Stability by Linearization of Einstein's Field Equation [electronic resource] /
Bruna, Lluís.author.
Girbau, Joan.author.
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.
Description based upon print version of record.
Cover; Title Page; Copyright Page; Table of Contents; Introduction; Chapter I Pseudo-Riemannian Manifolds; I.1 Connections; I.2 First results on pseudo-Riemannian manifolds; I.2.1 Associate connection; I.2.2 Curvature; I.2.3 Covariant differentiation and divergence; I.2.4 Divergence of the Ricci tensor; I.2.5 Lie derivative and infinitesimal isometries; I.3 Laplacians; I.4 Sobolev spaces of tensors on Riemannian manifolds; I.5 Lorentzian manifolds; I.5.1 Definitions; I.5.2 Specific notation for Lorentzian manifolds; Chapter II Introduction to Relativity; II.1 Classical fluid mechanics
II.1.1 A lemma on derivation of integralsII.1.2 Mass of a fluid. Continuity equation; II.1.3 Total force; II.1.4 Cauchy principle; II.1.5 Differential expression of the motion equations; II.2 Kinematics of special relativity; II.2.1 Inertial systems; II.2.2 Postulates of special relativity; II.2.3 Lorentz transformations; II.2.4 Inertial systems and the Minkowski space; II.2.5 Contraction of lengths; II.2.6 Proper time of a particle; II.2.7 Time dilation; II.3 Dynamics of special relativity; II.3.1 Mass and momentum; II.3.2 Collision laws. Equivalence of mass and energy
II.3.3 Minkowski forceII.3.4 Relativistic fluid dynamics; II.3.5 Stress-energy tensor of a fluid; II.4 General relativity; II.4.1 Fundamentals; II.4.2 Einstein's field equation; II.5 Cosmological models; II.6 Appendix: a theorem in affine geometry; Chapter III Approximation of Einstein's Equation by the Wave Equation; III.1 Perturbations of the Ricci tensor; III.2 Einstein's equation for small perturbations of the Minkowski metric; III.3 Action on metrics of diffeomorphisms close to the identity; III.4 Continuing the calculation of Section III.2; III.5 Comparison with classical gravitation
Chapter IV Cauchy Problem for Einstein's Equation with MatterIV.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; Chapter V Stability by Linearization of Einstein's Equation, General Concepts
V.1 Classical concept of stability by linearization of Einstein's equation in the vacuumV.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 Φ; V.6 Tangent linear map of Φ; Chapter VI General Results on Stability by Linearization when the Submanifold M of V is Compact; VI.1 Adjoint of D(g,k)Φ; 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
Includes bibliographies and index.
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.
Mathematical physics.
Partial differential equations.
Theoretical, Mathematical and Computational Physics.
Partial Differential Equations.