This week I am at a conference on mathematical relativity in Oberwolfach. Today Jared Speck gave a talk on work of his with Igor Rodnianski which answers a question I have been interested in for a long time. The background is that it is expected that the presence of a stiff fluid (a fluid where the speed of sound is equal to the speed of light) leads to a simplification of the dynamics of the solutions of the Einstein equations near spacetime singularities. This leads to the hope that the qualitative properties of homogeneous and isotropic solutions of the Einstein equations coupled to a stiff fluid near their big bang singularities could be stable under small changes of the initial data. This should be independent of making any symmetry assumptions. As discussed in a previous post it has been proved by Fuchsian techniques that there is a large class of solutions consistent with this expectation. The size of the class is judged by the crude method of function counting. In other words there are solutions depending on as many free functions as the general solution. This is a weak indication of the stability of the singularity but it was clear that a much better statement would be that there is an open neighbourhood of the special data in the space of all initial data which has the conjectured singularity structure. This is what has now been proved by Rodnianski and Speck.

This result fits into a more general conceptual framework. Suppose we have an explicit solution of an evolution equation and we would like to investigate the stability of its behaviour in a certain limit . If we expect that solutions with data close to the data for have the same qualitative behaviour then we may try to prove this directly. Call this the forward method. If there is no evidence that this idea is false but it seems difficult to prove it then we can try another method as an interim approach to gain some insight. This is to attempt to construct solutions with the expected type of asymptotics which are as general as possible. I call this the backward method, since it means evolving away from the asymptotic regime of interest. The forward method is preferable to the backward if it can be done. In the case of singularities in Gowdy spacetimes Satyanad Kichenassamy and I applied the backward method and Hans Ringström later used the forward method. It is perhaps worth pointing out that while the forward method is more satisfactory than the backward one both together can sometimes be used to give a better total result than the forward method alone. There are also examples of this in the context of expanding cosmological methods with positive cosmological constant. I applied the backward method while Ringström, Rodnianski and Speck later used the forward method. The result for the stiff fluid with which I started this post also fits into this framework using the forward method. The corresponding result for the backward method was done by Lars Andersson and myself more than ten years ago.

There were two other talks at this conference which can be looked at from the point of view just introduced. One was a talk by Gustav Holzegel on his work with Dafermos and Rodnianski on the existence of asymptotically Schwarzschild solutions. The second was my talk on an apsect of Bianchi models I have discussed in a previous post. Both of these used the backward method.

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