When studying the Cauchy problem of general relativity we typically obtain L² bounds on the (Ricci) curvature tensor of spacelike hypersurfaces and its derivatives. In many situations it is useful to deduce from these H^{k} bounds that there exists coordinates on the spacelike hypersurface with (optimal) H^{k+2} bounds on the components of the induced Riemannian metric. The general idea is that this can be achieved using harmonic coordinates -- in which the principal terms of the Ricci curvature tensor are the Laplace-Beltrami operators of the metric components -- and standard elliptic regularity results. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with boundary, with curvature in L² and second fundamental form of the boundary in H^{1/2} both close to their respective Euclidean unit 3-disk values. The crux of the proof is a refined Bochner identity with boundary for harmonic functions. The cherry on the cake is that this result does not require any topology assumption on the Riemannian 3-manifold, and that we obtain -- as a conclusion -- that it must be diffeomorphic to the 3-disk. This talk is based on a result that I obtained in [Global nonlinear stability of Minkowski space for spacelike-characteristic initial data, Appendix A].
Perturbations of Kerr spacetime are typically studied with the Teukolsky formalism, in which a pair of gauge invariant components of the perturbed Weyl tensor are expressed in terms of separable modes that satisfy ordinary differential equations. However, for certain applications it is desirable to construct the full metric perturbation in the Lorenz gauge, in which the linearized Einstein field equations take a manifestly hyperbolic form. Directly solving the Lorenz gauge equations in Kerr spacetime is challenging for two reasons: (i) unlike the Teukolsky equation, the Lorenz gauge equations are not known to admit separable solutions; (ii) the equations for the ten components of the metric perturbation are coupled. In this talk, I will present a formalism in which the Lorenz gauge metric perturbation is obtained from a set of six decoupled and separable solutions to the spin-2, spin-1 and spin-0 Teukolsky equations. The formalism is ideally suited to hyperboloidal methods, which have been shown to provide a highly-efficient approach to solving Teukolsky equations. As a demonstration of the approach, I will give results for the Lorenz-gauge gravitational self-force problem in Kerr spacetime. This talk is based on work with Sam Dolan, Chris Kavanagh and Leanne Durkan.
Let (M,g) be a complete, connected, noncompact Riemannian three-manifold with nonnegative Ricci curvature. R. Hamilton has conjectured that if the largest eigenvalue of the Ricci curvature of (M,g) is less than the product of its smallest eigenvalue and a universal constant, then (M,g) is flat. This conjecture has recently been confirmed by A. Deruelle, F. Schulze and M. Simon using Ricci flow. In this lecture, I will present a short new proof of R. Hamilton's conjecture based on inverse
mean curvature flow. This is joint work with Gerhard Huisken.
In this talk, I will present research in the direction of understanding the asymptotic behaviour of gravitational radiation in spacetimes that are dynamically constructed from physically motivated assumptions. More precisely, I will describe how to set up and solve the scattering problem around spacelike infinity for the linearised Einstein vacuum equations around Schwarzschild with characteristic scattering data posed towards and along past null infinity. I will then describe how to pick scattering data that model the exterior of $N$ infalling bodies from the infinite past, and I will show how to compute the global asymptotic behaviour of the resulting solutions. In particular, this will show that the latter fail to admit a smooth future null infinity or the usual expansions of Bondi coordinates, while at the same time providing an alternative framework for the asymptotics of gravitational radiation, which, in particular, gives constructive corrections to both Bondi coordinates and the notion of a smooth null infinity.
I will discuss a recent work with Gustav Holzegel, in which we prove integrated decay bounds for solutions of the geometric wave equation with small linear perturbations on Kerr black hole spacetimes. Our proof adapts the framework introduced by Dafermos, Rodnianski, and Shlapentokh-Rothman for the homogeneous wave equation on Kerr spacetimes. When adding the perturbative term one must also compensate for obstructions caused by the necessary degeneration of Morawetz-type estimates in these spacetimes, which is due to the presence of trapped null geodesics. Mathematically, the key mechanism to our approach is the construction of a pseudodifferential commutator W, such that for the commuted equation one may obtain a nondegenerate Morawetz-type estimate.
In cosmology, the equation of state of a perfect fluid is considered to be p = c_s^2 \rho, where c_s is the speed of sound. The simplest solution to the Einstein-Euler system, known as FLRW, representing a cosmological fluid, was discovered by Friedmann already in 1922. There is an extensive literature in physics concerning the dynamics of cosmological fluids. However, rigorous mathematical works proving the stability of homogeneous backgrounds are so far restricted to small sound speeds, up to the radiation threshold. Interesting bifurcation phenomena and instabilities are predicted for larger sound speeds. I will discuss ongoing work proving the global stability of homogeneous solutions with so-called extreme tilt, whose fluid vector field becomes asymptotically null, beyond the radiation case.
To a first approximation, objects in general relativity move along geodesics. Looked at more closely, a body's internal structure affects its motion, causing different objects to fall in different ways. This talk will explore what is possible and what is not in that context. For example, it is possible for a suitably-engineered spacecraft to change its orbit purely by changing its shape. Still, there are constraints. I will discuss how all such constraints arise from a very weak type of "local symmetry." Constraints arise from the presence of Killing vectors and from conformal Killing-Yano tensors, but from much more as well.
In the past 20 years the mathematical study of dynamical properties of black holes has greatly accelerated. Indeed, the first mathematical description of a dynamical black hole is from 2013. Many of the works focus on symmetry reduced settings, or toy problems that ought to capture important phenomenology of the Einstein equations. Outside a few exceptions, these works are focused on the solution in a neighbourhood of a single black hole, resulting in almost spherically symmetric setting in the far region. In this talk, I will present a model problem that I think captures some, albeit not all, important features and difficulties of studying multi black hole solutions in asymptotically flat spacetimes and report on progress in the understanding of this model. In particular, I will present the construction of a solution to the energy critical wave equation in a neighbourhood of timelike infinity that has a prescribed radiation field through null infinity.
In this talk, I will present a construction of regular initial data for the Einstein-Maxwell-charged scalar field system collapsing to exactly extremal Reissner-Nordström black holes within finite advanced time. In particular, our result can be viewed as a definitive disproof of “the third law of black hole thermodynamics.’’ I will further present results on black hole formation and event horizon gluing in vacuum for very slowly rotating Kerr black holes. This is based on joint work with Ryan Unger (Princeton).
After a brief discussion of the motivation for the study of massive scalar fields in general relativity and in relation to black holes, I will recall some old results concerning situations where one expects instabilities for massive scalar fields and then discuss a new result concerning the case of a spherically symmetric black hole where one can prove decay for the scalar field. This final decay result is joint with Federico Pasqualotto and Maxime Van de Moortel.
Often in a physical problem there are some degrees of freedom that we treat as unobservable because the energy required to excite them is much greater than the energy scale of the problem. For example, when modelling a pendulum with a rigid arm we ignore the vibrational modes of the arm. Or in quantum physics it may be that the energy scales available to us, for example in a collider, are not sufficient to create a particular heavy particle. These unobservable degrees of freedom can still affect the physics that we do observe at subleading order. A tool that is often used by physicists to study this is the machinery of effective field theory (EFT). In this talk I will discuss a paper with Harvey Reall in which we studied an EFT for a model problem and explained how it is possible to handle the troublesome PDEs that can arise.
We discuss the linear stability problem to gravitational and electromagnetic perturbations of the extremal, $ |\mathcal{Q}|=M, $ Reissner-Nordström spacetime, as a solution to the Einstein-Maxwell equations. Our work uses and extends the framework of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon $ \mathcal{H}^+ $. In particular, for associated quantities shown to satisfy generalized Regge-Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along $ \mathcal{H}^+ $, the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component $ \underline{\alpha} $ not decaying asymptotically along the event horizon $ \mathcal{H}^+, $ a result previously unknown in the literature.
The black hole stability conjecture is one of the outstanding open problems in the General Relativity, and an active area of research since the late 1950s. Key to this endeavor is a quantitative understanding of solutions to the so-called Teukolsky master equation in linearized gravity around Kerr. In this talk, we discuss a uniform-in-frequency analysis of separable solutions to the Teukolsky equations which captures superradiance and trapping phenomena. A corollary of our analysis is that general solutions to the Teukolsky equations are bounded and decay in time; this is a key first step in establishing linear stability of Kerr. Particular emphasis will be given to mode stability, that is non-existence of exponentially growing or bounded but non-decaying separable solutions. This talk contains joint work with Marc Casals (Leipzig) and Yakov Shlapentokh-Rothman (Toronto).
As gravitational-wave detectors become more sensitive to lower frequencies, they will increasingly detect binaries with smaller mass ratios, larger spins, and higher eccentricities. In this talk I describe how gravitational self-force theory, when combined with a method of multiscale expansions, provides an ideal framework for modelling these systems. The framework proceeds from first principles while simultaneously enabling rapid generation of waveforms on a timescale of milliseconds. I discuss the state of the art in this method: nonspinning, quasicircular waveforms at second perturbative order in the mass ratio. I present progress toward extending this second-order model to include spins and to include the final merger and ringdown. I also discuss the domain of validity of these models, focusing on their accuracy for mass ratios in the intermediate regime ~1:10 to 1:100.
Effective field theory provides a way of parameterizing strong-field deviations from General Relativity that might be observable in the gravitational waves emitted in a black hole merger. To perform numerical simulations of black hole mergers in such theories it is necessary that the equations be written in a form that admits a well-posed initial value formulation. In this talk, I will discuss recent progress in the initial value formulation of effective theories of gravity, as well as some open problems.
In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. One way to introduce confinement to the equations is by imposing a negative value for the cosmological constant and study the evolution of solutions with Anti-de Sitter asymptotics. In this talk, we will focus on two settings where turbulence emerges in the dynamics of the Einstein equations. First, in the case of small perturbations of Anti de-Sitter spacetime, we will establish the AdS instability conjecture for the spherically symmetric Einstein-scalar field system. We will, then, proceed to show how weak turbulence arises in a quasilinear toy model for the vacuum Einstein equations in the exterior of a Schwarzschild-AdS black hole. This is joint work with Christoph Kehle.
Penrose's proposal of smooth conformal compactification is not only of geometric elegance, it also has concrete, physical implications, such as the "peeling off" of gravitational radiation near infinity. One natural question to ask is then: Do physically relevant spacetimes admit a smooth conformal compactification? To answer this question, I will present a scattering construction of spacetimes describing the far region of $N$ infalling masses coming from the infinite past and following approximately Keplerian orbits, and prove that such spacetimes violate the "peeling property" both in the past and in the future. This violation is in principle measurable in the form of leading-order deviations from the usual late-time tails of gravitational radiation.
