in this work, we must therefore assume several abstract concepts that hardly need defending at this point in the history of mechanics. Most notably, these include the concept of the point particle and the concept of the inertial observer. The study of the relativistic particle system is undertaken here by means of a particular classical theory, which also exists on the quantum level, and which is especially suited to the many-body system in flat spacetime. In its fundamental postulates, the theory may be consid ered to be primarily the work of E.C.G. Stiickelberg in the 1940's, and of L.P. Horwitz and C. Piron in the 1970's, who may be said to have provided the generalization of Stiickelberg's theory to the many-body system. The references for these works may be found in Chapter 1. The theory itself may be legitimately called off-shell Hamiltonian dynamics, parameterized relativistic mechanics, or even classical event dynamics. The most important feature of the theory is probably the use of an invariant world time parameter, usually denoted T, which provides an evolution time for the system in such as way as to allow manifest co variance within a Hamiltonian formalism. In general, this parameter is neither a Lorentz-frame time, nor the proper time of the particles in the system."

2 Homogeneous superconducting state 210 3 Superconducting phases with broken space symmetries 213 4 Flavor asymmetric quark condensates 219 5 Concluding remarks 221 Acknowledgments 222 References 223 Neutral Dense Quark Matter 225 Mei Huang and Igor Shovkovy 1 Introduction 225 2 Local charge neutrality: homogeneous phase 226 3 Global charge neutrality: mixed phase 234 4 Conclusion 238 References 238 Possibility of color magnetic superconductivity 241 Toshitaka Tatsumi, Tomoyuki Maruyama, and Eiji Nakano 1 Introduction 241 2 What is ferromagnetism in quark matter? 243 3 Color magnetic superconductivity 248 4 Chiral symmetry and magnetism 253 5 Summary and Concluding remarks 258 Acknowledgments 260 References 260 Magnetic Fields of Compact Stars with Superconducting Quark Cores 263 David M. Sedrakian, David Blaschke, and Karen M. Shahabasyan 1 Introduction 263 2 Free Energy 265 3 Ginzburg-Landau equations 267 4 Vortex Structure 269 5 Solution of Ginzburg-Landau Equations 271 6 The Magnetic Field Components 273 7 Summary 275 Acknowledgments 275 References 275 Thermal Color-superconducting Fluctuations in Dense Quark Matter 277 D. N.

This volume introduces the application of two-component spinor calculus and fibre-bundle theory to complex general relativity. A review of basic and important topics is presented, such as two-component spinor calculus, conformal gravity, twistor spaces for Minkowski space-time and for curved space-time, Penrose transform for gravitation, the global theory of the Dirac operator in Riemannian four-manifolds, various definitions of twistors in curved space-time and the recent attempt by Penrose to define twistors as spin-3/2 charges in Ricci-flat space-time. Original results include some geometrical properties of complex space-times with nonvanishing torsion, the Dirac operator with locally supersymmetric boundary conditions, the application of spin-lowering and spin-raising operators to elliptic boundary value problems, and the Dirac and Rarita--Schwinger forms of spin-3/2 potentials applied in real Riemannian four-manifolds with boundary. This book is written for students and research workers interested in classical gravity, quantum gravity and geometrical methods in field theory. It can also be recommended as a supplementary graduate textbook.

Papers from the Discussion Conference on Recent Advances in General Relativity, held at the U. of Pittsburgh, May 1990, survey the interacting fields of classical general relativity, astrophysics, and quantum gravity. Some of the remarks made following the invited papers are also included. The conference also included three workshops on classical g

This book provides an easy and rewarding route into special relativity for first and second year undergraduates at university. The subjects that cause students difficulty, such as time-dilation, are delayed to allow the establishment of confidence so that the greater part of relativity is easy to understand and to apply correctly. This plan involves an early move towards the use of four-vectors and readers are guided gradually towards their use.
The book covers the principle topics of special relativity including: the conserved relatavistic energy and momentum; applications of the Lorentz transformation; developments in twentieth-century physics that depended on relativity; and a review of some of the early experiments relevant to the development of the subject are contained in the final chapter. This text should provide essential reading for all undergraduate physics students and their lecturers.

Time's 'Man of the Century', Albert Einstein is the unquestioned founder of modern physics. His theory of relativity is the most important scientific idea of the modern era. In this short book Einstein explains, using the minimum of mathematical terms, the basic ideas and principles of the theory which has shaped the world we live in today. Unsurpassed by any subsequent books on relativity, this remains the most popular and useful exposition of Einstein's immense contribution to human knowledge.

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This thesis focuses on understanding the growth and formation mechanism of supermassive black holes (SMBHs), an issue it addresses by investigating the dense interstellar medium that is assumed to be a crucial component of the fuel for SMBHs. The thesis also offers unique guidance on using the Atacama Large Millimeter/submillimeter Array (ALMA) in active galactic nuclei (AGN) research. The author presents the three major findings regarding SMBH formation and growth: (1) The development of a new diagnostic method for the energy sources in galaxies based on submillimeter spectroscopy, which allows identification of accreting SMBHs even in obscured environments, (2) the discovery that the circumnuclear dense gas disk (CND), with a typical size of a few tens of parsecs, which plays a crucial role in governing the growth of SMBHs, and (3) the discovery that the mass transfer budget from the CND to the central SMBHs can be quantitatively understood with a theoretical model incorporating the circumnuclear starburst as a driver of mass transfer. The thesis skillfully reviews these three findings, which have greatly improved our understanding of the growth mechanism of SMBHs.

New Edition: Introductory Quantum Physics and Relativity (2nd Edition)This book is based on the lecture courses taught by Dunningham and Vedral at the University of Leeds. The book contains all the necessary material for quantum physics and relativity in the first two years of a typical physics degree course. The choice of topics complies fully with the Institute of Physics guidelines, but the coverage also includes more interesting and up-to-date applications, such as Bose condensation and quantum teleportation.

After about a century of success, physicists feel the need to probe the limits of validity of special-relativity base theories. This book is the outcome of a special seminar held on this topic. The authors gather in a single volume an extensive collection of introductions and reviews of the various facets involved, and also includes detailed discussion of philosophical and historical aspects.

This book provides an introduction to classical celestial mechanics. It is based on lectures delivered by the authors over many years at both Padua University (MC) and V.N. Karazin Kharkiv National University (EB). The book aims to provide a mathematical description of the gravitational interaction of celestial bodies. The approach to the problem is purely formal. It allows the authors to write equations of motion and solve them to the greatest degree possible, either exactly or by approximate techniques, when there is no other way. The results obtained provide predictions that can be compared with the observations. Five chapters are supplemented by appendices that review certain mathematical tools, deepen some questions (so as not to interrupt the logic of the mainframe with heavy technicalities), give some examples, and provide an overview of special functions useful here, as well as in many other fields of physics. The authors also present the original investigation of torus potential. This book is aimed at senior undergraduate students of physics or astrophysics, as well as graduate students undertaking a master's degree or Ph.D.

Provides the essential principles and results of special relativity as required by undergraduates. The text uses a geometric interpretation of space-time so that a general theory is seen as a natural extension of the special theory. Although most results are derived from first principles, complex and distracting mathematics is avoided and all mathematical steps and formulae are fully explained and interpreted, often with explanatory diagrams.; The emphasis throughout the text is on understanding the physics of relativity. The structure of the book is designed to allow students of different courses to choose their own route through the short self-contained sections in each chapter. The latter part of the book shows how Einstein's theory of gravity is central to unravelling fundamental questions of cosmology.

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This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique

This book provides an accessible, yet thorough, introduction to special and general relativity, crafted and class-tested over many years of teaching. Suitable for advanced undergraduate and graduate students, this book provides clear descriptions of how to approach the mathematics and physics involved. It is also contains the latest exciting developments in the field, including dark energy, gravitational waves, and frame dragging. The table of contents has been carefully developed in consultation with a large number of instructors teaching courses worldwide, to ensure its wide applicability to modules on relativity and gravitation. Features: A clear, accessible writing style, presenting a sophisticated approach to the subject, that remains suitable for advanced undergraduate students and above Class-tested over many years To be accompanied by a partner volume on 'Advanced Topics' for students to further extend their learning

After an extensive introduction to the asymptotic safety approach to quantum gravity, this thesis explains recent key advances reported in four influential papers. Firstly, two exact solutions to the reconstruction problem (how to recover a bare action from the effective average action) are provided. Secondly, the fundamental requirement of background independence in quantum gravity is successfully implemented. Working within the derivative expansion of conformally reduced gravity, the notion of compatibility is developed, uncovering the underlying reasons for background dependence generically forbidding fixed points in such models. Thirdly, in order to understand the true nature of fixed-point solutions, one needs to study their asymptotic behaviour. The author carefully explains how to find the asymptotic form of fixed point solutions within the f(R) approximation. Finally, the key findings are summarised and useful extensions of the work are identified. The thesis finishes by considering the need to incorporate matter into the formalism in a compatible way and touches upon potential opportunities to test asymptotic safety in the future.

This proceedings volume gathers selected, revised papers presented at the X International Meeting on Lorentzian Geometry (GeLoCor 2021), virtually held at the University of Cordoba, Spain, on February 1-5, 2021. It includes surveys describing the state-of-the-art in specific areas, and a selection of the most relevant results presented at the conference. Taken together, the papers offer an invaluable introduction to key topics discussed at the conference and an overview of the main techniques in use today. This volume also gathers extended revisions of key studies in this field. Bringing new results and examples, these unique contributions offer new perspectives to the original problems and, in most cases, extend and reinforce the robustness of previous findings. Hosted every two years since 2001, the International Meeting on Lorentzian Geometry has become one of the main events bringing together the leading experts on Lorentzian geometry. In this volume, the reader will find studies on spatial and null hypersurfaces, low regularity in general relativity, conformal structures, Lorentz-Finsler spacetimes, and more. Given its scope, the book will be of interest to both young and experienced mathematicians and physicists whose research involves general relativity and semi-Riemannian geometry.

The thesis tackles two distinct problems of great interest in gravitational mechanics - one relativistic and one Newtonian. The relativistic one is concerned with the "first law of binary mechanics", a remarkably simple variational relation that plays a crucial role in the modern understanding of the gravitational two-body problem, thereby contributing to the effort to detect gravitational-wave signals from binary systems of black holes and neutron stars. The work reported in the thesis provides a mathematically elegant extension of previous results to compact objects that carry spin angular momentum and quadrupolar deformations, which more accurately represent astrophysical bodies than mere point particles. The Newtonian problem is concerned with the isochrone problem of celestial mechanics, namely the determination of the set of radial potentials whose bounded orbits have a radial period independent of the angular momentum. The thesis solves this problem completely in a geometrical way and explores its consequence on a variety of levels, in particular with a complete characterisation of isochrone orbits. The thesis is exceptional in the breadth of its scope and achievements. It is clearly and eloquently written, makes excellent use of images, provides careful explanations of the concepts and calculations, and it conveys the author's personality in a way that is rare in scientific writing, while never sacrificing academic rigor.

This is the second volume of textbooks on atomic, molecular and optical physics, aiming at a comprehensive presentation of this highly productive branch of modern physics as an indispensable basis for many areas in physics and chemistry as well as in state of the art bio- and material-sciences. It primarily addresses advanced students (including PhD students), but in a number of selected subject areas the reader is lead up to the frontiers of present research. Thus even the active scientist is addressed. This volume 2 introduces lasers and quantum optics, while the main focus is on the structure of molecules and their spectroscopy, as well as on collision physics as the continuum counterpart to bound molecular states. The emphasis is always on the experiment and its interpretation, while the necessary theory is introduced from this perspective in a compact and occasionally somewhat heuristic manner, easy to follow even for beginners.

This book investigates Lorentzian structures in the four-dimensional space-time, supplemented either by a covector field of the time-direction or by a scalar field of the global time. Furthermore, it proposes a new metrizable model of gravity. In contrast to the usual General Relativity theory, where all ten components of the symmetric pseudo-metric are independent variables, the gravity model presented here essentially depends only on a single four-covector field, and is restricted to have only three-independent components. However, the author proves that the gravitational field, governed by the proposed model and generated by some massive body, resting and spherically symmetric in some coordinate system, is given by a pseudo-metric that coincides with the well known Schwarzschild metric from General Relativity. The Maxwell equations and electrodynamics are also investigated in the framework of the proposed model. In particular, the covariant formulation of electrodynamics of moving dielectrics and para/diamagnetic media is derived.

This book presents a fresh, original exposition of the foundations of classical electrodynamics in the tradition of the so-called metric-free approach. The fundamental structure of classical electrodynamics is described in the form of six axioms: (1) electric charge conservation, (2) existence of the Lorentz force, (3) magnetic flux conservation, (4) localization of electromagnetic energy-momentum, (5) existence of an electromagnetic spacetime relation, and (6) splitting of the electric current into material and external pieces.
The first four axioms require an arbitrary 4-dimensional differentiable manifold. The fifth axiom characterizes spacetime as the environment in which the electromagnetic field propagates a" a research topic of considerable interest a" and in which the metric tensor of spacetime makes its appearance, thus coupling electromagnetism and gravitation. Repeated emphasis is placed on interweaving the mathematical definitions of physical notions and the actual physical measurement procedures.
The tool for formulating the theory is the calculus of exterior differential forms, which is explained in sufficient detail, along with the corresponding computer algebra programs. Prerequisites for the reader include a knowledge of elementary electrodynamics (with Maxwell's equations), linear algebra and elementary vector analysis; some knowledge of differential geometry would help. Foundations of Classical Electrodynamics unfolds systematically at a level suitable for graduate students and researchers in mathematics, physics, and electrical engineering.

This book addresses supergravity and supergravity-motivated effective field theories in the context of cosmological model building. Extracting information about quintessence from string theory has attracted much attention in the past few years. The question became more urgent very recently after the possibility of obtaining de Sitter space was called into question. Therefore, there is an interesting debate as to whether de Sitter space or, even, quintessence can be derived from a fundamental theory, string theory or otherwise. This is a very active field of research, and the topics covered in the book render this work very timely. Throughout the book, special care has been taken in demonstrating historical relevance of the field and describing the set of open questions motivating the state-of-the-art research. The first few chapters in each part provide a detailed review of standard perturbative and non-perturbative techniques in supergravity model building, as a way to prepare the reader for the more technical and original subsequent chapters. These early chapters also represent a self-contained review that would be useful for anyone planning to enter this challenging area of study. The subsequent chapters detail research in supergravity-motivated effective field theories, in the first part, and supergravity models, in the second part. One of the important conclusions in this book is that modelling quintessence in perturbative string theory is at least as challenging as modelling de Sitter, placing the wider programme on a collision course with observations.

This thesis presents several significant new results that shed light on two major puzzles of modern cosmology: the nature of inflation, the very early phase of the universe that is thought to have given rise to the large-scale structures that we observe today; and that of the current accelerated expansion. In particular, it develops a clean method for characterizing linear cosmological perturbations for general theories where gravity is modified and/or affected by a new component, called dark energy, responsible for the accelerated expansion. It proposes a new extension to what were long thought to be the most general scalar field theories devoid of instabilities, and demonstrates the robustness of the relation between the energy scale of inflation and the predicted amplitude of gravitational waves. Finally, it consolidates a set of consistency relations between correlation functions of the cosmological density field and investigates the phenomenological consequences of their potential violation. Presented in a clear, succinct and rigorous style, each of these original results is both profound and important and will leave a deep mark on the field.

This book offers a concise and coherent introduction to accelerator physics and technology at the fundamental level but still in connection to advanced applications ranging from high-energy colliders to most advanced light sources, i.e., Compton sources, storage rings and free-electron lasers. The book is targeted at accelerator physics students at both undergraduate and graduate levels, but also of interest also to Ph.D. students and senior scientists not specialized in beam physics and accelerator design, or at the beginning of their career in particle accelerators. The book introduces readers to particle accelerators in a logical and sequential manner, with paragraphs devoted to highlight the physical meaning of the presented topics, providing a solid link to experimental results, with a simple but rigorous mathematical approach. In particular, the book will turn out to be self-consistent, including for example basics of Special Relativity and Statistical Mechanics for accelerators. Mathematical derivations of the most important expressions and theorems are given in a rigorous manner, but with simple and immediate demonstration where possible. The understanding gained by a systematic study of the book will offer students the possibility to further specialize their knowledge through the wide and up-to-date bibliography reported. Both theoretical and experimental items are presented with reference to the most recent achievements in colliders and light sources. The author draws on his almost 20-years long experience in the design, commissioning and operation of accelerator facilities as well as on his 10-years long teaching experience about particle accelerators at the University of Trieste, Department of Engineering and of Physics, as well as at international schools on accelerator physics.