A general introduction to the initial value problem for Einstein's equations coupled to collisionless matter. The book contains a proof of future stability of models of the universe consistent with the current observational data and a discussion of the restrictions on the possible shapes of the universe imposed by observations.
The standard starting point in cosmology is the cosmological principle; the assumption that the universe is spatially homogeneous and isotropic. After imposing this assumption, the only freedom left, as far as the geometry is concerned, is the choice of one out of three permissible spatial geometries, and one scalar function of time. Combining the cosmological principle with an appropriate description of the matter leads to the standard models. It is worth noting that these models yield quite a successful description of our universe. However, even though the universe may, or may not, be almost spatially homogeneous and isotropic, it is clear that the cosmological principle is not exactly satisfied. This leads to several questions. The most natural one concerns stability: given initial data corresponding to an expanding model of the standard type, do small perturbations give rise to solutions that are similar to the future? Another question concerns the shape of the universe: what are the restrictions if we only assume the universe to appear almost spatially homogeneous and isotropic to every observer? The main purpose of the book is to address these questions. However, to begin with, it is necessary to develop the general theory of the Cauchy problem for the Einstein-Vlasov equations. In order to to make the results accessible to researchers who are not mathematicians, but who are familiar with general relativity, the book contains an extensive prologue putting the results into a more general context.
Accelerating Expansion explores some of the philosophical implications of modern cosmology, focused on the significance that the discovery of the accelerating expansion of the Universe has for our understanding of time, geometry, and physics. The appearance of the cosmological constant in the equations of general relativity allows one to model universes in which space has an inherent tendency towards expansion. This constant, introduced by Einstein but subsequently abandoned by him, returned to centre stage with the discovery of the accelerating expansion. This pedagogically-oriented essay begins with a study of the most basic and elegant relativistic world that involves a positive cosmological constant, de Sitter spacetime. It then turns to the relatives of de Sitter spacetime that dominate modern relativistic cosmology. Some of the topics considered include: the nature of time and simultaneity in de Sitter worlds; the sense in which de Sitter spacetime is a powerful dynamical attractor; the limited extent to which observation can give us information about the topology of space in a world undergoing accelerated expansion; and cosmologists' favourite sceptical worry about the reliability of evidence and the possibility of knowledge, the problem of Boltzmann brains.
The articles included in this Volume represent a broad and highly qualified view on the present state of general relativity, quantum gravity, and their cosmological and astrophysical implications. As such, it may serve as a valuable source of knowledge and inspiration for experts in these fields, as well as an advanced source of information for young researchers. The occasion to gather together so many leading experts in the field was to celebrate the centenary of Einstein's stay in Prague in 1911-1912. It was in fact during his stay in Prague that Einstein started in earnest to develop his ideas about general relativity that fully developed in his paper in 1915. Approaching soon the centenary of his famous paper, this volume offers a precious overview of the path done by the scientific community in this intriguing and vibrant field in the last century, defining the challenges of the next 100 years. The content is divided into four broad parts: (i) Gravity and Prague, (ii) Classical General Relativity, (iii) Cosmology and Quantum Gravity, and (iv) Numerical Relativity and Relativistic Astrophysics.
This book, dedicated to Roger Penrose, is a second, mathematically oriented course in general relativity. It contains extensive references and occasional excursions in the history and philosophy of gravity, including a relatively lengthy historical introduction. The book is intended for all students of general relativity of any age and orientation who have a background including at least first courses in special and general relativity, differential geometry, and topology. The material is developed in such a way that through the last two chapters the reader may acquire a taste of the modern mathematical study of black holes initiated by Penrose, Hawking, and others, as further influenced by the initial-value or PDE approach to general relativity. Successful readers might be able to begin reading research papers on black holes, especially in mathematical physics and in the philosophy of physics. The chapters are: Historical introduction, General differential geometry, Metric differential geometry, Curvature, Geodesics and causal structure, The singularity theorems of Hawking and Penrose, The Einstein equations, The 3+1 split of space-time, Black holes I: Exact solutions, and Black holes II: General theory. These are followed by two appendices containing background on Lie groups, Lie algebras, & constant curvature, and on Formal PDE theory.
This book is an updated version of the classic 1987 monograph "Spectral Theory and Differential Operators".The original book was a cutting edge account of the theory of bounded and closed linear operators in Banach and Hilbert spaces relevant to spectral problems involving differential equations. It is accessible to a graduate student as well as meeting the needs of seasoned researchers in mathematics and mathematical physics. This revised edition corrects various errors, and adds extensive notes to the end of each chapter which describe the considerable progress that has been made on the topic in the last 30 years.
This book is aimed at providing a coherent, essentially self-contained, rigorous and comprehensive abstract theory of Feynman's operational calculus for noncommuting operators. Although it is inspired by Feynman's original heuristic suggestions and time-ordering rules in his seminal 1951 paper An operator calculus having applications in quantum electrodynamics, as will be made abundantly clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman's work. Hence, the second part of the main title of this book. The basic properties of the operational calculus are developed and certain algebraic and analytic properties of the operational calculus are explored. Also, the operational calculus will be seen to possess some pleasant stability properties. Furthermore, an evolution equation and a generalized integral equation obeyed by the operational calculus are discussed and connections with certain analytic Feynman integrals are noted. This volume is essentially self-contained and we only assume that the reader has a reasonable, graduate level, background in analysis, measure theory and functional analysis or operator theory. Much of the necessary remaining background is supplied in the text itself.
This book contains several recent articles written about broken spacetime symmetry. The context is curved spacetime as used in General Relativity and the broken symmetry most discussed is Local Lorentz Symmetry. While there is currently no experimental evidence for broken Lorentz symmetry in nature, it is an object of great study from theoretical, phenomenological, and experimental perspectives. All three appear in this volume. There are three review articles in this volume: Fabian Kislat summarizes astrophysical probes of Lorentz violation, especially those using polarized light; Michael Seifert discusses a particular limit of the Standard-Model Extension that is useful for relating theoretical and experimental ideas; and Marco Schreck describes circumstances under which gravitational Cerenkov radiation could arise from Lorentz violation. The other three articles focus more on original research: Charles Lane and Quentin Bailey relate a particular theory of noncommutative geometry to the curved-spacetime Standard-Model Extension; Yuri Bonder and Christobal Corral consider the existence of spacetime symmetries in models with explicit Lorentz violation; and Pawel Gusin et al. study a spacetime transformation that relates the inside and outside of a nonrotating black hole.
The Future of Life: Meta-Evolution represents the first comprehensive formulation of the hypothesis that evolution is the unifying force underlying the dynamics of all processes in the universe, both organic and inorganic. These include all facets of human existence and civilisation- the sciences, technology, arts, humanities and religion. In essence, by applying quantum information, network and decision theory, it is demonstrated that an overarching evolutionary process shapes the spectrum of life and phenomena in the universe, as a generic paradigm beyond Darwin's original biology-based theory. The Theory of Evolution is undoubtedly the most powerful paradigm ever conceived by humans to explain their own existence. Since Darwin´s epoch-making treatise, Origin of Species', published in 1859, evolution has been centre-stage, universally recognised as the driving force in the emergence of modern humans from the genesis of life on this planet almost 4 billion years ago. However, despite its ubiquitous brilliance as the jewel in the crown of human intellectual achievement, the notion of evolution has never been developed to its full potential. It remains instead constrained within its biological cradle, often reduced in everyday connotation to its lowest common denominator of ´survival of the fittest´. The intention of this book to re-evaluate and expand the Darwinian model of evolution; to demonstrate that its current application is only the tip of the intellectual iceberg and that by combining its formidable biological principles with those of decision complexity, network, quantum and information theory, it emerges as an incalculably deeper and richer model than previously contemplated. It will be demonstrated that the evolutionary engine which drives biological development, also drives all other dynamic adaptive processes- the physical, social, cognitive, economic, political and technological and is in fact the major dynamic governing the Universe, past present and future. It is further proposed to demonstrate that recent developments in artificial intelligence and ubiquitous computing through the Internet, mark the next crucial stage in life's evolution, involving the inevitable symbiosis of vast computational intelligence with the human mind. The major hypothesis developed in this book, of a global all-encompassing Theory of Evolution, coupled with its potential for realising the emancipation of human intelligence and potential, provides a vastly more powerful paradigm for exploring the Future of Life than current scientific scenarios. The resulting Omega state of infinite knowledge and wisdom which is proposed, has been actively championed by a number of eminent 19th and 20th century philosophers such as Teillhard de Chardin, Henri Bergson, Schelling, Alfred Whitehead, Samuel Alexander and more recently by the leading physicist and futurist- Professor Frank Tipler. However to date no equivalent scientific framework for supporting such a hypothesis has been provided. In conclusion, The Future of Life: Meta-Evolution has been written not as an academic text but as primarily a non-technical review of the evidence to support such a hypothesis, in much the same vein as other recent publications in the popular science/philosophy genre. It is hoped that this approach will therefore provide a window into the wider evolutionary debate for the general reader interested in one of the most critical emerging paradigm shifts of the 21st century.
The book contains recent contributions in the field of waves propagation and stability in continuous media. In particular, the contributions consider discontinuity and shock waves, stability in fluid dynamics, small parameter problems, kinetic theories towards continuum models, non-equilibrium thermodynamics, and numerical applications.The volume is the fourth in a series published by World Scientific since 1999. The following distinguished authors contribute to the present book: S Bianchini, R Caflish, C Cercignani, Y Choquet-Bruhat, C Dafermos, L Desvillettes, V Giovangigli, H Gouin, I Muller, D Parker, B Straughan, M Sugiyama and W Weiss.
