Mihalis Dafermos

• University of Cambridge

I present a series of conjectures aiming to describe the general dynamics of the Einstein equations of classical general relativity in the vicinity of extremal black holes. I will reflect upon how these conjectures transcend older paradigms concerning extremality and near-extremality, in particular, the so-called “third law of black hole thermodyna...

General relativity is an area at the interface of partial differential equations, differential geometry, global analysis, mathematical physics and dynamical systems. It interacts with astrophysics, cosmology, high energy physics, and numerical analysis. The field is rapidly expanding and has witnessed remarkable developments and interconnections wi...

We prove global existence, boundedness and decay for small data solutions $\psi$ to a general class of quasilinear wave equations on Kerr black hole backgrounds in the full sub-extremal range $|a|<M$. The method extends our previous [DHRT22], which considered such equations on a wide class of background spacetimes, including Kerr, but restricted in...

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \...

We prove the non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region: general vacuum initial data, with no symmetry assumed, sufficiently close to Schwarzschild data evolve to a vacuum spacetime which (i) possesses a complete future null infinity $\mathcal{I}...

We consider the wave equation on Reissner Nordstrom 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 asymptotically flat λ = 0 case-the competition with...

We consider the wave equation on Reissner-Nordstr\"om-de Sitter and more generally Kerr-Newman-de Sitter black hole spacetimes with $\Lambda>0$. 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 $\Lambda=0$ case-the competition with the d...

We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild...

We prove boundedness and polynomial decay statements for solutions of the spin $\pm2$ Teukolsky equation on a Kerr exterior background with parameters satisfying $|a|\ll M$. The bounds are obtained by introducing generalisations of the higher order quantities $P$ and $\underline{P}$ used in our previous work on the linear stability of Schwarzschild...

In this paper, we provide an elementary, unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background: the celebrated blue-shift at the Cauchy horizon (familiar from the strong cosmic censorship conjecture) and the time-reversed red-shift at the event horizon (relevant in classical sc...

We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We...

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 "scattering matrix" mapping r...

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 backgrounds in the general subextremal...

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 in...

This paper contains the first two parts (I-II) of a three-part series concerning the scalar wave equation \Box_g{\psi} = 0 on a fixed Kerr background. We here restrict to two cases: (II1) |a| \ll M, general {\psi} or (II2) |a| < M, {\psi} axisymmetric. In either case, we prove a version of 'integrated local energy decay', specifically, that the 4-i...

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_g{\psi} = 0 on Kerr exterior backgrounds. We begin with the very slowly rotating case |a| \ll M, where first boundedness and then decay has been shown in rapid dev...

We present a new general method for proving global decay of energy through a suitable spacetime foliation, as well as pointwise decay, starting from an integrated local energy decay estimate. The method is quite robust, requiring only physical space techniques, and circumvents use of multipliers or commutators with weights growing in t. In particul...

We consider solutions to the linear wave equation on a (maximally extended) Schwarzschild spacetime, assuming only that the solution decays suitably at spatial infinity on a complete Cauchy hypersurface. (In particular, we allow the support of the solution to contain the bifurcate event horizon.) We prove uniform decay bounds for the solution in th...

Understanding the behaviour of linear waves on black hole backgrounds is a central problem in general relativity, intimately connected with the nonlinear stability of the black hole spacetimes themselves as solutions to the Einstein equations--a major open question in the subject. Nonetheless, it is only very recently that even the most basic bound...

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 spacetimes themselves as solution...

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 the plane spanned by the Killi...

In recent work, we have proven uniform decay bounds for solutions of the wave equation $\Box_g\phi=0$ on a Schwarzschild exterior, in particular, the uniform pointwise estimate $|\phi|\le Cv_+^{-1}$, which holds throughout the domain of outer communications, where $v$ is an advanced Eddington-Finkelstein coordinate, $v_+=\max\{v,1\}$, and $C$ is a...

We consider solutions to the linear wave equation $\Box_g\phi=0$ on a non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface. (In particular, no symmetry is assumed on initial data, and the support of the solutions may contain the sphere of bifurcat...

This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature $k$ of the symmetric surfaces and the cosmological constant $\Lambda$. With a suitable form...

We prove strong cosmic censorship for T^2-symmetric cosmological spacetimes (with spatial topology T^3 and vanishing cosmological constant Lambda) with collisionless matter. Gowdy symmetric spacetimes constitute a special case. The formulation of the conjecture is in terms of generic C^2-inextendibility of the metric. Our argument exploits a rigidi...

A new criterion for inextendibility of expanding cosmological models with symmetry is presented. It is applied to derive a number of new results and to simplify the proofs of existing ones. In particular, it shows that the solutions of the Einstein Vlasov system with T2 symmetry, including the vacuum solutions, are inextendible in the future. The t...

A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coo...

Let G(x) be a C0 function such that |G(x)|⩽Kp|x| for |x|⩽c, for constants K,c>0. We consider spherically symmetric solutions of g□ϕ=G(ϕ) where g is a Schwarzschild or more generally a Reissner–Nordström metric, and such that ϕ and ∇ϕ are compactly supported on a complete Cauchy surface. It is proven that for p>4, such solutions do not blow up in th...

We consider a spherically symmetric characteristic initial value problem for the Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heu...

Let G(x) be a C^0 function such that |G(x)|\le K|x|^{p} for |x|\le c, for constants K,c>0. We consider spherically symmetric solutions of \Box_g\phi=G(\phi) where g is a Schwarzschild or more generally a Reissner-Nordstrom metric, and such that \phi and \nabla \phi are compactly supported on a complete Cauchy surface. It is proven that for p> 4, su...

We prove that “first singularities” in the non-trapped region of the maximal development of spherically symmetric asymptotically flat data for the Einstein-Vlasov system must necessarily emanate from the center. The notion of “first” depends only on the causal structure and can be described in the language of terminal indecomposable pasts (TIPs). T...

In recent work of Allen at. al., heuristic and numerical arguments were put forth to suggest that boundary value problems for black hole evolution, where an appropriate Sommerfeld radiation condition is imposed, would fail to produce Price law tails. The interest in this issue lies in its possible implications for numerical relativity, where black...

Two aspects of the widely accepted heuristic picture of the final state of gravitational collapse are the so-called Price law tails, describing the asymptotics of the exterior region of the black hole that forms, and Israel-Poisson's mass inflation scenario, describing the internal structure of the black hole. (The latter scenario, if valid, would...

An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole in the future and yet past-causally geodesically complete, in particular, containing no white holes. We give such an example in this paper--in fact, a family of such examples, stable in a suitabl...

This talk describes some recent results [16] regarding the problem of uniqueness in the large (also known as strong cosmic censorship) for the initial value problem in general relativity. In order to isolate the essential analytic features of the problem from the complicated setting of gravitational collapse in which it arises, some familiarity wit...