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

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

This volume addresses the history and epistemology of early modern cosmology. The authors reconstruct the development of cosmological ideas in the age of 'scientific revolution' from Copernicus to Leibniz, taking into account the growth of a unified celestial-and-terrestrial mechanics. The volume investigates how, in the rise of the new science, cosmology displayed deep and multifaceted interrelations between scientific notions (stemming from mechanics, mathematics, geometry, astronomy) and philosophical concepts. These were employed to frame a general picture of the universe, as well as to criticize and interpret scientific notions and observational data. This interdisciplinary work reconstructs a conceptual web pervaded by various intellectual attitudes and drives. It presents an historical-epistemological unified itinerary which includes Copernicus, Kepler, Galileo, Descartes, Huygens, Newton and Leibniz. For each of the scientists and philosophers, a presentation and commentary is made of their cosmological views, and where relevant, outlines of their most relevant physical concepts are given. Furthermore, the authors highlight the philosophical and epistemological implications of their scientific works. This work is helpful both as a synthetic overview of early modern cosmology, and an analytical exposition of the elements that were intertwined in early-modern cosmology. This book addresses historians, philosophers, and scientists and can also be used as a research source book by post-graduate students in epistemology, history of science and history of philosophy.

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

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

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

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

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

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

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

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

All physicists would agree that one of the most fundamental problems of the 21st century physics is the dimensionality of the world. In the four-dimensional world of Minkowski (or Minkowski spacetime) the most challenging problem is the nature of the temporal dimension. In Minkowski spacetime it is merely one of the four dimensions, which means that it is entirely given like the other three spacial dimensions. If the temporal dimension were not given in its entirety and only one constantly changing moment of it existed, Minkowski spacetime would be reduced to the ordinary three-dimensional space. But if the physical world, represented by Minkowski spacetime, is indeed four-dimensional with time being the fourth dimension, then such a world is drastically different from its image based on our perceptions.

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

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

Bad Hofgastein who made the very successful Salzburger Abend with indi- nous music from Salzburg possible. Special thanks also to the former director of the Institute of Astronomy in Vienna, Prof. Paul Jackson for his generous private donation. We should not forget our hosts Mr. and Mrs. Winkler and their employees from the hotel who made the stay quite enjoyable. None of us will forget the very last evening, when the staff of kitchen under the le- ership of the cook himself came to offer us as farewell the famous Salzburger Nockerln, a traditional Austrian dessert. Everyone got a lot of scienti?c input during the lectures and the discussions and, to summarize, we all had a spl- did week in Salzburg in the Hotel Winkler. We all hope to come again in 2008 to discuss new results and new perspectives on a high level scienti?c standard in the Gasteinertal. Rudolf Dvorak and Sylvio Ferraz-Mello Celestial Mechanics and Dynamical Astronomy (2005) 92:1-18 (c) Springer 2005 DOI 10. 1007/s10569-005-3314-7 FROM ASTROMETRY TO CELESTIAL MECHANICS: ORBIT DETERMINATION WITH VERY SHORT ARCS (Heinrich K. Eichhorn Memorial Lecture) 1 2 ? ' ANDREA MILANI and ZORAN KNEZEVIC 1 Department of Mathematics, University of Pisa, via Buonarroti 2, 56127 Pisa, Italy, e-mail: milani@dm. unipi. it 2 Astronomical Observatory, Volgina 7, 11160 Belgrade 74, Serbia and Montenegro, e-mail: zoran@aob. bg. ac.

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

A IUTAM symposium on 'Waves in Liquid/Gas and Liquid/Vapor Two-Phase Systems' was held in Kyoto, Japan, 9-13 May 1994. Sixty-three scientists partici pated coming from ten countries, and forty-two lectures were presented. The list of participants and the program are included in this volume. The symposium was held in response to the request of the participants in the IUTAM symposium 'Adiabatic Waves in Liquid-Vapor System' held at Gottingen in 1989. At that time, the need for another symposium in about five years had been indicated by all the participants. This symposium intends to develop the subject of wave properties in more general liquid-gas two-phase systems. Topics in this symposium may be classified as (1) waves in liquid-gas bubble systems including interfacial effects, (2) waves in gas( vapor )-droplets systems, (3) waves in films or stratified systems, (4) waves with liquid-vapor transition, (5) waves with vapor-liquid transition, (6) wave propagation near the critical point and (7) waves with low pressure effect. As for topic (1), experiments, numerical simulations and analytical approaches to waves in bubly liquids were discussed. The importance of interbubble interactions through the liquid-field is now well established at least in terms of potential theory. There was also a progress concerning the well-posedness of governing equations for void waves. For pressure waves there were some new phenomena, such as bubble cluster formation and the occurrence of three-dimensional structures, in addition to a progress from more qualitative studies to quantitative ones."

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

This thesis sheds valuable new light on the second-order cosmological perturbation theory, extensively discussing it in the context of cosmic microwave background (CMB) fluctuations. It explores the observational consequences of the second-order vector mode, and addresses magnetic field generation and the weak lensing signatures, which are key phenomena of the vector mode. The author demonstrates that the second-order vector mode, which never appears at the linear-order level, naturally arises from the non-linear coupling of the first-order scalar modes. This leads to the remarkable statement that the vector-order mode clearly contributes to the generation of cosmological magnetic fields. Moreover, the weak lensing observations are shown to be accessible to the vector mode. On the basis of ongoing and forthcoming observations, the thesis concludes that the second-order vector mode is detectable.

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

This volume is a collection of scholarly articles on the Mach Principle, the impact that this theory has had since the end of the 19th century, and its role in helping Einstein formulate the doctrine of general relativity. 20th-century physics is concerned with the concepts of time,space, motion, inertia and gravity. The documentation on all of these makes this book a reference for those who are interested in the history of science and the theory of general relativity.

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

Devoted to the history of general relativity, this text provides reviews from scholars all over the world. Many of the papers originated at the Third International Conference on the History of General Relativity, held at the University of Pittsburgh in the summer of 1991. Topics covered include: disputes with Einstein; the empirical basis of general relativity; variational principles in general relativity; the reception and development of general relativity; and cosmology and general relativity.