In a previous post I mentioned the Field-Noyes model which is a three-dimensional dynamical system which gives a description of the Belousov-Zhabotinski oscillatory chemical reaction. I want to discuss it here in the context of methods to show the existence or non-existence of periodic solutions of a dynamical system. For two-dimensional systems there are special possibilities due to the availability of Poincaré-Bendixson theory. Here I want to concentrate on what can be done in higher dimensions.

A task which is often simpler than that of finding periodic solutions is that of finding stationary solutions. This reduces to algebra and is in many cases tractable. Another algebraic task is to linearize about the stationary solutions and compute the corresponding eigenvalues. This gives information about the stability of the stationary solutions. Suppose we have a dynamical system with a compact invariant set where the stationary solutions, if any, are all such that no solution can converge to them at late times. Then the -limit set of a non-stationary solution must contain a periodic solution or must be associated with some more complicated kind of attractor. Under the weaker condition that each stationary point is unstable it could be hoped that a similar conclusion will hold for generic solutions. (This might fail in the presence of heteroclinic cycles.) In the Field-Noyes model there are precisely two stationary solutions, one of which is always unstable. The stability of the other depends on the values of the parameters in the system, which are related to reaction rates. When the second is unstable we are in the situation described above. So are there periodic solutions corresponding to the oscillations observed experimentally?

One way of proving the existence of periodic solutions is to use a bifurcation analysis. If a dynamical system depends on parameters and a stationary solution loses stability at some point of parameter space it may be possible to do a local analysis to determine the behaviour of the system for parameter values just beyond the bifurcation. The simplest case of this is a Hopf bifurcation where a pair of eigenvalues passes through the imaginary axis in a sufficiently non-degenerate way. In this case the existence of periodic solutions in a certain (a priori small) region of parameter space follows. This kind of analysis can be done in the case of the Field-Noyes model. This is described in some detail in Murray’s book ‘Mathematical Biology’, where it is also indicated that the experimantally observed oscillations cannot plausibly be thought of as corresponding to parameter values belonging to this small region. There is an interesting further analysis due to Hastings and Murray (SIAM J. Appl. Math. 28, 678). They show that almost all solutions are oscillatory by the following procedure. It is shown that all solutions enter a certain box in state space at late times. This box is the union of eight smaller boxes. It is proved that, apart from some rare exceptions, a given solution must visit each box repeatedly in a certain order at late times. This shows that generic solutions are oscillatory and gives some information about the qualitative properties of the oscillations. In addition it is shown that there is at least one periodic solution for given values of the parameters. This is proved using the Brouwer fixed point theorem.

Having recently spent some time learning about influenza vaccines I had the idea of looking back at other vaccinations I have had. Quite a few of these were done for a trip to Cameroon. I have no way of finding out exactly what vaccinations I had as a child. The only two I can remember is that at school we were given the BCG vaccination against tuberculosis and the oral vaccine against polio. The latter disease was more than an abstraction for me because of my very first schoolteacher (when I was five and six). She had a serious physical handicap and we knew why – she had had polio. I do not expect that I had many other vaccinations since a number of the diseases which were common in children before vaccinations against them were common (and which are still common in developing countries) rolled through our school. There were times when more than half the class was off school. I had measles when I was four and so had got one disease behind me even before school. Remarkably I was not affected (at least on the symptomatic level) by any of the series which were caught by most of my schoolmates (chicken pox, German measles, mumps, whooping cough). In most cases I do not know if I completely avoided the disease or if I had a subclinical infection. I never bothered to have a test for antibodies so as to find out. There is one exception. I got chicken pox in my mid twenties and so that is one I definitely missed as a child.

Now I come to the vaccinations I had in later years. Here I have a certificate of vaccination which I could consult. I will start with hepatitis A and B since that involves some themes which I mentioned in a previous post on influenza vaccines. A point I want to make is that the swine flu vaccines about which there has been so much public discussion recently are not so different from lots of other vaccines, including ones which I have had without thinking about it. The vaccine I had is called Twinrix. The hepatitis A vaccine contains inactivated virus, aluminium hydroxide (alum) as an adjuvant and is produced in cell culture (MRC5 cells).These are cells which are not immortal but can survive for 50-100 generations in culture. The hepatitis B vaccine contains surface antigens and is produced in cells of the yeast Saccharomyces cerevisiae. It contains aluminium phosphate as an adjuvant. In the manufacturer’s description of the vaccine it is mentioned that it contains traces of thiomersal. This is the organic compound containing mercury which has been a subject of discussion in the context of influenza vaccines. I have no idea if the concentrations involved in the two cases are comparable. Another vaccine I had was a polio vaccine with inactivated virus. This is produced in cell culture using Vero cells. The vaccination against tetanus (tetanol) contains toxoid (a modified form of the toxin produced by the bacterium causing the disease) and an adjuvant containing aluminium. As far as I know I was not vaccinated against diphtheria as a child and the vaccination I got more recently is analogous to that against tetanus and is actually effective against both diseases. I have no particular information about the vaccination I had against typhoid fever. Finally, there was Stamaril against yellow fever. This contains attenuated live virus. Of all the vaccinations I have mentioned this is the only one where I can remember having had even minor side effects. In that case I had a light headache about seven or eight days after the injection. Since I almost never have headaches and this is a known side effect I presume there was a causal connection. It is natural with a live vaccine and shows adaptive immunity getting activated.

I recently discovered a blog called WeiterGen (in German) with a lot of valuable information relevant to the public discussion of the influenza vaccine in Germany and elsewhere.There is also information about a notorious skin cream whose claimed medicinal properties have been the subject of some very dubious reporting on German TV recently. In Germany if you have a television set you have to pay the considerable licence fees for the public TV service. A justification often given for this is that it is important that the public have a trustworthy source of objective information. After this skin cream business I will be a lot more sceptical about the information on public TV, expecially things concerned with medical themes.

On a more cheerful note, the blog I just quoted also has an interesting post about the bacterium Mycoplasma pneumoniae as a model organism for use in systems biology.

A bootstrap argument is an analogue of mathematical induction where the natural numbers are replaced by the non-negative real numbers. This type of argument is a powerful tool for proving long-time existence theorems for evolution equations. For instance, it plays a central role in the proof of the stability of Minkowski space by Christodoulou and Klainerman and the theorem on formation of trapped surfaces by Christodoulou discussed in previous posts. The name comes from a story where someone pulls himself up by his bootstraps, leather attachments to the back of certain boots. This story is often linked to the name of Baron Münchhausen. In another variant he pulls himself out of a bog by his pigtail. This was a person who really lived and was known for telling tall tales. In later years people wrote various books about him and incorporated many other tall tales from various sources. The word ‘booting’ applied to computers is derived from ‘bootstrapping’ in the sense of this story. There are also bootstrap methods used in statistics. They involve analysing new samples drawn from a fixed sample. In some sense this means obtaining more knowledge about a system without any further input of information. It is this aspect of ‘apparently getting something for nothing’ which is typical of the bootstrap. In French the procedure in statistics has been referred to as ‘méthode Cyrano’. Unfortunately is seems that in PDE theory the French have just adopted the English term. I say ‘unfortunately’ because of a fondness for Cyrano de Bergerac. As in the case of Münchhausen there was a real person of this name, this time a writer. However the name is much better known as that of a fictional character, the hero of a play by Edmond Rostand. The non-fictional Cyrano wrote among other things about a trip to the moon. There is also a Münchhausen story where he uses a kind of inverse bootstrap (could there be a PDE analogue here?) to return from the moon. He constructs a rope which he attaches to one of the horns of the moon but it is much too short to reach down to the ground. He climbs to the bottom of the rope, reaches up and cuts off and detaches the part ‘which he does not need any more’ and ties it onto the bottom. He then repeats this process. Returning to Cyrano, he describes seven methods for getting to the moon of which the sixth is the one relevant to the bootstrap. He stands on an iron plate and throws a magnet into the air. The iron plate is attracted by the magnet and starts to rise. Then he rapidly catches the magnet and throws it into the air again. I should point out that Cyrano does not believe in the nonsensical stories he is telling – his aim is a practical one, holding the attention of the Duc de Guiche so as to delay him for a very specific reason.

Now I return to the topic of bootstrap arguments for evolution equations. I have given a discussion of the nature of these arguments in Section 10.3 of my book. Another description can be found in section 1.3 of Terry Tao’s book ‘Nonlinear dispersive equations: local and global analysis‘. A related and more familiar concept is that of the method of continuity. Consider a statement depending on a parameter belong to the interval . Let be the subset consisting of those for which the statement is true on the interval . If it can be shown that is non-empty, open and closed then it can be concluded that the statement holds for all , by the connectedness of the interval. The special feature of a bootstrap argument is the way in which openness is obtained. Suppose that, starting from , we can prove a string of implications which ends again with . This is nothing other than a circular argument and proves nothing. Suppose, however, that in addition this can be improved so that the statement at the end of the string is slightly stronger than that at the beginning. This improvement is something to work with and is a typical way of proving the openness needed to apply the continuity argument. It is more convenient here to work with the open interval since we want to look at properties of solutions of an evolution equation defined on the interval . Let be the statement that a certain inequality (1) holds on the interval and suppose that implies the statment that a stronger inequality (2) holds on the same interval. Things are usually set up so that implies by continuity that (2) holds at and that the the property of being ’stronger’ then shows that holds for slightly greater than . This shows the openness property. I think the best way to really understand what a bootstrap argument means is to write out a known example explicitly or, even better, to invent a new one to solve a problem which interests you. The key thing is to find the right choice of and . What I have described here is only the simplest variant. In the work of Christodoulou mentioned above he uses a continuity argument on two-dimensional sets.

A recent press release by the Max Planck Institute for Neurobiology in Martinsried reports on a paper (‘Effector T cell interactions with meningeal vascular structures in nascent autoimmune CNS lesions’, Nature 462, 94) where detailed information is given on certain aspects of the way that activated T cells cross the blood-brain barrier during the development of the disease EAE in rats. In fact the authors were able to film the behaviour of the cells in living rats over extended time periods. In the best-known type of interaction between white blood cells and blood vessels through which they pass, the cells roll along the wall of the vessels until at some point they stop and exit the vessel by squeezing between the cells forming the wall. In this work the disease is provoked by introducing activated T cells which recognize myelin basic protein. In the case of most blood vessels they roll as just described but in the case of certain blood vessels belonging the BBB they instead crawl along the walls, often against the direction of the blood flow. This type of behaviour has apparently not been seen before in T cells although it is known from some other types of leukocytes. It looks as if the cells are searching for something particular although it is not clear what. Some of them eventually cross the BBB into the central nervous system while others let go and return to the bloodstream.

Once the cells get into the CNS they encounter phagocytic cells which activate them and cause them to produce substances such as interferon and interleukin 17. This then causes further T cells to be recruited to the CNS, thus leading to the full development of the disease. The identity of these phagocytic cells seems a bit mysterious. They are described in the paper as being intermediate between macrophages and dendritic cells. They are said to be constantly probing the region just outside the vessel walls. What I find particularly interesting about this work is that instead of just obtaining indirect information on what is going on it shows very directly what the cells are doing. The information presented in the paper is much more extensive that what I have just indicated. It has been possible to follow the cells on their way to deeper levels of the brain and to compare these particular T cells to other activated T cells which recognize a different antigen having nothing to do with CNS tissue.

I have recently been reading about influenza vaccines and I am summarizing some of the information I found here. I start with some remarks on the classification of influenza viruses. The first distinction is between influenza A and influenza B viruses. The former are classified further into subtypes HnNm for numbers and . Well-known examples are H5N1 (which includes the recent ‘bird flu’) and H1N1 (which includes the pandemic of 1918 and the current ’swine flu’.) Influenza B does not carry a pandemic threat and will not be considered further here. Every year a vaccine is produced for the seasonal flu epidemic (in fact two – one for the southern and one for the northern hemisphere). It is trivalent, being directed against three types of virus. In recent years this has always been of the form H3N2 + H1N1 + B. In particular this is the case for the present vaccine for seasonal flu. It is not expected that this vaccine will be effective against the pandemic H1N1 swine flu. Thus a separate type of vaccine has been developed for that. In the classification H and N stand for haemagglutinin and neuraminidase, two proteins which occur on the surface of the virus and come in different forms in different strains. These are the main molecules of the virus recognized by antibodies. They are involved in the processes by which the virus enters and leaves host cells, respectively.

Next I come to some details concerning the vaccines themselves. I concentrate on those being applied in Germany since this is what would be relevant for me if I got vaccinated myself. I get the impression that there are a lot of unreliable and misleading statements on this subject in the media and so some care is necessary in judging the information available. On the web page of the Paul Ehrlich Institute there is a list of vaccines against seasonal flu approved in Germany in this season. Twenty products are listed. All are classified as inactivated. This means that if manufactured successfully the vaccine cannot lead to any reproduction of the virus. In other words the vaccine uses (parts of) ‘dead’ virus particles. Three of the vaccines are described as ‘virosomal’ which means that they can be administered as a nasal spray. Presumably all the others are administered by injection. Two of them include an adjuvant, a substance which is intended to amplify the immune response. This is one theme which has led to recent controversy in connection with swine flu vaccines and I will return to it later. One vaccine (Optaflu) is said to be produced in cell culture. This is connected to another theme of recent controversy, with discussion in the media about vaccines produced using cancer cells.

Having looked at the vaccines for seasonal flu I now come to swine flu vaccines.The web page of the Paul Ehrlich Institute lists three vaccines approved in Germany for the new H1N1 influenza. These are called Celvapan, Focetria and Pandemrix. All three are inactivated. The second and third include an adjuvant. The first is produced in cell culture. Apparently Pandemrix is intended to be the main vaccine used in Germany. There is a statement on the web page of the PEI that, contrary to some claims in the media, this is also what has been used to immunize the employees of that institute. There has been discussion of the fact that apparently politicians and the army are to get Celvapan so that a debate about ’second class citizens’ has taken place. This is likely to obscure the real issues. Consider next the topic of adjuvants, substances which have recently been getting some bad press. An adjuvant is a substance which increases the reaction of the immune system to an antigen given as a vaccine. A stronger immune response can lead to better immunity for a given amount of antigen. It could also in principle lead to an excessive and damaging immune reaction although I have not seen any convincing evidence that this has happened in the context of the swine flu vaccine. It would be wrong to think that the name adjuvant denotes a particular class of substances. Many different things can act as adjuvants. What they have in common is that they activate some part of the immune system. Given that the immune system is so interconnected this can lead to a stronger immune response on a wider basis. An interesting example is that in the combination vaccination for diphtheria, whooping cough and tetanus the diphtheria toxoid acts as an adjuvant for the other two vaccinations. What has just been said about the nature of adjuvants makes it clear that it is nonsensical to say that all adjuvants are bad. Each one must be considered on its own merits. In the case of Pandemrix the adjuvant is called AS03. I am unable to give any judgement on it, since I have not spent enough time studying the question. In any case my basic assumption is that what has been approved by the relevant medical authorities is OK. In other words, for these things my default attitude is trust, not mistrust.

Now to the question of the cancer cells. Celvapan is produced using a cell line called Vero cells which is derived from monkeys. It seems that these cells arose from kidneys of normal monkeys and have nothing to do with cancer. Usually normal cells can only undergo a limited number of divisions while cancer cells can be immortal. Vero cells are not cancer cells but they do seem to be able to survive in cell culture for an unlimited time. I do not understand how this works. It may be noted that Vero cells have been used in a routine way to produce millions of doses of polio vaccine and so there has been ample opportunity to discover any possible dangers associated to their use. In the production of Optaflu a cell line called Madin-Darby canine kidney cells is used. This looks like the same kind of tissue as with Vero cells, except derived from a different animal. I have not found a reference anywhere claiming the use of cancer cells in this context which looks trustworthy.

To finish, here is a piece of news from the web site of the ECDC in Stockholm. In the Ukraine there have now been 500 000 reported cases of acute respiratory illness and it seems that the expert opinion (cf. also the WHO website) is that most of these are related to the new H1N1 influenza. There have been 86 deaths reported from there. So it seems that the pandemic is alive and well.

In what follows I describe another subject which was a theme in the talk of Orion Weiner mentioned in the previous post. In the meantime I am familiar with the fact that there are techniques which allow us to see details of what is going on in cells. Here the most prominent protagonist is the green fluorescent protein (GFP) which was honoured by Nobel prizes in 2008. It allows information to be exported from the cell. This is a passive process in the sense that once the system has been prepared we just watch what happens. A more active process which is sometimes shown on video is that where a neutrophil follows the moving tip of a micropipette which is releasing a substance to which the cell is chemotactic. The subject of the present post is how it is possible to actively manipulate cells by sending in light of certain wavelengths. This may mean bathing the cell in light, illuminating certain precisely defined areas with a laser or a combination of the two.

The first type of experiment involves proteins which can be located either at the cell membrane or in the cytosol and which are fluorescently labelled so that their position can be monitored. It is possible to cause these molecules to move rapidly from the one localization to the other. This can be done on a time scale of a couple of seconds and it looks likes switching on and off a light. This can be done many times in a row. Here the effect on the cell is global. The second type of experiment has to do with localizing this type of effect. It allows patterns chosen by the experimenter to be projected onto the cell. Here coloured patches are visible. Their interpretation is that concentrations of a certain substance have been fixed according to the pattern. The third type of experiment is the most striking. Here a spot of light is moved over the cell and away from it in a certain direction. There results a long projection of the cell in that direction. On the video it looks as if the the cell is being pulled by a sticky object. All these things are done by switching on certain proteins which have been made light-sensitive.The sensitivity to light is achieved by incorporating elements which are responsible for allowing certain plants to react to light. One of the plants which acts as a source here is the favourite model organism among plants, Arabidopsis thaliana. The reference to the paper describing these results is ‘Spatiotemporal control of cell signalling using a light-switchable protein interaction’, Nature 461, 997-1001 (15 October, 2009).

A few weeks ago I heard an interesting talk by Orion Weiner from the University of California at San Francisco. This contained a lot of information and it has taken me some time to get around to processing it. One of the things he talked about establishes a surprising link between two topics I have discussed before, chemotaxis and spiral waves. The idea is that the motion of the leading edge of cells such as neutrophils are driven by spiral waves in the concentrations of certain proteins. These waves have been filmed using sophisticated techniques of microscopy. The proteins involved belong to something called the WAVE complex. The name has nothing to do with waves. The WA in the name comes from ‘Wiskott-Aldrich syndrome protein (WASp)’. More specifically the protein whose concentration shows the wavelike phenomena is hematopoietic protein 1 (Hem-1). This protein interacts with the actin which is involved in the mechanics of the motion. However the waves are not visible in the concentration of the actin itself. More information about this and an interview with Weiner can be found here.

The last post was sparked off by a talk I heard at a conference in Oberwolfach. Here I will write about a topic where another talk at that conference looks like a big step forward. This was by Jared Speck. He was describing work of his with Igor Rodnianski which is not yet fully written up.

These days there is a wide consensus among astrophysicists that there is strong observational evidence to indicate that the expansion of the universe is accelerated. In other words it is not only the case that all distant galaxies are moving away from us (and from each other) but the velocity of recession is actually increasing. In the standard view this is only consistent with general relativity if there is a positive cosmological constant or some exotic matter called dark energy. For convenience I will not distinguish between these two in what follows. Dark energy leads to accelerated expansion and accelerated expansion causes spatial irregularities to be damped. The
geometry of spacetime and the matter distribution are smoothed. This kind of idea can be turned into a precise mathematical statement (maybe not uniquely) called the cosmic no hair theorem. From a mathematical point of view this is rather a conjecture than a theorem – at least it has been that way for most of the time it has existed. The name originates from a phrase of John Wheeler, ‘a black hole has no hair’. The idea of this was that a particular solution of the Einstein equations describing a black hole, the Kerr solution, should be attractor for the evolution of more general solutions containing a black hole. In other words a general class of solutions should evolve so as to look more and more like the Kerr solution. The Kerr solution depends only on two parameters. Thus in this scenario all the details get lost dynamically, leaving a very simple object with no complicated features, no hair. In models for an expanding universe with positive cosmological constant the smoothing process mentioned above also seems to drive all solutions towards an attractor, the de Sitter solution. It is this analogy which gave rise to the name ‘cosmic no hair theorem’.

The mathematical formalization of the cosmic no hair theorem says that a solution of the Einstein-matter equations with positive cosmological constant converges to the de Sitter solution at late times in a suitable sense. A weaker statement is that this should be true for solutions which start close to the de Sitter solution. The latter version can also be thought of as a kind of stability statement for de Sitter space. In the case of the vacuum Einstein equations the stability of de Sitter space was proved by Helmut Friedrich in 1986. Since our universe is certainly not empty the relevance of this result to cosmology is not immediately obvious. It turns out, however, that there are reasons to believe that the cosmological constant can often have a dominant effect on the late-time cosmological expansion which tends to make the effect of the matter into a higher-order correction. It is important to confirm these ideas by a theorem which includes the effect of matter. The most commonly used matter model in cosmology is a perfect fluid with linear equation of state. It contains a parameter which is often restricted by an inequality corresponding to perfect fluids which are less stiff than radiation. The result of Rodnianski and Speck is a form of the cosmic no hair theorem for precisely this class of matter models. The proofs build on previous work of Hans Ringström. Friedrich’s proof uses a technique (the conformal method) which is very powerful but rather rigid. It is difficult to see how to modify the proof to include matter such as a fluid, or indeed to replace the cosmological constant by some other kind of dark energy, such as a nonlinear scalar field. Ringström introduced more flexible methods which allowed him to obtain a version of the cosmic no hair theorem for dark energy modelled by certain types of nonlinear scalar field. His methods open up the perspective of including matter and this is what Rodnianski and Speck have now done. These methods use energy estimates, the workhorse of the theory of nonlinear hyperbolic equations, in a clever way. (I might say more clever than I am, since I once tried very hard to do this, without success.)

The result of Rodnianski and Speck is restricted to the case of irrotational fluids. I see no fundamental reason why this should be necessary. Nevertheless there is a clear technical reason – in the irrotational case the Euler equation of the fluid is equivalent to a nonlinear wave equation. On the level of formal power series the case with rotation works out, as shown in a paper of mine (Ann. H. Poincare 5, 1041). Another question is what happens for large data. In that case there are various restrictions.For sufficiently large data it is to be expected that black holes would be formed (even in the vacuum case). Moreover, the fluid can be expected to form shocks which means that the solution cannot be continued, at least in the realm of smooth solutions. I find it remarkable that the expansion caused by a positive cosmological constant is strong enough to suppress formation of shocks in a small data regime. There is just one result available on this subject for large initial data and inhomogeneous solutions. In this work, due to Blaise Tchapnda and myself (arising from Blaise’s PhD thesis, Class. Quantum Grav. 20, 3037) we treated plane-symmetric solutions of the Einstein-Vlasov system with positive cosmological constant. In this case the symmetry prevents formation of black holes and the choice of matter model allows any analogue of shocks to be avoided.

This week I am at a workshop on mathematical relativity at the Mathematical Research Institute in Oberwolfach which I am organizing together with Piotr Chrusciel and Jim Isenberg. I was a co-organizer of similar conferences here in 2000, 2003 and 2006. The institute organizes workshops fifty weeks in the year on all areas of mathematics and participation is generally by invitation only. The isolated setting of the institute in the Black Forest tends to create an intense research environment. Work is also stimulated by the fact that the institute has the best mathematics library in Germany which is no doubt also one of the best in the world.

There have been a lot of excellent talks here. One of these which was of particular interest to me personally was by Todd Oliynyk. His subject was connected with the Newtonian limit of general relativity. I mentioned this topic in a previous post as having been something whose importance was emphasized by Jürgen Ehlers. Unfortunately Jürgen did not live long enough to see some of his questions answered by the work which Oliynyk has been doing recently.

So what is the Newtonian limit? General relativity is, among other things, a theory of gravity which is fully relativistic. In standard textbooks on the subject we can read that Newtonian physics arises as a limit of general relativity when typical velocities in the system are small compared with the speed of light. Unfortunately it is quite unclear what this means mathematically. For instance, in general relativity gravity is described by the metric, a tensor with ten components, while in Newtonian gravity it is described by a scalar function. How can the former converge to the latter? The conceptual basis of the Newtonian limit was elucidated in work by many people over many years and these ideas were synthesized by Ehlers. On this basis I was able to prove a theorem about convergence to the Newtonian limit in 1994. This concerned asymptotically flat spacetimes (in physical terms isolated systems) and the matter was described by kinetic theory (Vlasov equation). I chose this type of matter since a more commonly used description, the perfect fluid, suffers from technical difficulties. This is because the equations degenerate when the fluid density becomes small and in an isolated system the density has to become small somewhere. The Vlasov equation is immune to these difficulties.

The key mathematical problems involved in the analysis of the Newtonian limit should be independent of the details of the matter model chosen. We just need some matter model which does not place obstacles in our way. In a way Oliynyk took this more literally than I did myself. I had proved an existence theorem for certain types of fluid bodies in general relativity by extending ideas introduced by Tetu Makino in the Newtonian case. These were far away from the generality which would be desirable from a physical point of view but they are good enough to play the role of matter sources when studying the Newtonian limit. This has been exploited by Oliynyk who used ‘Makino fluids’ as matter source in his results. The formulation of the Newtonian limit involves a family of solutions of he Einstein-matter equations depending on a parameter , roughly corresponding to where is the speed of light. The Newtonian limit is then the limit . What I proved in 1994 was the existence of families which are continuous in at . It is also interesting to know how smooth the family is at . The derivatives, when they exist, define higher order approximations to general relativity called the post-Newtonian approximations (PN for derivatives). I only got 0PN. Oliynyk has in the meantime reached 2PN. Results have been obtained for the asymptotically flat case which is the one most frequently considered in physics. It is well known that after 2PN the simple expansion breaks down. How can this be understood? My explanation (a little vague) is as follows. We are trying to approximate something which is bounded in its dependence on the spatial variables. Unfortunately above the 2PN level the approximation is not uniform and the coefficients in the expansion want to be unbounded. If you try to force them to be bounded by assumption the expansion breaks down. These coefficients are supposed to solve Poisson equations but the right hand sides have poor decay. The physicists typically try to represent the solutions of the Poisson equations in terms of the fundamental solution and get divergent integrals.

In his talk here Oliynyk reported that he has been able to treat post-Newtonian expansions of arbitrary order in the cosmological (spatially compact) case. It was a pleasant surprise for me that this works at all. When solving the Poisson equations in the cosmological case the right hand sides must have integral zero. It is remarkable that this works out at all. Once it is known that the procedure works at all, even for low orders, the intuition presented above makes it plausible that the obstructions familiar from the asymptotically flat case will not come up.

Returning to the asymptotically flat case, methods based on the post-Newtonian approximations are used to do the theoretical modelling for gravitational wave detectors whose cost is of the order of a billion dollars. It is an interesting comment on the role of mathematics in applications that nobody seems to worry too much about the almost entire lack of a rigorous mathematical foundation for these methods. In any case, the work I have been reporting on here represents the first steps on the road to changing this.

I have just returned from a conference at the Mathematical Sciences Research Institute (MSRI) in Berkeley with the title ‘Hot topics: black holes in relativity‘. The central theme of this conference was the work of Demetrios Christodoulou on the formation of black holes in vacuum which I discussed in a previous post

On the first day of the conference I gave a talk on the characteristic initial value problem in general relativity. This was based on a paper I wrote more than twenty years ago (Proc. R. Soc. Lond. A427, 221 – I find it difficult to believe that it has been so long). The result of this paper is used in Christodoulou’s work and this was the main justification for the talk. In the ordinary initial value problem (Cauchy problem) for a hyperbolic system, or for the Einstein equations, initial data are prescribed on a spacelike hypersurface. The idea of the characteristic initial value problem is to instead prescribe data on a characteristic hypersurface. In fact it is necessary to use a singular characteristic hypersurface (such as a cone) or a pair of smooth hypersurfaces which intersect transversely. The result of Christodoulou is formulated in terms of the first of these possibilities, with data prescribed on a light cone. However he assumes that these data coincide with flat space data near the vertex of the cone, which allows the problem to be reduced to the second, easier possibility and it is the latter which I treated in my paper. In the result of that paper, which applies to smooth initial data, existence and uniqueness results for the characteristic initial value problem are deduced from the corresponding results for the Cauchy problem. In the ordinary Cauchy problem for the Einstein equations it is necessary to solve the constraint equations, which means solving an elliptic problem. In the characteristic case the constraints reduce to a hierarchical system of ordinary differential equations, which can be a big advantage.

During the conference Christodoulou gave five talks about his theorem and its proof and I found these very enlightening. I feel I have a much better understanding of the basics of this work now that I did before. One aspect of the result is that the data used are in one sense small (close to flat space data) and in another sense large. If they were small in a sufficiently strong sense then this should lead to a global existence result which in particular rules out the formation of black holes due to the theorem of Christodoulou and Klainerman on the stability of Minkowski space. On the other hand the interpretation of the result (formation of a trapped surface starting from a weak-field situation) requires that the data be small in some sense. Combining these two requirements (smallness and largeness in different senses) is a key feature of the theorem. It is also the case that the data are in some sense close to being spherically symmetric but in another sense far from spherical symmetry. Intuitively, it is necessary to have data which represent a sufficiently strong pulse of gravitational radiation. Spherical symmetry rules out gravitational radiation and this might be extrapolated to say that being close to spherical symmetry means restricting to a small amount of radiation.

In the proof of the theorem the solution is parametrized in the following way. The initial hypersurface is a null cone . It can be foliated by surfaces which are of constant affine distance from the vertex. Through each of these there is a null hypersurface transverse to which is taken to be a level hypersurface of a function . This function agrees with the affine distance (suitably normalized) on . A function is defined to be constant on the null cones of the points on a timelike curve passing through the vertex of the cone. Things are always set up so that these null hypersurfaces have no caustics. The two functions define a foliation by spheres by means of the intersections of their level hypersurfaces. This foliation is in a sense the analogue of that by symmetry orbits in a spherically symmetric problem. The fact that the problem is almost spherically symmetric is witnessed by the fact that the Gaussian curvature of these spheres is almost constant. Note that the gradients of the functions and do not commute as vector fields in general. Thus they are not tangent to surfaces and this is an important difference from spherical symmetry.

The initial data is such that a suitable energy density on the cone changes suddenly from being zero to being sufficiently large. This is the basis of the short pulse method, which is the central new technique in the proof of the theorem. What is this energy density? It is the norm squared of the trace-free part of the second fundamental form of the spheres in the direction along the cone.

When Christodoulou had completed his last lecture someone in the audience asked, ‘What’s next?’ In reply he announced that this had been his last project in general relativity, which came as quite a shock to the audience. The word ‘announced’ is perhaps not appropriate since it sounds too formal – he just said it spontaneously. This is sad news for the field of mathematical relativity but perhaps it is less sad in a wider context. After all, Christodoulou has a number of fascinating projects he is working on in other areas. At the same time the theorem I have been talking about here will probably be a beginning rather than an end. At the conference Igor Rodnianski gave a talk on work he has been doing with Sergiu Klainerman aimed at generalizing this result while understanding it more deeply. I look forward to seeing where that will lead.