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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…mehr
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
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 Einstein's Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einstein's 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 Einstein's 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 Einstein's 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 Einstein's Equation, General Concepts: V.1 Classical concept of stability by linearization of Einstein's equation in the vacuum / V.2 A new concept of stability by linearization of Einstein's equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einstein's 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 Einstein's Equation at Minkowski's 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 Minkowski's metric / VII.3 Some proofs on topological isomorphisms in Rn / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's 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 Einstein's 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
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 Einstein's Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einstein's 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 Einstein's Equation with Matter: IV.1 1. Differential operators in an open set of R n+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 Einstein's 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 Einstein's Equation, General Concepts: V.1 Classical concept of stability by linearization of Einstein's equation in the vacuum / V.2 A new concept of stability by linearization of Einstein's equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einstein's 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 Einstein's Equation at Minkowski's 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 Minkowski's metric / VII.3 Some proofs on topological isomorphisms in R n / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's 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 Einstein's 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
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 Einstein's Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einstein's 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 Einstein's 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 Einstein's 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 Einstein's Equation, General Concepts: V.1 Classical concept of stability by linearization of Einstein's equation in the vacuum / V.2 A new concept of stability by linearization of Einstein's equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einstein's 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 Einstein's Equation at Minkowski's 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 Minkowski's metric / VII.3 Some proofs on topological isomorphisms in Rn / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's 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 Einstein's 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
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 Einstein's Equation by the Wave Equation: III.1 Perturbations of Ricci tensor / III.2 Einstein's 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 Einstein's Equation with Matter: IV.1 1. Differential operators in an open set of R n+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 Einstein's 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 Einstein's Equation, General Concepts: V.1 Classical concept of stability by linearization of Einstein's equation in the vacuum / V.2 A new concept of stability by linearization of Einstein's equation in the presence of matter / V.3 How to apply the definition of stability by linearization of Einstein's 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 Einstein's Equation at Minkowski's 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 Minkowski's metric / VII.3 Some proofs on topological isomorphisms in R n / VII.4 Stability of the Minkowski metric: Y. Choquet-Bruhat and S. Deser's 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 Einstein's 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
Rezensionen
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 Einstein's 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)
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