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General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems. One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether's fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations. Another method, the study and systematic use of foliations by characteristic (null) hypersurfaces, is in the spirit of Roger Penrose's approach in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media. The book is intended for advanced students and researchers seeking an introduction to the methods and applications of general relativity.
Book Synopsis Mathematical Problems of General Relativity I by : Demetrios Christodoulou
Download or read book Mathematical Problems of General Relativity I written by Demetrios Christodoulou and published by European Mathematical Society. This book was released on 2008 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation. It is based on and extends Newton's theory of gravitation as well as Newton's equations of motion. It is thus fundamentally rooted in classical mechanics. The theory can be seen as a development of Riemannian geometry, itself an extension of Gauss' intrinsic theory of curved surfaces in Euclidean space. The domain of application of the theory is astronomical systems. One of the mathematical methods analyzed and exploited in the present volume is an extension of Noether's fundamental principle connecting symmetries to conserved quantities. This is involved at a most elementary level in the very definition of the notion of hyperbolicity for an Euler-Lagrange system of partial differential equations. Another method, the study and systematic use of foliations by characteristic (null) hypersurfaces, is in the spirit of Roger Penrose's approach in his incompleteness theorem. The methods have applications beyond general relativity to problems in fluid mechanics and, more generally, to the mechanics and electrodynamics of continuous media. The book is intended for advanced students and researchers seeking an introduction to the methods and applications of general relativity.
Book Synopsis Problem Book in Relativity and Gravitation by : Alan P. Lightman
Download or read book Problem Book in Relativity and Gravitation written by Alan P. Lightman and published by . This book was released on 1975 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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.
Book Synopsis The Evolution Problem in General Relativity by : Sergiu Klainerman
Download or read book The Evolution Problem in General Relativity written by Sergiu Klainerman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Unsolved Problems in Special and General Relativity by : Florentin Smarandache
Download or read book Unsolved Problems in Special and General Relativity written by Florentin Smarandache and published by Infinite Study. This book was released on 2013 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Mathematical Problems of General Relativity by : Dēmētrēs Ch Christodulu
Download or read book Mathematical Problems of General Relativity written by Dēmētrēs Ch Christodulu and published by . This book was released on 20?? with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein's equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann-Lemaitre-Robertson-Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein's equations as an initial value problem allows a closer study of their solutions. This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those without prior background in the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.
Book Synopsis The Cauchy Problem in General Relativity by : Hans Ringström
Download or read book The Cauchy Problem in General Relativity written by Hans Ringström and published by European Mathematical Society. This book was released on 2009 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: The general theory of relativity is a theory of manifolds equipped with Lorentz metrics and fields which describe the matter content. Einstein's equations equate the Einstein tensor (a curvature quantity associated with the Lorentz metric) with the stress energy tensor (an object constructed using the matter fields). In addition, there are equations describing the evolution of the matter. Using symmetry as a guiding principle, one is naturally led to the Schwarzschild and Friedmann-Lemaitre-Robertson-Walker solutions, modelling an isolated system and the entire universe respectively. In a different approach, formulating Einstein's equations as an initial value problem allows a closer study of their solutions. This book first provides a definition of the concept of initial data and a proof of the correspondence between initial data and development. It turns out that some initial data allow non-isometric maximal developments, complicating the uniqueness issue. The second half of the book is concerned with this and related problems, such as strong cosmic censorship. The book presents complete proofs of several classical results that play a central role in mathematical relativity but are not easily accessible to those without prior background in the subject. Prerequisites are a good knowledge of basic measure and integration theory as well as the fundamentals of Lorentz geometry. The necessary background from the theory of partial differential equations and Lorentz geometry is included.
Based on a course taught for years at Oxford, this book offers a concise exposition of the central ideas of general relativity. The focus is on the chain of reasoning that leads to the relativistic theory from the analysis of distance and time measurements in the presence of gravity, rather than on the underlying mathematical structure. Includes links to recent developments, including theoretical work and observational evidence, to encourage further study.
Book Synopsis General Relativity by : N.M.J. Woodhouse
Download or read book General Relativity written by N.M.J. Woodhouse and published by Springer Science & Business Media. This book was released on 2007-03-06 with total page 218 pages. Available in PDF, EPUB and Kindle. Book excerpt: Based on a course taught for years at Oxford, this book offers a concise exposition of the central ideas of general relativity. The focus is on the chain of reasoning that leads to the relativistic theory from the analysis of distance and time measurements in the presence of gravity, rather than on the underlying mathematical structure. Includes links to recent developments, including theoretical work and observational evidence, to encourage further study.
This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.
Book Synopsis An Introduction to General Relativity by : L. P. Hughston
Download or read book An Introduction to General Relativity written by L. P. Hughston and published by Cambridge University Press. This book was released on 1990 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.
A self-contained introduction to advanced general relativity.
Book Synopsis Advanced General Relativity by : John Stewart
Download or read book Advanced General Relativity written by John Stewart and published by Cambridge University Press. This book was released on 1993-11-26 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: A self-contained introduction to advanced general relativity.
The book presents state-of-the-art results on the analysis of the Einstein equations and the large scale structure of their solutions. It combines in a unique way introductory chapters and surveys of various aspects of the analysis of the Einstein equations in the large. It discusses applications of the Einstein equations in geometrical studies and the physical interpretation of their solutions. Open problems concerning analytical and numerical aspects of the Einstein equations are pointed out. Background material on techniques in PDE theory, differential geometry, and causal theory is provided.
Book Synopsis The Einstein Equations and the Large Scale Behavior of Gravitational Fields by : Piotr T. Chrusciel
Download or read book The Einstein Equations and the Large Scale Behavior of Gravitational Fields written by Piotr T. Chrusciel and published by Birkhäuser. This book was released on 2012-12-06 with total page 487 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book presents state-of-the-art results on the analysis of the Einstein equations and the large scale structure of their solutions. It combines in a unique way introductory chapters and surveys of various aspects of the analysis of the Einstein equations in the large. It discusses applications of the Einstein equations in geometrical studies and the physical interpretation of their solutions. Open problems concerning analytical and numerical aspects of the Einstein equations are pointed out. Background material on techniques in PDE theory, differential geometry, and causal theory is provided.
