Stability by Linearization of Einstein's Field Equation
Stability by Linearization of Einstein's Field Equation

Author: Lluís Bruna

Publisher: Springer Science & Business Media

Published: 2013-11-25

Total Pages: 220

ISBN-13: 3034603045

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V ? V ?K? , 3 2 2 R ? /?x K i i g V T G g ?T , ? G g g 4 ? R ? ? G ? T g g ? h h ? 2 2 2 2 ? ? ? ? ? ? ? h ?S , ?? ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 3 S T S T? T?. ? ̃ T S 2 2 2 2 ? ? ? ? ? ? ? h . ?? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 3 g h h ?? g T T g vacuum M n R n R Acknowledgements n R Chapter I Pseudo-Riemannian Manifolds I.1 Connections M C n X M C M F M C X M F M connection covariant derivative M ? X M ×X M ?? X M X,Y ?? Y X ? Y ? Y ? Y X +X X X 1 2 1 2 ? Y Y ? Y ? Y X 1 2 X 1 X 2 ? Y f? Y f?F M fX X ? fY X f Y f? Y f?F M X X ? torsion ? Y?? X X,Y X,Y?X M . X Y localization principle Theorem I.1. Let X, Y, X , Y be C vector ?elds on M.Let U be an open set

The Einstein Equations and the Large Scale Behavior of Gravitational Fields
The Einstein Equations and the Large Scale Behavior of Gravitational Fields

Author: Piotr T. Chrusciel

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 487

ISBN-13: 3034879539

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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.

Theoretical Foundations of Cosmology
Theoretical Foundations of Cosmology

Author: Micha? Heller

Publisher: World Scientific

Published: 1992

Total Pages: 170

ISBN-13: 9789810207564

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The book's principal aim is to clarify fundamental concepts, decipher mathematical structures used to model space-time and relativistic worlds, and to disclose their physical meaning. After each chapter, philosophical implications of the presented material are commented upon.Both special and general theories of relativity are presented in the book with the stress on their global aspects. Although global mathematical methods are extensively used throughout the book, the definitions of new concepts, short comments and examples make reading smooth without the need to consult other textbooks or review papers.

Nonlinear Wave Equations
Nonlinear Wave Equations

Author: Satyanad Kichenassamy

Publisher: CRC Press

Published: 2021-05-30

Total Pages: 297

ISBN-13: 1000444724

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This work examines the mathematical aspects of nonlinear wave propagation, emphasizing nonlinear hyperbolic problems. It introduces the tools that are most effective for exploring the problems of local and global existence, singularity formation, and large-time behaviour of solutions, and for the study of perturbation methods.

Accelerating Expansion
Accelerating Expansion

Author: Gordon Belot

Publisher: Oxford University Press

Published: 2023-07-25

Total Pages: 241

ISBN-13: 0192691562

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Accelerating Expansion explores some of the philosophical implications of modern cosmology, focused on the significance that the discovery of the accelerating expansion of the Universe has for our understanding of time, geometry, and physics. The appearance of the cosmological constant in the equations of general relativity allows one to model universes in which space has an inherent tendency towards expansion. This constant, introduced by Einstein but subsequently abandoned by him, returned to centre stage with the discovery of the accelerating expansion. This pedagogically-oriented essay begins with a study of the most basic and elegant relativistic world that involves a positive cosmological constant, de Sitter spacetime. It then turns to the relatives of de Sitter spacetime that dominate modern relativistic cosmology. Some of the topics considered include: the nature of time and simultaneity in de Sitter worlds; the sense in which de Sitter spacetime is a powerful dynamical attractor; the limited extent to which observation can give us information about the topology of space in a world undergoing accelerated expansion; and cosmologists' favourite sceptical worry about the reliability of evidence and the possibility of knowledge, the problem of Boltzmann brains.

Differential Geometry, Part 2
Differential Geometry, Part 2

Author: Shiing-Shen Chern

Publisher: American Mathematical Soc.

Published: 1975

Total Pages: 455

ISBN-13: 0821802488

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Contains sections on Complex differential geometry, Partial differential equations, Homogeneous spaces, and Relativity.

Global Nonlinear Stability of Schwarzschild Spacetime Under Polarized Perturbations
Global Nonlinear Stability of Schwarzschild Spacetime Under Polarized Perturbations

Author: Sergiu Klainerman

Publisher: Princeton University Press

Published: 2020-12-15

Total Pages: 858

ISBN-13: 0691212430

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Essential mathematical insights into one of the most important and challenging open problems in general relativity—the stability of black holes One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. In this book, Sergiu Klainerman and Jérémie Szeftel take a first important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes—or Schwarzschild spacetimes—under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, Klainerman and Szeftel introduce a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, this book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.