The Cauchy Problem In General Relativity

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The Cauchy Problem In General Relativity

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Author : Hans Ringström
language : en
Publisher: European Mathematical Society
Release Date : 2009

The Cauchy Problem In General Relativity written by Hans Ringström and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.

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 Einstein Equations And The Large Scale Behavior Of Gravitational Fields

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Author : Piotr T. Chrusciel
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06

The Einstein Equations And The Large Scale Behavior Of Gravitational Fields written by Piotr T. Chrusciel and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Science categories.

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.

Cauchy Problem In General Relativity

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Author : HANS RINGSTROM.
language : en
Publisher:
Release Date :

Cauchy Problem In General Relativity written by HANS RINGSTROM. and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.

Global Aspects Of The Cauchy Problem In General Relativity

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Author : Yvonne Choquet-Bruhat
language : en
Publisher:
Release Date : 1969

Global Aspects Of The Cauchy Problem In General Relativity written by Yvonne Choquet-Bruhat and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1969 with categories.

Studies In The History Of General Relativity

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Author : Jean Eisenstaedt
language : en
Publisher: Springer Science & Business Media
Release Date : 1992-02-07

Studies In The History Of General Relativity written by Jean Eisenstaedt and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992-02-07 with Science categories.

Among the considerations of the two dozen papers are the reception and development of Einstein's theory of general relativity in various institutions around the world; conceptual issues of the theory, especially themes, concepts, and principles associated with his theory of gravity; a number of tech

Gravitoelectromagnetism And The Cauchy Problem In General Relativity

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Author : Mark Andrew Spanier Aarons
language : en
Publisher:
Release Date : 2000

Gravitoelectromagnetism And The Cauchy Problem In General Relativity written by Mark Andrew Spanier Aarons and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000 with Cauchy problem categories.

Constraint Damping In The Cauchy Problem In General Relativity

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Author : Allen Juntao Fang
language : en
Publisher:
Release Date : 2019

Constraint Damping In The Cauchy Problem In General Relativity written by Allen Juntao Fang and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with categories.

The Cauchy Problem in General Relativity is an area of active research in partial differential equations (PDEs). Much progress has been made on the Cauchy Problem in General Relativity via the local existence theory of symmetric hyperbolic systems and the choice of generalized harmonic gauge. The choice of generalized harmonic gauge also allows for the introduction of constraint damping terms, which allow for control over any violation of the general harmonic gauge constraint. In this thesis, we first go through the symmetric hyperbolic theory of partial differential equations since it will establish the basis for our local existence result for Einstein's vacuum equations. Afterwards, we look at the 3+1 formalism as well as generalized harmonic coordinates, before finally proving stability for the damped constraint evolution equations in Minkowski space.

Notes On A Formulation Of The Problem Of Initial Conditions In The Cauchy Problem Of General Relativity

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Author : Elliot Paul Belasco
language : en
Publisher:
Release Date : 1967

Notes On A Formulation Of The Problem Of Initial Conditions In The Cauchy Problem Of General Relativity written by Elliot Paul Belasco and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1967 with categories.

General Relativity And The Einstein Equations

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Author : Yvonne Choquet-Bruhat
language : en
Publisher: Oxford University Press
Release Date : 2009

General Relativity And The Einstein Equations written by Yvonne Choquet-Bruhat and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Mathematics categories.

General Relativity has passed all experimental and observational tests to model the motion of isolated bodies with strong gravitational fields, though the mathematical and numerical study of these motions is still in its infancy. It is believed that General Relativity models our cosmos, with a manifold of dimensions possibly greater than four and debatable topology opening a vast field of investigation for mathematicians and physicists alike. Remarkable conjectures have been proposed, many results have been obtained but many fundamental questions remain open. In this monograph, aimed at researchers in mathematics and physics, the author overviews the basic ideas in General Relativity, introduces the necessary mathematics and discusses some of the key open questions in the field.

Hyperbolic Equations And General Relativity

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Author : Marica Minucci
language : en
Publisher:
Release Date : 2019

Hyperbolic Equations And General Relativity written by Marica Minucci and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with Differential equations, Hyperbolic categories.

This work is divided into three parts. In the first part, the hyperbolic equations' theory is analysed, the second part concerns the Cauchy problem in General Relativity, whereas the third part gives a modern perspective of General Relativity.In the first part, the study of systems of partial differential equations allows the introduction of the concept of wave-like propagation and the definition of hyperbolic equation is given. Thus, once the definition of Riemann kernel is given, Riemann's method to solve a hyperbolic equation in two variables is shown. The discussion moves on the fundamental solutions and its relation to Riemann kernel is pointed out. Therefore, the study of the fundamental solutions concludes by showing how to build them providing some examples of solution with odd and even number of variables. Moreover, the fundamental solution of the scalar wave equation with smooth initial conditions is studied.In the second part, following the work of Fourès-Bruhat, the problem of finding a solution to the Cauchy problem for Einstein field equations in vacuum with non-analytic initial data is presented by first studying under which assumptions second-order systems of partial differential equations, linear and hyperbolic, with n functions and four variables admit a solution. Hence, it is shown how to turn non-linear systems of partial differential equations into linear systems of the same type for which the previous results hold. These considerations allow us to prove the existence and uniqueness of the solution to the Cauchy problem for Einstein's vacuum field equations with non-analytic initial data. Eventually, the causal structure of space-time is studied. The definitions of strong causality, stable causality and global hyperbolicity are given and the relation between the property of global hyperbolicity and the existence of Cauchy surfaces is stressed. In the third part, Riemann's method is used to study the news function describing the gravitational radiation produced in axisymmetric black hole collisions at the speed of light. More precisely, since the perturbative field equations may be reduced to equations in two independent variables, as was proved by D'Eath and Payne, the Green function can be analysed by studying the corresponding second-order hyperbolic operator with variable coefficients. Thus, an integral representation of the solution in terms of the Riemann kernel function can be given.