We study asymptotics for solutions of Maxwell's equations, in fact of the Hodge-de Rham equation @math without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions @math . We prove that solutions decay exponentially to @math or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell's equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum.
Non-linear results for wave equations on black hole spacetimes include @cite_34 , see also the references therein, Luk's work @cite_28 on semilinear forward problems on Kerr, and the scattering construction of dynamical black holes by Dafermos, Holzegel and Rodnianski @cite_4 . Fully general stability results for Einstein's equations specifically are available for de Sitter space by the works of Friedrich @cite_11 , Anderson @cite_12 , Rodnianski and Speck @cite_14 and Ringstr "om @cite_24 , and for Minkowski space by the work of Christodoulou and Klainerman @cite_15 , partially simplified and extended by Lindblad and Rodnianski @cite_35 @cite_36 , Bieri and Zipser @cite_50 and Speck @cite_3 .
@cite_35 @cite_14 @cite_4 @cite_28 @cite_36 @cite_3 @cite_24 @cite_50 @cite_15 @cite_34 @cite_12 @cite_11
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We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are “unstable in the large” and provides an alternative approach to the stability problem originally solved ( for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. In this article, we study small perturbations of the family of Friedmann-Lema ^itre-Robertson-Walker cosmological background solutions to the Euler-Einstein system with a positive cosmological constant in 1 + 3 dimensions. The background solutions describe an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing accelerated expansion. We show that under the equation of state p = c_s^2*(energy density), 0 < c_s^2 < 1 3, the background solutions are globally future asymptotically stable under small irrotational perturbations. In particular, we prove that the perturbed spacetimes, which have the topological structure [0,infinity) x T^3, are future causally geodesically complete. We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full "functional" degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift. We study a semilinear equation with derivatives satisfying a null condition on slowly rotating Kerr spacetimes. We prove that given sufficiently small initial data, the solution exists globally in time and decays with a quantitative rate to the trivial solution. The proof uses the robust vector field method. It makes use of the decay properties of the linear wave equation on Kerr spacetime, in particular the � ! We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time. In this article, we study the coupling of the Einstein field equations of general relativity to a family of models of nonlinear electromagnetic fields. The family comprises all covariant electromagnetic models that satisfy the following criteria: they are derivable from a sufficiently regular Lagrangian, they reduce to the linear Maxwell model in the weak-field limit, and their corresponding energy-momentum tensors satisfy the dominant energy condition. Our main result is a proof of the global nonlinear stability of the 1 + 3-dimensional Minkowski spacetime solution to the coupled system for any member of the family, which includes the linear Maxwell model. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in our wave coordinate gauge. Our analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows us to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. Our analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor developed in the article. We work directly with the electromagnetic fields, and thus avoid the use of electromagnetic potentials. This book consists of two independent works: Part I is 'Solutions of the Einstein Vacuum Equations', by Lydia Bieri. Part II is 'Solutions of the Einstein-Maxwell Equations', by Nina Zipser. A famous result of Christodoulou and Klainerman is the global nonlinear stability of Minkowski spacetime. In this book, Bieri and Zipser provide two extensions to this result. In the first part, Bieri solves the Cauchy problem for the Einstein vacuum equations with more general, asymptotically flat initial data, and describes precisely the asymptotic behavior. In particular, she assumes less decay in the power of @math and one less derivative than in the Christodoulou-Klainerman result. She proves that in this case, too, the initial data, being globally close to the trivial data, yields a solution which is a complete spacetime, tending to the Minkowski spacetime at infinity along any geodesic. In contrast to the original situation, certain estimates in this proof are borderline in view of decay, indicating that the conditions in the main theorem on the decay at infinity on the initial data are sharp. In the second part, Zipser proves the existence of smooth, global solutions to the Einstein-Maxwell equations. A nontrivial solution of these equations is a curved spacetime with an electromagnetic field. To prove the existence of solutions to the Einstein-Maxwell equations, Zipser follows the argument and methodology introduced by Christodoulou and Klainerman. To generalize the original results, she needs to contend with the additional curvature terms that arise due to the presence of the electromagnetic field @math ; in her case the Ricci curvature of the spacetime is not identically zero but rather represented by a quadratic in the components of @math . In particular the Ricci curvature is a constant multiple of the stress-energy tensor for @math . Furthermore, the traceless part of the Riemann curvature tensor no longer satisfies the homogeneous Bianchi equations but rather inhomogeneous equations including components of the spacetime Ricci curvature. Therefore, the second part of this book focuses primarily on the derivation of estimates for the new terms that arise due to the presence of the electromagnetic field. We consider quasilinear wave equations on manifolds for which infinity has a structure generalizing that of Kerr-de Sitter space; in particular the trapped geodesics form a normally hyperbolic invariant manifold. We prove the global existence and decay, to constants for the actual wave equation, of solutions. The key new ingredient compared to earlier work by the authors in the semilinear case [33] and by the first author in the non-trapping quasilinear case [30] is the use of the Nash-Moser iteration in our framework. A new proof of Friedrich’s theorem on the existence and stability of asymptotically de Sitter spaces in 3 + 1 dimensions is given, which extends to all even dimensions. In addition we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future. It is demonstrated that initial data sufficiently close to De-Sitter data develop into solutions of Einstein's equations Ric[g]=Λg with positive cosmological constant Λ, which are asymptotically simple in the past as well as in the future, whence null geodesically complete. Furthermore it is shown that hyperboloidal initial data (describing hypersurfaces which intersect future null infinity in a space-like two-sphere), which are sufficiently close to Minkowskian hyperboloidal data, develop into future asymptotically simple whence null geodesically future complete solutions of Einstein's equations Ric[g]=0, for which future null infinity forms a regular cone with vertexi+ that represents future time-like infinity.