In one ofthe fundamental notions is that any physical theory of of energy the hand: in mechanics at one considers the objects energy of, say, moving in fieldtheories is one interested inthe offield masses; energy configurations. A unified treatment of this which both to mechanics question, applies and to field Hamiltonian a formalism. We will theory, proceeds through shortly reviewbelowhowsuch iscarried aprocedure out inthe ofscalarfields theory Minkowski let at this on mention that an space time; us, stage, important often inthe isthat ofthe issue, ignored conditions sat textbooks, boundary isfied the set of fields under consideration. While by this issuecanbe safely for when the ignored usual field many purposes considering theories, such scalar fields or the = as on electromagnetism, ft constj hypersurfaces, where t is a it sometimes critical Minkowski time, a rolewhen other plays of classes are considered. Inthe of the hypersurfaces case situation is gravity for t= worse: even Minkowskian slicesthe f constj asymptotically boundary terms crucial. is ofthe are one main differencesbetweentheArnowitt (This Deser Misner for Sect. 5. 4 mass which is (ADM) gravity (cf. below), given a andthe usual for by boundary integral, field theories in energyexpression Minkowski wheretheHamiltonian is volume space time, a usually integral. ) in field the itsmost role in Now, theory plays important theradiation energy where it can be radiated the field.

Like a river, the progress of science has a tendency to run tast or slow. Once the water meets a dam, it may stop for a while, but eventually it will flow over the top and run fast again. In scientific research, a breakthrough to overcome a simile>r barrier is often made by a small number of scientists, or perhaps by a single person of special creativity, extraordinary talent and unusual perseverance. Through such individuals science can proceed in great strides. No one can deny that Professor Kazuo Takayanagi is one of these special individuals who have played a leading role in the field of atomic and molecular physics, as well as space physics. This book is dedicated to Professor Takayanagi on the occasion of his retirement from the Institute of Space and Astronautical Science. Professor Takayanagi was born in 1926 and grew up in Tomakomai in Hokkaido, the northern island of Japan. In his boyhood, he was interested in natural sciences, particularly astronomy. On 5th February, 1943, when he was attending secondary school, a solar eclipse was seen in his town. He organized a group of students from his school to observe the eclipse. He still remembers the scene: it grew so dark during the eclipse that two stars, Vega and Arcturus, could be seen. After graduation from the University of Tokyo in 1948, he entered the graduate school there.

The visible universe is a small perturbation on the material universe. Zwicky and Sinclair Smith in the 1930s gave evidence of invisible mass in the Coma and Virgo Clusters of Galaxies. Better optical data has only served to confound their critics and the X-ray data confirms that the gravitational potentials are many times larger than those predicted on the basis of the observed stars. Dynamical analyses of individual galaxies have found that significant extra mass is needed to explain their rotational velocities. On much larger scales, tens of megaparsecs, there is suggestive evidence that there is even more mass per unit luminosity. What is this non-luminous stuff of which the universe is made'? How much of it is there? Need there be only one kind of stuff? There are three basic possi bili ties:- all of it is ordinary (baryonic) matter, all of it is some other kind of (non-baryonic) matter, or some of it is baryonic and some is non-baryonic.

A small country builds a world-class telescope in its backyard and lives happily ever after (or at least for a quarter century). That in a nutshell is the story told in this collection of essays. The country of course is the Netherlands, and the telescope is the Westerbork Synthesis Radio Tele scope (WSRT), brainchild of Jan Oort. Living happily in this context is a continuing record of discovery and as such also a continuing basis for se curing observing time on facilities in other countries and operating at other frequencies. As our community celebrates the Silver Anniversary of the radio tele scope at Westerbork, it is fitting that we pause to take account of the scientific discoveries and insights it made possible. Initially the instrument represented the very significant step away from university-run, specialist facilities to a well-supported, common-user radio imager also having spec tral and polarization capabilities. It pioneered the mode of operation now common for satellite observatories, in which data is taken and calibrated by technicians and provided to researchers ready for analysis. It has been a major source of discovery in, among other areas, research on neutral hy drogen and studies of dark matter in galaxies.

edited by c. H. Lineweaver, J. G. Bartlett, A. Blanchard Observatoire Astronomique de Strasbourg, Strasbourg, France M. Signore Ecole normale superieure, Paris, France and J. Silk Departments of Astronomy and Physics, University of California at Berkeley, Berkeley, CA, U. S. A. Kluwer Academic Publishers Dordrecht / Boston / London Published in cooperation with NATO Scientific Affairs Division ISBN-13:978-94-010-6512-2 e-ISBN-13:978-94-009-0051-6 DOI:10. 107/ 978-94-009-0051-6 Softcover reprint of the hardcover 1st edition 1997 Dedication We dedicate these proceedings to the people who paid for it: taxpayers of the NATO alliance. Table of Contents PREFACE IX LIST OF PARTICIPANTS XI LISTOF CONTRIBUTORS Xlll I. Introduction, Mathematical Tools and Background An Introduction to CBR Studies: Spectrum, Degree-Scale Fluctuations, Foregrounds and Interferometry R. B. Partridge ElementsofGeneral Relativity, Cosmology and the Cosmic Microwave Background Jose L. Sanz 33 Statisticsand Random Functions in Astrophysics BernhardIT. Jones 67 Structure Formation Joseph Silk III II. eMB Anistropies CalculationofCosmic Background Radiation Anisotropies and Implications Emory F. Bunn 135 The CMB Anistropy Experiments: Cosmic Microwave Background George F. Smoot 185 viii III. CMB Spectrum The CMBR-Spectrum: ATheoretical Introduction Albert Stebbins 241 The CMB Spectrum George F. Smoot 271 IV. Astroparticle Physics Inflation and the Cosmic Background Radiation: What Every Cosmologist Needs to Know Michael S. Turner 309 Primordial Chemistry and Cosmic Background Radiation M. Signori, P. Encrenaz, R. MaoJi, B. Melchiorri, F. Melchiorri and D. Puy 345 The Cosmic Background Radiation and Elementary Particles Pierre Salati 365 V.

The main goal of this work is to revisit the proof of the global stability of Minkowski space by D. Christodoulou and S. Klainerman, [Ch-KI]. We provide a new self-contained proof of the main part of that result, which concerns the full solution of the radiation problem in vacuum, for arbitrary asymptotically flat initial data sets. This can also be interpreted as a proof of the global stability of the external region of Schwarzschild spacetime. The proof, which is a significant modification of the arguments in [Ch-Kl], is based on a double null foliation of spacetime instead of the mixed null-maximal foliation used in [Ch-Kl]. This approach is more naturally adapted to the radiation features of the Einstein equations and leads to important technical simplifications. In the first chapter we review some basic notions of differential geometry that are sys tematically used in all the remaining chapters. We then introduce the Einstein equations and the initial data sets and discuss some of the basic features of the initial value problem in general relativity. We shall review, without proofs, well-established results concerning local and global existence and uniqueness and formulate our main result. The second chapter provides the technical motivation for the proof of our main theorem.

Dr. KURT GODEL'S sixtieth birthday (April 28, 1966) and the thirty fifth anniversary of the publication of his theorems on undecidability were celebrated during the 75th Anniversary Meeting of the Ohio Ac ademy of Science at The Ohio State University, Columbus, on April 22, 1966. The celebration took the form of a Festschrift Symposium on a theme supported by the late Director of The Institute for Advanced Study at Princeton, New Jersey, Dr. J. ROBERT OPPENHEIMER: "Logic, and Its Relations to Mathematics, Natural Science, and Philosophy." The symposium also celebrated the founding of Section L (Mathematical Sciences) of the Ohio Academy of Science. Salutations to Dr. GODEL were followed by the reading of papers by S. F. BARKER, H. B. CURRY, H. RUBIN, G. E. SACKS, and G. TAKEUTI, and by the announcement of in-absentia papers contributed in honor of Dr. GODEL by A. LEVY, B. MELTZER, R. M. SOLOVAY, and E. WETTE. A short discussion of "The II Beyond Godel's I" concluded the session."

In a distilled and pedagogical fashion, the contributions to this volume of the famous summer school in Les Houches cover the recent developments in supersymmetric string theory, the gauge theory/string theory correspondence and string duality. Further chapters deal with quantum gravity and D-brane geometry. Black hole mechanics and cosmology are treated too, as well as the AdS-CFT correspondence. The book is a comprehensive introduction to the recent developments in string/M-theory and quantum gravity. It addresses graduate students in physics and astrophysics.

First published in 1973, this influential work discusses Einstein's General Theory of Relativity to show how two of its predictions arise: first, that the ultimate fate of many massive stars is to undergo gravitational collapse to form 'black holes'; and second, that there was a singularity in the past at the beginning of the universe. Starting with a precise formulation of the theory, including the necessary differential geometry, the authors discuss the significance of space-time curvature and examine the properties of a number of exact solutions of Einstein's field equations. They develop the theory of the causal structure of a general space-time, and use it to prove a number of theorems establishing the inevitability of singularities under certain conditions. A Foreword contributed by Abhay Ashtekar and a new Preface from George Ellis help put the volume into context of the developments in the field over the past fifty years.

This volume is dedicated to the one hundredth anniversary of the publication of Hermann Minkowski's paper "Raum und Zeit" in 1909 [1]. The paper presents the textofthetalkMinkowskigaveatthe80thMeetingoftheGermanNaturalScientists and Physicians in Cologne on September 21, 1908. Minkowski's work on the spacetime representation of special relativity had a huge impact on the twentieth century physics, which can be best expressed by merely stating what is undeniable - that modern physics would be impossible wi- out the notion of spacetime. It is suf cient to mention as an example only the fact that general relativity would be impossible without this notion; Einstein succeeded to identifygravitywith the curvatureofspacetime onlywhen he overcamehis initial hostile reactionto Minkowski'sfour-dimensionalrepresentationof special relativity and adopted spacetime as the correct relativistic picture of the world. While there exists an unanimous consensus on the mathematical signi cance of spacetime for theoretical physics, for a hundred years there has been no consensus on the nature of spacetime itself. The rst sign of this continuing controversy was Sommerfeld's remark in his notes on Minkowski's article [2]: "What will be the epistemologicalattitudetowardsMinkowski'sconceptionofthetime-spaceproblem is another question, but, as it seems to me, a question which does not essentially touch his physics".

The third volume in Leonard Susskind's one-of-a-kind physics series cracks open Einstein's special relativity and field theory In the first two books in his wildly popular The Theoretical Minimum series, world-class physicist Leonard Susskind provided a brilliant first course in classical and quantum mechanics, offering readers not an oversimplified introduction, but the real thing - everything you need to start doing physics, and nothing more. Now, thankfully, Susskind and his former student Art Friedman are back, this time to introduce readers to special relativity and classical field theory. At last, waves, forces and particles will be demystified. Using their typical brand of relatively simple maths, enlightening sketches and the same fictional counterparts, Art and Lenny, Special Relativity and Classical Field Theory takes us on an enlightening journey through a world now governed by the laws of special relativity. Starting in their new watering hole, Hermann's Hideaway, with a lesson on relativity, Art and Lenny walk us through the complexities of Einstein's famous theory. Combining rigor with humour, Susskind and Friedman guarantee that Special Relativity and Classical Field Theory will become part of the reader's physics toolbox.

The Tum of the Tide During centuries physicists were supposed to be studying the physical world. Since the turn of the century this assumption has often been challenged as naive: it was proclaimed that physics is not about the external world but about observers and their manipUlations: that it is meaningless to talk of anything else than observation devices and opera tions: that the laws of physics concern our knowledge rather than the external world. This view of the nature of physical science has old roots in philo sophy but it was independently reinvented by a number of philosophi cally inclined physicists, notably ERNST MACH. These scientists were disgusted with the school philosophies and they were alarmed by the increasing number of physical concepts which they regarded as meta physical or beyond experimental control, such as those of absolute motion, ether, electromagnetic field, and molecule. Reasonably enough, they wished to keep physics testable. To accomplish this goal they adopted the safe method, namely to banish every idea that could not be closely tied to observation. In this way they certainly avoided the risks of untestable speculation but they also failed to enjoy the benefits of theoretical invention. Furthermore they instituted unawares a new meta physics that was to dominate the philosophy of physics for half a century: the metaphysics according to which the world is made of sense experience."

The reader will find in this volume the Proceedings of the NATO Advanced Study Institute held in Cortina d'Ampezzo, Italy between August 6 and August 17, 1990 under the title "Predictability, Stability, and Chaos in N-Body Dynamical Systems". The Institute was the latest in a series held at three-yearly inter vals from 1972 to 1987 in dynamical astronomy, theoretical mechanics and celestial mechanics. These previous institutes, held in high esteem by the international community of research workers, have resulted in a series of well-received Proceedings. The 1990 Institute attracted 74 participants from 16 countries, six outside the NATO group. Fifteen series of lectures were given by invited speakers; additionally some 40 valuable presentations were made by the younger participants, most of which are included in these Proceedings. The last twenty years in particular has been a time of increasingly rapid progress in tackling long-standing and also newly-arising problems in dynamics of N-body systems, point-mass and non-point-mass, a rate of progress achieved because of correspondingly rapid developments of new computer hardware and software together with the advent of new analytical techniques. It was a time of exciting progress culminating in the ability to carry out research programmes into the evolution of the outer Solar 8 System over periods of more than 10 years and to study star cluster and galactic models in unprecedented detail.

After some decades of work a satisfactory theory of quantum gravity is still not available; moreover, there are indications that the original field theoretical approach may be better suited than originally expected. There, to first approximation, one is left with the problem of quantum field theory on Lorentzian manifolds. Surprisingly, this seemingly modest approach leads to far reaching conceptual and mathematical problems and to spectacular predictions, the most famous one being the Hawking radiation of black holes. Ingredients of this approach are the formulation of quantum physics in terms of C*-algebras, the geometry of Lorentzian manifolds, in particular their causal structure, and linear hyperbolic differential equations where the well-posedness of the Cauchy problem plays a distinguished role, as well as more recently the insights from suitable concepts such as microlocal analysis. This primer is an outgrowth of a compact course given by the editors and contributing authors to an audience of advanced graduate students and young researchers in the field, and assumes working knowledge of differential geometry and functional analysis on the part of the reader.

Puts the emphasis on conceptual questions: Why is there no such thing as absolute motion? What is the physical meaning of relativity of simultaneity? But, the most important question that is addressed in this book is "what is the nature of spacetime?" or, equivalently, "what is the dimensionality of the world at the macroscopic level?"

Develops answers to these questions via a thorough analysis of relativistic effects and explicitly asking whether the objects involved in those effects are three-dimensional or four-dimensional.

Discusses the implication of the result (this analysis clearly shows that if the world and the physical objects were three-dimensional, none of the kinematic relativistic effects and the experimental evidence supporting them would be possible) for physics, philosophy, and our entire world view are discussed.

All kind ofinformationfrom distant celestialbodies comestous intheform of radiation. Inmost the of electromagnetic cases thisradiation propagation be can as a reasonable in described, terms of This is approximation, rays. truenot inthe but also inthe radio only ofthe electro opticalrange range Forthis the of reason laws areoffundamental magneticspectrum. optics ray for importance and astronomy, astrophysics, cosmology. to According arethe of general relativity, light light likegeodesics a rays Lorentzianmetric whichthe is described. how by spacetimegeometry This, is true as as the under the ever, only long light are rays freelypropagating influence ofthe fieldwhich is coded in the only gravitational spacetime ge If a is in an medium ometry. light ray influenced, addition, by optical (e. g. , then it will not follow ofthe by a a plasma), light like geodesic spacetime metric. It is true that for radiation the electromagnetic traveling through universe the influence ofamedium on the ofthe and usually path on ray the is small. there are several cases inwhich this influ frequency However, ence is well in inthe radio For measurable, particular very range. example, the deflection ofradio inthe field oftheSun is consider gravitational rays influenced the Solarcorona. currentand ably by Moreover, planned Doppler with microwaves in the Solar reach in the an experiments system accuracy 5 of 10 15 whichmakes it totakethe influence frequency Awlw necessary ofthe medium into account.

This is a book about physics, written for mathematicians. The readers we have in mind can be roughly described as those who: I. are mathematics graduate students with some knowledge of global differential geometry 2. have had the equivalent of freshman physics, and find popular accounts of astrophysics and cosmology interesting 3. appreciate mathematical elarity, but are willing to accept physical motiva tions for the mathematics in place of mathematical ones 4. are willing to spend time and effort mastering certain technical details, such as those in Section 1. 1. Each book disappoints so me readers. This one will disappoint: 1. physicists who want to use this book as a first course on differential geometry 2. mathematicians who think Lorentzian manifolds are wholly similar to Riemannian ones, or that, given a sufficiently good mathematical back ground, the essentials of a subject !ike cosmology can be learned without so me hard work on boring detaiis 3. those who believe vague philosophical arguments have more than historical and heuristic significance, that general relativity should somehow be "proved," or that axiomatization of this subject is useful 4. those who want an encyclopedic treatment (the books by Hawking-Ellis [1], Penrose [1], Weinberg [1], and Misner-Thorne-Wheeler [I] go further into the subject than we do; see also the survey article, Sachs-Wu [1]). 5. mathematicians who want to learn quantum physics or unified fieId theory (unfortunateIy, quantum physics texts all seem either to be for physicists, or merely concerned with formaI mathematics).

This Brief presents a new way of introducing relativity theory, in which perplexing relativistic effects such as time dilation and Lorentz contraction are explained prior to the discussion of Lorentz-transformation. The notion of relativistic mass is shown to contradict the spirit of relativity theory and the true significance of the mass-energy relation is contrasted with the popular view of it. The author discusses the twin paradox from the point of view of both siblings. Last but not least, the fundamentals of general relativity are described, including the recent Gravity Probe B experiment.

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.

Based on class-tested notes, this text offers an introduction to Conformal Field Theory with a special emphasis on computational techniques of relevance for String Theory. It introduces Conformal Field Theory at a basic level, Kac-Moody algebras, one-loop partition functions, Superconformal Field Theories, Gepner Models and Boundary Conformal Field Theory. Eventually, the concept of orientifold constructions is explained in detail for the example of the bosonic string. In providing many detailed CFT calculations, this book is ideal for students and scientists intending to become acquainted with CFT techniques relevant for string theory but also for students and non-specialists from related fields.

This volume contains notes based on the lectures delivered at the fourth New Zealand Symposium in Laser Physics, held at the University of Waikato, Hamilton, February 10-15, 1986. At this meeting, about 80 physicists work ing in many parts of the world met to discuss topics of current interest in contemporary laser physics and quantum optics. These symposia, which have been held triennially since 1977, have evolved into an important meet ing ground for experimentalists and theoreticians working in a very rapidly developing field. As the format has evolved, the number of participants, in cluding the number from overseas, has grown steadily, and this year a poster session was included for the first time, enabling a far greater range of topics to be discussed than was possible in the limited lecture time available. At this meeting the major interest of the participants concerned the the oretical investigation of squeezed states of the radiation field and the very recently reported experimental observations of such states. Other related ar eas of work reported here include bistability and chaotic behaviour of optical systems, the quantum theory of measurements, optical tests of general rel ativity, and the current technological limitations governing the stabilization of lasers. The editors would like to thank the participants for providing detailed notes for publication shortly after the meeting, and the various organisa tions that have provided financial support."

One of the most of exciting aspects is the general relativity pred- tion of black holes and the Such Big Bang. predictions gained weight the theorems through Penrose. singularity pioneered In various by te- books on theorems general relativity singularity are and then presented used to that black holes exist and that the argue universe started with a To date what has big been is bang. a critical of what lacking analysis these theorems predict-' We of really give a proof a typical singul- theorem and this ity use theorem to illustrate problems arising through the of possibilities violations" and "causality weak "shell very crossing These singularities." add to the problems weight of view that the point theorems alone singularity are not sufficient to the existence of predict physical singularities. The mathematical theme of the book In order to both solid gain a of and intuition understanding good for any mathematical theory, one, should to realise it as model of try a a fam- iar non-mathematical theories have had concept. Physical an especially the important on of and impact development mathematics, conversely various modern theories physical rather require sophisticated mathem- ics for their formulation. both and mathematics Today, physics are so that it is often difficult complex to master the theories in both very s- in the of jects. However, case differential pseudo-Riemannian geometry or the general relativity between and mathematics relationship physics is and it is therefore especially close, to from interd- possible profit an ciplinary approach.

The aim of this book is to give graduate students an overview of quantum gravity but it also covers related topics from astrophysics. Some well-written contributions can serve as an introduction into basic conceptual concepts like time in quantum gravity or the emergence of a classical world from quantum cosmology. This makes the volume attractive to philosophers of science, too. Other topics are black holes, gravitational waves and non-commutative extensions of physical theories.

This volume contains the Proceedings of the Fifth Scientific Meeting of the Spanish Astronomical Society (Sociedad Espanola de Astronomfa, SEA). The meeting was held at the Universidad de Castilla La Mancha in Toledo, from September 9 to 13, 2002. The event brought together 219 participants who pre sented their latest results in many different subjects. In comparison with the previous scientific meetings of the Society, the numbers of oral talks and poster contributions (122 and 64, respectively) are rapidly increasing, confirming that the SEA conferences are becoming a point of reference to assess the interests and achievements of astrophysical research in Spain. During the meeting, the SEA made public the granting of the Prize to the Best Spanish Ph. D. Thesis in As tronomy and Astrophysics for the period 2000-2001 ex aequo to Dr. A. Zurita and Dr. E. Villaver. This is the second time that the SEA is awarding this prize, which aim is to encourage young spanish astrophysicists to pursue a high level scientific career. The Society is indebted to the Universidad de Castilla La Mancha, and, in particular, to the San Pedro Martir staff, for its hospitality. It is also indebted to the Local Organizing Committee for its dedication and the good atmosphere that prevailed at any moment, and to the Scientific Organizing Committee for its excel lent work."

Since the dawn of mankind, observers of the sky have wondered at the sudden appearance of new stars on the seemingly unchanging heavens and, for at least 2000 years, have recorded these phenomena in their annals and archives. Even in more modern times, since the discovery of SN1885A in S Andromeda which ?gured in the important "island universe" discussions of the 1920's, the puzzle of supernovae (SNe) has played an important role in astrophysics. Only with the seminal work of Fritz Zwicky and Walter Baade in the 1930's did we begin to understand the di?erences between novae and SNe and the importance of SNe as the fonts of energy for the interstellar medium and as drivers of chemical evolution in galaxies. As recently as the 1940's and 1950's the early days of radio astronomy were heavily in?uenced by the familiar names of Cassiopeia A and Taurus A, two young supernova remnants, and two Nobel prizes have been awarded for discovery and study of a related phenomenon, pulsars. In spite of the great age of the study of SNe, since at least the Chinese records of SN185and probably earlier, the ?eld is, in fact, very young having only attracted a large devoted following since the spectacular Type II SN1987A in the Large Magellanic Cloud, the ?rst naked-eye SN in more than 400 years.