Michael Tsamparlis. Special Relativity. An Introduction with 200 Problems and Solutions

Transcription

1 Special Relativity

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3 Michael Tsamparlis Special Relativity An Introduction with 200 Problems and Solutions 123

4 Dr. Michael Tsamparlis Department of Astrophysics, Astronomy and Mechanics University of Athens Panepistimiopolis GR ZOGRAFOS Athens Greece Additional material to this book can be downloaded from Password: ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: estudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Omnia mea mecum fero Whatever I possess I bear with me

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7 Preface Writing a new book on the classic subject of Special Relativity, on which numerous important physicists have contributed and many books have already been written, can be like adding another epicycle to the Ptolemaic cosmology. Furthermore, it is our belief that if a book has no new elements, but simply repeats what is written in the existing literature, perhaps with a different style, then this is not enough to justify its publication. However, after having spent a number of years, both in class and research with relativity, I have come to the conclusion that there exists a place for a new book. Since it appears that somewhere along the way, mathematics may have obscured and prevailed to the degree that we tend to teach relativity (and I believe, theoretical physics) simply using heavier mathematics without the inspiration and the mastery of the classic physicists of the last century. Moreover current trends encourage the application of techniques in producing quick results and not tedious conceptual approaches resulting in long-lasting reasoning. On the other hand, physics cannot be done á la carte stripped from philosophy, or, to put it in a simple but dramatic context A building is not an accumulation of stones! As a result of the above, a major aim in the writing of this book has been the distinction between the mathematics of Minkowski space and the physics of relativity. This is necessary for one to understand the physics of the theory and not stay with the geometry, which by itself is a very elegant and attractive tool. Therefore in the first chapter we develop the mathematics needed for the statement and development of the theory. The approach is limited and concise but sufficient for the purposes it is supposed to serve. Having finished with the mathematical concepts we continue with the foundation of the physical theory. Chapter 2 sets the framework on the scope and the structure of a theory of physics. We introduce the principle of relativity and the covariance principle, both principles being keystones in every theory of physics. Subsequently we apply the scenario first to formulate Newtonian Physics (Chap. 3) and then Special Relativity (Chap. 4). The formulation of Newtonian Physics is done in a relativistic way, in order to prepare the ground for a proper understanding of the parallel formulation of Special Relativity. Having founded the theory we continue with its application. The approach is systematic in the sense that we develop the theory by means of a stepwise introduction vii

8 viii Preface of new physical quantities. Special Relativity being a kinematic theory forces us to consider as the fundamental quantity the position four-vector. This is done in Chap. 5 where we define the relativistic measurement of the position four-vector by means of the process of chronometry. To relate the theory with Newtonian reality, we introduce rules, which identify Newtonian space and Newtonian time in Special Relativity. In Chaps. 6 and 7 we introduce the remaining elements of kinematics, that is, the four-velocity and the four-acceleration. We discuss the well-known relativistic composition law for the three-velocities and show that it is equivalent to the Einstein relativity principle, that is, the Lorentz transformation. In the chapter of fouracceleration we introduce the concept of synchronization which is a key concept in the relativistic description of motion. Finally, we discuss the phenomenon of acceleration redshift which together with some other applications of four-acceleration shows that here the limits of Special Relativity are reached and one must go over to General Relativity. After the presentation of kinematics, in Chap. 8 we discuss various paradoxes, which play an important role in the physical understanding of the theory. We choose to present paradoxes which are not well known, as for example, it is the twin paradox. In Chap. 9 we introduce the (relativistic) mass and the four-momentum by means of which we distinguish the particles in massive particles and luxons (photons). Chapter 10 is the most useful chapter of this book, because it concerns relativistic reactions, where the use of Special Relativity is indispensible. This chapter contains many examples in order to familiarize the student with a tool, that will be necessary to other major courses such as particle physics and high energy physics. In Chap. 11 we commence the dynamics of Special Relativity by the introduction of the four-force. We discuss many practical problems and use the tetrahedron of Frenet Serret to compute the generic form of the four-force. We show how the wellknown four-forces comply with the generic form. In Chap. 12 we introduce the concept of covariant decomposition of a tensor along a vector and give the basic results concerning the decomposition in Minkowski space. The mathematics of this chapter is necessary in order to understand properly the relativistic physics. It is used extensively in General Relativity but up to now we have not seen its explicit appearance in Special Relativity, even though it is a powerful and natural tool both for the theory and the applications. Chapter 13 is the next pillar of Special Relativity, that is, electromagnetism. We present in a concise way the standard vector form of electromagnetism and subsequently we are led to the four formalism formulation as a natural consequence. After discussing the standard material on the subject (four-potential, electromagnetic field tensor, etc.) we continue with lesser known material, such as the tensor formulation of Ohm s law and the decomposition of Maxwell s equations. The reason why we introduce these more advanced topics is that we wish to prepare the student for courses on important subjects such as relativistic magnetohydrodynamics (RMHD). The rest of the book concerns topics which, to our knowledge, cannot be found in the existing books on Special Relativity yet. In Chap. 14 we discuss the concept

9 Preface ix of spin as a natural result of the generalization of the angular momentum tensor in Special Relativity. We follow a formal mathematical procedure, which reveals what the spin is without the use of the quantum field theory. As an application, we discuss the motion of a charged particle with spin in a homogeneous electromagnetic field and recover the well-known results in the literature. Chapter 15 deals with the covariant Lorentz transformation, a form which is not widely known. All four types of Lorentz transformations are produced in covariant form and the results are applied to applications involving the geometry of threevelocity space, the composition of Lorentz transformations, etc. Finally, in Chap. 16 we study the reaction A + B C + D in a fully covariant form. The results are generic and can be used to develop software which will solve such reactions directly, provided one introduces the right data. The book includes numerous exercises and solved problems, plenty of which supplement the theory and can be useful to the reader on many occasions. In addition, a large number of problems, carefully classified in all topics accompany the book. The above does not cover all topics we would like to consider. One such topic is relativistic waves, which leads to the introduction of De Broglie waves and subsequently to the foundation of quantum mechanics. A second topic is relativistic hydrodynamics and its extension to RMHD. However, one has to draw a line somewhere and leave the future to take care of things to be done. Looking back at the long hours over the many years which were necessary for the preparation of this book, I cannot help feeling that, perhaps, I should not have undertaken the project. However, I feel that it would be unfair to all the students and colleagues, who for more that 30 years have helped me to understand and develop the relativistic ideas, to find and solve problems, and in general to keep my interest alive. Therefore the present book is a collective work and my role has been simply to compile these experiences. I do not mention specific names the list would be too long, and I will certainly forget quite a few but they know and I know, and that is enough. I close this preface, with an apology to my family for the long working hours; that I was kept away from them for writing this book and I would like to thank them for their continuous support and understanding. Athens, Greece October 2009 Michael Tsamparlis

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11 Contents 1 Mathematical Part Introduction Elements From the Theory of Linear Spaces CoordinateTransformations Inner Product Metric Tensors OperationsofTensors The Case of Euclidean Geometry TheLorentzGeometry LorentzTransformations Algebraic Determination of the General Vector Lorentz Transformation The Kinematic Interpretation of the General Lorentz Transformation Relativistic Parallelism of Space Axes The Kinematic Interpretation of Lorentz Transformation TheGeometryoftheBoost CharacteristicFramesofFour-Vectors Proper Frame of a Timelike Four-Vector Characteristic Frame of a Spacelike Four-Vector ParticleFour-Vectors The Center System (CS) of a System of Particle Four-Vectors The Structure of the Theories of Physics Introduction TheRoleofPhysics The Structure of a Theory of Physics Physical Quantities and Reality of a Theory of Physics InertialObservers GeometrizationofthePrincipleofRelativity PrincipleofInertia The Covariance Principle Relativity and the Predictions of a Theory xi

12 xii Contents 3 Newtonian Physics Introduction NewtonianKinematics MassPoint Space Time NewtonianInertialObservers DeterminationofNewtonianInertialObservers Measurement of the Position Vector Galileo Principle of Relativity Galileo Transformations for Space and Time Newtonian Physical Quantities Galileo Covariant Principle: Part I Galileo Principle of Communication Newtonian Physical Quantities. The Covariance Principle Galileo Covariance Principle: Part II Newtonian Composition Law of Vectors Newtonian Dynamics Law of Conservation of Linear Momentum The Foundation of Special Relativity Introduction Light and the Galileo Principle of Relativity The Existence of Non-Newtonian Physical Quantities The Limit of Special Relativity to Newtonian Physics The Physical Role of the Speed of Light The Physical Definition of Spacetime TheEvents The Geometry of Spacetime Structures in Minkowski Space TheLightCone WorldLines Curves in Minkowski Space Geometric Definition of Relativistic Inertial Observers(RIO) Proper Time The Proper Frame of a RIO Proper or Rest Space Spacetime Description of Motion The Physical Definition of a RIO Relativistic Measurement of the Position Vector The Physical Definition of an LRIO TheEinsteinPrincipleofRelativity TheEquationofLorentzIsometry...106

13 Contents xiii 4.8 The Lorentz Covariance Principle RulesforConstructingLorentzTensors Potential Relativistic Physical Quantities Universal Speeds and the Lorentz Transformation The Physics of the Position Four-Vector Introduction The Concepts of Space and Time in Special Relativity Measurement of Spatial and Temporal Distance in Special Relativity Relativistic Definition of Spatial and Temporal Distances Timelike Position Four-Vector Measurement of Temporal Distance Spacelike Position Four-Vector Measurement of Spatial Distance The General Case The Reality of Length Contraction and Time Dilation TheRigidRod Optical Images in Special Relativity How to Solve Problems Involving Spatial and Temporal Distance A Brief Summary of the Lorentz Transformation Parallel and Normal Decomposition of Lorentz Transformation Methodologies of Solving Problems Involving Boosts The Algebraic Method TheGeometricMethod Relativistic Kinematics Introduction RelativisticMassPoint Relativistic Composition of Three-Vectors RelativeFour-Vectors The three-velocity Space ThomasPrecession Four-Acceleration Introduction The Four-Acceleration Calculating Accelerated Motions Hyperbolic Motion of a Relativistic Mass Particle Geometric Representation of Hyperbolic Motion Synchronization Einstein Synchronization RigidMotionofManyRelativisticMassPoints...205

14 xiv Contents 7.7 Rigid Motion and Hyperbolic Motion The Synchronization of LRIO Synchronization of Chronometry The Kinematics in the LCF Σ The Case of the Gravitational Field General One-Dimensional Rigid Motion The Case of Hyperbolic Motion Rotational Rigid Motion The Transitive Property of the Rigid Rotational Motion TheRotatingDisk TheKinematicsofRelativisticObservers Chronometry and the Spatial Line Element TheRotatingDisk Definition of the Rotating Disk for a RIO The Locally Relativistic Inertial Observer (LRIO) The Accelerated Observer The Generalization of Lorentz Transformation and the Accelerated Observers The Generalized Lorentz Transformation The Special Case u 0 (l, x ) = u 1 (l, x ) = u(x ) Equation of Motion in a Gravitational Field The Limits of Special Relativity Experiment 1: The Gravitational Redshift Experiment 2: The Gravitational Time Dilation Experiment 3: The Curvature of Spacetime Paradoxes Introduction Various Paradoxes Mass Four-Momentum Introduction The(Relativistic)Mass TheFour-MomentumofaReMaP The Four-Momentum of Photons (Luxons) TheFour-MomentumofParticles TheSystemofNaturalUnits Relativistic Reactions Introduction RepresentationofParticleReactions RelativisticReactions TheSumofParticleFour-Vectors The Relativistic Triangle Inequality

15 Contents xv 10.4 WorkingwithFour-Momenta Special Coordinate Frames in the Study of Relativistic Collisions The Generic Reaction A + B C The Physics of the Generic Reaction ThresholdofaReaction TransformationofAngles RadiativeTransitions Reactions With Two-Photon Final State Elastic Collisions Scattering Four-Force Introduction TheFour-Force InertialFour-ForceandFour-Potential TheVectorFour-Potential The Lagrangian Formalism for Inertial Four-Forces MotioninaCentralPotential MotionofaRocket The Frenet Serret Frame in Minkowski Space ThePhysicalBasis The Generic Inertial Four-Force Irreducible Decompositions Decompositions Writing a Tensor of Valence (0,2) as a Matrix The Irreducible Decomposition wrt a Non-null Vector Decomposition in a Euclidean Space E n Decomposition in Minkowski Space Decomposition wrt a Pair of Timelike Vectors The Electromagnetic Field Introduction MaxwellEquationsinNewtonianPhysics The Electromagnetic Potential TheEquationofContinuity The Electromagnetic Four-Potential The Electromagnetic Field Tensor F ij TheTransformationoftheFields Maxwell Equations in Terms of F ij The Invariants of the Electromagnetic Field The Physical Significance of the Electromagnetic Invariants The Case Y = The Case Y

16 xvi Contents 13.8 Motion of a Charge in an Electromagnetic Field The Lorentz Force Motion of a Charge in a Homogeneous Electromagnetic Field The Case of a Homogeneous Electric Field The Case of a Homogeneous Magnetic Field The Case of Two Homogeneous Fields of Equal Strength and Perpendicular Directions The Case of Homogeneous and Parallel Fields E B The Relativistic Electric and Magnetic Fields TheLevi-CivitaTensorDensity The Case of Vacuum The Electromagnetic Theory for a General Medium The Electric and Magnetic Moments Maxwell Equations for a General Medium The Decomposition of Maxwell Equations The Four-Current of Conductivity and Ohm s Law The Continuity Equation J a ;a = 0 foranisotropicmaterial The Electromagnetic Field in a Homogeneous andisotropicmedium Electric Conductivity and the Propagation Equation for E a The Generalized Ohm s Law The Energy Momentum Tensor of the Electromagnetic Field The Electromagnetic Field of a Moving Charge TheInvariants The Fields E i, B i The Liénard Wiechert Potentials and the Fields E, B Special Relativity and Practical Applications The Systems of Units SI and Gauss in Electromagnetism Relativistic Angular Momentum Introduction MathematicalPreliminaries Decomposition of a Bivector X ab The Derivative of X ab Along the Vector p a The Angular Momentum in Special Relativity The Angular Momentum in Newtonian Theory The Angular Momentum of a Particle in Special Relativity The Intrinsic Angular Momentum The Spin Vector The Magnetic Dipole TheRelativisticSpin Motion of a Particle with Spin in a Homogeneous Electromagnetic Field Transformation of Motion in Σ...517

17 Contents xvii 15 The Covariant Lorentz Transformation Introduction TheCovariantLorentzTransformation Definition of the Lorentz Transformation Computation of the Covariant Lorentz Transformation The Action of the Covariant Lorentz Transformation onthecoordinates The Invariant Length of a Four-Vector The Four Types of the Lorentz Transformation Viewed as Spacetime Reflections Relativistic Composition Rule of Four-Vectors ComputationoftheCompositeFour-Vector The Relativistic Composition Rule for Three-Velocities Riemannian Geometry and Special Relativity The Relativistic Rule for the Composition of Three-Accelerations The Composition of Lorentz Transformations Geometric Description of Relativistic Interactions Collisions and Geometry Geometric Description of Collisions in Newtonian Physics GeometricDescriptionofRelativisticReactions The General Geometric Results The 1+3 Decomposition of a Particle Four-Vector wrtatimelikefour-vector TheSystemofTwotoOneParticleFour-Vectors The Triangle Function of a System of Two Particle Four-Vectors Extreme Values of the Four-Vectors (A ± B) The System A a, B a, (A + B) a of Particle Four-VectorsinCS The System A a, B a, (A + B) a inthelab The Relativistic System A a + B a C a + D a The Reaction B C + D Bibliography Index...591