Workshop: Advances in General Relativity
Abstracts
Thomas Bäckdahl (Golm)
Title: Symmetries and conservation laws for linearized gravity
Abstract: In this talk I will review recent results on the structure of the linearized gravity equations. The results apply to yield new symmetry operators and conservation laws for linearized gravity on the Kerr spacetime, as well as new hyperbolic systems governing the linearized gravitational field.
Beverly Berger
Title: When black holes collide: a new window on the universe
Abstract: LIGO’s detection of gravitational waves from a binary black hole merger inaugurates a completely new mode of observational astronomy and represents the culmination of a quest lasting half a century. After a brief review of gravitational waves in general relativity, I will discuss the detection itself. How do the LIGO instruments work? How do we know the signal was caused by a binary black hole merger? What does this detection tell us about binary black holes? Then I will focus on how this moment came to pass. The detection required many ingredients to be in place including (1) developments in theoretical relativity to allow proof that gravitational waves were not coordinate artifacts; (2) a bold vision to recognize that gravitational wave detection was not impossible; (3) technological developments of novel vacuum systems, lasers, optical coatings, active seismic isolation, etc.; (4) the successful conclusion of a 35 year effort to simulate binary black holes on the computer; (5) development of sophisticated, new data analysis methods to tease a waveform from noisy data; (5) the growth of the field of gravitational wave science from a handful of practitioners to the more than 1000 authors on the detection paper; and finally (6) the (nearly) unwavering support of the National Science Foundation. The first detection was followed by a second one in this first "science run" and soon another science run will begin. I will end with discussion of the future — more binary black holes, other sources of gravitational waves and what we might learn, instrument upgrades, new facilities — and other ways to detect gravitational waves — from space and from monitoring millisecond pulsars.
Sebastiano Bernuzzi (Parma U & INFN )
Title: Modeling neutron star binaries and gravitational waves with numerical relativity
Abstract: Neutron stars in binary systems are among the strongest sources of gravitational waves and among the main targets for ground-based gravitational-wave interferometers Advanced LIGO and Virgo. The observation of these events in the gravitational-wave window can provide us with unique information on neutron stars' masses, radii, and spins, including the possibility to set strong constraints on the unknown equation of state of matter at supranuclear densities. A crucial and necessary step for the development of gravitational-wave astronomy is the precise knowledge of the dynamics of the sources and of the emitted waveforms. I will talk about recent developments on the modeling of gravitational waves from neutron star mergers using numerical simulations in general relativity.
Lydia Bieri (Michigan)
Title: Gravitational Radiation in Cosmological Spacetimes
Abstract: Some of the most interesting solutions of the Einstein equations are spacetimes exhibiting gravitational radiation. A major breakthrough of General Relativity happened in 2015 with LIGO's first detection of gravitational waves. So far, most studies have been devoted to asymptotically flat systems, which applies perfectly to gravitational wave sources whose distance to the detector is small compared to the Hubble radius. However, some of the most powerful sources are at cosmological distances, and we have to study what happens in an expanding universe. In this talk, we investigate the geometric-analytic properties of various spacetimes with gravitational radiation, in particular of cosmological spacetimes. This is joint work with D. Garfinkle and N. Yunes.
Piotr Bizoń (Cracow)
Title: From AdS to BEC
Abstract: The long-time behavior of nonlinear dispersive waves subject to spatial confinement can be very rich and complex because, in contrast to unbounded domains, waves cannot disperse to infinity and keep self-interacting for all times. If, in addition, the linear spectrum around the ground state is fully resonant, then the nonlinearity can produce significant effects for arbitrarily small perturbations. The weak field dynamics of such systems can be approximated by solutions of the corresponding infinite-dimensional time-averaged Hamiltonian systems, which govern resonant interactions between the modes.
A major mathematical challenge in this context is to describe the energy transfer between the modes. I will discuss this problem for three different models of confinement: the Einstein equation with negative cosmological constant describing weakly turbulent behavior of small perturbations of the anti-de Sitter (AdS) spacetime, a nonlinear wave equation on a compact manifold (like the cubic wave equation on the 3-sphere), and the nonlinear Schroedinger equation with a trapping potential describing the dynamics of Bose-Einstein condensates (BEC). Some intriguing parallels between these systems will be emphasized.
Carla Cederbaum (Tübingen)
Title: Rigidity properties of the Schwarzschild manifold in all dimensions
Abstract: We generalize the static black hole uniqueness theorem by Bunting and Masood-ul-Alam to higher dimensions, beyond staticity and vacuum, and to more general boundary conditions than static horizons. Our result in particular implies static vacuum black hole uniqueness in n+1 spacetime dimensions – reproving a result by Gibbons, Ida, and Shiromizu – and extends it beyond the static vacuum setting. It also generalizes static vacuum photon sphere uniqueness – proved by Galloway and the speaker – from 3+1 to n+1 spacetime dimensions and beyond the static vacuum setting. A key ingredient of the proof is the rigidity statement of the positive mass theorem for Riemannian manifolds.
Otis Chodosh (Cambridge)
Title: Global uniqueness for CMC foliations of asymptotically flat 3-manifolds
Abstract: I'll describe recent work with M. Eichmair concerning the global uniqueness of large stable CMC surfaces in asymptotically flat 3-manifolds.
Mihalis Dafermos (Cambridge and Princeton)
Title: Boundedness and polynomial decay for the Teukolsky equation on Kerr spacetimes
Abstract: This is joint work with Igor Rodnianski and Gustav Holzegel. As part of our previous proof of linear stability of Schwarzschild, we showed both boundedness and polynomial decay estimates for solutions of the spin ±2 Teukolsky equation by exploiting a physical space transformation to solutions of the Regge-Wheeler equation. We show how this procedure generalises to yield similar results for the Teukolsky equation on Kerr.
Cécile Huneau (Grenoble)
Title: High frequency back reaction for the Einstein equations
Abstract: It has been observed by physicists (Isaacson, Burnett, Green-Wald) that metric perturbations of a background solution, which are small amplitude but with high frequency, yield at the limit to a non trivial contribution which corresponds to the presence of an energy impulsion tensor in the equation for the background metric. This non trivial contribution is of due to the nonlinearities in Einstein equations, which involve products of derivatives of the metric. It has been conjectured by Burnett that the only tensors which can be obtained this way are massless Vlasov, and it has been proved by Green and Wald that the limit tensor must be traceless and satisfy the dominant energy condition. The known exemples of this phenomena are constructed under symmetry reductions which involve two Killing fields and lead to an energy impulsion tensor which consists in at most two dust fields propagating in null directions. In this talk, I will explain our construction, under a symmetry reduction involving one Killing field, which leads to an energy impulsion tensor consisting in N dust fields propagating in arbitrary null directions. This is a joint work with Jonathan Luk (Stanford).
Christos Mantoulidis (Stanford)
Title: Nonnegative scalar curvature fill-ins and quasi-local mass
Abstract: In this talk we'll introduce the procedure of "filling in" prescribed boundary data (surface, metric) with compact domains of nonnegative scalar curvature. We'll explain how it relates to various notions of quasilocal mass in the time-symmetric initial data set approach to general relativity, and discuss implications on the relationships between the Brown-York mass and the Hawking mass of compact regions, and the boundary capacity of an asymptotically flat initial data set.
Lorenzo Mazzieri (Trento)
Title: On the mass of static vacuum Einstein metrics with positive cosmological constant
Abstract: We introduce and discuss a notion of mass for static vacuum Einstein metrics with positive cosmological constant. In this context, we provide a positive mass statement as well as sharp area bounds for both cosmological horizons and black hole type horizons. In the first case, these area bounds represent the natural extension of a well known result by Boucher, Gibbons and Horowitz, whereas for black hole type horizons they can be seen as the analogue of the celebrated Riemannian Penrose Inequality. As an application, we deduce a uniqueness statement for the Schwarzschild-de Sitter static black hole.
Ettore Minguzzi (Firenze)
Title: Causality theory for cone structures
Abstract: I introduce the notion of cone structure and present some results on their causality properties. The theory is applied to the problem of characterizing Lorentzian manifolds isometrically embeddable in Minkowski spacetime.
Georgios Moschidis (Princeton)
Title: A proof of the instability of AdS spacetime for the Einstein–null dust system
Abstract: The AdS instability conjecture is a conjecture related to the initial-boundary value problem for the vacuum Einstein equations with a negative cosmological constant in general relativity. It states that generic, arbitrarily small perturbations to the initial data of the AdS spacetime, under evolution by the vacuum Einstein equations with reflecting boundary conditions on conformal infinity, lead to the formation of black holes. This conjecture was introduced in 2006 by Dafermos and Holzegel and, since then, it has attracted a vast amount of numerical and heuristic works by several authors, starting from the seminal work of Bizon and Rostworowski in 2011. These works have been focused mainly on the simpler setting of the spherically symmetric Einstein--scalar field system.
In this talk, we will provide the first rigorous proof of the AdS instability conjecture in the simplest possible setting, namely for the spherically symmetric Einstein--massless Vlasov system, in the case when the Vlasov field is moreover supported only on radial geodesics. This system is equivalent to the Einstein--null dust system, allowing for both ingoing and outgoing dust. In order to overcome the "trivial" break down occuring once the null dust reaches the centre r=0, we will study the evolution of the system in the exterior of an inner mirror with positive radius r0 and prove the conjecture in this setting. After presenting our proof, we will briefly explain how the main ideas can be extended to more general matter fields.
Roger Penrose (Oxford)
Title: The equations of conformal cyclic cosmology: implications as to the nature of dark matter, its decay, and possible observational tests
Abstract: In the cosmological scheme of conformal cyclic cosmology (CCC), the equations governing the crossover form each aeon to the next demand the creation of a dominant new scalar material that is postulated to be dark matter. In order that this material does not build up from aeon to aeon, it is taken to decay away completely over the history of the aeon. The dark matter particles (erebons) may be expected to behave almost as classical particles, though with bosonic properties, being probably of around a Planck mass, and interacting only gravitationally. Their decay would be to gravitational signals, and responsible for the (~scale invariant) temperature fluctuations in the CMB of the succeeding aeon. In our own aeon, erebon decay might be detectable by gravitational wave detectors.
Harvey Reall (Cambridge)
Title: On the local well-posedness of Lovelock and Horndeski theories
Abstract: Lovelock theories of gravity are the most general diffeomorphism covariant theories of a metric tensor with second order equations of motion. Horndeski theories are the most general four-dimensional diffeomorphism covariant theories of a metric tensor and scalar field with second order equations of motion. I will discuss local well-posedness of the initial value problem for these theories. A necessary condition for local well-posedness is strong hyperbolicity of the equations of motion. Even weak hyperbolicity can fail for strong fields so we restrict to weak fields. We study Lovelock theories in harmonic gauge. We show that the equation of motion is always weakly hyperbolic for weak fields but, in a generic weak-field background, it is not strongly hyperbolic. For Horndeski theories, we prove that, for weak fields, the equation of motion is always weakly hyperbolic in any generalized harmonic gauge. For some Horndeski theories there exists a generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a weak-field background. This includes "k-essence" like theories. However, for more general Horndeski theories, there is no generalized harmonic gauge for which the equation of motion is strongly hyperbolic in a generic weak-field background. Our results show that the standard method used to establish local well-posedness of the Einstein equation does not extend to Lovelock or general Horndeski theories. This raises the possibility that these theories may not admit a well-posed initial value problem even for weak fields.
Anna Sakovich (Uppsala)
Title: On geometric foliations and center of mass for isolated systems in general relativity
Abstract: While the concepts of mass and linear momentum are by now well-established in mathematical general relativity, it is not entirely clear whether the existing notions of center of mass are the ultimate ones. In this talk we introduce a novel geometric foliation of an initial data set and discuss its connections to and\ advantages over the existing approaches to defining the center of mass of an isolated system. This is joint work with Carla Cederbaum and Julien Cortier.
Jan Sbierski (Cambridge)
Title: The wave equation in the interior of black holes
Yakov Shlapentokh-Rothman (Princeton)
Title: The asymptotically self-similar regime for the Einstein vacuum equations
Abstract: We will dynamically construct singular solutions to the Einstein vacuum equations which are asymptotically self-similar in that successive rescalings around the singularity converge to a self-similar solution. Connections both to Chrisotodulou’s bounded variation solutions of the spherically symmetric Einstein-scalar field system and to the ambient metric construction of Fefferman and Graham will be elaborated on. This is joint work with Igor Rodnianski
László Szabados (Budapest)
Title: Gravity, as a classical regulator for the Higgs field, and the origin of rest masses and electric charge
Abstract: The classical Einstein-Standard Model system with conformally invariant coupling of the Higgs field to gravity is investigated. We show that the energy-momentum tensor may have two singularities: In cosmological spacetimes the usual Big Bang type singularity with diverging matter field variables, and a second, less violent one (`Small Bang'), in which it is only the geometry that is singular but the matter field variables remain finite. The latter provides a finite, universal upper bound for the pointwise norm of the Higgs field in terms of Newton's gravitational constant.
As a consequence of this structure of the energy-momentum tensor, we also show that, in the presence of Friedman-Robertson-Walker or Kantowski-Sachs symmetries the energy density can have finite local minimum only if the transitivity hypersurfaces of the spacetime symmetries are locally hyperboloidal and their mean curvature is less than a finite critical value. In particular, in the very early era of an expanding universe or in a nearly spherically symmetric black hole near the central singularity, the Higgs sector does not have any instantaneous (symmetric or symmetry breaking) vacuum state, and hence the (zero or non-zero) rest mass of the matter fields is not defined. For smaller mean curvature instantaneous symmetry breaking vacuum states emerge, yielding non-zero rest mass and electric charge for some of the gauge and spinor fields via the Brout-Englert-Higgs mechanism. These rest masses are decreasing with decreasing mean curvature, but the charge remains constant. It is also shown that globally defined instantaneous symmetry breaking vacuum states do not exist at all in the k=1 cosmological model and inside spherically symmetric black holes.
Jérémie Szeftel (Paris)
Title: On the stability of black holes
András Vasy (Stanford)
Title: The stability of Kerr-de Sitter black holes
Abstract: In this lecture, based on joint work with Peter Hintz, I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.
I will first discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In the second part of the talk I will discuss analytic aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.