Hydrobates

A mathematician thinks aloud

Archive for November, 2010

Me on TV

November 26, 2010

Recently I was interviewed by TV journalists for a documentary of the channel 3Sat called “Rätsel Dunkle Materie” [The riddle of dark matter]. It was broadcast yesterday. Before I say more about my experience with this let me do a flashback to the only other time in my life I appeared on TV. On that occasion the BBC visited our school. I guess I was perhaps twelve at the time although I do not know for sure. I was filmed reading a poem which I had written myself. I was seen sitting in a window of the Bishops’ Palace in Kirkwall, looking out. I suppose only my silhouette was visible. I no longer have the text of the poem. All I know is that the first line was ‘Björn, adventuring at last’ and that later on there was some stuff about ravens. At that time I was keen on Vikings. The poem was no doubt very heroic, so that the pose looking out the window was appropriate.

Coming back to yesterday, the documentary consisted of three main elements. There was a studio discussion with three guests – the only one I know personally is Simon White. There were some clips illustrating certain ideas. Thirdly there were short sequences from interviews with some other people. I was one of these people. They showed a few short extracts of the interview with me and I was quite happy with the selection they made. This means conversely that they nicely cut out things which I might not have liked so much. I was answering questions posed by one of the journalists and which were not heard on TV. They told me in advance that this would be the case. They told me that for this reason I should not refer to the question during my answers. I found this difficult to do and I think I would need some practice to do it effectively. Fortunately it seems that they efficiently cut out these imperfections. I did not know the questions in advance of the filming and this led to some hesitant starts in my answers. This also did not come through too much in what was shown. Summing up, it was an interesting experience and I would do it again if I had the chance. Of course being a studio guest would be even more interesting …

I found the documentary itself not so bad. I could have done without the part about religion at the end. Perhaps the inclusion of this is connected with the fact that the presenter of the series, Gert Scobel, studied theology and also has a doctorate in hermeneutics. (I had to look up that word to have an idea what it meant.) An aspect of the presentation which was a bit off track was that it gave the impression that the idea of a theory unifying general relativity and quantum theory was solely due to Stephen Hawking. Before ending this post I should perhaps say something about my own point of view on dark matter and dark energy. Of course they are symptoms of serious blemishes in our understanding of reality. I believe that dark matter and dark energy are better approaches to explaining the existing observational anomalies than any other alternative which is presently available. In the past I have done some work related to dark energy myself. The one thing that I do not like about a lot of the research in this area is that while people are very keen on proposing new ‘theories’ (which are often just more or less vague ideas for models) there is much less enthusiasm for working out these ideas to obtain a logically sound proposal. Of course that would be more difficult. A case study in this direction was carried out in the diploma thesis of Nikolaus Berndt which was done under my supervision. The theme was to what extent the so-called Cardassian models (do not) deserve to be called a theory. We later produced a joint publication on this. It has not received much attention in the research community and as far as I know has only been cited once.

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Twisting Gowdy

November 11, 2010

I have previously written about the Gowdy solutions as the simplest model case of spatially inhomogeneous solutions of the Einstein vacuum equations and about the generalization of this to the Einstein-Maxwell equations. I have also written about spatially homogeneous solutions (Bianchi models). I have just posted a paper where I explore connections between these two classes of solutions (Gowdy and Bianchi) in the hope of learning more about both. The basic idea is simple. A Bianchi model has Killing vectors which form a three-dimensional Lie algebra. Most of these (all except types VIII and IX) have a two-dimensional Abelian subalgebra. When this is the case we can simply forget about one Killing vector and end up with a spacetime with two commuting Killing vectors. This looks very like a Gowdy spacetime. There are, however, two potential problems. The first is that Gowdy spacetimes have a certain discrete symmetry in addition to their two Killing vectors. Almost all Bianchi types do satisfy this condition, the exception being type ${\rm VI}{}_{\frac19}$. In what follows I ignore this exceptional type. The other problem is that a Gowdy spacetime is supposed to have a compact Cauchy surface and the Killing vectors are assumed to exist globally. If these conditions are weakened then more alternatives are allowed. Let me call this situation a locally Gowdy spacetime. (This is not standard terminology.) Assume that the Killing vector fields have no zeroes. Then in the Gowdy case the Cauchy surface has the topology of a three-torus $T^3$. In the locally Gowdy case the existence of a compact Cauchy surface is closely linked to those Bianchi types belonging to Bianchi Class A. I now restrict to that class. Then the additional Bianchi types which are possible are II, ${\rm VI}{}_0$ and ${\rm VII}{}_0$.

In Gowdy spacetimes there are two of the Einstein equations for functions $P(t,\theta)$ and $Q(t,\theta)$ which are of central importance. I call them the Gowdy equations. In order to describe Gowdy spacetimes the functions $P$ and $Q$ should be periodic in $\theta$. To accomodate Bianchi models of types II and ${\rm VI}{}_0$ more complicated boundary conditions are required. The twisted Gowdy solutions are solutions of the Gowdy equations which satisfy these boundary conditions but need not be homogeneous. They represent finite inhomogeneous perturbations of the corresponding Bianchi models. Periodic boundary conditions can accommodate type ${\rm VII}{}_0$, which then corresponds to what are called circular loop spacetimes. In the twisted cases the topology of a compact Cauchy surface is more complicated than that of $T^3$. For type II and type ${\rm VI}{}_0$ it corresponds to a manifold admitting geometric structures of type Nil and Sol in the sense of Thurston, respectively. These are twisted topologies which are torus bundles over a circle.

A lot is understood about the global dynamics of Gowdy solutions, mainly due to the work of Hans Ringström. By uncovering certain connections between different classes of solutions I have been able to transfer some of these results to the twisted Gowdy case. Unfortunately I was not able to obtain a general analysis of the late-time behaviour of twisted Gowdy solutions. The analytical techniques which were successful in the untwisted case do not seem to adapt well. This leaves a challenge for the future.
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