Study axisymmetric waves on extremal Kerr spacetime using physical-space estimates.
problem Obtain integrated local energy decay estimates for axisymmetric waves on extremal Kerr backgrounds.
method Use physical-space analysis and a method introduced by Stogin, simplifying Aretakis' derivation.
result Extend Morawetz estimates to extremal Kerr spacetime using purely classical currents.
Study proves curvature estimates for Kerr spacetime's linearized perturbations.
problem Proving elliptic L2(S2)-estimates for linearised curvature quantities in Kerr spacetime. method Applies linearised system from doctoral thesis, covers full sub-extremal range of Kerr parameters.
result Elliptic L2(S2)-estimates for linearised curvature quantities in the full sub-extremal range of Kerr parameters. Massless Majorana spinors cannot exist in Kerr spacetime under certain conditions.
problem Existence of massless Majorana spinors in Kerr spacetime.
method Analyzing Dirac equation and separating it into radial and angular equations; proving nonexistence under specific conditions.
result Massless Majorana spinors can only exist if angular momentum a=0 and spacetime reduces to Schwarzschild spacetime. New framework for gravitational perturbations of Kerr spacetimes, focusing on stability.
problem Stability of Kerr spacetimes to gravitational perturbations.
method New geometric framework with tailored null frames and gauge, reformulating Einstein equations.
result Derivation of linearised vacuum Einstein equations in the new framework.
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.
No closed timelike geodesics in Kerr spacetimes, proving absence of closed causal geodesics.
problem Proving the nonexistence of closed timelike geodesics in Kerr spacetimes.
method Analyzing the Kerr-star spacetime, excluding closed null geodesics and proving the nonexistence of closed timelike geodesics.
result No closed timelike geodesics in Kerr spacetimes.
Characterizes Kerr spacetimes using conformal methods.
problem Understanding Kerr spacetimes through conformal transformations.
method Conformally covariant characterization approach.
result Ideal characterization of Kerr spacetimes.
No time-periodic Majorana fermions found in Kerr-Newman spacetimes with nontrivial charge.
problem Existence of Majorana fermions in Kerr-Newman spacetimes with nontrivial charge.
method Analysis of Dirac equation in Kerr-Newman spacetimes, leading to algebraic identities.
result No differentiable time-periodic Majorana fermions in Kerr-Newman spacetimes with nontrivial charge.
Null geodesics in Kerr spacetimes cannot be closed or bounded.
problem Existence of closed null geodesics in Kerr spacetimes.
method Analytical proof of non-existence of closed null geodesics in Kerr spacetimes.
result Null geodesics in Kerr spacetimes cannot be closed or contained in a compact subset.
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. Study constructs scattering theory for massless Dirac field on Kerr spacetime.
problem Scattering theory for massless Dirac field on Kerr spacetime.
method Conformal geometric method, pointwise decay assumption.
result Valid construction in Schwarzschild and slowly rotating black hole spacetimes.
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.
Kerr spacetimes without closed null geodesics for non-zero rotation.
problem Analyzing closed geodesics in Kerr spacetimes.
method Analytic extension and geodesic analysis.
result Kerr spacetimes do not admit closed null geodesics for any non-zero rotation parameter.
Study of charged scalar fields on Reissner-Nordström spacetimes via energy estimates.
problem Understanding the behavior and stability of charged scalar fields on near-extremal Reissner-Nordström spacetimes.
method Global integrated energy decay and boundedness estimates for solutions to the charged scalar field equation.
result Established global, weighted integrated energy decay and boundedness estimates for solutions on (near-)extremal Reissner-Nordström(--de Sitter) spacetimes.
We generalize Penrose's notion of conformal infinity of spacetime, to situations with anisotropic scaling. This is relevant not only for Lifshitz-type anisotropic gravity models, but also in standard general relativity and string theory, for spacetimes exhibiting a natural asymptotic anisotropy. Examples include the Li…
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.
Carter tensor analysis aids wave equation on Kerr-Newman spacetime.
problem Analyzing perturbations of Kerr-Newman spacetime using wave equation.
method Physical-space analysis adapted to Kerr-Newman spacetime, leveraging Carter operator commutation.
result Carter operator commutes with wave equation on Kerr-Newman spacetime, enabling wave equation analysis.
The higher-dimensional Kerr-NUT-de Sitter spacetime describes the general rotating asymptotically de Sitter black hole with NUT parameters. It is known that such a spacetime possesses a rank-2 closed conformal Killing-Yano (CKY) tensor as a ``hidden'' symmetry which provides the separation of variables for the geodesic…
The paper proves smoothness of event horizons in Kerr spacetime perturbations.
problem Smoothness of event horizons in Kerr spacetime perturbations.
method Proof of smoothness using stability of slowly rotating Kerr spacetimes.
result Smooth null hypersurfaces of event horizons are proven for Kerr spacetime perturbations.
The Levi-Civita connection and geodesic equations for a stationary spacetime are studied in depth. General formulae which generalize those for warped products are obtained. These results are applicated to some regions of Kerr spacetime previously studied by using variational methods. We show that they are neither space…
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.
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…
Paper glues characteristic data to Kerr spacetime, proving spacelike gluing.
problem Solving characteristic gluing problem for Einstein vacuum equations.
method Detailed characteristic gluing of strongly asymptotically flat data to Kerr spacetime.
result Alternative proof of spacelike gluing construction for strongly asymptotically flat spacelike initial data.
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.
In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor Mab which solves the…
Study finds necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
problem Understanding the asymptotic behavior of black hole geometries.
method Used hidden symmetry and conformal geometry technology to find necessary conditions.
result Necessary conditions for black hole geometries to asymptotically approach Kerr-de Sitter spacetime.
Study of Dirac fields on Kerr spacetimes using peeling method.
problem Understanding decay and regularity of Dirac fields on Kerr spacetimes.
method Penrose conformal compactification and geometric energy estimates.
result Optimal initial data spaces for peeling of Dirac fields on Kerr spacetimes.
This paper constructs GCM hypersurfaces in Kerr spacetimes.
problem Extending the Kerr family stability proof to full stability.
method Concatenating a 1-parameter family of GCM spheres by solving an ODE system.
result Removes symmetry restrictions in GCM procedure.
Closed and broken electromagnetic orbits in Kerr-Newman spacetime
problem Constructing closed and broken electromagnetic orbits in the Kerr-Newman spacetime
method Constructing smooth closed electromagnetic orbits tangent to the axial Killing field and proving the existence of spherical electromagnetic orbits
result Proving the existence of spherical electromagnetic orbits and constructing closed broken electromagnetic orbits
Researchers found a way to measure energy in black hole perturbations.
problem Lack of positive-definite and conserved energy in black hole stability.
method Dimensional reduction and construction of a positive-definite energy functional.
result Conserved Hamiltonian energy for axially symmetric perturbations of Kerr black holes.
Proves rigidity of extremal Kerr-Newman horizons.
problem Classifying near-horizon geometries of extremal Kerr-Newman horizons.
method Proves intrinsic geometry constraints leading to rigidity.
result Proves extremal Kerr-Newman horizons are unique.
We study the Cauchy problem for the wave equation on extreme Kerr backgrounds under axisymmetry. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface S which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain …
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 construct one-parameter families of solutions to the Einstein--Klein--Gordon equations bifurcating off the Kerr solution such that the underlying family of spacetimes are each an asymptotically flat, stationary, axisymmetric, black hole spacetime, and such that the corresponding scalar fields are non-zero and time-p…
New proof of stability for expanding Kerr-de Sitter spacetimes with smoothness at the boundary.
problem Stability of expanding region of Kerr-de Sitter spacetimes.
method Modified generalized harmonic gauge, local stability near conformal boundary, smoothness down to future conformal boundary.
result Smoothness of conformally rescaled metric down to future conformal boundary with mild singularity.
Second part of series studying charged scalar fields on Reissner--Nordström spacetimes.
problem Analyzing late-time behavior and stability of charged scalar fields on black hole backgrounds.
method Purely physical-space based methods, energy estimates, inverse-power laws.
result First pointwise decay estimates for charged scalar fields on black hole backgrounds.
Using black-hole inequalities and the increase of the horizon's areas, we show that there are arbitrarily small electro-vacuum perturbations of the standard initial data of the extreme Reissner-Nordstrom black-hole that, (by contradiction), cannot decay in time into any extreme Kerr-Newman black-hole. This proves the e…
We prove a perturbative result concerning the uniqueness of Kerr-Newman family of black holes: given an asymptotically flat space-time with bifurcate horizons, if it agrees with a non-extremal Kerr-Newman space-time asymptotically flat at infinity and it is sufficiently close to the Kerr-Newman family, then the space-t…
We consider Kerr spacetimes with parameters a and M such that |a|<< M, Kerr-Newman spacetimes with parameters |Q|<< M, |a|<< M, and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M>0, with appropriate geometric assumptions on …
We classify all spacetimes with a closed rank-2 conformal Killing-Yano tensor. They give a generalization of Kerr-NUT-de Sitter spacetimes. The Einstein condition is explicitly solved and written as an indefinite integral. It is characterized by a polynomial in the integrand. We briefly discuss the smoothness condition…
This work proves Kerr black holes are dynamically stable under certain perturbations.
problem Dynamical stability of Kerr black holes under axially symmetric perturbations.
method Dimensional reduction to 2+1 Einstein-wave map system, construction of positive-definite energy functional, proving boundary terms vanish.
result Strictly conserved positive energy for axially symmetric linear perturbations of Kerr black holes.
We prove that extreme Kerr initial data set is a unique absolute minimum of the total mass in a (physically relevant) class of vacuum, maximal, asymptotically flat, axisymmetric data for Einstein equations with fixed angular momentum. These data represent non-stationary, axially symmetric, black holes. As a consequence…
A characterization of the Kerr-NUT-(A)de Sitter metric among four dimensional Λ-vacuum spacetimes admitting a Killing vector is obtained in terms of the proportionality of the self-dual Weyl tensor and a natural self-dual double two-form constructed from the Killing vector. This result recovers and extends a previous c…
New memory effect discovered in gravitational wave behavior.
problem Understanding gravitational wave behavior in spacetimes with angular momentum.
method Mathematical analysis of Minkowski spacetime and Kerr black holes.
result Angular momentum memory effect observed at future null infinity.
Higher-dimensional Schwarzschild spacetimes violate the Penrose property.
problem Causal behavior of higher-dimensional Schwarzschild spacetimes.
method Analyzing causal properties in (2+1), (3+1), and (d+1) dimensions. result The Penrose property does not hold for (d+1) dimensional Schwarzschild if d>3. Researchers extend microlocal analysis across event horizons of rotating black holes.
problem Incomplete microlocal theory of fields across black hole event horizons.
method Extended microlocal theory for extremal rotating black holes, showing null covectors form an involutive double characteristic manifold.
result Mathematical basis for asymptotic oscillatory solutions near event horizons.
These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spa…
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.