Radiation fields

Abstract

Abstract. -We study the "hyperboloidal Cauchy problem" for linear and semilinear wave equations on Minkowski space-time, with initial data in weighted Sobolev spaces allowing singular behavior at the boundary, or with polyhomogeneous initial data. Specifically, we consider nonlinear symmetric hyperbolic systems of a form which includes scalar fields with a λφ p nonlinearity, as well as wave maps, with initial data given on a hyperboloid; several of the results proved apply to general space-times admitting conformal completions at null infinity, as well to a large class of equations with a similar non-linearity structure. We prove existence of solutions with controlled asymptotic behavior, and asymptotic expansions for solutions when the initial data have such expansions. In particular we prove that polyhomogeneous initial data (satisfying compatibility conditions) lead to solutions which are polyhomogeneous at the conformal boundary I + of the Minkowski space-time. Résumé (Champs rayonnants). -Nousétudions le « problème de Cauchy hyperboloïdal » pour deséquations d'ondes linéaires et semi-linéaires sur l'espace-temps de Minkowski, avec des données initiales, singulières au bord, dans des espaces de Sobolevà poids, où polyhomogènes. Plus précisement, nous considérons une classe de systèmes symétriques hyperboliques non-linéaires, compatibles avec l'équation d'onde scalaire λφ p , ainsi qu'avec des applications d'onde, avec données initiales prescrites sur un hyperboloide. Plusieurs de nos résultats restent valables pour une classe gé-nérale d'espace-temps avec complétions conformesà l'infini isotrope, ainsi que pour une large classe d'équations avec une certaine structure des termes non-linéaires. Nous démontrons l'existence de solutions avec comportement asymptotique contrôlé, ainsi que des développements asymptotiques si les données initiales en possèdent. En particulier nous démontrons, sous une condition de compatibilité, que les données initiales polyhomogènes conduisentà des solutions polyhomogènes près du bord conforme I + de l'espace-temps de Minkowski.

Superradiant instabilities of asymptotically anti-de Sitter black holes

“…(table A1). In a general gauge compatible with our gauge conditions we would have more generally £ K γ ab = £ δK g ab + £ K £ X g ab , where X a must satisfy (18). Commuting the Lie-derivative operators in the last term and using that K a Lie-derives the background then implies that a perturbation to another stationary black hole is one such that…”

“…Following [17,18], the idea is to make a particular ansatz for the correction to (δq ab , δp ab ). The linearized constraints C may be viewed as the result of acting on the perturbed initial data by a linear operator which maps the pair (δq ab , δp ab ) consisting of a symmetric tensor, δq ab , and a symmetric tensor density, δp ab , on Σ into a pair (u, X a ) consisting of a scalar density and dual vector density on Σ.…”

“…(ii) The WKB solution is not an exact solution, and so its initial data do not satisfy the linearized constraints. But we can, following [16], fix this up by an application of the Corvino-Schoen method [17,18]. For ω 1, the correction is small and it remains true that E K (γ) < 0 for the corrected perturbation.…”

Stationarity of extremum entropy fluid bodies in general relativity

“…which, given (ii), is equivalent toδ MS s = 0. These three conditions uniquely determine δ (1) MS n, δ (1) MS s, δ (1) MS u i , and formulas for these can be calculated explicitly from the definition of P i , (19), and from the formula for p, (5), which in particular implies that…”

“…with the boundary condition ψ → 0 at infinity. As the quantity in square brackets multiplying ψ is nonnegative, by standard arguments [18,19] there exists a unique solution of (56) satisfying this boundary condition. It immediately follows that the perturbation…”