Håkan Andreasson (Chalmers): "Global existence of solutions to the Einstein equations with symmetry"
The cosmic censorship conjecture was proposed by Roger Penrose
in the sixties and is considered to be one of the most important
open questions in general relativity. I will try to explain what
this conjecture is all about (it has to do with the nature of
spacetime singularities) and we will see that an important step
in proving cosmic censorship is to prove global (geometric)
existence of solutions to the Einstein equations. This is a very
hard problem in general and a way to simplify it is to make
symmetry assumptions. I will present a global existence result
for matter spacetimes with a certain kind of symmetry (Gowdy
symmetry). As opposed to spherical symmetry this symmetry class
is sufficiently non restrictive to admit for gravitational
waves.
Alan Barnes (Aston University): "Some restrictions on the
symmetry groups of axisymmetric spacetimes"
The symmetry groups of axisymmetric spacetimes are investigated.
It is shown that the form of the symmetry group is considerably
restricted by the existence of a one-parameter subgroup with
circular orbits. For example, if the complete group is
two-dimensional, then it is necessarily Abelian whereas if it is
three-dimensional; then it must be of Bianchi type I, II, III,
VII0, VIII or IX. Some results for the general m-dimensional
case are also obtained and applied to classify all axisymmetric
spacetimes with a complete four-dimensional symmetry group.
In spacetimes which admit a foliation by non-null 3-D
conformally flat hypersurfaces it is shown how to obtain all the
Killing and conformal Killing vectors of the full spacetime (if
any exist) by 'lifting' the conformal Killing vectors in the 3-D
hypersurfaces. A number of applications of this technique are
indicated: the isometries and conformal motions of Stephani's
conformally flat perfect fluid solutions are obtained; this
class of metrics includes the Friedmann-Robertson-Walker and
interior Schwarzschild metrics as important special cases. The
technique has also been applied recently by the author and U
Camci to obtain all Ricci collineations and Ricci inheriting
collineations of the Friedmann-Robertson-Walker metrics.
Johan Brännlund (Stockholms Unviersitet): "Geometry of
spin systems in quantum mechanics"
Quantum mechanics has a natural geometric formulation if one
works in complex projective space CP^n rather than in Hilbert
space. A few finite-dimensional examples will be discussed, in
particular submanifolds of constant entanglement. Scattered
connections to general relativity will also be made.
Ray d'Inverno (University of Southampton): "The current
status of the Computer Database of Exact Solutions"
The area of Computer Algebra in General Relativity
revolutionised the field of exact solutions of Einstein's
equations. It made calculations possible which would have
previously been prohibitive in time. This in turn led to a
renewed attack on the classic equivalence problem of general
relativity, namely: given two metrics is there a local
coordinate transformation which transforms one metric into the
other? The Cartan-Karlhede algorithm for classifying metrics
allied with Computer Algebra systems (predominantly CLASSI and
GRTENSOR) have led to a significant advance in solving the
problem. This in turn has led to the setting up of the Computer
Database of Exact Solutions, a collaborative effort to put all
of the solutions of Einstein's equations into a computer
database for easy access by the relativity community. This talk
will look at the current status of the database.
Anders Höglund (Linköpings Universitet): "Energy-momentum
tensors and Algebraic Rainich conditions in higher
dimensions"
The Rainich conditions tells us when a given energy-momentum
tensor is given by a 2-form in four dimensions. In this talk we
examine the situation in higher dimensions regarding the
algebraic conditions. We will in turn look at energy-momentum
tensors of simple forms, 2-forms of rank four, 2-forms of higher
rank and finally p-forms where p>2. An important tool in this
examination is obtaining identities through antisymmetrisation.
Hans Lundmark (Linköpings Universitet): "Killing tensors
of cofactor type and separation of variables"
I will try to explain some recent developments in separability
theory, concentrating on the following result: if a Riemannian
manifold (of dimension n) admits a conformal Killing tensor J
with vanishing Nijenhuis torsion, then it admits n Killing
tensors, and the geodesic equations can be integrated by
separation of variables in the Hamilton-Jacobi equation. The
separation coordinates are given by the eigenvalues of J. I will
also show how to construct potentials that can be added to the
geodesic Hamiltonian without destroying separability. Examples
will be given for flat space and for the sphere.
Claes Uggla (Karlstads Universitet): "Dynamical Systems in
Cosmology"
The evolution of the Universe is ruled by gravity. Our best
theory of gravity is general relativity (GR), and hence one uses
Einstein's field equations to produce cosmological models. What
scenarios are possible for the early Universe and what is the
eventual fate of the Universe according to GR? To address issues
like these, it has turned out to be fruitful to use a dynamical
systems approach to Einstein's field equations. To illustrate
how dynamical systems ideas are applied in GR, I will start by
discussing spatially homogeneous and isotropic
Friedmann-Lemaitre models, which have been remarkably successful
in explaining many cosmological observations. First I will give
a qualitative picture of the possible features such models
exhibit by using simple potential diagrams. Subsequently I will
use a dynamical systems approach and give a comparison between
the two pictures. Thereafter I will outline how GR exhibits a
hierarchical structure that allows one to build increasingly
complex models using dynamical systems methods. I will finish by
discussing some recent results and speculate about the generic
This page is maintained by Anders Höglund anhog@mai.liu.se and Jonas
Bergman jober@mai.liu.se.