This book provides a number of combinatorial tools that allow a systematic study of very general discrete spaces involved in the context of discrete quantum gravity. In any dimension D, we can discretize Euclidean gravity in the absence of matter over random discrete spaces obtained by gluing families of polytopes together in all possible ways. These spaces are then classified according to their curvature. In D=2, it results in a theory of random discrete spheres, which converge in the continuum limit towards the Brownian sphere, a random fractal space interpreted as a quantum random space-time. In this limit, the continuous Liouville theory of D=2 quantum gravity is recovered. Previous results in higher dimension regarded triangulations, converging towards a continuum random tree, or gluings of simple building blocks of small sizes, for which multi-trace matrix model results are recovered in any even dimension. In this book, the author develops a bijection with stacked two-dimensional discrete surfaces for the most general colored building blocks, and details how it can be used to classify colored discrete spaces according to their curvature. The way in which this combinatorial problem arrises in discrete quantum gravity and random tensor models is discussed in detail.

Yi-Shi Duan (1927-2016) was one of the world-renowned pioneers in the study of gauge field theory and general relativity. Trained in the former Soviet Union, Prof. Duan returned to China in 1957 to work in Lanzhou University for 60 years. In 1963, he came up with a general co-variant form of the conservation law of the energy-momentum tensor in general relativity. In 1979, he suggested that the gauge potential could be decomposed, which has important implications to gauge field theory. He trained in China a big team of talents in theoretical physics. His contributions to theoretical physics in China have earned him praise from both Professor Shiing-Shen Chern and Professor Chen-Ning Yang.

This book is devoted to the Einstein's field equations of general relativity for self-gravitating massive scalar fields. We formulate the initial value problem when the initial data set is a perturbation of an asymptotically flat, spacelike hypersurface in Minkowski spacetime. We then establish the existence of an Einstein development associated with this initial data set, which is proven to be an asymptotically flat and future geodesically complete spacetime.

The aim of this two-volume title is to give a comprehensive review of one hundred years of development of general relativity and its scientific influences. This unique title provides a broad introduction and review to the fascinating and profound subject of general relativity, its historical development, its important theoretical consequences, gravitational wave detection and applications to astrophysics and cosmology. The series focuses on five aspects of the theory: The first three topics are covered in Volume 1 and the remaining two are covered in Volume 2. While this is a two-volume title, it is designed so that each volume can be a standalone reference volume for the related topic.

The aim of this two-volume title is to give a comprehensive review of one hundred years of development of general relativity and its scientific influences. This unique title provides a broad introduction and review to the fascinating and profound subject of general relativity, its historical development, its important theoretical consequences, gravitational wave detection and applications to astrophysics and cosmology. The series focuses on five aspects of the theory: The first three topics are covered in Volume 1 and the remaining two are covered in Volume 2. While this is a two-volume title, it is designed so that each volume can be a standalone reference volume for the related topic.

This textbook provides an introduction to gravitational lensing, which has become an invaluable tool in modern astrophysics, with applications that range from finding planets orbiting distant stars to understanding how dark matter and dark energy conspired to form the cosmic structures we see today. Principles of Gravitational Lensing begins with Einstein's prediction that gravity bends light, and shows how that fundamental idea has spawned a rich field of study over the past century. The gravitational deflection of light was first detected by Eddington during a solar eclipse in May 1919, launching Einstein and his theory of relativity into public view. Yet the possibility of using the phenomenon to unlock mysteries of the Universe seemed remote, given the technology of the day. Theoretical work was carried out sporadically over the next six decades, but only with the discovery of the system Q0957+561 in 1979 was gravitational lensing transformed from a curiosity of general relativity into a practical observational tool. This book describes how the three subfields known as strong lensing, weak lensing, and microlensing have grown independently but become increasingly intertwined. Drawing on their research experience, Congdon and Keeton begin with the basic physics of light bending, then present the mathematical foundations of gravitational lensing, building up to current research topics in a clear and systematic way. Relevant background material from physics and mathematics is included, making the book self-contained. The derivations and explanations are supplemented by exercises designed to help students master the theoretical concepts as well as the methods that drive current research. An extensive bibliography guides those wishing to delve more deeply into particular areas of interest. Principles of Gravitational Lensing is ideal for advanced students and seasoned researchers looking to penetrate this thriving subject and even contribute research of their own.

'The content of the SaulsonaEURO (TM)s book remains valid and offers a versatile introduction to gravitational wave astronomy. The book is appropriate for undergraduate students and can be read by graduate students and researchers who want to be involved in either the theoretical or the experimental traits of the study of gravitational waves.'Contemporary PhysicsLIGO's recent discovery of gravitational waves was headline news around the world. Many people will want to understand more about what a gravitational wave is, how LIGO works, and how LIGO functions as a detector of gravitational waves.This book aims to communicate the basic logic of interferometric gravitational wave detectors to students who are new to the field. It assumes that the reader has a basic knowledge of physics, but no special familiarity with gravitational waves, with general relativity, or with the special techniques of experimental physics. All of the necessary ideas are developed in the book.The first edition was published in 1994. Since the book is aimed at explaining the physical ideas behind the design of LIGO, it stands the test of time. For the second edition, an Epilogue has been added; it brings the treatment of technical details up to date, and provides references that would allow a student to become proficient with today's designs.

'The content of the SaulsonaEURO (TM)s book remains valid and offers a versatile introduction to gravitational wave astronomy. The book is appropriate for undergraduate students and can be read by graduate students and researchers who want to be involved in either the theoretical or the experimental traits of the study of gravitational waves.'Contemporary PhysicsLIGO's recent discovery of gravitational waves was headline news around the world. Many people will want to understand more about what a gravitational wave is, how LIGO works, and how LIGO functions as a detector of gravitational waves.This book aims to communicate the basic logic of interferometric gravitational wave detectors to students who are new to the field. It assumes that the reader has a basic knowledge of physics, but no special familiarity with gravitational waves, with general relativity, or with the special techniques of experimental physics. All of the necessary ideas are developed in the book.The first edition was published in 1994. Since the book is aimed at explaining the physical ideas behind the design of LIGO, it stands the test of time. For the second edition, an Epilogue has been added; it brings the treatment of technical details up to date, and provides references that would allow a student to become proficient with today's designs.

It has been over 100 years since the presentation of the Theory of General Relativity by Albert Einstein, in its final formulation, to the Royal Prussian Academy of Sciences. To celebrate 100 years of general relativity, World Scientific publishes this volume with a dual goal: to assess the current status of the field of general relativity in broad terms, and discuss future directions. The volume thus consists of broad overviews summarizing major developments over the past decades and their perspective contributions.

"The General Theory of Relativity: A Mathematical Exposition" will serve readers as a modern mathematical introduction to the general theory of relativity. Throughout the book, examples, worked-out problems, and exercises (with hints and solutions) are furnished. Topics in this book include, but are not limited to:
tensor analysis
the special theory of relativity
the general theory of relativity and Einstein's field equations
spherically symmetric solutions and experimental confirmations
static and stationary space-time domains
black holes
cosmological models
algebraic classifications and the Newman-Penrose equations
the coupled Einstein-Maxwell-Klein-Gordon equations
appendices covering mathematical supplements and special topics
Mathematical rigor, yet very clear presentation of the topics make this book a unique text for both university students and research scholars.

Anadijiban Das has taught courses on Relativity Theory at The University College of Dublin, Ireland, Jadavpur University, India, Carnegie-Mellon University, USA, and Simon Fraser University, Canada. His major areas of research include, among diverse topics, the mathematical aspects of general relativity theory.

Andrew DeBenedictis has taught courses in Theoretical Physics at Simon Fraser University, Canada, and is also a member of The Pacific Institute for the Mathematical Sciences. His research interests include quantum gravity, classical gravity, and semi-classical gravity."

This book is the first serious, book-length study of the philosophical implications of relativity. It explains the relevant technical issues of general relativity theory and discusses how these issues bear upon philosophical problems about the nature of space and time, causality and laws of nature. Features include an analysis of time travel, The Big Bang and the Horizon Problem, Eternal Recurrence and Cyclic Time amongst other much debated issues today which have been brought to light in the the face of a series of theorems, due largely to Stephen Hawking and Roger Penrose. Written by a leading philosopher, this book provides an overview of the technical literature as well as analytical commentary on its philosophical significance.

This book is the second edition of an excellent undergraduate-level overview of classical and modern physics, intended for students of physics and related subjects, and also perfectly suited for the education of physics teachers. The twelve-chapter book begins with Newton's laws of motion and subsequently covers topics such as thermodynamics and statistical physics, electrodynamics, special and general relativity, quantum mechanics and cosmology , the standard model and quantum chromodynamics. The writing is lucid, and the theoretical discussions are easy to follow for anyone comfortable with standard mathematics. An important addition in this second edition is a set of exercises and problems, distributed throughout the book. Some of the problems aim to complement the text, others to provide readers with additional useful tools for tackling new or more advanced topics. Furthermore, new topics have been added in several chapters; for example, the discovery of extra-solar planets from the wobble of their mother stars, a discussion of the Landauer principle relating information erasure to an increase of entropy, quantum logic, first order quantum corrections to the ideal gas equation of state due to the Fermi-Dirac and Bose-Einstein statistics. Both gravitational lensing and the time-correction in geo-positioning satellites are explained as theoretical applications of special and general relativity. The discovery of gravitational waves, one of the most important achievements of physical sciences, is presented as well. Professional scientists, teachers, and researchers will also want to have this book on their bookshelves, as it provides an excellent refresher on a wide range of topics and serves as an ideal starting point for expanding one's knowledge of new or unfamiliar fields. Readers of this book will not only learn much about physics, they will also learn to love it.

This book presents the author's personal historical perspective and conceptual analysis on symmetry and geometry. The author enlightens with modern views the historical process which led to the contemporary vision of space and symmetry that are used in theoretical physics and in particular in such abstract and advanced descriptions of the physical world as those provided by supergravity. The book is written intertwining storytelling and philosophical argumentation with some essential technical material. The author argues that symmetry and geometry are inextricably entangled and their current meaning is the result of a long process of abstraction which was determined through history and can be understood within the analytic system of thought of western civilization that started with the Ancient Greeks. The evolution of geometry and symmetry theory in the last forty years has been deeply and constructively influenced by supersymmetry/supergravity and the allied constructions of strings and branes. Further advances in theoretical physics cannot be based simply on the Galilean method of interrogating nature and then formulating a testable theory to explain the observed phenomena. One ought to interrogate human thought, meaning frontier-line mathematics concerned with geometry and symmetry in order to find there the threads of so far unobserved correspondences, reinterpretations and renewed conceptions.

This book tells the story of how, over the past century, dedicated observers and pioneering scientists achieved our current understanding of the universe. It was in antiquity that humankind first attempted to explain the universe often with the help of myths and legends. This book, however, focuses on the time when cosmology finally became a true science. As the reader will learn, this was a slow process, extending over a large part of the 20th century and involving many astronomers, cosmologists and theoretical physicists. The book explains how empirical astronomical data (e.g., Leavitt, Slipher and Hubble) were reconciled with Einstein's general relativity; a challenge which finally led Friedmann, De Sitter and Lemaitre, and eventually Einstein himself, to a consistent understanding of the observational results. The reader will realize the extraordinary implications of these achievements and how deeply they changed our vision of the cosmos: From being small, static, immutable and eternal, it became vast and dynamical - originating from (almost) nothing, and yet now, nearly 14 billion years later, undergoing accelerated expansion. But, as always happens, as well as precious knowledge, new mysteries have also been created where previously absolute certainty had reigned.

Clarity, readability, and rigor combine in the third edition of this widely used textbook to provide the first step into general relativity for advanced undergraduates with a minimal background in mathematics. Topics within relativity that fascinate astrophysics researchers and students alike are covered with Schutz's characteristic ease and authority, from black holes to relativistic objects, from pulsars to the study of the Universe as a whole. This third edition contains discoveries by astronomers that require general relativity for their explanation; two chapters on gravitational waves, including direct detections of gravitational waves and their observations' impact on cosmological measurements; new information on black holes and neutron stars; and greater insight into the expansion of the Universe. Over 300 exercises, many new to this edition, give students the confidence to work with general relativity and the necessary mathematics, while the informal writing style and worked examples make the subject matter easily accessible.

Introduction to General Relativity and Cosmology gives undergraduate students an overview of the fundamental ideas behind the geometric theory of gravitation and spacetime. Through pointers on how to modify and generalise Einstein's theory to enhance understanding, it provides a link between standard textbook content and current research in the field.Chapters present complicated material practically and concisely, initially dealing with the mathematical foundations of the theory of relativity, in particular differential geometry. This is followed by a discussion of the Einstein field equations and their various properties. Also given is analysis of the important Schwarzschild solutions, followed by application of general relativity to cosmology. Questions with fully worked answers are provided at the end of each chapter to aid comprehension and guide learning. This pared down textbook is specifically designed for new students looking for a workable, simple presentation of some of the key theories in modern physics and mathematics.

This textbook is suitable for a one-semester introduction to General Relativity for advanced undergraduates in physics and engineering. The book is concise so that the entire material can be covered in the one-semester time frame. Besides, the readers are introduced to the subject easily without the need for advanced mathematics. Though concise, the theory development is lucid and the readers are exposed to possible analytic calculations. Full solutions to some important problems are provided, and the experimental evidence is discussed in detail.Resources are provided to instructors who adopt this textbook for their courses. Adopting instructors can print and copy portions of these resources solely for their teaching needs. All instructional resources are furnished for informational use only, and are subject to change without notice.

This book highlights the review of articles in theoretical physics by the students of Professor K. Babu Joseph, as a Festschrift for his 80th Birthday. This book is divided into four sections based on the contributions of Babu Joseph and his students. The four sections are Cosmology, High Energy Physics, Mathematical Physics and Non-linear Dynamics and its applications.

This compact yet informative Guide presents an accessible route through Special Relativity, taking a modern axiomatic and geometrical approach. It begins by explaining key concepts and introducing Einstein's postulates. The consequences of the postulates - length contraction and time dilation - are unravelled qualitatively and then quantitatively. These strands are then tied together using the mathematical framework of the Lorentz transformation, before applying these ideas to kinematics and dynamics. This volume demonstrates the essential simplicity of the core ideas of Special Relativity, while acknowledging the challenges of developing new intuitions and dealing with the apparent paradoxes that arise. A valuable supplementary resource for intermediate undergraduates, as well as independent learners with some technical background, the Guide includes numerous exercises with hints and notes provided online. It lays the foundations for further study in General Relativity, which is introduced briefly in an appendix.

These are the proceedings of an international meeting hosted by the Royal Observatory, Edinburgh, to commemorate the 30th anniversary of the dedication of the UKIRT, the United Kingdom InfraRed Telescope.

The volume comprises 31 professional level papers. The first part of the book has 10 thorough reviews of the conception, design and build of the telescope, as well as accounts of some its key instruments such as IRCAM (the common-user infrared camera), CGS4 (the fourth Cooled Grating Spectrometer) and the Wide Field Camera. The second part of the book comprises 14 reviews of scientific achievements during its twenty years of visitor mode operations. The final part of the book is a series of 7 reviews of the results from the multiple surveys being done as part of UKIDSS (UKIRT Infrared Deep Sky Survey).

The authors are all experts in their respective fields, for example instrument scientists, operations staff and leading astronomers.

This book is composed of two parts: First part describes basics in numerical relativity, that is, the formulations and methods for a solution of Einstein's equation and general relativistic matter field equations. This part will be helpful for beginners of numerical relativity who would like to understand the content of numerical relativity and its background. The second part focuses on the application of numerical relativity. A wide variety of scientific numerical results are introduced focusing in particular on the merger of binary neutron stars and black holes.

This book is composed of two parts: First part describes basics in numerical relativity, that is, the formulations and methods for a solution of Einstein's equation and general relativistic matter field equations. This part will be helpful for beginners of numerical relativity who would like to understand the content of numerical relativity and its background. The second part focuses on the application of numerical relativity. A wide variety of scientific numerical results are introduced focusing in particular on the merger of binary neutron stars and black holes.

In 1908 Hermann Minkowski gave the four-dimensional(spacetime) formulationof special relativity 1]. In fact, HenriPoincare 2] rst noticedin1906that the Lorentz transformations had a geometric interpretation as rotations in a four-dimensional space with time as the fourth dimension. However it was Minkowski, who succe- fully decoded the profound message about the dimensionality of the world hidden in the relativity postulate, which re ects the experimental fact that natural laws are the same in all inertial reference frames. Unlike Poincare, Minkowski did not regardspacetime - the uni cation of space and time - as a convenientmathematical space, but insisted that this absolute four-dimensional world, as Minkowski called it, represents physical phenomena and the world more adequately than the relativity postulate: "the word relativity-postulate. . . seems to me very feeble. Since the pos- late comes to mean that only the four-dimensional world in space and time is given by the phenomena. . . I prefer to call it the postulate of the absolute world" 3]. The impact of Minkowski's ideas on the twentieth century physics has been so immense that one cannot imagine modern physics without the notion of spacetime. It would hardly be an exaggeration to say that spacetime has been the greatest discoveryinphysicsofall times. Theonlyotherdiscoverythatcomesclosetospa- time is Einstein's general relativity, which revealed that gravity is a manifestation of the curvature of spacetime. But it was the discovery of spacetime, which paved the way for this deep understanding of what gravity really is. Einstein saw the link betweenthegeometryofspacetimeandgravitationonlyafterheovercamehis initial hostile attitude toward the notion of spacetime."