Stability of black holes proven in full subextremal range with positive cosmological constant.
problem Stability of Kerr-de Sitter black holes in the full subextremal range.
method Similar to previous proof in slowly rotating case, with implementation of constraint damping and verification of subprincipal symbol condition.
result Stability of Kerr-de Sitter black holes proven in the full subextremal range.
The paper analyzes Teukolsky equations on Kerr backgrounds, proving boundedness and decay of solutions.
problem Analyzing boundedness and decay of solutions to Teukolsky equations on Kerr backgrounds.
method Frequency space analysis of transformed Teukolsky equations on Kerr backgrounds.
result Fixed frequency solutions remain bounded and decay in time for subextremal Kerr backgrounds.
Proves wave equation solutions in Kerr-de Sitter spacetime have specific asymptotic expansions.
problem Analyzing solutions to wave equations in Kerr-de Sitter spacetime.
method Developed a Fredholm setup for quasinormal modes and analyzed trapping of lightlike geodesics.
result Proves asymptotic expansions of wave equation solutions up to a decay order.
The paper proves boundedness and decay of Teukolsky equations on Kerr backgrounds.
problem Analyzing boundedness and decay of Teukolsky equations on Kerr backgrounds.
method Adapting techniques from scalar waves, uniform-in-frequency estimates for Teukolsky PDEs were obtained.
result Solutions of Teukolsky equation on subextremal Kerr backgrounds remain bounded and decay in time.
Characterizes photon surfaces in static spacetimes, proving uniqueness.
problem Understanding photon surfaces in static spacetimes of arbitrary dimension.
method Complete characterization and new insights into spacetime geometry.
result Proves uniqueness of certain electrostatic spacetimes.
Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.
problem Analyzing decay and non-decay rates of solutions to the massless Vlasov equation on Reissner-Nordström spacetimes.
method Quantitative analysis of geodesic flow and comparison to wave equation instability results.
result Exponential decay rates in subextremal cases and polynomial rates in extremal cases, with non-decay of transversal derivatives in extremal cases.
We show a uniqueness result for the n-dimensional spatial Reissner-Nordström manifold: a static, electrovacuum, asymptotically flat system which is asymptotically Reissner-Nordström is a subextremal Reissner-Nordström manifold with positive mass, provided that its inner boundary is a (possibly disconnected) photon sphe…
Constructs solutions of Einstein equations for black holes gluing along timelike geodesics.
problem Constructing solutions of Einstein equations for black holes gluing along timelike geodesics.
method Constructs solutions gε of the Einstein equations describing a mass ε Kerr black hole traveling along a timelike geodesic C. result Constructs true solutions gε of the Einstein equations for black holes gluing along timelike geodesics. Formally constructs metrics near timelike geodesics in vacuum spacetimes.
problem Constructing metrics near timelike geodesics in spacetimes.
method Constructs a family of metrics depending on a small parameter ε, solving the Einstein vacuum equations modulo O(ε^∞).
result The rescalings near the geodesic tend to a fixed subextremal Kerr metric.
New methods show quasinormal modes can be defined using various stationary Killing vectors.
problem Proving asymptotic expansions for wave equations in Kerr-de Sitter spacetimes.
method New definition of quasinormal modes using different stationary Killing vectors.
result Horizon Killing vector fields work for analysis, simplifying the problem.
The paper proves the existence of a horizon for certain spacetimes.
problem Existence of horizons in asymptotically stationary spacetimes.
method General unstable manifold theorem for perturbations of flows.
result Existence of a unique null hypersurface asymptotic to a horizon.
The paper classifies electrovacuum spaces in higher dimensions, proving several key results.
problem Classifying regular static black hole solutions of the static Einstein-Maxwell equations.
method Analytical proofs and geometric analysis of electrovacuum spaces.
result An n-dimensional locally conformally flat extremal electrovacuum space must be in the Majumdar-Papapetrou class.
This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgroun…
We show that axisymmetric extremal horizons are unstable under linear scalar perturbations. Specifically, we show that translation invariant derivatives of generic solutions to the wave equation do not decay along such horizons as advanced time tends to infinity, and in fact, higher order derivatives blow up. This resu…
This paper describes the black hole threshold in a moduli space of spherically symmetric spacetimes.
problem Understanding the black hole threshold in a moduli space of spherically symmetric spacetimes.
method Complete description and analysis of the black hole threshold in the moduli space M. result The black hole threshold is the extremal leaf of a C1 foliation of the moduli space, separating black hole solutions from non-collapsing solutions. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting…
Proves inextendibility of weak null singularities from curvature blow-up.
problem Inextendibility of weak null singularities in the context of curvature blow-up.
method Introduces a new strategy to infer Cloc0,1-inextendibility from curvature blow-up. result Expected to contribute to the resolution of strong cosmic censorship conjecture.
We prove that a large class of smooth solutions ψ to the linear wave equation □gψ=0 on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along t…
Proves analyticity of quasinormal modes in Kerr and Kerr-de Sitter spacetimes.
problem Analyticity of quasinormal modes in extreme Kerr and Kerr-de Sitter spacetimes.
method Observation of stable radial point source/sink structure in bicharacteristic flow; recent microlocal analysis result by Galkowski and Zworski.
result Quasinormal modes are real analytic in subextremal Kerr and Kerr-de Sitter spacetimes.
We review our recent work on linear stability for scalar perturbations of Kerr spacetimes, that is to say, boundedness and decay properties for solutions of the scalar wave equation \Box_gψ = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay ha…
In this global study of solutions to the linear wave equation on Schwarzschild de Sitter spacetimes we attend to the cosmological region of spacetime which is bounded in the past by cosmological horizons and to the future by a spacelike hypersurface at infinity. We prove an energy estimate capturing the expansion of th…
Study on instability of extreme Reissner-Nordström spacetime perturbations.
problem Linear stability of gravitational and electromagnetic perturbations in extreme Reissner-Nordström spacetime.
method Extends Giorgi's framework to prove instability results for a set of gauge invariant quantities along the event horizon.
result Proves decay, non-decay, and polynomial blow-up estimates for certain quantities along the event horizon, depending on the number of derivatives.
The paper examines gravitational singularities in spacetimes and proves inextendibility.
problem Investigating gravitational singularities in spacetimes.
method Analyzing local holonomy and using it to prove inextendibility.
result Proves the Cloc0,1-inextendibility of certain spacetimes. Study shows formation of Kerr black holes with complete apparent horizons and proves Penrose inequalities.
problem Formation of Kerr black holes and Penrose inequalities.
method Combining gravitational-collapse and Kerr stability results with new coordinate changes and elliptic arguments.
result Proves dynamical and spacetime Penrose inequalities in black hole formation spacetimes.
Proves stability of Schwarzschild black holes without symmetry assumptions.
problem Stability of Schwarzschild black holes under general conditions.
method Teleologically normalised double null gauges, analysis of linear stability, and control of non-linearities.
result Proves non-linear asymptotic stability of Schwarzschild family as solutions to Einstein vacuum equations.
Global existence and boundedness proved for quasilinear wave equations on Kerr black holes.
problem Global existence and boundedness for quasilinear wave equations on Kerr black holes.
method Combines linear inhomogeneous estimates on Kerr backgrounds and tailored physical space currents.
result Global existence, boundedness and decay for small data solutions to quasilinear wave equations on Kerr black holes.
We consider the wave equation on Reissner-Nordström-de Sitter and more generally Kerr-Newman-de Sitter black hole spacetimes with Λ>0. The strength of the blue-shift instability associated to the Cauchy horizon of these spacetimes has been the subject of much discussion, since-in contrast to the Λ=0 case-the compet…