Archive for April, 2010

Mixmaster in motion

April 21, 2010

In a previous post I wrote about the Mixmaster model. I mentioned that not much had happened in this area for many years and that I was involved in an attempt to change this. I was happy to see that two papers have recently appeared on the arXiv which show that this area of research is getting moving again. To describe the results of these papers I first need some background. The Wainwright-Hsu system is a system of ODE which describes solutions of the Einstein equations of several Bianchi types, including types IX (which is Mixmaster) and I. The \alpha-limit set of a solution of type IX contains points of type I and it is interesting to know what subsets of the set of all vacuum type I solutions (the Kasner circle K) can occur as \alpha-limit sets. Apart from three exceptional points T_i, i=1,2,3 (the Taub points) each point X of the Kasner circle is the \omega-limit set of a unique solution of Bianchi type II. The \alpha-limit set of that solution of Bianchi type II is also a point of the Kasner circle, call it \phi(X). This defines a mapping \phi from the complement of the Taub points in K to K, the Kasner mapping. It is suggestive (and widely assumed in the physics literature) that the dynamics of Bianchi IX solutions at early times should be approximated by the discrete dynamics defined by the Kasner map. On a rigorous level essentially nothing was known about this.

The simplest orbit of the Kasner mapping is an orbit of period three which is unique up to symmetry transformations. In a paper by Marc Georgi, Jörg Härterich, Stefan Liebscher and Kevin Webster (arXiv:1004.1989) they studied this orbit. The three points in this orbit are joined by three orbits of type II, forming a heteroclinic cycle. The question is if there are solutions of Bianchi type IX which have this heteroclinic orbit as \alpha-limit set and if so how many. In the paper just mentioned the authors show that there is a Lipschitz submanifold invariant under the Bianchi IX flow with the properties that all solutions starting at points on this manifold converge to the heteroclinic cycle in the past time direction. They do this for the Einstein vacuum equations and for the Einstein equations coupled to a perfect fluid with linear equation of state. If the equation of state in the latter case is p=(\gamma-1)\rho then the result is restricted to the range \gamma<\frac{5-\sqrt{5}}{2}. They also indicate various possible generalizations of this result. The method of proof is a careful study of a return map which describes the behaviour of solutions starting near the heteroclinic cycle. In the end it is shown that this return map defines a contraction on a space of functions whose graphs are candidates for the invariant manifold of interest.

The other paper by François Béguin (arXiv:1004.2984) concentrates on orbits of the Kasner map which, while keeping away from the Taub points, are otherwise as general as possible. The reason for the genericity assumption is to avoid resonances. To put this into context I recall the following facts. Let p be a stationary solution of a dynamical system. Then the system can be linearized about that point. It can be asked under what circumstances the system can be transformed into its linearization by a homeomorphism, in a local neighbourhood of p. The Hartman-Grobman theorem says that this is the case provided no eigenvalue of the linearization is purely imaginary. The corresponding statement for a diffeomorphism of class C^1 is not true. There is an analogue of the Hartman-Grobman theorem in the C^1 case, due to Sternberg, but it requires extra assumptions. These assumptions are conditions on linear combinations of eigenvalues with integer coefficients. The assumptions of Béguin are related to these conditions. In the Mixmaster problem there is an extra complication. The presence of the Kasner circle means that the linearization about any point automatically has a zero eigenvalue. Thus what is needed is an analogue of Sternberg’s theorem in certain circumstances where a zero eigenvalue is present. There is a theorem of this type, due to Takens. I heard about it for the first term when Béguin gave a talk on this subject in out institute. In any case, Takens’ theorem is at the heart of the proof of the paper. It allows certain things to be linearized in a way which is differentiable.

Shock waves, part 3

April 6, 2010

In a previous post I wrote about shock waves in fluids, including the case that they are described by the Einstein-Euler equations for a self-gravitating fluid in general relativity. I mentioned there a result of Fredrik Ståhl and myself proving that smooth solutions of the Einstein-Euler system can lose regularity in the course of their time evolution. This was done in the framework of spacetimes with plane symmetry. Here I want to describe some complementary results which were recently obtained by Philippe LeFloch and myself. These new results concern the existence of global weak solutions in situations where shocks may be present. This work is done under the assumption of Gowdy symmetry, which is weaker than plane symmetry. It allows the presence of gravitational waves, which plane symmetry does not. It uses time coordinates different from the constant mean curvature (CMC) coordinate used in the work with Ståhl. This difference in the time coordinates makes it difficult to relate the results of the two papers directly. It would be interesting to adapt the results of either of these papers to the time coordinates of the other.

In the paper with LeFloch we use coordinates (areal, conformal) which have previously been used in analysing analogous problems for vacuum spacetimes or spacetimes where the matter content is described by collisionless kinetic theory. A big difference is the weak regularity. One effect of this is that while in the given context it has been possible to prove global existence theorems for the initial value problem, nothing is known about the uniqueness of the solutions in terms of initial data. It should, however, be noted that in the corresponding analytical framework uniqueness is not even known for a one-dimensional non-relativistic fluid without gravity. Another new element introduced by the use of weak solutions is that it is only possible to evolve in one time direction. This model is not reversible, a fact implemented mathematically by the imposition of entropy inequalities. One of the results obtained concerns a forever expanding cosmological model. The other one concerns a contracting model which ends in a singularity. The second is not a global existence result in the conventional sense but it can be thought of as saying that the solution can be extended until certain specific things happen (a big crunch singularity).

To finish this post I want to indicate the type of regularity of the solutions obtained. I only state this roughly – more precise information can be found in the paper. The energy density and momentum density of the fluid is integrable in space, with the L^1 norms locally bounded in time. The quantities parametrizing the spacetime metric have first order derivatives which are square integrable in space. These conditions allow for jump discontinuities in the energy density which is what comes up in shock waves. It also allows singularities of Dirac \delta type in the metric, corresponding to what are often called impulsive gravitational waves.


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