Mathematical Aspects of General Relativity

List of talks

18. The interface between mathematics and astrophysics in the study of cosmic acceleration.

Alan Rendall (AEI Golm)

4/7/08, 11:15 AM

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...

16. An Introduction to the Geometry of Black Holes

Hubert Bray (Duke)

4/7/08, 2:15 PM

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...

28. On the Penrose Conjecture for Arbitrary Slices of a Spacetime

Hubert Bray (Duke)

4/8/08, 11:30 AM

The proofs of the Riemannian Penrose Conjecture by Huisken-Ilmanen 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,...

17. Quantum Field Theory in Curved Spacetime

Robert Wald (Chicago)

4/8/08, 3:15 PM

13. When can one extend the conformal metric through a space-time singularity?

Poul Tod (Oxford)

4/9/08, 2:00 PM

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...

19. Asymptotic Stability of the five-dimensional Schwarzschild metric under biaxial perturbations

Gustav Holzegel (Cambridge)

4/9/08, 3:15 PM

4. Gluing construction for asymptotically hyperbolic manifolds with constant scalar curvature

Erwann Delay (Avignon)

4/10/08, 2:00 PM

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 Schwarzschild-AdS slice to a new constant scalar curvature metric. This produce initial data for the vacuum...

26. The Motion of `Point Particles'

Robert Wald (Chicago)

4/10/08, 3:15 PM

11. U(1) symmetric gravitationnal collapse in Iwasawa variable

Yvonne Choquet-Bruhat (Paris Jussieu)

4/11/08, 9:30 AM

3. Ricci flow as an algorithm to construct black hole metrics

Tobias Wiseman (Imperial College)

4/11/08, 10:20 AM

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...

22. Stability of marginally trapped surfaces, and applications to black holes

Greg Galloway (Miami)

4/11/08, 11:35 AM

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 2-spheres. Recent interest and developments in the study of higher dimensional black holes has drawn...

5. Open issues in the classification and characterization of higher- dimensional black holes

Roberto Emparan (Barcelona)

4/11/08, 2:15 PM

After a review of our current knowledge about higher-dimensional 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...

8. Rod-structure of stationary and axisymmetric solutions in General Relativity

Troels Harmark (Niels Bohr Institute)

4/11/08, 3:05 PM

We consider stationary and axisymmetric solutions in General Relativity, primarily in five dimensions. We motivative the introduction of the Rod-structure for any given solution and give examples of Rod-structures for various five-dimensional exact solutions of General Relativity. We consider the questions of uniqueness and existence of a solution given the Rod-structure. Finally we...

14. Negative Point Mass Singularities (NPMS)

Hubert Bray (Duke)

4/11/08, 4:20 PM

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...

29. Rate of change of widths under flows

Tobias Colding

4/12/08, 9:30 AM

10. Classical Effective Field Theory and Non-Relativistic Gravitation

Barak Kol (Jerusalem)

4/12/08, 10:20 AM

I shall discuss an improvement to the (Classical) Effective Field Theory approach to the non-relativistic or Post-Newtonian approximation of General Relativity. The "potential metric field" is decomposed through a temporal Kaluza-Klein ansatz into three NRG-fields: a scalar identified with the Newtonian potential, a 3-vector corresponding to the gravito-magnetic vector potential and...

2. One-parameter families of conformally related asymptotically flat, static vacuum data

Helmut Friedrich (AEI Golm)

4/12/08, 11:35 AM

We give a complete description of the asymptotically flat, conformally non-flat, static vacuum data which admit conformal mappings onto other such data which extend smoothly to space-like infinity. These data form a 3-parameter family which decomposes into 1-parameter families of data which are conformal to each other

12. Inverse scattering in gravity

Henriette Elvang (MIT)

4/12/08, 12:25 PM

21. Regularity conditions at spatial infinity revisited

Juan A Valiente (Queen Mary, London)

4/14/08, 11:00 AM

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...

15. Future stability of the Einstein-non-linear scalar field system, power law expansion

Hans Ringstrom (Stockholm (KTH))

4/14/08, 2:00 PM

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.

9. The trapped region

Lars Andersson (AEI Golm)

4/15/08, 11:00 AM

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.

23. The non-linear stability problem for black hole spacetimes

Mihalis Dafermos (Cambridge)

4/15/08, 1:45 PM

30. A uniqueness theorem for the Kerr metric

Piotr Chrusciel

4/16/08, 2:00 PM

25. The non-linear stability problem for black hole spacetimes

Mihalis Dafermos (Cambridge)

4/16/08, 3:15 PM

7. Rigidly Rotating Bodies in General Relativity

Robert Beig (Vienna)

4/17/08, 11:00 AM

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.