Bad Honnef seminar

Alan Rendall, Albert Einstein Institute

A seminar with the title Mathematical Problems in General Relativity took place in Bad Honnef, Germany, from 7th to 11th September 1998. This seminar, with about sixty participants from eighteen countries, was organized by Herbert Pfister (University of Tübingen) and Bernd Schmidt (AEI Potsdam) and financially supported by the Heraeus Foundation. It was primarily aimed at graduate students and so strong emphasis was put on the pedagogical nature of the talks. At the same time it was intended to give a point of entry to recent results in the area of mathematical relativity.

A mathematical understanding of general relativity requires knowledge of the solution theory of the Einstein constraints and evolution equations and the corresponding mathematical background. Robert Beig gave an introduction to the constraints together with a discussion of identifying spacetime Killing vectors in terms of initial data and giving a four-dimensional characterization of special solutions of the constraints, such as the multiple black hole solutions of interest in numerical relativity. Oscar Reula treated the basic theory of the evolution equations. He also presented a more general account of the nature of hyperbolicity of systems of partial differential equations and new results on writing the Einstein equations expressed in Ashtekar variables in symmetric hyperbolic form.

A subject which was given particular attention was that of asymptotically flat vacuum spacetimes with Helmut Friedrich talking for four hours on his program to investigate the consistency of the classical conformal picture with the field equations and Alan Rendall talking for four hours on the theorem of Christodoulou and Klainerman on the nonlinear stability of Minkowski space. Friedrich presented his results on the stability of de Sitter and anti-de Sitter spacetimes as well as new developments concerning restrictions on asymptotically flat initial data related to smoothness of null infinity. As explained in talks by Gabriel Nagy, insights from the anti-de Sitter case were important in the recent existence theorem for the initial boundary value problem for the vacuum Einstein equations by him and Friedrich. Rendall explained some of the analytical techniques used in the Christodoulou-Klainerman proof such as energy estimates (also prominent in Reula's talk), the Bel-Robinson tensor, bootstrap arguments and the null condition. Lars Andersson showed how some of these techniques, in particular the Bel-Robinson tensor, have been applied to a class of cosmological spacetimes in his work with Vincent Moncrief on the stability of the Milne model. This opens up the possibility that the Christodoulou-Klainerman result may not stand in splendid isolation much longer. Yvonne Choquet-Bruhat, in a talk on geometrical optics expansions for the Einstein equations, told how this reveals an almost linear property of these equations, exceptional among hyperbolic systems, which is related to the null condition.

Matter was not neglected at the conference either. Herbert Pfister and Urs Schaudt described their progress towards constructing solutions of the Einstein-Euler equations with given equation of state representing rotating stars. Lee Lindblom talked on the inverse problem of reconstructing the equation of state given data on masses and radii of corresponding fluid bodies. In constructing fluid bodies it is always wise to keep an an eye on the corresponding Newtonian problem. Jürgen Ehlers gave an introduction to his mathematical formulation of the Newtonian limit which can be used to give a conceptually clear approach to this. Gerhard Rein summarized our present knowledge on the gravitational collapse of collisionless matter, including his recent numerical work with Rendall and Jack Schaeffer on the boundary between dispersion and black hole formation for this matter model. From the point of view of exact solutions, Gernot Neugebauer spoke on the inverse scattering method and Dietrich Kramer described an approach to producing null dust solutions.

The fact that the whole seminar took place in the Physics Centre of the German Physical Society in Bad Honnef and that all participants were accomodated in that building provided ample opportunity for formal and (particularly on the evening with free beverages) less formal discussions.

Jorge Pullin