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This book offers a comprehensive examination of ongoing advancements in theoretical and experimental approaches to Maxwell's equations. It focuses on three key aspects: quantum effects, scale effects, and kinetic effects, which exert subtle influences at microscopic levels. Addressing pressing challenges for future progress, the text explores the interplay of these phenomena within classical electromagnetism. The evolution of data communication and information technology has led to a demand for high-density, minimization, ultra-compact nano-photo-electronic integration. As electronic devices scale down to nanometer and sub-nanometer levels, classical Maxwell's equations reveal quantum effects. This book provides insights into these advancements, focusing on potential applications in nano-scale electronic and optic devices. Tailored for physicists, engineering scientists, electronics engineers, and developers, this text serves as a valuable resource. It guides readers from classical Maxwell's equations to their quantum-affected counterparts, providing essential insights for electromagnetic simulation and the design of nano-scale electronic and optic systems. With its blend of theoretical foundations and practical applications, this book equips professionals with the knowledge needed to apply these advancements in real-world scenarios.

This book is dedicated to Einstein's personal and scientific relationships with Italy, which began as early as adolescence and continued over various stages of his life. It collects together a wealth of historical documentation including the letters in which Einstein recalls having lived as a young man with his family in Lombardy, Pavia and Milan; the texts of three lectures delivered by Einstein in Italian in October 1921 in Bologna at the invitation of the mathematician Federigo Enriques, the only lectures he held in Italy; the correspondence with Enriques and other Italian intellectuals, together with the letter that Einstein sent in 1931 to the Minister of Justice Alfredo Rocco to try to avoid the infamy of the oath of allegiance to the fascist regime imposed by Mussolini on university professors. The book closes with some writings on relativity from 1907 to 1914 (Einstein, Abraham, Corbino, Castelnuovo) and the 1920s debate between pro-relativists and anti-relativists publishedin the international journal “Scientia”, founded in 1907 by Enriques.

The 2nd edition of this textbook features more than 100 pages of new material, including four new chapters, as well as an improved discussion of differential geometry concepts and their applications. The textbook aims to provide a comprehensive geometric description of Special and General Relativity, starting from basic Euclidean geometry to more advanced non-Euclidean geometry and differential geometry. Readers will learn about the Schwarzschild metric, the relativistic trajectory of planets, the deflection of light, the black holes, and the cosmological solutions like de Sitter, Friedman-Lemaître-Robertson-Walker, and Gödel ones, as well as the implications of each of them for the observed physical world. In addition, the book provides step-by-step solutions to problems and exercises, making it an ideal introduction for undergraduate students and readers looking to gain a better understanding of Special and General Relativity. In this new edition, a wide discussion on metric-affine theories of gravity and equivalent formulations of General Relativity is reported. The aim is presenting also topics which could be useful for PhD students and researchers studying General Relativity from an advanced point of view.

In most undergraduate physics classes Special Relativity is taught from a simplistic point of view using Newtonian concepts rather than the relativistic way of thinking. This results in students often finding it difficult to understand properly the new approach/new ideas, and consequently to solve relativistic problems. Furthermore, a number of books treat the theory using advanced mathematics which is not necessary for the first approach to the theory. This book is intended to serve two roles: a. To treat a student in a systematic constructive way to the basic structure of the theory and b. To provide a large number of solved in-detail problems in the kinematics and dynamics of Special Relativity. Concerning the first aim the book introduces the basics of four-dimensional mathematics, i.e., Lorentz metric, relativistic tensors, and prepares, through working examples, the transition to General Relativity, which requires, besides the relativistic concepts, the use of Differential Geometry and tensor analysis. The presentation is concise and does not replace a book on Special Relativity. Concerning the second intention the large number of problems provides the necessary material which can be used in order to familiarize the student with the relativistic “world”. These problems can be used in the class by the teachers either as working examples or as problem sheets. It will be our pleasure if the book will be useful to both students and teachers.

1905 is probably the best-known year in physics, since it was the year of the discovery of the special theory of relativity. For decades, historiography has told us that Albert Einstein, then a patent examiner in Bern, succeeded in developing this theory on his own, overcoming all the difficulties that the greatest scientists of his time had not been able to solve. However, some have pointed out that, before Einstein's first publication in this field, the French mathematician and physicist Henri Poincaré had obtained the same results, which he had published several months before Einstein. Yet today, this theory is known as Einstein's special theory of relativity. Thus, considering the indisputable anteriority of Poincaré's contributions, there is only one real question that needs to be answered:Why didn't Poincaré claim the authorship of special theory of relativity?After recapping on the ideas and concepts of the special theory of relativity in a manner accessibleto non-specialists and recalling the historical context of the discovery of this theory, we will answer this question and thus put finally an end to this long-running controversy.

This book deals with theoretical mechanics. Newton published the'Philosophiæ Naturalis Principia Mathematica'in 1687. In it, he sets out the basic principles of physics that are required to understand the motion of the planets, their moons, and the comets in the solar system. It includes the gravitational (inverse square) law, the inertial principle, and the basic elements of mechanics. Since its publication, a large number of refinements and reformulations have been introduced, thereby adding enormous insight into the structure of mechanics, which is commonly known as “classical mechanics”. All these have in common that by taking a suitable limit, Newton's original principles re-appear. Thus, physicists and mathematicians who work on the subject always have a notion that if their theories do not return to Newton's foundations, then there is something wrong. Newton himself acknowledged that'if I have seen further (than others), it is by standing on the shoulders of giants'. One of these giants was undoubtedly Galileo who died in the year Newton was born. So, Newton himself adhered to the'classical limit'.

This book is an English translation from a Hungarian book designed for graduate and postgraduate students about the use of variational principles in theoretical physics. Unlike many academic textbooks, it dashes across several lecture disciplines taught in physics courses. It emphasizes and demonstrates the use of the variational technique and philosophy behind the basic laws in mechanics, relativity theory, electromagnetism, and quantum mechanics. The book is meant for advanced students and young researchers in theoretical physics but, also, more experienced researchers can benefit from its reading.

Quantentheorie und Relativitätstheorie haben das Weltbild der Physik revolutioniert. Beide Theorien gelten jedoch als unanschaulich und schwer verständlich. Dieses Sachbuch schafft neue Zugänge und lädt zum Mitdenken ein. Renommierte Experten aus Physik und Philosophie erläutern Grundbegriffe, Erkenntnisfortschritte und Deutungsfragen zu Raum, Zeit und Materie.Dabei kommen typische Themen aus der Philosophie der Physik zur Sprache, wie etwa die Interpretationsdebatte der Quantentheorie oder Modellbildungen in der Kosmologie. Weltbildrelevante Fragen nach dem Verhältnis von Mathematik, Empirie und Anschauung oder nach der Verlässlichkeit physikalischer Erkenntnis erfordern die Verbindung von Physik und Philosophie, von fachwissenschaftlicher Grundlagenforschung und methodenkritischer Reflexion. Der thematische Bogen geht zurück auf ein hochkarätig besetztes Symposium mit populärwissenschaftlicher Ausrichtung. Das Buch enthält Beiträge von Andreas Bartels, RobertHarlander, Paul Hoyningen-Huene, Gert-Ludwig Ingold, Claus Kiefer, Meinard Kuhlmann, Klaus Mainzer, Oliver Passon, Manfred Stöckler, Rüdiger Vaas und Reinhard Werner.

Despite successes of modern physics, the existence of dark energy and matter is indicative that conventional mechanical accounting is lacking. The most basic of all mechanical principles is Newton's second law, and conventionally, energy is just energy whether particle or wave energy. In this monograph, Louis de Broglie's idea of simultaneous existence of both particle and associated wave is developed, with a novel proposal to account for mass and energy through a combined particle-wave theory. Newton's second law of motion is replaced by a fully Lorentz invariant reformulation inclusive of both particles and waves. The model springs from continuum mechanics and forms a natural extension of special relativistic mechanics. It involves the notion of “force in the direction of time” and every particle has both particle and wave energies, arising as characteristics of space and time respectively. Dark matter and energy then emerge as special or privileged states occurring for alignments of spatial forces with the force in the direction of time. Dark matter is essentially a backward wave and dark energy a forward wave, both propagating at the speed of light. The model includes special relativistic mechanics and Schrödinger's quantum mechanics, and the major achievements of mechanics and quantum physics. Our ideas of particles and waves are not yet properly formulated, and are bound up with the speed of light as an extreme limit and particle-wave demarcation. Sub-luminal particles have an associated superluminal wave, so if sub-luminal waves have an associated superluminal particle, then there emerges the prospect for faster than light travel with all the implications for future humanity. Carefully structured over special relativity and quantum mechanics, Mathematics of Particle-Wave Mechanical Systems is not a completed story, but perhaps the first mechanical model within which such exalted notions might be realistically and soberly examined. If ultimately the distant universe become accessible, this will necessitate thinking differently about particles, waves and the role imposed by the speed of light. The text constitutes a single proposal in that direction and a depository for mathematically related results. It will appeal to researchers and students of mathematical physics, applied mathematics and engineering mechanics.

This book presents an overview of Classical Electrodynamics. Its second edition includes new chapters that pick up where the material from the first edition left off. The image method introduced in the first edition is expanded to series of images, using simple examples like a point charge or a charged wire between two grounded plates, as well as more relevant examples such as two charged conducting spheres and the force between them. The topic of complex functions is broadened with the introduction of conformal mapping. One new chapter introduces the method of separation of variables, including in Cartesian coordinates (box with sides at fixed voltages), in spherical coordinates (dielectric and conducting sphere, potential of a charged ring), in cylindrical coordinates (conducting wedge, cylinder in uniform field). It also presents the potentials and the fields for a point charge in motion, radiation by a point charge and by a dipole, radiation reaction. Two other chapters present updated lessons on the mass of the photon and search for monopoles. Examples and/or solvable problems are provided throughout.

This book examines the topics of magnetohydrodynamics and plasma oscillations, in addition to the standard topics discussed to cover courses in electromagnestism, electrodynamics, and fundamentals of physics, to name a few. This textbook on electricity and magnetism is primarily targeted at graduate students of physics. The undergraduate students of physics also find the treatment of the subject useful. The treatment of the special theory of relativity clearly emphasises the Lorentz covariance of Maxwell's equations. The rather abstruse topic of radiation reaction is covered at an elementary level, and the Wheeler–Feynman absorber theory has been dwelt upon briefly in the book.

This textbook expounds the major topics in the special theory of relativity. It provides a detailed examination of the mathematical foundation of the special theory of relativity, relativistic mass, relativistic mechanics, and relativistic electrodynamics. As well as covariant formulation of relativistic mechanics and electrodynamics, the text discusses the relativistic effect on photons. A new chapter on electromagnetic waves as well as several new problems and examples have been included in the second edition of the book. Using the mathematical approach, the text offers graduate students a clear, concise view of the special theory of relativity. Organized into 15 chapters and two appendices, the content is presented in a logical order, and every topic has been dealt with in a simple and lucid manner. To aid understanding of the subject, the text provides numerous relevant worked-out examples in every chapter. The mathematical approach of the text helps students in theirindependent study and motivates them to research the topic further.

This book offers an introduction to statistical mechanics, special relativity, and quantum physics. It is based on the lecture notes prepared for the one-semester course of'Quantum Physics'belonging to the Bachelor of Science in Material Sciences at the University of Padova.The first chapter briefly reviews the ideas of classical statistical mechanics introduced by James Clerk Maxwell, Ludwig Boltzmann, Willard Gibbs, and others. The second chapter is devoted to the special relativity of Albert Einstein. In the third chapter, it is historically analyzed the quantization of light due to Max Planck and Albert Einstein, while the fourth chapter discusses the Niels Bohr quantization of the energy levels and the electromagnetic transitions. The fifth chapter investigates the Schrodinger equation, which was obtained by Erwin Schrodinger from the idea of Louis De Broglie to associate to each particle a quantum wavelength. Chapter Six describes the basic axiomsof quantum mechanics, which were formulated in the seminal books of Paul Dirac and John von Neumann. In chapter seven, there are several important application of quantum mechanics: the quantum particle in a box, the quantum particle in the harmonic potential, the quantum tunneling, the stationary perturbation theory, and the time-dependent perturbation theory. Chapter Eight is devoted to the study of quantum atomic physics with special emphasis on the spin of the electron, which needs the Dirac equation for a rigorous theoretical justification. In the ninth chapter, it is explained the quantum mechanics of many identical particles at zero temperature, while in Chapter Ten the discussion is extended to many quantum particles at finite temperature by introducing and using the quantum statistical mechanics. The four appendices on Dirac delta function, complex numbers, Fourier transform, and differential equations are a useful mathematical aid for the reader.

This textbook introduces the special theory of relativity at a level which is accessible to undergraduate students and even high school students with a strong foundation in algebra. The presentation emphasizes clean algebraic and geometrical methods, visualized with plenty of illustrations, resulting in a textbook that is modern and serious yet accessible. Replete with many solved exercises and copious spacetime diagrams, this book will help students develop relativistic intuition when encountering the subject for the first time. The emphasis on geometric methods, combined with the pedagogically appealing k-calculus approach, makes this book ideal for a self-contained course on special relativity or as supplementary reading for modern physics courses. It will also appeal to high schoolers with a strong math background who want to get ahead.

This book presents Special Relativity in a language accessible to students while avoiding the burdens of geometry, tensor calculus, space-time symmetries, and the introduction of four vectors. The search for clarity in the fundamental questions about Relativity, the discussion of historical developments before and after 1905, the strong connection to current research topics, many solved examples and problems, and illustrations of the material in colloquial discussions are the most significant and original assets of this book. Importantly for first-time students, Special Relativity is presented such that nothing needs to be called paradoxical or apparent; everything is explained. The content of this volume develops and builds on the book Relativity Matters (Springer, 2017). However, this presentation of Special Relativity does not require 4-vector tools. The relevant material has been extended and reformulated, with additional examples and clarifications. Thisintroduction of Special Relativity offers conceptual insights reaching well beyond the usual method of teaching relativity. It considers relevant developments after the discovery of General Relativity (which itself is not presented), and advances the reader into contemporary research fields. This presentation of Special Relativity is connected to present day research topics in particle, nuclear, and high intensity pulsed laser physics and is complemented by the current cosmological perspective. The conceptual reach of Special Relativity today extends significantly further compared even to a few decades ago. As the book progresses, the qualitative and historical introduction turns into a textbook-style presentation with many detailed results derived in an explicit manner. The reader reaching the end of this text needs knowledge of classical mechanics, a good command of elementary algebra, basic knowledge of calculus, and introductory know-how of electromagnetism.

This book takes a fresh, systematic approach to determining the equation of motion for the classical model of the electron introduced by Lorentz 130 years ago. The original derivations of Lorentz, Abraham, Poincaré, and Schott are modified and generalized for the charged insulator model of the electron to obtain an equation of motion consistent with causal solutions to the Maxwell-Lorentz equations and the equations of special relativity. The solutions to the resulting equation of motion are free of pre-acceleration and pre-deceleration. The generalized method is applied to obtain the causal solution to the equation of motion of a charge accelerating in a uniform electric field for a finite time interval. Alternative derivations of the Landau-Lifshitz approximation are given as well as necessary and sufficient conditions for the Landau-Lifshitz approximation to be an accurate solution to the exact Lorentz-Abraham-Dirac equation of motion. Binding forces and a total stress-momentum-energy tensor are derived for the charged insulator model. Appendices provide simplified derivations of the self-force and power at arbitrary velocity. In this third edition, some of the history has been made more accurate and some of the derivations have been simplified and clarified. A detailed three-vector exact solution to the Landau-Lifshitz approximate equation of motion is given for the problem of an electron traveling in a counterpropagating plane-wave laser-beam pulse. Semi-classical analyses are used to derive the conditions that determine the significance of quantum effects not included in the classical equation of motion. The book is a valuable resource for students and researchers in physics, engineering, and the history of science.

This book seeks to present a new way of thinking about the interaction of gravitational fields with quantum systems. Despite the massive amounts of research and experimentation, the myriad meetings, seminars and conferences, all of the articles, treatises and books, and the seemingly endless theorization, quantization and just plain speculation that have been engaged in regarding our evolving understanding of the quantum world, that world remains an enigma, even to the experts. The usefulness of general relativity in this regard has proven to be imperfect at best, but there is a new approach. We do not simply have to accept the limitations of Einstein's most celebrated theorem in regard to quantum theory; we can also embrace them, and thereby utilize them, to reveal new facts about the behavior of quantum systems within inertial and gravitational fields, and therefore about the very structure of space–time at the quantum level. By taking existing knowledge of the essential functionality of spin (along with the careful identification of the omnipresent inertial effects) and applying it to the quantum world, the book gives the reader a much clearer picture of the difference between the classical and quantum behaviors of a particle, shows that Einstein's ideas may not be as incompatible within this realm as many have come to believe, sparks new revelations of the way in which gravity affects quantum systems and brings a new level of efficiency—quantum efficiency, if you will—to the study of gravitational theory.

This book treats various aspects of the quantum theory of measurement, partially in a relativistic framework. Measurement(-like) processes in quantum theory are identified and analysed; and the quantum operator formalism is derived in full generality without postulating operators as observables. Consistency conditions are derived, expressing the requirement of Lorentz-frame independence of outcomes of spacelike separated measurements and implying the impossibility of using quantum nonlocality to send signals faster than light. Local commutativity is scrutinized. The localization problem of relativistic quantum theory is studied, including comprehensive derivation of the theorems of Hegerfeld, Malament and Reeh-Schlieder. Finally, the quantum formalism is derived from the dynamics of particles with definite positions in Bohmian mechanics.

This book presents an alternative representation of Einstein's Special Theory of Relativity, which makes Special Relativity much more comprehensible. Moreover, one will come across a fundamental relationship between the Special Theory of Relativity and the mechanics of space lattice. In all previous formulations, the Einsteinian special principle of relativity, in one or the other form is used as the starting point for Special Relativity. In correspondence to this principle, one takes it as granted apriori, that all observers independent of their uniform motion to each other measure one and the same propagation velocity of a light signal. This book is thought of as a lecture for physicists, mathematicians and computer scientists and concentrates on the students of these fields. The book should reach a broad circle of interested readers from the fields of natural sciences and philosophy and provide and invigorating experience for engineers.

This textbook develops Special Relativity in a systematic way and offers problems with detailed solutions to empower students to gain a real understanding of this core subject in physics. This new edition has been thoroughly updated and has new sections on relativistic fluids, relativistic kinematics and on four-acceleration. The problems and solution section has been significantly expanded and short history sections have been included throughout the book.The approach is structural in the sense that it develops Special Relativity in Minkowski space following the parallel steps as the development of Newtonian Physics in Euclidian space. A second characteristic of the book is that it discusses the mathematics of the theory independently of the physical principles, so that the reader will appreciate their role in the development of the physical theory.The book is intended to be used both as a textbook for an advanced undergraduate teaching course in Special Relativity but also as a reference book for the future.

This book presents a detailed study of the Lanczos potential in general relativity by using tetrad formalisms. It demonstrates that these formalisms offer some simplifications over the tensorial methods, and investigates a general approach to finding the Lanczos potential for algebraic space–time by translating all the tensorial relations concerning the Lanczos potential into the language of tetrad formalisms and using the Newman–Penrose and Geroch–Held–Penrose formalisms. In addition, the book obtains the Lanczos potential for perfect fluid space–time, and applies the results to cosmological models of the universe. In closing, it highlights other methods, apart from tetrad formalisms, for finding the Lanczos potential, as well as further applications of the Newman–Penrose formalism. Given its scope, the book will be of interest to pure mathematicians, theoretical physicists and cosmologists, and will provide common ground for communication among these scientific communities.

This book presents the Projective approach to de Sitter Relativity. It traces the development of renewed interest in models of the universe at constant positive curvature such as'vacuum'geometry. The De Sitter Theory of Relativity, formulated in 1917 with Willem De Sitter's solution of the Einstein equations, was used in different fields during the 1950s and 1960s, in the work of H. Bacry, J.M. LevyLeblond and F.Gursey, to name some important contributors. From the 1960s to 1980s, L. Fantappié and G. Arcidiacono provided an elegant group approach to the De Sitter universe putting the basis for special and general projective relativity. Today such suggestions flow into a unitary scenario, and this way the De Sitter Relativity is no more a'missing opportunity'(F. Dyson, 1972), but has a central role in theoretical physics. In this volume a systematic presentation is given of the De Sitter Projective relativity, with the recent developments in projective general relativity and quantum cosmology.

This thesis describes one of the most precise experimental tests of Lorentz symmetry in electrodynamics by light-speed anisotropy measurement with an asymmetric optical ring cavity. The author aims to answer the fundamental, hypothetical debate on Lorentz symmetry in the Universe. He concludes that the symmetry is protected within an error of 10-15, which means providing one of the most stringent upper limits on the violation of the Lorentz symmetry in the framework of the Standard Model Extension. It introduces the following three keys which play an important role in achieving high-precision measurement: (1) a high-index element (silicon) interpolated into part of the light paths in the optical ring cavity, which improves sensitivity to the violation of the Lorentz symmetry, (2) double-pass configuration of the interferometer, which suppresses environmental noises, and (3) continuous data acquisition by rotating the optical ring cavity, which makes it possible tosearch for higher-order violations of Lorentz symmetry. In addition to those well-described keys, a comprehensive summary from theoretical formulations to experimental design details, data acquisition, and data analysis helps the reader follow up the experiments precisely.

This book offers a comprehensive, university-level introduction to Einstein's Special Theory of Relativity. In addition to the purely theoretical aspect, emphasis is also given to its historical development as well as to the experiments that preceded the theory and those performed in order to test its validity.The main body of the book consists of chapters on Relativistic Kinematics and Dynamics and their applications, Optics and Electromagnetism. These could be covered in a one-semester course. A more advanced course might include the subjects examined in the other chapters of the book and its appendices.As a textbook, it has some unique characteristics: It provides detailed proofs of the theorems, offers abundant figures and discusses numerous examples. It also includes a number of problems for readers to solve, the complete solutions of which are given at the end of the book.It is primarily intended for use by university students of physics, mathematics and engineering. However, as the mathematics needed is of an upper-intermediate level, the book will also appeal to a more general readership.

This book offers an essential bridge between college-level introductions and advanced graduate-level books on special relativity. It begins at an elementary level, presenting and discussing the basic concepts normally covered in college-level works, including the Lorentz transformation. Subsequent chapters introduce the four-dimensional worldview implied by the Lorentz transformations, mixing time and space coordinates, before continuing on to the formalism of tensors, a topic usually avoided in lower-level courses. The book's second half addresses a number of essential points, including the concept of causality; the equivalence between mass and energy, including applications; relativistic optics; and measurements and matter in Minkowski space-time. The closing chapters focus on the energy-momentum tensor of a continuous distribution of mass-energy and its co-variant conservation; angular momentum; a discussion of the scalar field of perfect fluids and the Maxwell field;and general coordinates.Every chapter is supplemented by a section with numerous exercises, allowing readers to practice the theory. These exercises constitute an essential part of the textbook, and the solutions to approximately half of them are provided in the appendix.

Special relativity is the basis of many fields in modern physics: particle physics, quantum field theory, high-energy astrophysics, etc. This theory is presented here by adopting a four-dimensional point of view from the start. An outstanding feature of the book is that it doesn't restrict itself to inertial frames but considers accelerated and rotating observers. It is thus possible to treat physical effects such as the Thomas precession or the Sagnac effect in a simple yet precise manner. In the final chapters, more advanced topics like tensorial fields in spacetime, exterior calculus and relativistic hydrodynamics are addressed. In the last, brief chapter the author gives a preview of gravity and shows where it becomes incompatible with Minkowsky spacetime. Well illustrated and enriched by many historical notes, this book also presents many applications of special relativity, ranging from particle physics (accelerators, particle collisions, quark-gluon plasma) to astrophysics (relativistic jets, active galactic nuclei), and including practical applications (Sagnac gyrometers, synchrotron radiation, GPS). In addition, the book provides some mathematical developments, such as the detailed analysis of the Lorentz group and its Lie algebra. The book is suitable for students in the third year of a physics degree or on a masters course, as well as researchers and any reader interested in relativity. Thanks to the geometric approach adopted, this book should also be beneficial for the study of general relativity. “A modern presentation of special relativity must put forward its essential structures, before illustrating them using concrete applications to specific dynamical problems. Such is the challenge (so successfully met!) of the beautiful book by Éric Gourgoulhon.” (excerpt from the Foreword by Thibault Damour)

From the reviews:'This attractive book provides an account of the theory of special relativity from a geometrical viewpoint, explaining the unification and insights that are given by such a treatment. […] Can be read with profit by all who have taken a first course in relativity physics.'ASLIB Book Guide

he present volume is part of a larger project, which is the attempt to draft a T coherent doctrine of divine eternity and God's relationship to time. In my God, l Time, and Eternity, I argued that whether one construes divine eternity in terms of timelessness or of omnitemporality will depend'''crucially upon one's views about the objectivity of tensed facts and temporal becoming. If one adopts a tensed, or in McTaggart's terminology, an A-Theory of time, then a coherent doctrine of divine eternity requires that one construe God, at least since the moment of creation, to exist temporally, which implies that divine timelessness can be successfully maintained only if a tenseless or B-Theory of time is correct. Accordingly in my companion volumes The Tensed Theory of Time: a Critical Examination and The Tenseless Theory of Time: a Critical Examination I set for myself. the task of 2 adjudicating the A- vs. B-Theory of time. In the former volume; r examine arguments for and against the A-Theory of time, and in the latter l'tum to an examination of arguments for and against the B-Theory. This inquiry tookme into a study of relativity theory,. its presuppositions and implications. The paucity of integrative literature dealing with the concept of. God and· relativity theory is striking.

In this concise primer it is shown that, with simple diagrams, the phenomena of time dilatation, length contraction and Lorentz transformations can be deduced from the fact that in a vacuum one cannot distinguish physically straight and uniform motion from rest, and that the speed of light does not depend on the speed of either the source or the observer. The text proceeds to derive the important results of relativistic physics and to resolve its apparent paradoxes. A short introduction into the covariant formulation of electrodynamics is also given. This publication addresses, in particular, students of physics and mathematics in their final undergraduate year.

The aim of this book is to provide a short but complete exposition of the logical structure of classical relativistic electrodynamics written in the language and spirit of coordinate-free differential geometry. The intended audience is primarily mathematicians who want a bare-bones account of the foundations of electrodynamics written in language with which they are familiar and secondarily physicists who may be curious how their old friend looks in the new clothes of the differential-geometric viewpoint which in recent years has become an important language and tool for theoretical physics. This work is not intended to be a textbook in electrodynamics in the usual sense; in particular no applications are treated, and the focus is exclusively the equations of motion of charged particles. Rather, it is hoped that it may serve as a bridge between mathemat­ ics and physics. Many non-physicists are surprised to learn that the correct equation to describe the motion of a classical charged particle is still a matter of some controversy. The most mentioned candidate is the Lorentz-Dirac equation t. However, it is experimentally unverified, is known to have no physically reasonable solutions in certain circumstances, and its usual derivations raise serious foundational issues. Such difficulties are not extensively discussed in most electrodynamics texts, which quite naturally are oriented toward applying the well-verified part of the subject to con­ crete problems.

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, mathem- ics 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 a ´ 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 r- ativity.

After A. Ungar had introduced vector algebra and Cartesian coordinates into hyperbolic geometry in his earlier books, along with novel applications in Einstein's special theory of relativity, the purpose of his new book is to introduce hyperbolic barycentric coordinates, another important concept to embed Euclidean geometry into hyperbolic geometry. It will be demonstrated that, in full analogy to classical mechanics where barycentric coordinates are related to the Newtonian mass, barycentric coordinates are related to the Einsteinian relativistic mass in hyperbolic geometry. Contrary to general belief, Einstein's relativistic mass hence meshes up extraordinarily well with Minkowski's four-vector formalism of special relativity. In Euclidean geometry, barycentric coordinates can be used to determine various triangle centers. While there are many known Euclidean triangle centers, only few hyperbolic triangle centers are known, and none of the known hyperbolic triangle centers has been determined analytically with respect to its hyperbolic triangle vertices. In his recent research, the author set the ground for investigating hyperbolic triangle centers via hyperbolic barycentric coordinates, and one of the purposes of this book is to initiate a study of hyperbolic triangle centers in full analogy with the rich study of Euclidean triangle centers. Owing to its novelty, the book is aimed at a large audience: it can be enjoyed equally by upper-level undergraduates, graduate students, researchers and academics in geometry, abstract algebra, theoretical physics and astronomy. For a fruitful reading of this book, familiarity with Euclidean geometry is assumed. Mathematical-physicists and theoretical physicists are likely to enjoy the study of Einstein's special relativity in terms of its underlying hyperbolic geometry. Geometers may enjoy the hunt for new hyperbolic triangle centers and, finally, astronomers may use hyperbolic barycentric coordinates in the velocityspace of cosmology.

This volume provides a detailed discussion of the mathematical aspects and the physical applications of a new geometrical structure of space-time, based on a generalization ('deformation') of the usual Minkowski space, as supposed to be endowed with a metric whose coefficients depend on the energy.Such a formalism (Deformed Special Relativity, DSR) allows one to account for breakdown of local Lorentz invariance in the usual, special-relativistic meaning (however, Lorentz invariance is recovered in a generalized sense) to provide an effective geometrical description of the four fundamental interactions (electromagnetic, weak, strong and gravitational) Moreover, the four-dimensional energy-dependent space-time is just a manifestation of a larger, five-dimensional space in which energy plays the role of a fifth (non-compactified) dimension. This new five-dimensional scheme (Deformed Relativity in Five Dimensions, DR5) represents a true generalization of the usual Kaluza-Klein (KK) formalism.The mathematical properties of such a generalized KK scheme are illustrated. They include the solutions of the five-dimensional Einstein equations in vacuum in most cases of physical relevance, the infinitesimal symmetries of the theory for the phenomenological metrics of the four interactions, and the study of the five-dimensional geodesics.The mathematical results concerning the geometry of the deformed five-dimensional spacetime (like its Killing symmetries) can be applied also to other multidimensional theories with infinite extra dimensions.Some experiments providing preliminary evidence for the hypothesized deformation of space-time for all the four fundamental interactions are discussed. Audience:Graduate students and researchers in mathematics and physics; researchers and engineers working in nuclear and space industries, space agencies, governmental scientific (including military and defense) institutions.

Between 1994 and 1999, I had the pleasure of lecturing Special and General Relativity in the Facultad de Ciencias Exactas y Naturales of the Universidad de Buenos Aires. These lectures were targeted to undergraduate and graduate students of Physics. However, it is increasingly apparent that interest in Relativity extends beyond these academic circles. Because of this reason, this book intends to become useful to students of related disciplines and to other readers interested in Einstein's work, who will be able to incorporate entirely the fundamental ideas of Relativity starting from the very basic concepts of Physics. To understand the Theory of Relativity it is necessary to give up our intuitive notions of space and time, i. e., the notions used in our daily relation with the world. These classical notions of space and time are also the foundations of Newtonian mechanics, which dominated Physics for over two centuries until they clashed with Maxwell's electromagnetism. Classical physics assumed that space is immutable and its geometry obeys the Euclidean postulates. Furthermore, distances and time intervals are believed invariant, i. e., independent of the state of motion. Both preconceptions about the nature of space and time rely firmly on our daily experience, in such a way that the classical notions are imprinted in our thought with the status of “true.

Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (Möbius') addition forms the setting for the Beltrami (Poincaré) ball model of hyperbolic geometry. The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.