Mathematical Aspects of General Relativity
from
Monday, 7 April 2008 (08:00)
to
Thursday, 17 April 2008 (18:00)
Monday, 7 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: Bohr Institute Cafeteria
11:00
Opening of workshop
Opening of workshop
11:00  11:15
Room: Aud. A
11:15
The interface between mathematics and astrophysics in the study of cosmic acceleration.

Alan Rendall
(
AEI Golm
)
The interface between mathematics and astrophysics in the study of cosmic acceleration.
Alan Rendall
(
AEI Golm
)
11:15  12:15
Room: Aud. A
For about the last ten years cosmic acceleration has been a subject of wide interest in cosmology. By now there are a number of interesting mathematical results in this area. A closer examination reveals that while the mathematical theorems are often of greater generality than what is considered in the astrophysical literature there are topics of astrophysical interest which fail to be addressed at all by the mathematical developments up to now. This talk will discuss possibilities of improving the interface between the two subjects in this context, concentrating on the case of the massive scalar field as a source for the Einstein equations. Other aspects of the question will be illuminated by consideration of a modification of the Einstein equations given by Cardassian models, following work of Nikolaus Berndt.
12:15
Lunch break
Lunch break
12:15  14:15
Room: 
14:15
An Introduction to the Geometry of Black Holes

Hubert Bray
(
Duke
)
An Introduction to the Geometry of Black Holes
Hubert Bray
(
Duke
)
14:15  15:15
Room: Aud. A
In this lecture we will study the Schwarzschild spacetime, which represents a nonrotating black hole in vacuum, from a variety of perspectives. After considering the more intuitive coordinate chart representations of Schwarzschild, we will then focus on Kruskal coordinates which is a global coordinate chart on the whole spacetime. From this introductory material, we will then transistion into a discussion about what the correct, or most geometric, statement of the Penrose Conjecture for black holes should be. Time permiting, we'll prove the Penrose Conjecture in a very special case, discuss white holes as compared to black holes, and define a new notion of horizon, called a generalized apparent horizon, which may be an important notion useful for proving the Penrose Conjecture.
Tuesday, 8 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: Bohr Institute Cafeteria
11:30
On the Penrose Conjecture for Arbitrary Slices of a Spacetime

Hubert Bray
(
Duke
)
On the Penrose Conjecture for Arbitrary Slices of a Spacetime
Hubert Bray
(
Duke
)
11:30  12:30
Room: Aud. A
The proofs of the Riemannian Penrose Conjecture by HuiskenIlmanen in 1997 (for one black hole) and by the speaker in 1999 (for any number of black holes) describe the geometric relationships between the total mass of a slice of a spacetime and the size and number of black holes in the slice, in the special case that the slice has zero second fundamental form in the spacetime. However, Penrose's original 1973 conjecture concerns any asymptotically flat, spacelike slice of a spacetime and, consequently, is still open in its most general form. In this talk, the speaker will describe a joint effort with Marcus Khuri to reduce the general case of the Penrose Conjecture to the known case using a generalization of Jang's equation (used to prove the general case of the positive mass theorem) and a new geometric identity, which we are calling the generalized SchoenYau identity, which is designed to recognize arbitrary spacelike slices of static spacetimes (like the Schwarzschild spacetime, which is the case of equality of the Penrose Conjecture), and hence is ideally suited for our purposes. We will then discuss three different systems of p.d.e.s whose solutions, when they exist, imply the Penrose Conjecture.
12:30
Lunch
Lunch
12:30  14:00
Room: 
14:45
Coffee and cake in the Mathematics Department
Coffee and cake in the Mathematics Department
14:45  15:15
Room: Math dept. 4th floor
15:15
Quantum Field Theory in Curved Spacetime

Robert Wald
(
Chicago
)
Quantum Field Theory in Curved Spacetime
Robert Wald
(
Chicago
)
15:15  16:15
Room: Mathematics Colloquium HC Ørsted Institute Aud 8
Wednesday, 9 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: Bohr Institute Cafeteria
12:00
Lunch
Lunch
12:00  14:00
Room: 
14:00
When can one extend the conformal metric through a spacetime singularity?

Poul Tod
(
Oxford
)
When can one extend the conformal metric through a spacetime singularity?
Poul Tod
(
Oxford
)
14:00  15:00
Room: Aud. A
One knows, for example by proving wellposedness 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.)
15:15
Asymptotic Stability of the fivedimensional Schwarzschild metric under biaxial perturbations

Gustav Holzegel
(
Cambridge
)
Asymptotic Stability of the fivedimensional Schwarzschild metric under biaxial perturbations
Gustav Holzegel
(
Cambridge
)
15:15  16:15
Room: Aud. A
Thursday, 10 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: Bohr Institute Cafeteria
12:00
Lunch
Lunch
12:00  14:00
Room: 
14:00
Gluing construction for asymptotically hyperbolic manifolds with constant scalar curvature

Erwann Delay
(
Avignon
)
Gluing construction for asymptotically hyperbolic manifolds with constant scalar curvature
Erwann Delay
(
Avignon
)
14:00  15:00
Room: Aud. A
We will see that the Corvino gluing technique can be adapted to the asymptotically hyperbolic context. More precisely, if a Riemannian metric with constant negative scalar curvature is asymptotic to a hyperbolic model at infinity, it can be glued on an "annulus" with a SchwarzschildAdS slice to a new constant scalar curvature metric. This produce initial data for the vacuum Einstein equation that are exactly SchwarzschildAdS slice outside of a compact set. This is joint work with Piotr Chrusciel.
15:15
The Motion of `Point Particles'

Robert Wald
(
Chicago
)
The Motion of `Point Particles'
Robert Wald
(
Chicago
)
15:15  16:15
Room: Aud. A
Friday, 11 April 2008
09:30
U(1) symmetric gravitationnal collapse in Iwasawa variable

Yvonne ChoquetBruhat
(
Paris Jussieu
)
U(1) symmetric gravitationnal collapse in Iwasawa variable
Yvonne ChoquetBruhat
(
Paris Jussieu
)
09:30  10:15
Room: Aud. A
10:20
Ricci flow as an algorithm to construct black hole metrics

Tobias Wiseman
(
Imperial College
)
Ricci flow as an algorithm to construct black hole metrics
Tobias Wiseman
(
Imperial College
)
10:20  11:05
Room: Aud. A
Ricci flow potentially provides a tool to allow explicit numerical construction of black hole metrics of interest in physics. Whilst in 4 dimensions stationary black holes of interest are known analytically, I will discuss how black holes in theories with extra dimensions (such as string theory) are generally not known analytically. Then a numerical construction of the metric is likely the only way to examine them. I will show that black holes (of interest) are actually unstable fixed points of Ricci flow and I will discuss how to use the flow in practice for this application.
11:05
Coffee break
Coffee break
11:05  11:35
Room: Aud. A
11:35
Stability of marginally trapped surfaces, and applications to black holes

Greg Galloway
(
Miami
)
Stability of marginally trapped surfaces, and applications to black holes
Greg Galloway
(
Miami
)
11:35  12:20
Room: Aud. A
A basic result in the theory of black holes is Hawking's theorem on the topology of black holes, which asserts that cross sections of the event horizon in (3+1)dimensional asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2spheres. Recent interest and developments in the study of higher dimensional black holes has drawn attention to the question of what are the allowable black hole topologies in higher dimensions. We have addressed this question in two recent papers, the first with Rick Schoen, resulting in a natural generalization of Hawking's theorem to higher dimensions. In this talk we discuss these works and some further related developments. The results we describe are based on properties of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces, and which form the focus of our talk.
12:20
Lunch break
Lunch break
12:20  14:15
Room: 
14:15
Open issues in the classification and characterization of higher dimensional black holes

Roberto Emparan
(
Barcelona
)
Open issues in the classification and characterization of higher dimensional black holes
Roberto Emparan
(
Barcelona
)
14:15  15:00
Room: Aud. A
After a review of our current knowledge about higherdimensional black holes, I will discuss some problems that arise due to novel types of black holes that do not arise in four dimensions. In particular, the absence of uniqueness, and the existence of horizons with the geometry of a product space seem to require new ideas and tools in order to achieve a better characterization and classification of black holes.
15:05
Rodstructure of stationary and axisymmetric solutions in General Relativity

Troels Harmark
(
Niels Bohr Institute
)
Rodstructure of stationary and axisymmetric solutions in General Relativity
Troels Harmark
(
Niels Bohr Institute
)
15:05  15:50
Room: Aud. A
We consider stationary and axisymmetric solutions in General Relativity, primarily in five dimensions. We motivative the introduction of the Rodstructure for any given solution and give examples of Rodstructures for various fivedimensional exact solutions of General Relativity. We consider the questions of uniqueness and existence of a solution given the Rodstructure. Finally we review briefly the uniqueness theorem of Hollands and Yazadjiev which proves that fivedimensional stationary and axisymmetric solutions are unique given the Rodstructure, and the asymptotic charges.
15:50
Coffee break
Coffee break
15:50  16:20
Room: Aud. A
16:20
Negative Point Mass Singularities (NPMS)

Hubert Bray
(
Duke
)
Negative Point Mass Singularities (NPMS)
Hubert Bray
(
Duke
)
16:20  17:05
Room: Aud. A
In this talk we will discuss a geometric inequality which is in the same spirit as the Positive Mass Theorem and the Penrose Inequality for black holes. Whereas the cases of equality of these first two theorems are respectively Minkowski space (which can be thought of as Schwarzschild with zero mass) and the Schwarzschild spacetime with positive mass, the case of equality for the inequality we will discuss is the Schwarzschild spacetime with negative mass. Physically speaking, when positive amounts of energy are concentrated as much as possible, black holes results. However, when negative amounts of energy are "concentrated" as much as possible, it is in fact possible to form point singularities in each spacelike slice (which form a timelike curve of singularities in the spacetime). As usual we will focus on maximal, spacelike slices of spacetimes as a first step. The assumption of nonnegative energy density on these slices implies that these Riemannian 3manifolds have nonnegative scalar curvature. However, we will allow these 3manifolds to have singularities which contribute negatively to the total mass. The standard example is the negative Schwarzschild metric on R^3 minus a ball of radius m/2, (1  m/2r)^4 \delta_{ij}. This metric (which has total mass m) has zero scalar curvature everywhere but has a singularity at r = m/2. We will propose a definition for the mass of a singularity, and prove a sharp lower bound on the ADM mass in terms of the masses of the singularities in the 3manifold, modulo an interesting geometric conjecture.
Saturday, 12 April 2008
09:30
Rate of change of widths under flows

Tobias Colding
Rate of change of widths under flows
Tobias Colding
09:30  10:15
Room: Aud. A
10:20
Classical Effective Field Theory and NonRelativistic Gravitation

Barak Kol
(
Jerusalem
)
Classical Effective Field Theory and NonRelativistic Gravitation
Barak Kol
(
Jerusalem
)
10:20  11:05
Room: Aud. A
I shall discuss an improvement to the (Classical) Effective Field Theory approach to the nonrelativistic or PostNewtonian approximation of General Relativity. The "potential metric field" is decomposed through a temporal KaluzaKlein ansatz into three NRGfields: a scalar identified with the Newtonian potential, a 3vector corresponding to the gravitomagnetic vector potential and a 3tensor. The derivation of the EinsteinInfeldHoffmann Lagrangian simplifies such that each term corresponds to a single Feynman diagram providing a clear physical interpretation. Spin interactions are associated with the gravitomagnetic field. Leading correction diagrams corresponding to the 3PN correction to the spinspin interaction and the 2.5PN correction to the spinorbit interaction will be presented.
11:05
Coffee break
Coffee break
11:05  11:35
Room: Aud. A
11:35
Oneparameter families of conformally related asymptotically flat, static vacuum data

Helmut Friedrich
(
AEI Golm
)
Oneparameter families of conformally related asymptotically flat, static vacuum data
Helmut Friedrich
(
AEI Golm
)
11:35  12:20
Room: Aud. A
We give a complete description of the asymptotically flat, conformally nonflat, static vacuum data which admit conformal mappings onto other such data which extend smoothly to spacelike infinity. These data form a 3parameter family which decomposes into 1parameter families of data which are conformal to each other
12:25
Inverse scattering in gravity

Henriette Elvang
(
MIT
)
Inverse scattering in gravity
Henriette Elvang
(
MIT
)
12:25  13:10
Room: Aud. A
13:10
Lunch
Lunch
13:10  15:10
Room: 
19:00
Conference dinner
Conference dinner
19:00  22:30
Room: Restaurant Zeleste Store Strandstræde 6 DK 1255 København K, www.zeleste.dk
Sunday, 13 April 2008
08:00
08:00  18:00
Room: NONE
Monday, 14 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: NBIA Coffee room
11:00
Regularity conditions at spatial infinity revisited

Juan A Valiente
(
Queen Mary, London
)
Regularity conditions at spatial infinity revisited
Juan A Valiente
(
Queen Mary, London
)
11:00  12:00
Room: Aud. A
H. Friedrich has shown that if one considers a time symmetric initial data set for the Einstein vacuum equations admitting an analytic compactification at infinity, then necessary conditions for the solutions to the transport system implied by the conformal Einstein equations at the cylinder at spatial infinity to extend smoothly to the critical sets where null infinity touches spatial infinity is that the CottonBach tensor of the conformal metric, and its tracefree symmetrised higher order derivatives vanish at spatial infinity. In this talk the generalisation of this regularity condition to data with nonvanishing second fundamental forms is examined. It is discussed how these regularity conditions can be phrased in terms of the vanishing at infinity of a pair of tensors and their higher order symmetrised derivatives. It is shown that these "generalised regularity conditions" are only a restriction on the freely specifiable data. The relation of these "generalised regularity conditions" to stationary data is considered. Finally, it is also discussed how these regularity conditions can be used to construct purely radiative data at "past null infinity".
12:00
Lunch
Lunch
12:00  14:00
Room: 
14:00
Future stability of the Einsteinnonlinear scalar field system, power law expansion

Hans Ringstrom
(
Stockholm (KTH)
)
Future stability of the Einsteinnonlinear scalar field system, power law expansion
Hans Ringstrom
(
Stockholm (KTH)
)
14:00  15:00
Room: Aud. A
In the case of Einstein's equations coupled to a nonlinear 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 nonlinear stability of such models. The results generalize those of Mark Heinzle and Alan Rendall obtained using different methods.
Tuesday, 15 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: NBIA Coffee room
11:00
The trapped region

Lars Andersson
(
AEI Golm
)
The trapped region
Lars Andersson
(
AEI Golm
)
11:00  12:00
Room: Aud. A
I will discuss recent results on existence and regularity of marginal surfaces, blowup of Jang's equation, and regularity of the trapped region. This is joint work with Jan Metzger.
12:00
Lunch
Lunch
12:00  13:45
Room: 
13:45
The nonlinear stability problem for black hole spacetimes

Mihalis Dafermos
(
Cambridge
)
The nonlinear stability problem for black hole spacetimes
Mihalis Dafermos
(
Cambridge
)
13:45  14:45
Room: Aud. A
Wednesday, 16 April 2008
12:00
Lunch
Lunch
12:00  14:00
Room: 
14:00
A uniqueness theorem for the Kerr metric

Piotr Chrusciel
A uniqueness theorem for the Kerr metric
Piotr Chrusciel
14:00  15:00
Room: Aud. A
15:15
The nonlinear stability problem for black hole spacetimes

Mihalis Dafermos
(
Cambridge
)
The nonlinear stability problem for black hole spacetimes
Mihalis Dafermos
(
Cambridge
)
15:15  16:15
Room: Aud. A
Thursday, 17 April 2008
10:00
Coffee
Coffee
10:00  10:30
Room: NBIA coffee room
11:00
Rigidly Rotating Bodies in General Relativity

Robert Beig
(
Vienna
)
Rigidly Rotating Bodies in General Relativity
Robert Beig
(
Vienna
)
11:00  12:00
Room: Aud. A
The only rigorous result known so far on the existence of isolated bodies in GR in rigid rotation are the ones by Heilig of 1995 on perfect fluids. 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.
12:30
Lunch and goodbye
Lunch and goodbye
12:30  14:30
Room: 