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The Evolution Problem in General Relativity

2012-12-06 Science
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Release : 2012-12-06
Genre : Science
Kind : eBook
Book Rating : 84X/5 ( reviews)

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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. This book was released on 2012-12-06. 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.

The Evolution Problem in General Relativity

2002-12-13 Science
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Release : 2002-12-13
Genre : Science
Kind : eBook
Book Rating : 549/5 ( reviews)

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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. This book was released on 2002-12-13. 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.

The Cauchy Problem in General Relativity

2009 Mathematics
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Release : 2009
Genre : Mathematics
Kind : eBook
Book Rating : 531/5 ( reviews)

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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. This book was released on 2009. 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.

The Evolution Problem of General Relativity

2001
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Release : 2001
Genre :
Kind : eBook
Book Rating : /5 ( reviews)

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Book Synopsis The Evolution Problem of General Relativity by : Jakob Hansen

Download or read book The Evolution Problem of General Relativity written by Jakob Hansen. This book was released on 2001. Available in PDF, EPUB and Kindle. Book excerpt:

Global Structure and Evolution in General Relativity

2014-03-12 Science
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Release : 2014-03-12
Genre : Science
Kind : eBook
Book Rating : 710/5 ( reviews)

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Book Synopsis Global Structure and Evolution in General Relativity by : Spiros Cotsakis

Download or read book Global Structure and Evolution in General Relativity written by Spiros Cotsakis. This book was released on 2014-03-12. Available in PDF, EPUB and Kindle. Book excerpt: The five lectures presented in this volume address very timely mathematical problems in relativity and cosmology. Part I is devoted to the initial value and evolution problems of the Einstein equations. Especially it deals with the Einstein-Yang-Mills-Boltzmann system, fluid models with finite or infinite conductivity, global evolution of a new (two-phase) model for gravitational collapse and the structure of maximal, asymptotically flat, vacuum solutions of the constraint equations which have the additional property of containing trapped surfaces. Part II focuses on geometrical-topological problems in relativity and cosmology: on the role of cosmic censorship for the global structure of the Einstein-Maxwell equations and on the mathematical structure of quantum conformal superspace.

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