The Physics of the Early Universe is an edited and expanded version of the lectures given at a recent summer school of the same name. Its aim is to present an advanced multi-authored textbook that meets the needs of both postgraduate students and young researchers interested in, or already working on, problems in cosmology and general relativity, with emphasis on the early universe. A particularly strong feature of the present work is the constructive-critical approach to the present mainstream theories, the careful assessment of some alternative approaches, and the overall balance between theoretical and observational considerations. As such, this book will also benefit experienced scientists and nonspecialists from related areas of research.

Cosmology has undergone a revolution in recent years. The exciting interplay between astronomy and fundamental physics has led to dramatic revelations, including the existence of the dark matter and the dark energy that appear to dominate our cosmos. But these discoveries only reveal themselves through small effects in noisy experimental data. Dealing with such observations requires the careful application of probability and statistics.
But it is not only in the arcane world of fundamental physics that probability theory plays such an important role. It has an impact in many aspects of our everyday life, from the law courts to the lottery.
Why then do so few people understand probability? And why do so few people understand why it is so important for science? Why do so many people think that science is about absolute certainty when, at its core, it is actually dominated by uncertainty?
This book attempts to explain the basics of probability theory, and illustrate their application across the entire spectrum of science.

Motion is always relative to some thing. Is this thing a concrete body like the earth, is it an abstract space, or is it an imagined frame? Do the laws of physics depend on the choice of reference? It there a choice for which the laws are simplest? Is this choice unique? Is there a physical cause for the choice made? These questions traverse the history of modern physics from Galileo to Einstein. The answers involved Galilean relativity, Newton's absolute space, the purely relational concepts of Descartes, Leibniz, and Mach, and many forgotten uses of relativity principles in mechanics, optics, and electrodynamics - until the relativity theories of Poincare, Einstein, Minkowksi, and Laue radically redefined space and time to satisfy universal kinds of relativity. Accordingly, this book retraces the emergence of relativity principles in early modern mechanics, documents their constructive use in eighteenth- and nineteenth-century mechanics, optics, and electrodynamics, and gives a well-rooted account of the genesis of special and general relativity in the early twentieth century. As an exercise in long-term history, it demonstrates the connectivity of issues and approaches across several centuries, despite enormous changes in context and culture. As an account of the genesis of relativity theories, it brings unprecedented clarity and fullness by broadening the spectrum of resources on which the principal actors drew.

Applications of quantum field theoretical methods to gravitational physics, both in the semiclassical and the full quantum frameworks, require a careful formulation of the fundamental basis of quantum theory, with special attention to such important issues as renormalization, quantum theory of gauge theories, and especially effective action formalism. The first part of this graduate textbook provides both a conceptual and technical introduction to the theory of quantum fields. The presentation is consistent, starting from elements of group theory, classical fields, and moving on to the effective action formalism in general gauge theories. Compared to other existing books, the general formalism of renormalization in described in more detail, and special attention paid to gauge theories. This part can serve as a textbook for a one-semester introductory course in quantum field theory. In the second part, we discuss basic aspects of quantum field theory in curved space, and perturbative quantum gravity. More than half of Part II is written with a full exposition of details, and includes elaborated examples of simplest calculations. All chapters include exercises ranging from very simple ones to those requiring small original investigations. The selection of material of the second part is done using the "must-know" principle. This means we included detailed expositions of relatively simple techniques and calculations, expecting that the interested reader will be able to learn more advanced issues independently after working through the basic material, and completing the exercises.

Im Alter von 21 Jahren hat W. Pauli einen Handbuchartikel zur Relativitatstheorie verfasst, der bis heute gelesen und zitiert wird. Er ist wohl der beruhmteste Text zum Thema und wurde nicht zuletzt von A. Einstein begeistert gewurdigt. Die vorliegende Neuausgabe enthalt den Originalartikel sowie weitere, teilweise recht ausfuhrliche Erganzungen, die Pauli im Jahre 1956 fur die englische Ausgabe schrieb. Eine Reihe von Anmerkungen des Herausgebers dienen daruber hinaus als Lesehilfen und zeigen Verbindungen zu modernen Entwicklungen auf."

This volume of original articles, collected papers and commentaries by contemporary scholars illustrates the work of Tullio Regge, a giant in the panorama of theoretical physics in the second half of the 20th century, probably the most influential Italian physicist after Enrico Fermi. His brilliant contributions to quantum theory and to general relativity have marked significant turning points in the development of scientific knowledge: Regge poles, Regge behaviour, Regge calculus and his geometric approach to general relativity and its extensions, and they continue to have a profound impact on the work of the large theoretical community today. Moreover, his public engagement for the dissemination of scientific culture, his mastering of multimedia technology for outreach and play, and his support for important social causes such as the fight against pseudosciences and the rights of the disabled make him a charismatic character across time, space and disciplines.

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects, from primordial inflation and cosmic microwave anisotropy to the dark energy that propels an accelerating universe.
The author presents the subject with an emphasis on physical examples and simple applications without the full tensor apparatus. The reader first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation, cosmic microwave anisotropy to the "dark energy" that propels an accelerating universe.
Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own.

Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, in the formalism of Minkowski's four-dimensional space-time, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects, from primordial inflation and cosmic microwave anisotropy to the dark energy that propels an accelerating universe.
The author presents the subject with an emphasis on physical examples and simple applications without the full tensor apparatus. The reader first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, precession of Mercury's perihelion, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Many modern topics in cosmology are discussed in this book: from inflation, cosmic microwave anisotropy to the "dark energy" that propels an accelerating universe.
Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own.

With a focus on modified gravity this book presents a review of the recent developments in the fields of gravity and cosmology, presenting the state of the art, high-lighting the open problems, and outlining the directions of future research. General Relativity and the CDM framework are currently the standard lore and constitute the concordance paradigm of cosmology. Nevertheless, long-standing open theoretical issues, as well as possible new observational ones arising from the explosive development of cosmology in the last two decades, offer the motivation and lead a large amount of research to be devoted in constructing various extensions and modifications. In this review all extended theories and scenarios are first examined under the light of theoretical consistency, and are then applied in various geometrical backgrounds, such as the cosmological and the spherical symmetric ones. Their predictions at both the background and perturbation levels, and concerning cosmology at early, intermediate and late times, are then confronted with the huge amount of observational data that astrophysics and cosmology has been able to offer in the last two decades. Theories, scenarios and models that successfully and efficiently pass the above steps are classified as viable and are candidates for the description of Nature, allowing readers to get a clear overview of the state of the art and where the field of modified gravity is likely to go. This work was performed in the framework of the COST European Action "Cosmology and Astrophysics Network for Theoretical Advances and Training Actions" - CANTATA.

This book is a new edition of "Tensors and Manifolds: With Applications to Mechanics and Relativity" which was published in 1992. It is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialised courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other's discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics. Mathematical rigour combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics.

Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself.

These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large.

Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions.

These conditions are shown to be satisfied in the vicinity of stars of more than twice the solar mass near the endpoint of their nuclear evolution, and in a time-reversed sense for the universe as a whole. In the first case, the singularity in our past. A discussion of the Cauchy problem for General Relativity is also included in the book.

In this title, the bestselling author of How to Teach Quantum Physics to Your Dog tackles Einsten's most famous theory - with the help of his faithful Alsatian, Emmy In this highly anticipated follow-up book, popular physicist Chad Orzel explains relativity theory in a decidedly unscientific fashion - through a series of conversations with his dog, Emmy. It goes something like this: Emmy, the curious scamp that she is, typically pursues some half-understood aspect of science or maths as a means to catching bunnies or squirrels. You know, as dogs do. When she fails - or sometimes, even before she does - Orzel gently explains the flaws in her plan, followed by a more detailed explanation of the real physics for interested humans. Emmy, in true puppy fashion, provides some occasional (and always helpful) interjections, asking for clarification or commenting on the explanation offered. Readers can expect to learn such concepts as how length contraction causes fast-moving objects to shrink and how relativity causes moving clocks to run slow, along with some of the cosmological consequences of general relativity and what it can tell us about where the universe came from and how it will end. From relative motion and time dilation, to the unification of forces and extra dimensions, Orzel will have even the biggest physics-phobe hooked. An entertaining ride through one of the great theories of modern physics, How to Teach Relativity to Your Dog will teach you about space, time, and many other things you may have slept through in high school physics.

The book begins with a brief review of supersymmetry, and the construction of the minimal supersymmetric standard model and approaches to supersymmetry breaking. General non-perturbative methods are also reviewed leading to the development of holomorphy and the Affleck-Dine-Seiberg superpotential as powerful tools for analysing supersymmetric theories. Seiberg duality is discussed in detail, with many example applications provided, with special attention paid to its use in understanding dynamical supersysmmetry breaking. The Seiberg-Witten theory of monopoles is introduced through the analysis of simpler N=1 analogues. Superconformal field theories are described along with the most recent development known as "amaximization". Supergravity theories are examined in 4, 10, and 11 dimensions, allowing for a discussion of anomaly and gaugino mediation, and setting the stage for the anti- de Sitter/conformal field theory correspondence. This book is unique in containing an overview of the important developments in supersymmetry since the publication of "Suppersymmetry and Supergravity" by Wess and Bagger. It also strives to cover topics that are of interest to both formal and phenomenological theorists.

Relativistic hydrodynamics is a very successful theoretical framework to describe the dynamics of matter from scales as small as those of colliding elementary particles, up to the largest scales in the universe. This book provides an up-to-date, lively, and approachable introduction to the mathematical formalism, numerical techniques, and applications of relativistic hydrodynamics. The topic is typically covered either by very formal or by very phenomenological books, but is instead presented here in a form that will be appreciated both by students and researchers in the field. The topics covered in the book are the results of work carried out over the last 40 years, which can be found in rather technical research articles with dissimilar notations and styles. The book is not just a collection of scattered information, but a well-organized description of relativistic hydrodynamics, from the basic principles of statistical kinetic theory, down to the technical aspects of numerical methods devised for the solution of the equations, and over to the applications in modern physics and astrophysics. Numerous figures, diagrams, and a variety of exercises aid the material in the book. The most obvious applications of this work range from astrophysics (black holes, neutron stars, gamma-ray bursts, and active galaxies) to cosmology (early-universe hydrodynamics and phase transitions) and particle physics (heavy-ion collisions). It is often said that fluids are either seen as solutions of partial differential equations or as "wet". Fluids in this book are definitely wet, but the mathematical beauty of differential equations is not washed out.

General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

This book introduces the modern field of 3+1 numerical relativity. The book has been written in a way as to be as self-contained as possible, and only assumes a basic knowledge of special relativity. Starting from a brief introduction to general relativity, it discusses the different concepts and tools necessary for the fully consistent numerical simulation of relativistic astrophysical systems, with strong and dynamical gravitational fields. Among the topics discussed in detail are the following; the initial data problem, hyperbolic reductions of the field equations, guage conditions, the evolution of black hole space-times, relativistic hydrodynamics, gravitational wave extraction and numerical methods. There is also a final chapter with examples of some simple numerical space-times. The book is aimed at both graduate students and researchers in physics and astrophysics, and at those interested in relativistic astrophysics.

This thesis is devoted to the systematic study of non-local theories that respect Lorentz invariance and are devoid of new, unphysical degrees of freedom. Such theories are attractive for phenomenological applications since they are mostly unconstrained by current experiments. Non-locality has played an increasingly important role in the physics of the last decades, appearing in effective actions in quantum field theory, and arising naturally in string theory and non-commutative geometry. It may even be a necessary ingredient for quantum theories of gravity. It is a feature of quantum entanglement, and may even solve the long-standing black hole information loss problem. "Non-locality" is a broad concept with many promising and fruitful applications in theoretical and mathematical physics. After a historical and pedagogical introduction into the concept of non-locality the author develops the notion of non-local Green functions to study various non-local weak-field problems in quantum mechanics, quantum field theory, gravity, and quantum field theory in curved spacetime. This thesis fills a gap in the literature by providing a self-contained exploration of weak-field effects in non-local theories, thereby establishing a "non-local intuition" which may serve as a stepping stone for studies of the full, non-linear problem of non-locality.

Owing to the increased accuracy requirements in fields such as astrometry and geodesy the general theory of relativity must be taken into account for any mission requiring highly accurate orbit information and for practically all observation and measurement techniques. This book highlights the confluence of Applied Mathematics, Physics and Space Science as seen from Einstein's general theory of relativity and aims to bridge the gap between theoretical and applied domains. The book investigates three distinct areas of general relativity: Exact solutions of the Einstein field equations of gravitation. Dynamics of near-Earth objects and solar system bodies. Relativistic orbitography. This book is an updated and expanded version of the author's PhD thesis which was awarded the International Astronomical Union PhD prize in Division A: Fundamental Astronomy. Included is a new introduction aimed at graduate students of General Relativity and extended discussions and results on topics in post-Newtonian dynamics and general relativistic spacecraft propagation.

The book is about exact space-time models of the gravitational fields produced by gravitational radiation. The authors' extensive work in the field is reviewed in order to stimulate the study of such models, that have been known for a long time, and to highlight interesting physical aspects of the existing models in some novel detail. There is an underlying simplicity to the gravitational radiation studied in this book. Apart from the basic assumption that the radiation has clearly identifiable wave fronts, the gravitational waves studied are directly analogous to electromagnetic waves. The book is meant for advanced students and researchers who have a knowledge of general relativity sufficient to carry out research in the field.

The ancient Greeks believed that everything in the Universe should be describable in terms of geometry. This thesis takes several steps towards realising this goal by introducing geometric descriptions of systems such as quantum gravity, fermionic particles and the origins of the Universe itself. The author extends the applicability of previous work by Vilkovisky, DeWitt and others to include theories with spin 1/2 and spin 2 degrees of freedom. In addition, he introduces a geometric description of the potential term in a quantum field theory through a process known as the Eisenhart lift. Finally, the methods are applied to the theory of inflation, where they show how geometry can help answer a long-standing question about the initial conditions of the Universe. This publication is aimed at graduate and advanced undergraduate students and provides a pedagogical introduction to the exciting topic of field space covariance and the complete geometrization of quantum field theory.

Dark energy, the mysterious cause of the accelerating expansion of the universe, is one of the most important fields of research in astrophysics and cosmology today. Introducing the theoretical ideas, observational methods and results, this textbook is ideally suited to graduate courses on dark energy, and will also supplement advanced cosmology courses. Providing a thorough introduction to this exciting field, the textbook covers the cosmological constant, quintessence, k-essence, perfect fluid models, extra-dimensional models, and modified gravity. Observational research is reviewed, from the cosmic microwave background to baryon acoustic oscillations, weak lensing and cluster abundances. Every chapter ends with problems, with full solutions provided, and any calculations are worked through step-by-step.

This thoroughly revised 5th edition of Zeh's classic text investigates irreversible phenomena and their foundation in classical, quantum and cosmological settings. It includes new sections on the meaning of probabilities in a cosmological context, irreversible aspects of quantum computers, and various consequences of the expansion of the Universe. In particular, the book offers an analysis of the physical concept of time.

Monte Carlo methods are a class of computational algorithms for simulating the behavior of a wide range of various physical and mathematical systems (with many variables). Their utility has increased with general availability of fast computers, and new applications are continually forthcoming. The basic concepts of Monte Carlo are both simple and straightforward and rooted in statistics and probability theory, their defining characteristic being that the methodology relies on random or pseudo-random sequences of numbers. It is a technique of numerical analysis based on the approximate solution of a problem using repeated sampling experiments and observing the proportion of times a given property is satisfied.
The term Monte Carlo was first used to describe calculational methods based on chance in the 1940s, but the methods themselves preceded the term by as much as a century. Quantum Monte Carlo (QMC) first appeared in 1982 and similarly was preceded by development of the related calculational methodology. The success of QMC methods over the past few decades has been remarkable, and this book will clearly demonstrate that success in its discussion of applications. For isolated molecules, the basic material of chemistry, QMC methods have produced exact solutions of the Schroedinger equation for very small systems and the most accurate solutions available for very large systems. The range of applications is impressive: folding of protein molecules, interactions in liquids, structure modeling in crystals and enzymes, quantum dots, designing heat shields and aerodynamic forms, architecture, design, business and economics, and even cinema and video games (3D modeling).
This booktakes a similar approach to Henry Schaefers classic book Quantum Chemistry (OUP, 1984 now a Dover edition), collecting summaries of some of the most important papers in the quantum Monte Carlo literature, tying everything together with analysis and discussion of applications. Quantum Monte Carlo is a reference book for quantum Monte Carlo applications, belonging near the desk of every quantum chemist, physicist, and a wide range of scientists and engineers across many disciplines, destined to become a classic.

A Physics Today Best Book of the Year A Forbes "For the Physics and Astronomy Lover in Your Life" Selection "Succinct, accessible, and remarkably timely... This book is a rare find." -Physics Today "Belongs on the shelf of anyone interested in learning the scientific, historical, and personal stories behind some of the most incredible scientific advances of the 21st century." -Forbes The detection of gravitational waves has already been called the scientific breakthrough of the century. Einstein predicted these tiny ripples in the fabric of spacetime over a hundred years ago, but they were only recently perceived directly for the first time. Ripples in Spacetime is an engaging account of the international effort to complete Einstein's project, capture his elusive ripples, and launch an era of gravitational-wave astronomy that promises to explain, more vividly than ever before, our universe's structure and origin. "Schilling's deliciously nerdy grand tour takes us through compelling backstory, current research, and future expectations." -Nature "A lively and readable account... Schilling underlines that this discovery is the opening of a new window on the universe, the beginning of a new branch of science." -Graham Farmelo, The Guardian