 Illustration with Galileo’s Famous Sailboat Scenario, and Further Comments
1.2.3 Case of Non-Inertial Frames: Inertial Forces
1.2.3.1 Generalization
1.2.3.2 Fundamental Explanation of Inertial Forces: The Mach Principle
1.3 Methodological Considerations
1.3.1 The Observer’s Role in Relativity: Possibilities and Limits of Contradictory Statements
1.3.1.1 Further Axiomatic Consideration: The Event Coincidence Invariance Postulate
1.3.2 “Intrinsic” Notions: Definition and Properties
1.3.2.1 Definition of an Intrinsic Notion
1.3.2.2 Properties of Intrinsic Notions
1.3.3 “Primitive” Notions: Definition and Properties
1.3.3.1 Primitive Notions Reflect the State of
Knowledge at a Given Time
1.3.3.2 Quasi-Primitive Notions
1.3.3.3 Primitive and Quasi-Primitive Notions Are Intrinsic
1.4 Characterization of the New Coordinate Transformation Function
1.4.1 The New Transformation Must Be Linear
1.4.2 Invariance of Transverse Distances: Complete Demonstration
1.4.2.1 The Image of the OX and OY Axes of K
Must Be the O′X′ and O′Y′ Axes of K′
1.4.2.2 There Is No Relativity of Simultaneity along the Transverse Directions
1.4.2.3 Distance Invariance along the Transverse Directions
1.4.2.4 Comment
1.4.3 Length Contraction Law Demonstration Using Basic Relativistic Schemes
1.4.4 The Inertial Frames Velocity Reciprocity Law
1.4.5 Time Dilatation: Beware of Possible Misinterpretations
1.4.5.1 Further Comments
1.5 Supplements
1.5.1 The Fizeau Experiment: Light in a Moving Medium
1.5.2 The Gamma Factor: Approximation with the Taylor Polynomial Method
1.5.2.1 The Taylor Polynomial Method
1.5.2.2 Gamma Factor Approximation with the Taylor Polynomial Method
1.5.3 The Derivatives of the Gamma Factor
1.5.3.1 The Derivative of the Gamma Factor
when β Is a Function of the Time
1.6 Questions and Problems
Notes
2 Lorentz Transformation, New Metric and Accelerated Objects
Introduction
2.1 The Lorentzian Norm Invariance
2.1.1 Mathematical Demonstration of the Lorentzian Norm Invariance (ds[sub(2)])
2.1.2 Important Consequences of the Lorentzian Norm Invariance
2.2 Lorentz Transformation: Two Demonstrations
2.2.1 Lorentz Transformation Demonstration with the Symmetry Argument
2.2.2 Lorentz Transformation Demonstration Using Hyperbolic Trigonometry
2.3 Minkowski Diagrams
2.3.1 Why Do the Combined Minkowski Diagrams Work Like Magic?
2.3.2 The Minkowski–Loedel Diagram
2.3.2.1 The Angle α between the Two Axes: OX
and OX′ (or OcT and OcT′)
2.3.2.2 Examples of Application: The Time Dilatation Law
2.3.2.3 Second Example of Application: The Distance Contraction Law
2.4 The Minkowski Space with Complex Numbers
2.4.1 The Lorentz Transformation Is a Rotation in Minkowski’s Complex Space
2.4.2 Consequences
2.4.2.1 The Reverse Minkowski–Lorentz Matrix Is Its Transpose
2.4.2.2 The Velocity Composition Law
2.4.2.3 After a Boost, the Basis Vectors Are Complex
2.5 From Complex Numbers to Hyperbolic Trigonometry
2.5.2 Examples of Application
2.5.2.1 The Velocity Composition Law Obtained with Hyperbolic Trigonometry
2.5.2.2 The Invariance of the Lorentzian Norm: A Property of Hyperbolic Rotations
2.6 Special Relativity in the Real World
2.6.1 Demonstration of the ds[sub(2)] Triangle Inequality with the Mixed-Mode Graph
2.6.2 Accelerated Trajectories, and Problems for Time Synchronization
2.6.2.1 Langevin’s Twins Paradox: General Case
2.6.3 Mathematical Relations between the Acceleration in
K and in K′
2.6.3.1 Case Where the Acceleration Is Measured in the Inertial Tangent Frame of the Moving Object
2.6.4 Another Surprising Effect: The Event Horizon
2.6.5 The Proper Acceleration
2.7 Supplements
2.7.1 Geometrical Demonstration of the Lorentzian Distance Triangle Inequality
2.7.2 Twins Scenario Using the Minkowski Diagram: The Line of Simultaneity
2.7.2.1 Interpretation of This Scenario
2.7.3 The Gamma Factor Composition Law
2.7.4 Langevin’s Travelers Paradox, and Problems for Synchronizing the Time
2.7.4.1 Problem with Time Synchronization
2.7.5 Minkowski’s Complex Space: Lorentz Transformation is a Rotation, Lorentzian Norm is Invariant
2.8 Mathematical Supplements
2.8.1 A Classical Rotation Conserves the Classical Norm of Any Vector
2.8.2 A Classical Rotation Conserves the Scalar Product
2.8.3 Reminder on the Main Hyperbolic Functions
2.8.3.1 Sinh, Cosh and Tanh Curves
2.8.3.2 Artanh Curve
2.8.3.3 Useful Hyperbolic Trigonometric Relations
2.9 Questions and Problems
Notes
3 The Relativistic Laws of Dynamics
Introduction
3.1 Energy and Momentum from 3D to 4D
3.1.1 The Conservation of the 3D Quantity, p = mvγ[sub(v)], Implies the Conservation of a Scalar Quantity: Energy
3.1.2 Energy Conservation Implies the Conservation of the 3D Momentum p = mvγ[sub(v)]
3.1.3 Geometrical Representation of the System Momentum Conservation and of the CoM Frame
3.2 Nuclear Reactions
3.2.1 The Three Main Types of Decay
3.2.1.1 Alpha Decay
3.2.1.2 Beta Decay
3.2.1.3 Gamma Decay
3.2.2 Fusion and Fission: Theoretical Conditions
3.2.2.1 Nuclear Binding Energy
3.2.2.2 Fusion and Fission Conditions
3.2.3 The Proton–Proton Chain
3.2.4 Relations Concerning Mass and Energy in Disintegrations
3.2.4.1 Disintegration into Two Bodies
3.2.4.2 The Sum of the Resulting Masses Is Less Than E/c[sup(2)] in the CoM
3.2.5 Ultra-Relativistic Particles
3.3 The Force in Relativity
3.3.1 The Newtonian-Like Force and the Force of Minkowski
3.3.1.1 The Newtonian-Like Force
3.3.1.2 The Force of Minkowski
3.3.2 The Work of the Force in the Context of Constant Mass
3.3.3 Other Properties of the Newtonian-Like Force
3.3.3.1 The Power of the Newtonian-Like Force
3.3.3.2 The Acceleration Due to the Newtonian-Like Force
3.3.3.3 The Newtonian-Like Force Complies with Force Reciprocity
3.3.4 The Four-Force in Case of Mass Variation
3.3.5 Potential Energy Update and Applications
3.3.5.1 The Potential Energy of the Gravity Field in the General Case
3.4 The Unity between Electrical and the Magnetic Forces
3.5 The Lagrangian Approach for Momentum and Energy
3.5.1 Lagrangian History and Basic Considerations
3.5.1.1 The Lagrangian Theory
3.5.1.2 The Famous Noether Theorem
3.5.2 Application to Relativity
3.5.3 The Euler-Lagrange Equations
3.6 Complements and Case Studies
3.6.1 The Compton Effect
3.6.2 A Thought Experiment by Einstein Revisited: The Photon Box
3.6.2.1 First Analysis Based on Classical Physics Principles
3.6.2.2 This Scenario Revisited in the Light of Relativity
3.6.2.3 The Center of Gravity in Classical Physics and the Center of Inertia in Relativity
3.6.2.4 Consequence Regarding Our Photon Box Scenario and Alternative Method
3.6.2.5 Conclusive Comments
3.6.3 The Mossbauer Effect
3.6.4 Mathematical Demonstration of the 4D-Relativistic Momentum
3.6.4.1 Application to a Collision Seen from Two Inertial Frames
3.6.5 Demonstration of the Famous Energy Relation E = mc[sup(2)]γ from the Work of the Force
3.6.5.1 The Derivative of the Lorentzian Scalar Product
3.6.6 Supplements to the Potential Energy: The Poissons Equation and the Orbital Potential Energy
3.6.6.1 The Poisson’s Equation
3.6.6.2 The Orbital Potential
3.6.6.3 The Gravitational Time Dilation Effect in General Relativity
3.7 Questions and Problems
Notes
4 Introduction to General Relativity: Tensors, Affine Connection, Geodesic Equation
Introduction
4.1 Tensors
4.1.1 Tensors of Rank 1
4.1.1.1 Contravariant Tensors of Rank 1
4.1.1.2 Covariant Tensors of Rank 1
4.1.1.3 Relation with the Lorentzian Distance
4.1.1.4 Generalization to a Curved Surface
4.1.2 Tensors of Rank 2 and Remarkable Tensors
4.1.2.1 Rank 2 Tensors Definition and Main Properties
4.1.2.2 The Metric Tensor
4.1.2.3 The Lorentz Matrix
4.1.2.4 The Electromagnetic Tensor
4.1.2.5 The Stress–Energy–Tensor
4.1.3 The Dual of the Dual Space, and the Possibility to Raise or to Lower the Indices
4.1.3.1 The Dual of the Metric Tensor
4.1.3.2 Possibility to Raise or Lower the Indices
4.2 The Affine Connection, the Covariant Derivative and the Geodesic
4.2.1 The Riemannian Manifold and the Momentum Constancy Relation
4.2.1.1 The Pseudo-Riemannian Manifold
4.2.1.2 The Scheme for Characterizing the Momentum Constancy
4.2.2 The Affine Connection
4.2.3 The Covariant Derivative and the Parallel Transport
4.2.3.1 The Parallel Transport
4.2.4 Generalization to Our 4D Space–time Universe
4.2.4.1 The Affine Connection and Its Relation with the Space–Time Curvature
4.2.4.2 The Covariant Derivative and Parallel Transport
4.2.5 The Proper Time Is the Privileged Parameter for Indexing an Object Trajectory
4.2.6 The Geodesic Equation
4.2.6.1 Expression of the Property (A) in a Riemannian Space
4.2.6.2 Expression of the Property (B) in a Riemannian Space
4.3 Supplements
4.3.1 Case Study of the 2D Sphere: Christoffel Coefficients
and the Geodesic
4.3.1.1 Determination of the Christoffel
Coefficients on the Equator
4.3.1.2 Determination of the Christoffel
Coefficients at any Latitude
4.3.1.3 Application: The Geodesic on a 2D Sphere
4.3.2 Demonstration of the Relationship between the
Connection Coefficients and the Metric
4.4 Questions and Problems
Notes
5 Important Consequences: Relativistic Maxwell’s Laws, Schwarzschild’s Solution
Introduction
5.1 Relativistic Maxwell’s Laws
5.1.1 Introduction
5.1.2 The Electromagnetic Force of Minkowski and Faraday’s Tensor
5.1.2.1 The Frame-Changing Rule for Faraday’s Tensor
5.1.3 The Source of the Electromagnetic Field: The Current Density Tensor
5.1.3.1 The Divergence of the Current Vector J Is Null If the Charge Density Is Constant
5.1.4 The Relativistic Maxwell’s Laws
5.1.4.1 The First Approach
5.1.4.2 The Second Approach
5.1.5 Further Topics: The Covariant Faraday’s Tensor
5.1.6 Mathematical and Physical Supplements
5.1.6.1 The Vector Product
5.1.6.2 The Gradient
5.1.6.3 The Divergence
5.1.6.4 The d’Alembertian Operator
5.1.6.5 The Curl Operator
5.1.6.6 The Electromagnetic Tensor Must Be Antisymmetric
5.1.6.7 Demonstration That (A[sup(v)]) = μ[sub(0)]J[sup(v)] Using Classical Notations
5.2 Schwarzschild’s Solution
5.2.1 Polar Coordinates: Reminder and Adaptation to the 4D Space-Time Universe
5.2.1.1 Classical Polar Coordinates in 3D
5.2.1.2 Polar Coordinates in the 4D Space-Time Universe
5.2.2 Meaning of Schwarzschild’s Coordinates and the Possibility of Black Holes
5.2.2.1 The Radial Basis Vector i
5.2.2.2 The Time Basis Vector f and the Possibility of Black Holes
5.2.3 The Bending of Light Near a Great Mass
5.2.3.1 Other Consequences
5.2.4 Trajectories of Massive Objects and the ISCO Orbit
5.3 Questions and Problems
Notes
6 Introduction to Cosmological Models
Introduction
6.1 The Friedmann–Lemaître–Robertson–Walker Metric
6.1.1 The General Shape of a Homogeneous and Isotropic Universe
6.1.2 The FLRW Metric
6.1.2.1 Classical Distance in a 2D Homogeneous and Isotropic Curved Space
6.1.2.2 First Consequence: Limits in the Curvature of our Universe
6.1.2.3 Application to a 3D Homogeneous and Isotropic Curved Space and the FLRW Metric
6.1.3 Relation between the Redshift and the Scale Factor
6.1.3.1 The Proper Distance with the FLRW Metric, and the Apparent Recession Speed
6.1.3.2 Relation between the Redshift and the Scale Factor
6.2 The Friedmann Equation
6.2.1 Friedmann Equation with Classical Physics
6.2.2 Adaptation to General Relativity and First Consequences
6.2.2.1 The Current Situation and the Relation with the Hubble Constant
6.2.2.2 Relation between Space Curvature and Energy Density: The Critical Energy Density
6.2.3 The Fluid Equation
6.2.4 The Acceleration Equation
6.3 Cosmological Models
6.3.1 The Cosmological Constant
6.3.2 State Equations and Universe Evolution Models
6.3.2.1 The State Equations
6.3.2.2 Universe Evolution
6.3.3 The 4 Ideal Scenarios
6.3.3.1 Ideal Scenario of a Flat Universe with Lambda Only
6.3.3.2 Ideal Scenario of an Empty Universe
6.3.3.3 Ideal Scenario of a Flat Universe with Radiation Only
6.3.3.4 Ideal Scenario of a Flat Universe with Matter Only
6.3.3.5 Concluding Comments
6.4 Questions and Problems
Notes
7 Further Axiomatic Considerations
Introduction
7.1 Further Considerations on Time, Causality and Event
7.1.1 Time Revisited by Relativity
7.1.1.1 Restrictions Brought by Special Relativity on the Notion of Time
7.1.1.2 Further Restrictions Brought by General Relativity
7.1.2 The Causality Principle and the Impossibility of Greater Speeds than C
7.1.3 Axiomatic Considerations on the Concepts of Time and Event
7.1.3.1 The Proper Time and the Proper Distance as Primitive Notions?
7.1.3.2 The Event as a Primitive Notion?
7.2 Axiomatic Analysis of the Laws of Dynamics
7.2.1 The Root: The Law of Inertia
7.2.2 From the Law of Inertia to the Momentum
7.2.2.1 P Matches with the Relativistic Momentum
7.2.2.2 The Scalar mIs the Object’s Intrinsic Energy
7.2.2.3 The Inertia-Energy Equivalence
7.2.2.4 The Momenergy and the System Mass
7.2.3 Alternative Possibility: The Momenergy as a Primitive Concept
7.2.3.1 The Intimate Links between Proper Time, Proper Distance and Momenergy
7.2.3.2 Philosopher’s Views
7.3 Axiomatic Considerations Concerning the Speed of Light
7.3.1 The Invariance of Light Speed Is Not a Necessary Postulate
7.3.1.1 The Velocity Composition Law Must Still Be a Homographic Function
7.3.1.2 Characteristics of Homographic Functions Matching with the Velocity Composition Law
7.3.1.3 There Is One, and Only One, Invariant Speed; and It Is the Same in All Inertial Frames
7.3.1.4 Conclusion
7.3.2 Visit in the Hypothetical Tachyon’s World
7.3.2.1 The Coefficients of the Lorentz
Transformation Are Purely Imaginary
7.3.2.2 There Is No Correspondence between the Tachyon’s Proper Time and Our Time
7.3.2.3 The Energy of the Tachyon Is Purely Imaginary, and Consequence of the Causality Principle
Notes
8 Answers to the Questions and Problems
8.1 Answers to Problems of Chapter 1
Answers to Questions
Answers to Problems
8.2 Answers to Problems of Chapter 2
8.3 Answers to Problems of Chapter 3
8.4 Answers to Problems of Chapter 4
Answer to Questions
Answer to Problems
8.5 Answers to Problems of Chapter 5
8.6 Answers to Problems of Chapter 6
Note
Bibliography
Index
