Special Relativity: An Introduction With 200 Problems And .
Special Relativity
Michael TsamparlisSpecial RelativityAn Introduction with 200 Problemsand Solutions123
Dr. Michael TsamparlisDepartment of Astrophysics, Astronomy and MechanicsUniversity of AthensPanepistimiopolisGR 157 84 ZOGRAFOSAthensGreecemtsampa@phys.uoa.grAdditional material to this book can be downloaded from http://extra.springer.com.Password: 978-3-642-03836-5ISBN 978-3-642-03836-5e-ISBN 978-3-642-03837-2DOI 10.1007/978-3-642-03837-2Springer Heidelberg Dordrecht London New YorkLibrary of Congress Control Number: 2009940408c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, 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 publicationor 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. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.Cover design: eStudio Calamar S.L.Printed on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)
Omnia mea mecum feroWhatever I possess I bear with me
PrefaceWriting a new book on the classic subject of Special Relativity, on which numerousimportant physicists have contributed and many books have already been written,can be like adding another epicycle to the Ptolemaic cosmology. Furthermore, it isour belief that if a book has no new elements, but simply repeats what is writtenin the existing literature, perhaps with a different style, then this is not enoughto justify its publication. However, after having spent a number of years, both inclass and research with relativity, I have come to the conclusion that there exists aplace 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 theinspiration and the mastery of the classic physicists of the last century. Moreovercurrent trends encourage the application of techniques in producing quick resultsand not tedious conceptual approaches resulting in long-lasting reasoning. On theother hand, physics cannot be done á la carte stripped from philosophy, or, to put itin a simple but dramatic contextA building is not an accumulation of stones!As a result of the above, a major aim in the writing of this book has been thedistinction between the mathematics of Minkowski space and the physics of relativity. This is necessary for one to understand the physics of the theory and notstay 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 anddevelopment of the theory. The approach is limited and concise but sufficient for thepurposes it is supposed to serve. Having finished with the mathematical concepts wecontinue with the foundation of the physical theory. Chapter 2 sets the frameworkon the scope and the structure of a theory of physics. We introduce the principleof relativity and the covariance principle, both principles being keystones in everytheory of physics. Subsequently we apply the scenario first to formulate NewtonianPhysics (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 properunderstanding 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 introductionvii
viiiPrefaceof new physical quantities. Special Relativity being a kinematic theory forces usto consider as the fundamental quantity the position four-vector. This is done inChap. 5 where we define the relativistic measurement of the position four-vector bymeans of the process of chronometry. To relate the theory with Newtonian reality,we introduce rules, which identify Newtonian space and Newtonian time in SpecialRelativity.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 relativisticcomposition 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 inthe relativistic description of motion. Finally, we discuss the phenomenon of acceleration redshift which together with some other applications of four-accelerationshows that here the limits of Special Relativity are reached and one must go over toGeneral 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 chooseto 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 meansof which we distinguish the particles in massive particles and luxons (photons).Chapter 10 is the most useful chapter of this book, because it concerns relativisticreactions, where the use of Special Relativity is indispensible. This chapter containsmany examples in order to familiarize the student with a tool, that will be necessaryto other major courses such as particle physics and high energy physics.In Chap. 11 we commence the dynamics of Special Relativity by the introductionof the four-force. We discuss many practical problems and use the tetrahedron ofFrenet–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 tensoralong a vector and give the basic results concerning the 1 3 decomposition inMinkowski space. The mathematics of this chapter is necessary in order to understand properly the relativistic physics. It is used extensively in General Relativitybut up to now we have not seen its explicit appearance in Special Relativity, eventhough 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. Wepresent in a concise way the standard vector form of electromagnetism and subsequently we are led to the four formalism formulation as a natural consequence. Afterdiscussing the standard material on the subject (four-potential, electromagnetic fieldtensor, etc.) we continue with lesser known material, such as the tensor formulationof Ohm’s law and the 1 3 decomposition of Maxwell’s equations. The reason whywe introduce these more advanced topics is that we wish to prepare the student forcourses on important subjects such as relativistic magnetohydrodynamics (RMHD).The rest of the book concerns topics which, to our knowledge, cannot be foundin the existing books on Special Relativity yet. In Chap. 14 we discuss the concept
Prefaceixof spin as a natural result of the generalization of the angular momentum tensorin Special Relativity. We follow a formal mathematical procedure, which revealswhat “the spin is” without the use of the quantum field theory. As an application, wediscuss the motion of a charged particle with spin in a homogeneous electromagneticfield and recover the well-known results in the literature.Chapter 15 deals with the covariant Lorentz transformation, a form which is notwidely known. All four types of Lorentz transformations are produced in covariantform 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 covariantform. The results are generic and can be used to develop software which will solvesuch reactions directly, provided one introduces the right data.The book includes numerous exercises and solved problems, plenty of whichsupplement 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 thebook.The above does not cover all topics we would like to consider. One such topicis relativistic waves, which leads to the introduction of De Broglie waves and subsequently to the foundation of quantum mechanics. A second topic is relativistichydrodynamics 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 forthe preparation of this book, I cannot help feeling that, perhaps, I should not haveundertaken the project. However, I feel that it would be unfair to all the students andcolleagues, who for more that 30 years have helped me to understand and developthe relativistic ideas, to find and solve problems, and in general to keep my interestalive. Therefore the present book is a collective work and my role has been simplyto compile these experiences. I do not mention specific names – the list would betoo long, and I will certainly forget quite a few – but they know and I know, and thatis 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 themfor their continuous support and understanding.Athens, GreeceOctober 2009Michael Tsamparlis
Contents1 Mathematical Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2Elements From the Theory of Linear Spaces . . . . . . . . . . . . . . . . . .1.2.1Coordinate Transformations . . . . . . . . . . . . . . . . . . . . . . .1.3Inner Product – Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4.1Operations of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5The Case of Euclidean Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . .1.6The Lorentz Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.6.1Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .1.7Algebraic Determination of the General Vector LorentzTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.8The Kinematic Interpretation of the General LorentzTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.8.1Relativistic Parallelism of Space Axes . . . . . . . . . . . . . . .1.8.2The Kinematic Interpretation of Lorentz Transformation1.9The Geometry of the Boost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.10 Characteristic Frames of Four-Vectors . . . . . . . . . . . . . . . . . . . . . . .1.10.1Proper Frame of a Timelike Four-Vector . . . . . . . . . . . . .1.10.2Characteristic Frame of a Spacelike Four-Vector . . . . . .1.11 Particle Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.12 The Center System (CS) of a System of Particle Four-Vectors . . . .1122610131417184040424348484950522 The Structure of the Theories of Physics . . . . . . . . . . . . . . . . . . . . . . . . . .2.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2The Role of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.3The Structure of a Theory of Physics . . . . . . . . . . . . . . . . . . . . . . . .2.4Physical Quantities and Reality of a Theory of Physics . . . . . . . . .2.5Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6Geometrization of the Principle of Relativity . . . . . . . . . . . . . . . . . .2.6.1Principle of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.6.2The Covariance Principle . . . . . . . . . . . . . . . . . . . . . . . . .2.7Relativity and the Predictions of a Theory . . . . . . . . . . . . . . . . . . . .5555565960626363646626xi
xiiContents3 Newtonian Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2Newtonian Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.1Mass Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.2Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.3Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3Newtonian Inertial Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.3.1Determination of Newtonian Inertial Observers . . . . . . .3.3.2Measurement of the Position Vector . . . . . . . . . . . . . . . .3.4Galileo Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5Galileo Transformations for Space and Time – NewtonianPhysical Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.5.1Galileo Covariant Principle: Part I . . . . . . . . . . . . . . . . . .3.5.2Galileo Principle of Communication . . . . . . . . . . . . . . . .3.6Newtonian Physical Quantities. The Covariance Principle . . . . . . .3.6.1Galileo Covariance Principle: Part II . . . . . . . . . . . . . . . .3.7Newtonian Composition Law of Vectors . . . . . . . . . . . . . . . . . . . . .3.8Newtonian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.8.1Law of Conservation of Linear Momentum . . . . . . . . . .6767686869717475777879798081818283844 The Foundation of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2Light and the Galileo Principle of Relativity . . . . . . . . . . . . . . . . . . 884.2.1The Existence of Non-Newtonian Physical Quantities . . 884.2.2The Limit of Special Relativity to Newtonian Physics . . 894.3The Physical Role of the Speed of Light . . . . . . . . . . . . . . . . . . . . . . 924.4The Physical Definition of Spacetime . . . . . . . . . . . . . . . . . . . . . . . . 934.4.1The Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.4.2The Geometry of Spacetime . . . . . . . . . . . . . . . . . . . . . . . 944.5Structures in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.1The Light Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5.2World Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.5.3Curves in Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . 984.5.4Geometric Definition of Relativistic InertialObservers (RIO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.5Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.5.6The Proper Frame of a RIO . . . . . . . . . . . . . . . . . . . . . . . . 1004.5.7Proper or Rest Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6Spacetime Description of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.6.1The Physical Definition of a RIO . . . . . . . . . . . . . . . . . . . 1034.6.2Relativistic Measurement of the Position Vector . . . . . . 1044.6.3The Physical Definition of an LRIO . . . . . . . . . . . . . . . . . 1054.7The Einstein Principle of Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7.1The Equation of Lorentz Isometry . . . . . . . . . . . . . . . . . . 106
Contentsxiii4.8The Lorentz Covariance Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.8.1Rules for Constructing Lorentz Tensors . . . . . . . . . . . . . 1094.8.2Potential Relativistic Physical Quantities . . . . . . . . . . . . 110Universal Speeds and the Lorentz Transformation . . . . . . . . . . . . . 1104.95 The Physics of the Position Four-Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2The Concepts of Space and Time in Special Relativity . . . . . . . . . . 1175.3Measurement of Spatial and Temporal Distancein Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.4Relativistic Definition of Spatial and Temporal Distances . . . . . . . 1205.5Timelike Position Four-Vector – Measurementof Temporal Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.6Spacelike Position Four-Vector – Measurementof Spatial Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.7The General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.8The Reality of Length Contraction and Time Dilation . . . . . . . . . . 1305.9The Rigid Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325.10 Optical Images in Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . 1345.11 How to Solve Problems Involving Spatial and Temporal Distance 1415.11.1A Brief Summary of the Lorentz Transformation . . . . . . 1415.11.2Parallel and Normal Decomposition of LorentzTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.11.3Methodologies of Solving Problems Involving Boosts . 1435.11.4The Algebraic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.11.5The Geometric Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506 Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.2Relativistic Mass Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1556.3Relativistic Composition of Three-Vectors . . . . . . . . . . . . . . . . . . . . 1596.4Relative Four-Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666.5The three-Velocity Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1746.6Thomas Precession . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1777 Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1857.2The Four-Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1867.3Calculating Accelerated Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1937.4Hyperbolic Motion of a Relativistic Mass Particle . . . . . . . . . . . . . 1977.4.1Geometric Representation of Hyperbolic Motion . . . . . . 2007.5Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.5.1Einstein Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 2047.6Rigid Motion of Many Relativistic Mass Points . . . . . . . . . . . . . . . 205
xivContents7.77.87.97.107.117.12Rigid Motion and Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . . . . . 2067.7.1The Synchronization of LRIO . . . . . . . . . . . . . . . . . . . . . 2087.7.2Synchronization of Chronometry . . . . . . . . . . . . . . . . . . . 2097.7.3The Kinematics in the LCF Σ . . . . . . . . . . . . . . . . . . . . . . 2117.7.4The Case of the Gravitational Field . . . . . . . . . . . . . . . . . 214General One-Dimensional Rigid Motion . . . . . . . . . . . . . . . . . . . . . 2167.8.1The Case of Hyperbolic Motion . . . . . . . . . . . . . . . . . . . . 217Rotational Rigid Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2197.9.1The Transitive Property of the Rigid Rotational Motion 222The Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2247.10.1The Kinematics of Relativistic Observers . . . . . . . . . . . . 2247.10.2Chronometry and the Spatial Line Element . . . . . . . . . . . 2257.10.3The Rotating Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287.10.4Definition of the Rotating Disk for a RIO . . . . . . . . . . . . 2297.10.5The Locally Relativistic Inertial Observer (LRIO) . . . . . 2307.10.6The Accelerated Observer . . . . . . . . . . . . . . . . . . . . . . . . . 235The Generalization of Lorentz Transformationand the Accelerated Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.11.1The Generalized Lorentz Transformation . . . . . . . . . . . . 2407.11.2The Special Case u 0 (l , x ) u 1 (l , x ) u(x ) . . . . . . . 2427.11.3Equation of Motion in a Gravitational Field . . . . . . . . . . 247The Limits of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2487.12.1Experiment 1: The Gravitational Redshift . . . . . . . . . . . . 2497.12.2Experiment 2: The Gravitational Time Dilation . . . . . . . 2517.12.3Experiment 3: The Curvature of Spacetime . . . . . . . . . . 2528 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2538.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2538.2Various Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2549 Mass – Four-Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2659.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2659.2The (Relativistic) Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2669.3The Four-Momentum of a ReMaP . . . . . . . . . . . . . . . . . . . . . . . . . . . 2679.4The Four-Momentum of Photons (Luxons) . . . . . . . . . . . . . . . . . . . 2759.5The Four-Momentum of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2789.6The System of Natural Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27810 Relativistic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28310.2 Representation of Particle Reactions . . . . . . . . . . . . . . . . . . . . . . . . . 28410.3 Relativistic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28510.3.1The Sum of Particle Four-Vectors . . . . . . . . . . . . . . . . . . 28610.3.2The Relativistic Triangle Inequality . . . . . . . . . . . . . . . . . 288
Contents10.410.510.610.7xvWorking with Four-Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289Special Coordinate Frames in the Study of Relativistic Collisions 291The Generic Reaction A B C . . . . . . . . . . . . . . . . . . . . . . . . . . 29210.6.1The Physics of the Generic Reaction . . . . . . . . . . . . . . . . 29310.6.2Threshold of a Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 297Transformation of Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30410.7.1Radiative Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30810.7.2Reactions With Two-Photon Final State . . . . . . . . . . . . . 31210.7.3Elastic Collisions – Scattering . . . . . . . . . . . . . . . . . . . . . 31711 Four-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.2 The Four-Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32511.3 Inertial Four-Force and Four-Potential . . . . . . . . . . . . . . . . . . . . . . . 34011.3.1The Vector Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . 34211.4 The Lagrangian Formalism for Inertial Four-Forces . . . . . . . . . . . . 34311.5 Motion in a Central Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35011.6 Motion of a Rocket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35511.7 The Frenet–Serret Frame in Minkowski Space . . . . . . . . . . . . . . . . 36311.7.1The Physical Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36811.7.2The Generic Inertial Four-Force . . . . . . . . . . . . . . . . . . . . 37212 Irreducible Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.1 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37712.1.1Writing a Tensor of Valence (0,2) as a Matrix . . . . . . . . 37812.2 The Irreducible Decomposition wrt a Non-null Vector . . . . . . . . . . 37912.2.1Decomposition in a Euclidean Space En . . . . . . . . . . . . . 37912.2.21 3 Decomposition in Minkowski Space . . . . . . . . . . . 38312.3 1 1 2 Decomposition wrt a Pair of Timelike Vectors . . . . . . . . . . 38913 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39513.2 Maxwell Equations in Newtonian Physics . . . . . . . . . . . . . . . . . . . . 39613.3 The Electromagnetic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39913.4 The Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40513.5 The Electromagnetic Four-Potential . . . . . . . . . . . . . . . . . . . . . . . . . 41213.6 The Electromagnetic Field Tensor Fi j . . . . . . . . . . . . . . . . . . . . . . . . 41513.6.1The Transformation of the Fields . . . . . . . . . . . . . . . . . . . 41513.6.2Maxwell Equations in Terms of Fi j . . . . . . . . . . . . . . . . . 41713.6.3The Invariants of the Electromagnetic Field . . . . . . . . . . 41813.7 The Physical Significance of the Electromagnetic Invariants . . . . . 42113.7.1The Case Y 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42213.7.2The Case Y 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
3.1613.1713.18Motion of a Charge in an Electromagnetic Field – The LorentzForce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Motion of a Charge in a Homogeneous Electromagnetic Field . . . 42913.9.1The Case of a Homogeneous Electric Field . . . . . . . . . . 43013.9.2The Case of a Homogeneous Magnetic Field . . . . . . . . . 43413.9.3The Case of Two Homogeneous Fields of EqualStrength and Perpendicular Directions . . . . . . . . . . . . . . 43613.9.4The Case of Homogeneous and Parallel Fields E B . . 438The Relativistic Electric and Magnetic Fields . . . . . . . . . . . . . . . . . 44013.10.1 The Levi-Civita Tensor Density . . . . . . . . . . . . . . . . . . . . 44013.10.2 The Case of Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44213.10.3 The Electromagnetic Theory for a General Medium . . . 44513.10.4 The Electric and Magnetic Moments . . . . . . . . . . . . . . . . 44813.10.5 Maxwell Equations for a General Medium . . . . . . . . . . . 44813.10.6 The 1 3 Decomposition of Maxwell Equations . . . . . . 449The Four-Current of Conductivity and Ohm’s Law . . . . . . . . . . . . . 45413.11.1 The Continuity Equation J a ;a 0for an Isotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . 458The Electromagnetic Field in a Homogeneousand Isotropic Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459Electric Conductivity and the Propagation Equation for E a . . . . . . 463The Generalized Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465The Energy Momentum Tensor of the Electromagnetic Field . . . . 467The Electromagnetic Field of a Moving Charge . . . . . . . . . . . . . . . 47513.16.1 The Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47713.16.2 The Fields E i , B i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47813.16.3 The Liénard–Wiechert Potentials and the Fields E, B . . 478Special Relativity and Practical Applications . . . . . . . . . . . . . . . . . . 489The Systems of Units SI and Gauss in Electromagnetism . . . . . . . 49214 Relativistic Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.2 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49514.2.11 3 Decomposition of a Bivector X ab . . . . . . . . . . . . . 49514.3 The Derivative of X ab Along the Vector pa . . . . . . . . . . . . . . . . . . . 49814.4 The Angular M
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.
1 RELATIVITY I 1 1.1 Special Relativity 2 1.2 The Principle of Relativity 3 The Speed of Light 6 1.3 The Michelson–Morley Experiment 7 Details of the Michelson–Morley Experiment 8 1.4 Postulates of Special Relativity 10 1.5 Consequences of Special Relativity 13 Simultaneity and the Relativity
The theory of relativity is split into two parts: special and general. Albert Einstein came up with the spe-cial theory of relativity in 1905. It deals with objects mov-ing relative to one another, and with the way an observer's experience of space and time depends on how she is mov-ing. The central ideas of special relativity can be formu-
Introduction Special Relativity General Relativity Curriculum Books The Geometry of Special Relativity Tevi
Theory of Relativity. Einstein's General Theory of Relativity by Asghar Qadir. Einstein'sGeneralTheoryofRelativity ByAsgharQadir . Relativity: An Introduction to the Special Theory (World Scientific 1989) or equivalent, but do not have a sound background of Geometry. It can be used
Sean Carroll, “Spacetime and Geometry” A straightforward and clear introduction to the subject. Bob Wald, “General Relativity” The go-to relativity book for relativists. Steven Weinberg, “Gravitation and Cosmology” The go-to relativity book for particle physicists. Misner, Thorne and Wheeler, “Gravitation”
Modern readers turning to Einstein’s famous 1905 paper on special relativity may not find what they expect. Its title, “On the electrodynamics of moving bodies,” . special relativity could not be stopped. Its basic equations and notions . moving earth will proceed just as if the earth were a
a special subfamily: the inertial frames (inertial coordinates). The existence of these inertial frames is guaranteed by the principle of relativity: Special principle of relativity There exists a family of coordinate systems, which we call inertial frames of reference, w.r.t. which the laws of nature take one and the same form. In
entitled La fisica del '900: Henri Poincaré e la relatività, delivered at the Seminari di Storia delle Scienze, Almo Collegio Borromeo, Pavia 1995, on 30 March 1995. Partial results of this historiographical inquiry were discussed in: Henri Poincaré and the rise of special relativity, in Quanta Relativity
