12:00
Forthcoming events in this series
12:00
Hidden symmetries and higher-dimensional rotating black holes
Abstract
The 4D rotating black hole described by the Kerr geometry possesses many of what was called by Chandrasekhar "miraculous" properties. Most of them can be related to the existence of a fundamental hidden symmetry called the principal conformal Killing-Yano (PCKY) tensor. In my talk I shall demonstrate that, in this context, four dimensions are not exceptional and that the (spherical horizon topology) higher-dimensional rotating black holes are very similar to their four-dimensional cousins. Namely, I shall present the most general spacetime admitting the PCKY tensor and show that is possesses the following properties: 1) it is of the algebraic type D, 2) it allows a separation of variables for the Hamilton-Jacobi, Klein-Gordon, Dirac, gravitational, and stationary string equations, 3) the geodesic motion in such a spacetime is completely integrable, 4) when the Einstein equations with the cosmological constant are imposed the metric becomes the Kerr-NUT-(A)dS spacetime. Some of these properties remain valid even when one includes the electromagnetic field.
12:00
From the geometry of spacetime to the geometry of numbers
Abstract
One of the major open challenges in general relativity is the classification of black hole solutions in higher dimensional theories. I will explain a recent result in this direction in the context of Kaluza-Klein spacetimes admitting a sufficient number N of commuting U(1)-symmetries. It turns out that the black holes in such a theory are characterized by the usual asymptotic charges, together with certain combinatorical data and moduli. The combinatorial data characterize the nature of the U(1)^N-action, and its analysis is closely related to properties of integer lattices and questions in the area of geometric number theory. I will also explain recent results on extremal black holes which show that such objects display remarkable ``symmetry enhancement'' properties
12:00
A black hole uniqueness theorem.
Abstract
Klainerman on the black hole uniqueness problem. A classical result of
Hawking (building on earlier work of Carter and Robinson) asserts that any
vacuum, stationary black hole exterior region must be isometric to the
Kerr exterior, under the restrictive assumption that the space-time metric
should be analytic in the entire exterior region.
We prove that Hawking's theorem remains valid without the assumption of
analyticity, for black hole exteriors which are apriori assumed to be "close"
to the Kerr exterior solution in a very precise sense. Our method of proof
relies on certain geometric Carleman-type estimates for the wave operator.
12:00
Hidden symmetries and decay for the wave equation outside a Kerr black hole
Abstract
This is joint work with Lars Andersson.
12:00
A uniqueness theorem for charged rotating black holes in five- dimensional minimal supergravity
Abstract
We show that a charged rotating black hole in five-dimensional Einstein-Maxwell-Chern-Simons theory is uniquely characterized by the mass, charge, and two independent angular momenta, under the assumptions of the existence of two commuting axial isometries and spherical topology of horizon cross-sections. Therefore, such a black hole must be described by the Chong-Cveti\v{c}-L\"u-Pope metric.
12:00
Asymptotic Quasinormal Frequencies for d-Dimensional Black Holes
Abstract
I will explain what quasinormal modes are and how to obtain asymptotic formulae for the quasinormal frequencies of static, spherically symmetric black hole spacetimes in d dimensions in the limit of very large imaginary part.
12:00
Boundedness and decay of scalar waves on Kerr and more general black holes
Abstract
I will review our current mathematical understanding of waves on black hole backgrounds, starting with the classical boundedness theorem of Kay and Wald on Schwarzschild space-time and ending with recent boundedness and decay theorems on a wider class of black hole space-times.
12:00
Structure of singularities of spacetimes with toroidal or hyperbolic symmetry
Abstract
I will present recent results concerning the study of the global Cauchy problem in general relativity under symmetry assumptions.
More specifically, I will be focusing on the structure of singularities and the uniqueness in the large for solutions of the Einstein equations, the so-called strong cosmic censorship, under the assumption that the initial data is given on some compact manifold with prescribed symmetry.
In particular, I will present some results which concerned the asymptotic behaviour of the area of the orbits of symmetry, a quantity which plays in important role for the study of these solutions. From the point of view of PDE, this corresponds to a global existence theorem for a system of non-linear 1+1 wave equations.
On Mason's theorem: algebraically special metrics cannot be asymptotically simple
On the Extraction of Physical Content from Asymptotically Flat Space-times Metrics
Abstract
A major issue in general relativity, from its earliest days to the
present, is how to extract physical information from any solution or
class of solutions to the Einstein equations. Though certain
information can be obtained for arbitrary solutions, e.g., via geodesic
deviation, in general, because of the coordinate freedom, it is often
hard or impossible to do. Most of the time information is found from
special conditions, e.g., degenerate principle null vectors, weak
fields close to Minkowski space (using coordinates close to Minkowski
coordinates) or from solutions that have symmetries or approximate
symmetries. In the present work we will be concerned with
asymptotically flat space times where the approximate symmetry is the
Bondi-Metzner-Sachs (BMS) group. For these spaces the Bondi
four-momentum vector and its evolution, found from the Weyl tensor at
infinity, describes the total energy-momentum of the interior source
and the energy-momentum radiated. By generalizing certain structures
from algebraically special metrics, by generalizing the Kerr and the
charged-Kerr metric and finally by defining (at null infinity) the
complex center of mass (the real center of mass plus 'i' times the
angular momentum) with its transformation properties, a large variety
of physical identifications can be made. These include an auxiliary
Minkowski space viewed from infinity, kinematic meaning to the Bondi
momentum, dynamical equations of motion for the center of mass, a
geometrically defined spin angular momentum and a conservation law with
flux for total angular momentum.
12:00
Relativistic Figures of Equilibrium
Abstract
In this talk I shall review analytical and numerical results on equilibrium configurations of rotating fluid bodies within Einstein's theory of gravitation.
12:00
Asymptotic Stability of the five-dimensional Schwarzschild metric against biaxial perturbations
Abstract
I will start by reviewing the current status of the stability
problem for black holes in general relativity. In the second part of the
talk I will focus on a particular (symmetry) class of five-dimensional
dynamical black holes recently introduced by Bizon et al as a model to
study gravitational collapse in vacuum. In this context I state a recent
result establishing the asymptotic stability of the five dimensional
Schwarzschild metric with respect to vacuum perturbations in the given
class.
12:00
Existence of rough solutions to the Einstein constraint equations without CMC or near-CMC conditions
Abstract
> There is currently tremendous interest in geometric PDE, due in part
> to the geometric flow program used successfully to attack the Poincare
> and Geometrization Conjectures. Geometric PDE also play a primary
> role in general relativity, where the (constrained) Einstein evolution
> equations describe the propagation of gravitational waves generated by
> collisions of massive objects such as black holes.
> The need to validate this geometric PDE model of gravity has led to
> the recent construction of (very expensive) gravitational wave
> detectors, such as the NSF-funded LIGO project. In this lecture, we
> consider the non-dynamical subset of the Einstein equations called the
> Einstein constraints; this coupled nonlinear elliptic system must be
> solved numerically to produce initial data for gravitational wave
> simulations, and to enforce the constraints during dynamical
> simulations, as needed for LIGO and other gravitational wave modeling efforts.
>
> The Einstein constraint equations have been studied intensively for
> half a century; our focus in this lecture is on a thirty-year-old open
> question involving existence of solutions to the constraint equations
> on space-like hyper-surfaces with arbitrarily prescribed mean
> extrinsic curvature. All known existence results have involved
> assuming either constant (CMC) or nearly-constant (near-CMC) mean
> extrinsic curvature.
> After giving a survey of known CMC and near-CMC results through 2007,
> we outline a new topological fixed-point framework that is
> fundamentally free of both CMC and near-CMC conditions, resting on the
> construction of "global barriers" for the Hamiltonian constraint. We
> then present such a barrier construction for case of closed manifolds
> with positive Yamabe metrics, giving the first known existence results
> for arbitrarily prescribed mean extrinsic curvature. Our results are
> developed in the setting of a ``weak'' background metric, which
> requires building up a set of preliminary results on general Sobolev
> classes and elliptic operators on manifold with weak metrics.
> However, this allows us to recover the recent ``rough'' CMC existence
> results of Choquet-Bruhat
> (2004) and of Maxwell (2004-2006) as two distinct limiting cases of
> our non-CMC results. Our non-CMC results also extend to other cases
> such as compact manifolds with boundary.
>
> Time permitting, we also outline some new abstract approximation
> theory results using the weak solution theory framework for the
> constraints; an application of which gives a convergence proof for
> adaptive finite element methods applied to the Hamiltonian constraint.
12:00
Nonlinear spherical sound waves at the surface of a perfect fluid star
Abstract
Current numerical relativity codes model neutron star matter as a perfect fluid, with an unphysical "atmosphere" surrounding the star to avoid the breakdown of the equations at the fluid-vacuum interface at the surface of the star. To design numerical methods that do not require an unphysical atmosphere, it is useful to know what a generic sound wave looks near the surface. After a review of relevant mathematical methods, I will present results for low (finite) amplitude waves that remain smooth and, perhaps, for high amplitude waves that form a shock.
11:00
Future stability of the Einstein-non-linear scalar field system, power law expansion
Abstract
In the case of Einstein's equations coupled to a non-linear scalar field with a suitable exponential potential, there are solutions for which the expansion is accelerated and of power law type. In the talk I will discuss the future global non-linear stability of such models. The results generalize those of Mark Heinzle and Alan Rendall obtained using different methods.
11:00
Stationary rotating bodies in general relativity
Abstract
We outline a method to solve the stationary Einstein equations with source a body in rigid rotation consisting of elastic matter.
This is work in progress by R.B., B.G.Schmidt, and L.Andersson
11:00
On complete positive scalar curvature metrics (time symmetric initial data with positive cosmological constant)
Abstract
: I will review various constructions and properties of complete constant scalar curvature metrics. I will emphasize the role played by the so called "Fowler's solutions" which give rise to metrics with cylindrical ends. I will also draw the parallel between these constructions and similar constructions which surprisingly (or not) appear in a different context : constant mean curvature surfaces and more recently the Allen-Cahn equation and some equation in the biological theory of pattern formation.
11:00
When can one extend the conformal metric through a space-time singularity ?
Abstract
One knows, for example by proving well-posedness for an initial value problem with data at the singularity, that there exist many cosmological solutions of the Einstein equations with an initial curvature singularity but for which the conformal metric can be extended through the singularity. Here we consider a converse, a local extension problem for the conformal structure: given an incomplete causal curve terminating at a curvature singularity, when can one extend the conformal structure to a set containing a neighbourhood of a final segment of the curve?
We obtain necessary and sufficient conditions based on boundedness of tractor curvature components. (Based on work with Christian Luebbe: arXiv:0710.5552, arXiv:0710.5723.)
11:00
Quasi-local energy-momentum and flux for black holes
Abstract
In this talk I will look at a definition of the energy-momentum for the dynamical horizon of a black hole. The talk will begin by examining the role of a special class of observers at null infinity determined by Bramson's concept of frame alignment. It is shown how this is given in terms of asymptotically constant spinor fields and how this framework may be used together with the Nester-Witten two form to give a definition of the Bondi mass at null infinity.
After reviewing Ashtekar's concept of an isolated horizon we will look at the propagation of spinor fields and show how to introduce spinor fields for the horizon which play the role of the asymptotically constant spinor fields at null infinity, giving a concept of alignment of frames on the horizon. It turns out that the equations satisfied by these spinor fields give precisely the Dougan-Mason holomorphic condition on the cross sections of the horizon, together with a simple propagation equation along the generators. When combined with the Nester-Witten 2-form these equations give a quasi-local definition of the mass and momentum of the black hole, as well as a formula for the flux across the horizon. These ideas are then generalised to the case of a dynamical horizon and the results compared to those obtained by Ashtekar as well as to the known answers for a number of exact solutions.
11:00
Static vacuum data and their conformal classes
Abstract
Static vacuum data and their conformal classes play an important role in the discussion of the smoothness of gravitational fields at null infinity. We study the question under which conditions such data admit non-trivial conformal rescalings which lead again to such data. Some of the restrictions implied by this requirement are discussed and it is shown that there exists a 3-parameter family of static vacuum data which are not conformally flat and which admit non-trivial rescalings.
11:00
Towards a proof of a rigidity conjecture for asymptotically flat spacetimes
Abstract
I will discuss ongoing work to provide a proof for the following
conjecture: if the development of a time symmetric, conformally flat
initial data set admits a smooth null infinity, then the initial data
is Schwarzschildean in a neighbourhood of infinity. The strategy
to construct a proof consists in a detailed analysis of a
certain type of expansions that can be obtained using H. Friedrich's
"cylinder at infinity" formalism. I will also discuss a toy model for
the analysis of the Maxwell field near the
spatial infinity of the Schwarzschild spacetime