Stability by Linearization of Einstein's Field Equation by Joan Girbau (English)

Description: Stability by Linearization of Einstein's Field Equation by Joan Girbau, Lluís Bruna , 3 2 2 R ? /?x K i i g V T G g ?T , ? 2 2 2 2 ? 2 2 2 2 2 c ?t ?x ?x ?x 1 2 3 S T S T? ˜ T S 2 2 2 2 ? 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 ? Y X +X X X 1 2 1 2 ? Y X 1 2 X 1 X 2 ? FORMAT Hardcover LANGUAGE English CONDITION Brand New Publisher Description 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 Notes Contains classical results on stability of great beautyPresents the objects needed to prove the theorems and the Cauchy problem for Einsteins equation in a self-contained wayProvides introductory chapters on pseudo-Riemannian manifolds and relativity (both special and general) addressed to readers having no background in these topics and a new definition of stability of Einsteins equation adapted to the case of matter and gives results concerning stability in Robertson-Walker models Table of Contents Preface // I Pseudo-Riemannian Manifolds: I.1 Connections / I.2 Firsts results on pseudo-Riemannian manifolds / I.3 Laplacians / I.4 Sobolev spaces of tensors on Riemannian manifolds / I.5 Lorentzian manifolds // II Introduction to Relativity: II.1 Classical fluid mechanics / II.2 Kinematics of the special relativity / II.3 Dynamics of special relativity / II.4 General relativity / II.5 Cosmological models / II. 6 Appendix: a theorem in affine geometry // III. Approximation of Einsteins Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einsteins equation for small perturbations of the Minkowski metric / III.3 Action on metrics of diffeomorphisms close to identity / III.4 Continuing the calculation of Section 2 / III.5 Comparison with the classical gravitation // IV. Cauchy Problem for Einsteins Equation with Matter: IV.1 1. Differential operators in an open set of Rn+1 / IV.2 Differential operators in vector bundles / IV.3 Harmonic maps / IV.4 Admissible classes of stress-energy tensors / IV.5 Differential operator associated to Einsteins equation / IV.6 Constraint equations / IV.7 Hyperbolic reduction / IV.8 Fundamental theorem / IV.9 An example: the stress-energy tensor of holonomic media / IV.10 The Cauchy problem in the vacuum // V. Stability by Linearization of Einsteins Equation, General Concepts: V.1 Classical concept of stability by linearization of Einsteins equation in the vacuum / V.2 A new concept of stability by linearization of Einsteins equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einsteins equation in the presence of matter / V.4 Change of notation / V.5 Technical details concerning the map f / V.6 Tangent linear map of f // VI. General Results on Stability by Linearization when the Submanifold M of V is Compact: IV.1 1. Adjoint of D(g,k) f / VI.2 Results by A. Fischer and J. E. Marsden / VI.3 A result by V. Moncrief / VI.4 Appendix: general results on elliptic operators in compact manifolds // VII. Stability by Linearization of Einsteins Equation at Minkowskis Initial Metric: VII.1 A further expression of D(g,k) f / VII.2 The relation between Euclidean Laplacian and stability by linearization at the initial Minkowskis metric / VII.3 Some proofs on topological isomorphisms in Rn / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Desers result / VII.5 The Euclidean asymptotic case: generalization of a result by Y. Choquet-Bruhat, A. Fischer and J. E. Marsden // VIII. Stability by Linearization of Einsteins Equation in Robertson-Walker Cosmological Models: VIII.1 Euclidean model / VIII.2 Hyperbolic model / VIII.3 Sobolev spaces and hyperbolic Laplacian / VIII.4 Spherical model / VIII.5 Universes that are not simply connected // References Review From the reviews:"The authors of the book under review have contributed to this subject over the last ten years by studying the linearization stability for Einsteins equations with source terms and in cosmological solutions. Here they present the results in a systematic fashion accessible to a reader with some background in differential geometry and partial differential equations." (Hans-Peter KÜnzle, Mathematical Reviews, Issue 2011 h) Long Description 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 Review Quote From the reviews:The authors of the book under review have contributed to this subject over the last ten years by studying the linearization stability for Einsteins equations with source terms and in cosmological solutions. Here they present the results in a systematic fashion accessible to a reader with some background in differential geometry and partial differential equations. (Hans-Peter K Feature Contains classical results on stability of great beautyPresents the objects needed to prove the theorems and the Cauchy problem for Einsteins equation in a self-contained wayProvides introductory chapters on pseudo-Riemannian manifolds and relativity (both special and general) addressed to readers having no background in these topics and a new definition of stability of Einsteins equation adapted to the case of matter and gives results concerning stability in Robertson-Walker models Details ISBN3034603037 Year 2010 ISBN-10 3034603037 ISBN-13 9783034603034 Short Title STABILITY BY LINEARIZATION OF Language English Media Book Format Hardcover Edition Description Edition. DEWEY 510 Series Number 58 Author Lluís Bruna Publication Date 2010-02-19 Imprint Birkhauser Verlag AG Place of Publication Basel Country of Publication Switzerland Illustrations XV, 208 p. Pages 208 DOI 10.1007/978-3-0346-0304-1 Publisher Birkhauser Verlag AG Series Progress in Mathematical Physics Audience Professional & Vocational We've got this At The Nile, if you're looking for it, we've got it. With fast shipping, low prices, friendly service and well over a million items - you're bound to find what you want, at a price you'll love! TheNile_Item_ID:127849160;

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