Note that the mode analysis corresponding to formulation 1. described above yields necessary but not sufficient conditions for either statements (2a) and (2b) of true linear stability.( 1 ) In the case of the linear scalar wave equation g ϕ = 0, which can be thought of as a "poor man's" version of linearised gravity, the analogue of (2a) for Schwarzschild was proven by , and the analogue of (2a) and ( 2b) are shown now for the full subextremal Kerr family in [26], following a host of recent activity [24], [73], [1], [21]. See [25], [23] for a survey. See [6], [2] for generalisations to the Maxwell equations and [3] for a discussion of the extremal case |a|=M . Concerning the linearised Einstein equations themselves, work on the wave equation easily generalises to establish physical space decay on certain quantities, for instance those gauge-invariant quantities satisfying the Regge-Wheeler equation on Schwarzschild [32], [7], [29]. For the full system of linearised gravity however, both problems (2a) and ( 2b) have remained open until today. We note explicitly that even the question of uniform boundedness, let alone decay, for the gauge-invariant quantities satisfying the Teukolsky equation on Schwarzschild has remained open.(3) The full non-linear stability of Schwarzschild as a solution to the Cauchy problem for the non-linear Einstein vacuum equations (2). This is the definitive formulation of the fundamental question. See our previous [18] for a precise statement of the conjecture in the language of the Cauchy problem for (2). In analogy with 2. above, one could distinguish between questions of (3a) orbital stability and (3b) asymptotic stability. Experience from non-linear problems, however, in particular the proof of the non-linear stability of Minkowski space [14] referred to earlier (see also [47], [5]), indicates that (3a) and (3b) are naturally coupled.( 2 ) Since non-linear stability is thus necessarily a question of asymptotic stability, the "Schwarzschild" problem is more correctly re-phrased as the non-linear asymptotic stability of the Kerr family in a neighbourhood of Schwarzschild. For even if one restricts to small perturbations of Schwarzschild, it is expected that generically, spacetime dynamically asymptotes to a very slowly rotating Kerr solution with a =0. Since in the context of a non-linear stability proof, one effectively must "linearise" around the solution one expects to approach, this suggests that to resolve the full ( 1 ) Thus, the mode analysis can be an effective tool to show instability, but never, on its own, stability. For instability results for related problems proven via the existence of unstable modes, see [70], [28] and references therein. See also discussion in [78].( 2 ) This coupling arises from the super-criticality of the Einstein vacuum equations (2). Note that under spherical symmetry (where the vacuum equations must be replaced, however, by a suitable Einstein-matter system to restore a dynamical degree of freedom) this super-criticality is broken in the presence o...

In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

This paper investigates the decay properties of solutions to the massive linear wave equation left□gψ+αl2ψ=0 for g being the metric of a Kerr‐AdS spacetime satisfying | a |l

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

We prove boundedness and polynomial decay statements for solutions of the spin Teukolsky equation on a Kerr exterior background with parameters satisfying . The bounds are obtained by introducing generalisations of the higher order quantities P and used in our previous work on the linear stability of Schwarzschild. The existence of these quantities in the Schwarzschild case is related to the transformation theory of Chandrasekhar. In a followup paper, we shall extend this result to the general sub-extremal range of parameters . As in the Schwarzschild case, these bounds provide the first step in proving the full linear stability of the Kerr metric to gravitational perturbations.

We generalize our unique continuation results recently established for a class of linear and nonlinear wave equations □gϕ+σϕ = (ϕ,∂ϕ) on asymptotically anti-de Sitter (aAdS) spacetimes to aAdS spacetimes admitting nonstatic boundary metrics. The new Carleman estimates established in this setting constitute an essential ingredient in proving unique continuation results for the full nonlinear Einstein equations, which will be addressed in forthcoming papers. Key to the proof is a new geometrically adapted construction of foliations of pseudo-convex hypersurfaces near the conformal boundary

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet-, Neumann-or Robin-boundary conditions imposed on ψ at infinity. This class includes the regular Kerr-AdS black holes satisfying the Hawking Reall bound r 2 + > |a|l. We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of ψ and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr-AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions. ALBERTA THY 13-12

The massive wave equation g ψ − α Λ 3 ψ = 0 is studied on a fixed Kerr-anti de Sitter background (M, gM,a,Λ). We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that α < 9 4 , i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The total flux of the usual energy current arising from the timelike Killing vector field T (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to T , we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield T = ∂t with K = ∂t + λ∂ φ for an appropriate λ ∼ a, which is also Killing and-in contrast to the asymptotically flat case-everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field K which is null on the horizon.