June 14-15, 2006

Matematiska institutionen,  Linköpings universitet

Under the auspices of the Gravitation Section of the Swedish Physical Society

Program :  

All seminars in Glashuset (B-house, entrance 25). Some abstracts below.

June 14

13.00-13.40 Ingemar Bengtsson, Stockholms universitet: Extremal black holes --- some curious facts
13.40-14.20 Alfonso García-Parrado, Linköpings universitet: Isometric embeddings in the Schwarzschild spacetime

Coffee

14.50-15.30 Magnus Herberthson, Linköpings universitet: Calculation of, and bounds for, the multipole moments of stationary spacetimes
15.30-16.20 Claes Uggla, Karlstads universitet: Scalar curvature singularities
16.30-17.20 José M M Senovilla, University of the Basque Country (Bilbao): A new Laplacian leading to new potentials for curvature tensors

Dinner (time: 18.30, place: t.b.a.)

June 15

9.00-9.30 Tooba Feroze: The Connection between Symmetries of the Geodesic Equations and Isometries of the Underlying Spaces
9.30-10.00 Azad A Siddiqui: Classical Timelike Geodesics in a Naked Reissner-Nordström Singularity Background: Some Observations

Coffee

10.30-11.00 Jens Jonasson, Linköpings universitet: Multiplication of solutions for systems of partial differential equations
11.00-11.40 Jan Åman, Stockholms universitet: Information geometry of black hole thermodynamics

11.40-12.20  GR Sweden meeting

I Bengtsson, Extremal black holes --- some curious facts:
Extremal black holes have some very special properties. The near horizon geometry of the extremal Kerr black hole is built from squashed and stretched anti-de Sitter spaces. In the Reissner-Nordström case an extra discrete conformal symmetry appears, and there is a peculiar relation to Friedrich's treatment of conformal infinity.

A García-Parrado, Isometric embeddings in the Schwarzschild spacetime:
In General Relativity, a set of initial data consists of a 3-dimensional Riemannian manifold and a symmetric tensor field defined on it. If the symmetric tensor field and the Riemannian metric fulfill certain differential conditions (constraints) then it is a standard result that the Riemannian manifold can be isometrically embedded in a Lorentzian manifold (initial data development) which satisfies Einstein field equations. Developing methods to find such Lorentzian manifold has been a major topic of research during the years and they have found important applications in numerical studies of Einstein field equations. Alternatively, one may proceed the other way round, that is, one starts from a given solution of Einstein field equations and seeks to find out conditions under which a set of initial data can be isometrically embedded in the known Lorentzian manifold from which we started. This latter approach has been less researched by relativists. In this work we consider the maximal extension of Schwarzschild spacetime (Kruskal extension) and develop conditions ensuring that an initial data set can be isometrically embedded in this spacetime. This is achieved by writing a local invariant characterization of Schwarzschild spacetime in terms of longitudinal and transversal parts (3+1 decomposition) with respect to a unit timelike vector. To guarantee the existence of the embedding we resort to a set of conditions which are known to be an initial data set for a timelike Killing vector (Killing initial data or KID conditions). The KID conditions and the conditions arising from the 3+1 decomposition are shown to be sufficient conditions for the existence of the isometric embedding.

M Herberthson, Calculation of, and bounds for, the multipole moments of stationary spacetimes:
In this talk we present an efficient way of calculating of the multipole moments of stationary asymptotically flat spacetimes. With the use of normal coordinates the tensorial recursion of Geroch and Hansen be reduced to a scalar recursion on $R^2$, where the multipole moments are encoded in two real analytic functions. With a careful choice of conformal factor, the moments are given by the (direction dependent) radial derivatives at 0. We also discuss a bound for these moments, a condition which is related to a conjecture by Geroch.

C Uggla, Scalar curvature singularities:
The celebrated singularity theorems by Penrose, Hawking, and others, tell us that spacetime singularities exist under very general circumstances – they tell us that spacetime has an edge at the beginning of our universe and in the interior of black holes. But they do not tell us what happens when this edge is approached. In this talk I will discuss some aspects about scalar curvature singularities, i.e., the type of singularities you expect at the beginning of our universe and in black hole interiors. I will start with a brief review of the historically dominant approaches one has taken in order to understand the nature of scalar curvature singularities, and will subsequently introduce and discuss three complementary aspects of scalar curvature singularities: asymptotic Ricci and Weyl curvature, asymptotic spatial properties, and asymptotic causal properties. Thereafter I will focus on generic spacelike singularities and their nature. The talk will end with a discussion about open issues in this area.

José M M Senovilla, A new Laplacian leading to new potentials for curvature tensors:
We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2r potentials for any tensor field, where r is its form-structure number.  By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2,3)-form potential for the Weyl tensor due to its tracelessness.  This latter potential is the n-dimensional version of the double dual of the classical four dimensional (2,1)-form Lanczos potential.  We also introduce a new concept of harmonic tensor fields, demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, it is the natural operator to use in the Laplace-like equation for the Riemann tensor.

T Feroze, The Connection between Symmetries of the Geodesic Equations and Isometries of the Underlying Spaces:
A connection between the symmetries of manifolds and their geodesic equations, which are systems of second order ordinary differential equations, is sought through the geodesic equations of maximally symmetric spaces. Since such spaces have either constant positive, constant negative or zero curvature, three cases are considered. It is proved that for a space admitting so(n+1) or so(n,1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so(n+1) ⊕ d₂ or so(n,1) ⊕ d₂ (where d₂ is the 2-dimensional dilation algebra), while for those admitting so(n) ⊕_s Rⁿ the algebra is sl(n+2). It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h ⊕ d₂ provided that there is no cross-section of zero curvature.

A A Siddiqui, Classical Timelike Geodesics in a Naked Reissner-Nordström Singularity Background: Some Observations:
It is generally assumed that naked singularities must be physically excluded, as they could otherwise introduce unpredictable influences in their future null cones. Considering geodesics for a naked Reissner-Nordström singularity, it is found that it is effectively clothed by its repulsive nature. Regarding electrons as naked singularities, the size of the clothed un-accelerated electron turns out to be its classical electro-magnetic radius, and for accelerated electrons the size shrinks. For geodetic parameters corresponding to negative energy there are trapped geodesics. The similarity of this picture with that arising in the Quantum Theory is discussed.

J Jonasson, Multiplication of solutions for systems of partial differential equations:
Abstract: The Cauchy-Riemann equations is an example of a system of partial differential equations that is equipped with a multiplication (a bi-linear operation) on its solution set. This multiplication is an immediate consequence of the multiplication of holomorphic functions in one complex variable. Another, more sophisticated, example is the multiplication of cofactor pair systems, that provides a method for generating large families of dynamical systems that can be solved through the method of separation of variables. We will see that these two examples are special cases of a large class of systems of linear first order PDE's that allow a multiplication on the solution set. The multiplication can also be used as a tool for producing infinite families of non-trivial solutions by forming polynomials, and even power series, of elementary solutions.

Göran Bergqvist  (gober_at_mai.liu.se)
Brian Edgar (bredg_at_mai.liu.se)