The practical use of proofs

A question which has come up in this blog from time to time is that of the benefits of mathematical proofs. Here I want to return to it. The idea of a proof is central in mathematics. A key part of research papers in mathematics is generally stating and proving theorems. In many mathematics lectures for students a lot of the time is spent stating and proving theorems. In practical applications of mathematics, to theoretical parts of other sciences or to real life problems, proofs play a much less prominent role. Very often theoretical developments go more in the direction of computer simulations or use heuristic approaches. A scientist used to working in this way may be sceptical about what proofs have to contribute. Thus it is up to us mathematicians to understand what mathematics has to offer and then to explain it to a wider circle of scientists.

Thinking about this subject I remembered some examples from my earlier research area in mathematical relativity. Some years ago, at a time when dark energy and accelerated cosmological expansion were not yet popular research subjects, a paper was published containing numerical simulations of a model of the universe with oscillating volume. In other words in this model the expansion of the universe alternately increased and decreased. This can only happen if there is accelerated expansion at some time. In the model considered there was no cosmological constant and no exotic matter with negative pressure. In this situation it has been well known for a very long time that cosmic acceleration is impossible. Proving it is a simple argument using ordinary differential equations. Apparently this was not known to the authors since they presented numerical results blatantly contradicting this fact. This is an example of the fact that once something has been proved it can sometimes be used to show immediately that certain results must be wrong, although it does not always indicate what has gone wrong. So what was wrong in the example? In the problem under consideration there is an ordinary differential equation to be solved, but that is not all. To be physically relevant an extra algebraic condition (the Hamiltonian constraint) must also be satisfied. There is an analytical result which says that if a solution of the evolution equation satisfies the constraint at some time it satisfies it for ever. Numerically the constraint will only approximately be satisfied by the initial data and the theorem just mentioned does not say that, being non-zero, it cannot grow very fast so that at later times the constraint is far from being satisfied. Thus the situation was probably as follows: the calculation successfully simulated the solution of an ordinary differential equation, even the right ordinary differential equation, but it was not the physically correct solution. Incidentally the error in the constraint tends to act like exotic matter.

A similar problem came up in a more sophisticated type of numerical calculation. This was work by Beverly Berger and Vincent Moncrief on the dynamics of inhomogeneous cosmological models near the big bang. The issue was whether the approach of these solutions to the limit is monotone or oscillatory. The expectation was that it should be oscillatory but at one point the numerical calculations showed monotone behaviour. The authors were suspicious of the conclusion although they did not understand the problem. Being careful and conscientious they avoided publishing these results for a year (if I remember correctly). This was good since they found the explanation for the discrepancy. It was basically the same explanation as in the previous example. They were using a free evolution code where the constraint is used for the initial data and never again. Accumulating violation of the constraint was falsifying the dynamics. In this case the sign of the error was different so that it tended to damp oscillations rather than creating them. Changing the code so as to solve the constraint equation repeatedly at later times restored the correct dynamics.

Another example concerns a different class of homogeneous cosmological models. Here the equations, which are ordinary differential equations, can be formulated as a dynamical system on the interior of a rectangle. This system can be extended smoothly to the rectangle itself. Then the sides of the rectangle are invariant and the corners are stationary solutions. The solutions in the boundary tend to one corner in the past and another corner in the future. They fit together to form a cycle, a heteroclinic cycle. I proved together with Paul Tod that almost all solutions which start in the interior converge to the rectangle in the approach to the singularity, circling infinitely many times around it. The top right hand corner of the rectangle, call it , plays an important role in the story which follows. A relatively straightforward numerical simulation of this system suggested that there were solutions which started close to and converged to it directly, in contradiction to the theorem. A more sophisticated simulation, due to Beverly Berger, replaced this by the scenario where some solutions starting close to go once around the rectangle before converging to . Solutions which go around more than once could not be found, despite trying hard. (The results I am talking about here were never published – I heard about them by word of mouth.) So what is the problem? The point is non-hyperbolic. The linearization of the system there has a zero eigenvalue. The point is what is called a saddle node. On one side of a certain line (corresponding to one side of the rectangle) the solutions converge to . If, due to inaccuracies in the calculation, the solution gets on the wrong side of the line we get the first erroneous result. On the physical side of the line the point is a saddlle and all physical solutions pass it without converging to it. The problem is that, due the non-hyperbolic nature of a solution which starts near the rectangle and goes once around it comes back extremely close to it. It fact it is so close that a program using a fixed number of decimal digits can no longer resolve the behaviour. Thus, as was explained to me by Beverly, this type of problem can only be solved using a program which is capable of implementing an arbitrarily high number of digits. At the time it was known how to do this but Beverly did not have experience with that and as far as I know this type of technique has never been applied to the problem. (The reader interested in details of what was proved is referred to the paper in Class. Quantum Grav. 16, 1705 (1999).)

The general conclusions which are drawn here are simple and commonplace. The computer can be a powerful ally but the reliability of what comes out is strongly dependent on how appropriate the formulation of the problem was. This cannot always be known in advance and so continuous vigilance is highly recommended. Sometimes theorems which have been proved can be powerful tools in discovering flaws in the formulation. In the case of arguments which are analytical but heuristic the same is true but in an even stronger sense. The reliability of the conclusions depends crucially on the individual who did the work. At the same time even the best intuition should be subjected to careful scrutiny. The most prominent example of this in general relativity is that of the singularity theorems of Penrose and Hawking starting in the mid 1960’s which led spacetime singularities to be taken seriously on the basis of rigorous proofs which contradicted the earlier heuristic work of Khalatnikov and Lifshitz.

The general questions discussed here are ones I will certainly return to. Here I have highlighted two dangerous situations: simulations which do not exactly implement conservation laws in the system being modelled and non-hyperbolic steady states of a dynamical system.

The relativistic Boltzmann equation

In a previous post I wrote about the Einstein-Boltzmann system and some recent work on that subject by Ho Lee and myself. One of the things we realized as a consequence of this work is that the known local existence theorem for the Einstein-Boltzmann system requires very restrictive assumptions on the collision kernel. Now we have looked in more detail at other kinds of collision kernel which are closer to what is desirable from the point of view of the physical applications. As a result of this we have written a paper which is concerned with the hard potential type of collision kernel. The subject of the paper is the Boltzmann equation in special relativity or on a homogeneous and isotropic background. This is intended to prepare the ground for similar work on the coupled Einstein-Boltzmann system. The main results are global existence theorems for spatially homogeneous solutions of the Boltzmann equation without any small data restriction. They are analogous to results obtained previously by Norbert Noutchegueme and collaborators for a more restrictive type of collision kernel.

The collision kernel is a function of the relative momentum and the scattering angle . In the case of the classical (i.e. non-relativistic) Boltzmann equation a type of collision kernel which has often been studied is that arising from a power-law interaction between particles. The corresponding collision kernel has a power-law dependence on and a dependence on which cannot be determined explicitly. It has a non-integrable singularity in at . It has been observed that properties of solutions of the Boltzmann equation determine two different regimes for the exponent of . The cases between an inverse square and an inverse fifth power force between particles are known as soft potentials. The exponent of varies from to . Cases with powers of the force more negative than are known as hard potentials. As the power in the force ranges from to the exponent of varies from to zero. When the exponent of is equal to   there are simplifications in some calculations and this has led to this case being popular among theorists. It is called Maxwell molecules. The limit where the exponent of tends to zero corresponds to the case of collisions of hard spheres.

Given the importance of the distinction between the soft potential and hard potential cases in the theory of the classical Boltzmann equation it is natural to look for an analogous distinction in the relativistic case. This was done by Dudynski and Ekiel-Jezewska. The analogy seems to be not at all straightforward. This work was carried further by Robert Strain and collaborators, who were able to apply these concepts and obtain a variety of global existence results. In their work the data are not required to be symmetric but are assumed to be close to data for known solutions. Our work is at the opposite extreme with a very strong symmetry assumption (spatially homogeneous) but no smallness requirement. It is modelled on theorems for the classical Boltzmann equation due to Mischler and Wennberg in Annales IHP (Analyse non lineaire) 16, 467. There is an analogue of Maxwell molecules in the relativistic case called Israel molecules but the analogy is not simple. My global conclusion from my experience with this problem is that there are a lot of interesting and challenging open problems around in the study of the relativistic Boltzmann equation and the Einstein-Boltzmann system.

Stable big bang singularities

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.

Stability of heteroclinic cycles

A heteroclinic orbit is a solution of a dynamical system which converges to one stationary solution in the past and to another stationary solution in the future. A heteroclinic chain is a sequence of heteroclinic orbits where the past limit of each orbit is the future limit of the preceding one. If this sequence is periodic we get what is called a heteroclinic cycle. Given such an object it is of interest to ask about its stability. For an initial datum sufficiently close to the cycle, when does the corresponding solution converge to the cycle at late times? In particular, when is the -limit set of the solution of interest equal to the entire cycle? To obtain information about this question it is useful to consider the linearization of the system about the vertices of the cycle. For a solution of the kind we are looking for, if it exists, will spend most of its time near the stationary points which are the vertices. If during time periods near a vertex it tends to approach the cycle then this is a good sign that the whole solution will approach the cycle. The behaviour of the solution near the stationary solution is determined by the linearization, at least if the stationary solution is hyperbolic.

In a previous post I described a result of Stefan Liebscher and collaborators which provides detailed information on the nature of the initial singularities of some spatially homogeneous spacetimes which are vacuum or where the matter content is described by a perfect fluid with linear equation of state . In that situation the Einstein equations can be reduced to a system of ordinary differential equations, the Wainwright-Hsu system, which treats all Bianchi class A models in a unified way. In particular it includes the type IX models. There is a heteroclinic cycle consisting of three Bianchi type I vacuum solutions. The main theorem of the paper is that there is a codimension one submanifold of initial data for which the -limit set of the corresponding solution is the heteroclinic cycle just described. The qualitative nature of this result is just as in the general discussion above except that the direction of time has been reversed. The system for vacuum solutions is four-dimensional. The vertices of the cycle are embedded in a one-dimensional manifold of stationary solutions and so the linearization must have at least one zero eigenvalue. As a consequence these vertices are not hyperbolic but the problem can be overcome. Of the remaining eigenvalues one is negative and the others are positive. The theorem makes use of the fact that for . In the presence of a fluid there is additional positive eigenvalue . The same idea of proof applies provided . This inequality is equivalent to an inequality for the parameter in the equation of state of the fluid.

An analogue of the vacuum solutions of type IX is given by solutions of type with a magnetic field. The dynamics of these solutions near the singularity was studied a long time ago by Marsha Weaver. In this situation there is a heteroclinic cycle essentially identical to that in the vacuum case. It is then natural to ask whether an analogue of the known theorem in the vacuum case in the paper by Stefan and collaborators holds. Together with Stefan and Blaise Tchapnda we have now written a paper on this subject. It turns out that there is a closely analogous result but that it is a lot harder to prove. The reason is that the eigenvalues of the linearization are in a less favourable configuration. Fortunately a weaker condition on the eigenvalues suffices. Suppose that denotes the eigenvalue at a vertex corresponding to the outgoing orbit in the cycle at that point. Then it suffices to assume that without imposing conditions of the other , provided that another condition on the existence of invariant manifolds is satisfied. The existence of these manifolds is a consequence of the geometric nature of the problem which gives rise to the dynamical system being considered. In this way we get a result on the stability of the heteroclinic cycle in the model with magnetic field. We are also able to remove the undesirable restriction on in the case with fluid. This work gives rise to a number of new questions on possible generalizations of this result. For more information on this I refer to the discussion section of our paper.

The Einstein-Boltzmann system

The Boltzmann equation provides a description of the dynamics of a large number of particles undergoing collisions, such as the molecules of a gas. The classical Boltzmann equation belongs to Newtonian physics. It has a natural relativistic generalization. The Boltzmann model is adapted to capture the effects of short-range forces acting on short time scales during collisions. The model can be extended to also include the effects of long-range forces generated collectively by the particles. If the forces are gravitational and the description is made fully relativistic then the system of equations obtained is the Einstein-Boltzmann system. In any of the cases mentioned up to now the Boltzmann equation is schematically of the form . The term on the left is a transport term giving the rate of change of the function , the density of particles, along a vector field on phase space. The vector field is in general determined by the long-range forces. The term on the right is the collision term which, as its name suggests, models the effect of collisions. It is an integral term which is quadratic in . The function itself is a function of variables representing time, position and velocity (or momentum).

How is the collision term obtained? It is important to realize that it is in no sense universal – it contains information about the particular interaction between the particles due to collisions. This can be encoded in what is called the scattering kernel. In the classical case it is possible to do the following. Fix a type of interaction between individual particles and solve the corresponding scattering problem. Each specific choice of interaction gives a scattering kernel. Once various scattering kernels have been obtained in this way it is possible to abstract from the form of the kernels obtained to define a wider class. A similar process can be carried out in special relativity although it is more complicated. Any scattering kernel which has been identified as being of interest in special relativity can be taken over directly to general relativity using the principle of equivalence. Concretely this means that if the Boltzmann collision term is expressed in terms of the components of the momenta in an orthonormal frame
then the resulting expression also applies in general relativity.

For a system of evolution equations like the Einstein-Boltzmann system one of the most basic mathematical questions is the local well-posedness of the initial value problem. For the EB system this problem was solved in 1973 by Daniel Bancel and Yvonne Choquet-Bruhat (Commun. Math. Phys. 33, 83) for a certain class of collision terms. The physical interpretation of the unknown in the Boltzmann equation as a number density means that it should be non-negative. In the context of the initial value problem this means that it should be assumed that is initially non-negative and that it should then be proved that the corresponding solution is non-negative. In the existence proofs for many cases of the Boltzmann equation the solution is obtained as the limit of a sequence of iterates, each of which are by construction non-negative. The convergence to the limit is strong enough that that the non-negativity of the iterates is inherited by the solution. In the theorem of Bancel and Choquet-Bruhat the solution is also constructed as the limit of a sequence of iterates but no attention is paid to non-negativity. In fact that issue is not mentioned at all in their paper. To prove non-negativity of solutions of the EB system it is enough to prove the corresponding statement for solutions of the Boltzmann equation on a given spacetime background. The latter question has been addressed in papers of Bichteler and Tadmon. On the other hand it is not easy to see how their results relate to those of Bancel and Choquet Bruhat. This question has now been investigated in a paper by Ho Lee and myself . The result is that with extra work the desired posivity result can be obtained under the assumptions of the theorem of Bancel and Choquet-Bruhat. While working on this we obtained some other insights about the EB system. One is that the assumptions of the existence theorem appear to be very restrictive and that treating physically motivated scattering kernels will probably require more refined approaches. In the almost forty years since the local existence theorem there have been very few results on the initial value problem for the EB system (with non-vanishing collision term). We hope that our paper will set the stage for further progress on this subject.

Fuchsian equations

Fuchsian equations are a class of differential equations which have played a big role in my research and which have now found a niche in the mathematical relativity community. In fact they have a very wide range of applications. The story begins with Lazarus Fuchs in the mid nineteenth century. He was interested in the case of a higher order linear scalar ODE for a complex-valued function whose coefficients are analytic except for isolated singularities. The cases which I have mainly been concerned with are first order nonlinear systems of PDE for real-valued functions whose coefficients may only be smooth. It sounds like almost everything is different in the two cases and so what is the similarity? The basic idea is that a Fuchsian problem concerns a system of PDE with a singularity in the coefficients where the desire is to describe the behaviour of solutions near the singularity. If certain structural conditions are satisfied then for any formal power series solution of the problem there is a corresponding actual solution. In the original situation of Fuchs the series converged and the solution was its limit. More generally there need be no convergence and the series is only asymptotic. A related task is to find singular solutions of a regular system. By suitable changes of variables this can sometimes be transformed to the problem of finding a regular solution of a singular system.

With hindsight my first contact with a problem of Fuchsian type was in some work I did with Bernd Schmidt (Class. Quantum Grav. 8, 985) where we proved the existence of certain models for spherically symmetry stars in general relativity. Writing the equations in polar coordinates leads to a system of ODE with a singularity corresponding to the centre of symmetry. We proved the existence near the centre of solutions satisfying the appropriate boundary conditions by doing a direct iteration. In later years I was interested in the problem of mathematical models of the big bang in general relativity. I spent the academic year 1994-95 at IHES (I have written about this in a previous post) and in that period I often went to PDE seminars in Orsay. On one occasion the speaker was Satyanad Kichenassamy and his subject was Fuchsian equations. This is a long term research theme of his and he has written about it extensively in his books ‘Nonlinear Wave Equations’ and ‘Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics’. I found the talk very stimulating and after some time I realized that this technique might be useful for studying spacetime singularities. Kichenassamy and I cooperated on working out an application to Gowdy spacetimes and this resulted in a joint paper. A general theorem on Fuchsian systems proved there has since (together with some small later modifications) been a workhorse for investigating spacetime singularities in various classes of spacetimes.

One of the highlights of this development was the proof by Lars Andersson and myself (Commun. Math. Phys. 218, 479) that there are very general classes of solutions of the Einstein equations coupled to a massless scalar field whose initial singularities can be described in great detail. In later work with Thibault Damour, Marc Henneaux and Marsha Weaver (Ann. H. Poincare 3, 1049) we were able to generalize this considerably, in particular to the case of solutions of the vacuum Einstein equations in sufficiently high dimensions.  For these results no symmetry assumptions were necessary. More recently these results were generalized in another direction by Mark Heinzle and Patrik Sandin (arXiv:1105.1643). Fuchsian systems have also been applied to the study of the late-time behaviour of cosmological models with positive cosmological constant. In the meantime there are more satisfactory results on this question useing other methods (see this post) but this example does show that the Fuchsian method can be applied to problems in general relativity which have nothing to do with the big bang. In general this method is a kind of machine for turning heuristic calculations into theorems.

The Fuchsian method works as follows. Suppose that a system of PDE is given and write it schematically as . I consider the case that the equation itself is regular and the aim is to find singular solutions. Let be an explicit function which satisfies the equation up to a certain order in an expansion parameter and which is singular on a hypersurface defined by . Look for a solution of the form where is less singular than . The original equation can be rewritten as , where is singular. Now the aim is to show that there is a unique solution of the transformed equation which vanishes as . A theorem of this kind was proved in the analytic case in the paper mentioned above which I wrote with Kichenassamy. Results on the smooth case are harder to prove and there are less of them known. A variant of the procedure is to define as a solution (not necessarily explicit) of an equation which is a simplified version of the equation . In the cases of spacetime singularities which have been successfully handled the latter system is the so-called velocity-dominated system.

Twisting Gowdy

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 . 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 . 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, and .

In Gowdy spacetimes there are two of the Einstein equations for functions and which are of central importance. I call them the Gowdy equations. In order to describe Gowdy spacetimes the functions and should be periodic in . To accomodate Bianchi models of types II and 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 , 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 . For type II and type 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.

Self-similar solutions of the Einstein-Vlasov system

The Einstein-Vlasov system describes the evolution of a collisionless gas in general relativity. The unknown in the Vlasov equation is a function , the number density of particles with position and velocity at time . A regular solution is one for which the function is smooth (or at least ). These equations can be used to model gravitational collapse in general relativity, i.e. the process by which a concentration of matter contracts due to its own weight. I concentrate here on the case that the configuration is spherically symmetric since it is already difficult enough to analyse. It has been known for a long time that a solution of this system corresponding to a sufficiently small concentration of matter does not collapse. The matter spreads out at late times, with the matter density and the gravitational field tending to zero. More recently it has been proved that there is a class of data for which a black hole is formed. In particular singularities occur in these equations. It is of interest to know whether singularities can occur which are not contained in black holes. This is the question of cosmic censorship.

One way of trying to prove cosmic censorship involves investigating whether general initial data give rise to solutions which are global in a certain type of coordinate system. In spherical symmetry this has been looked at for the Schwarzschild coordinates. Here we can ask whether solutions are always global in Schwarzschild time. This statement is consistent with the presence of a black hole since in that case it can happen that the coordinate system being considered only covers a region outside the black hole. For general smooth spherically symmetric initial data this global existence question is open. A number of people, including myself, have put a lot of time and effort into proving global existence for this problem but this enterprise has not yet been successful. In view of the fact that research on this subject seems to be stuck it makes sense to think about trying to prove the opposite statement, in other words to prove that there are data for which global existence fails. This might in principle lead to either a positive or a negative answer to the global existence problem. An investigation of this type is being carried out by Juan Velázquez and myself and we have just written a paper on this. I will now explain what we were able to prove.

A type of matter which is frequently studied in general reletivity is dust, a perfect fluid with zero pressure. It has unpleasant mathematical properties and a strong tendency to form singularities even in the absence of gravity. Solutions of the Einstein-dust system can be interpreted as solutions of the Einstein-Vlasov system which instead of being regular have a Dirac dependence on the velocity variables. For fixed the support of in is a single point. In a smooth solution this support is three-dimensional. In the paper we look at a class of solutions where the support is two-dimensional. These solutions are self-similar. The construction of solutions of the type considerd in the paper can be reduced to the study of certain solutions of a four-dimensional dynamical system depending on two parameters. There is a point is the phase space defined by the application and a stationary solution depending on the parameters. What needs to be shown is that for suitable values of the parameters the solution which starts at converges to for large values of the independent variable. In more detail, when one of the parameters is fixed to be positive and sufficiently small there exists a value of the other paramater for which this statement holds. The proof is a shooting argument. This uses a family of initial data for a three-dimensional dynamical system depending on a parameter . (The relation of the three-dimensional system to the original four-dimensional one is too complicated to be described here.) It is proved that the solution has one type of asymptotic behaviour for small values of and another for large values of . It is shown that there must be at least one intermediate value of the parameter for which the asymptotics is of the type required to construct the solution of interest of the Einstein-Vlasov system. Shooting arguments are rather common when constructing solutions of ODE numerically. Here we have an example where similar ideas can be used to obtain an existence proof.

Black strings in Banff

This week I am attending a conference at the Banff International Research Station (BIRS) in the Canadian Rockies. This is an institution modelled to some extent on the Mathematical Research Institute in Oberwolfach which I wrote about in a previous post. Of course there are some differences, e.g. a slightly higher probability of meeting a grizzly bear if you go for a walk in the surroundings. The programme of talks is quite full but in spare moments I have had time to enjoy seeing common North American birds like American Robin and Red-winged Blackbird. Not surpisingly, this area bears no resemblance whatsoever to the town of Banff in Scotland which it is named after.

An interesting mathematical phenomenon is the breakup of a cylindrical configuration into to a row of spherical ones. This plays a role in the study of the collective motion of bacteria. It also comes up in the context of solutions of the vacuum Einstein equations in dimensions greater than four called black strings. A black string is obtained as the product of a spherical black hole (the Schwarzschild solution or its generalization to higher dimensions) with a circle of length . Today Frans Pretorius gave a talk here about the instability of black strings in five dimensions. Before I write about that I need to present some background.

The Schwarzschild solution is an explicit solution of the vacuum Einstein equations which represents a non-rotating black hole. As for any explicit solution of an equation in physics its physical significance depends on its stability to perturbations. Only configurations which exhibit some stability can be expected to be seen in reality. A complication related to the stability of the Schwarzschild solution is that one type of perturbations which certainly exist are those which make the black hole rotate a little. This results in explicit solution, the Kerr solution. Its existence means that if we want to prove the stability of the Schwarzschild solution we actually need to prove the stability of the Kerr solution. This is a very hard problem. Even the stability of empty space (Minkowski space) in general relativity is hard to prove. A natural strategy to make progress with black hole stability is to look at simpler related problems. One possibility is to linearize the equations around Kerr. This results in a system of linear hyperbolic equations. A further simplification is to replace this system by the linear wave equation. The linear wave equation on Kerr and the boundedness and decay of its solutions has been a focus of attention in mathematical relativity in the past few years. This problem can now be regarded as solved. In a talk yesterday Mihalis Dafermos gave a review of this including, in particular his recent work with Igor Rodnianski. Going from this to the linearized Einstein equations involves new difficulties. Even the formulation of the problem is subtle. Some progress on the problem of the linearized Einstein equations was reported on in a talk of Gustav Holzegel.

It is generally believed that the Schwarzschild solution is stable and linearized calculations supporting this go back a long way. The new mathematical results all tend to provide confirmation of this picture. In the case of black strings it was suggested in a 1988 paper of Gregory and Laflamme that they are also stable on the basis of a linearized analysis. This conclusion was reversed in a 1993 paper of the same authors and the ‘Gregory-Laflamme instability’ was born. It occurs when is too large in comparison with the mass . The defect in the analysis of the earlier paper seems to be related to the boundary conditions imposed on the perturbation and their interaction with the mode decomposition which was also used. Once this instability had been discovered it was natural to ask what the endpoint of the evolution of a solution is which originally starts as a growing small perturbation of the black string. The subject of the talk of Pretorius was this question in dimension five. One possibility would be a modification of the standard black string with a characterstic radius depending on the extra direction. This kind of object has been called a non-uniform black string (NUBS) and there is some evidence that objects of this kind exist. Another possibility is that the cylindrical configuration could break up into a row of black holes. Calculations from a few years ago showed this kind of thing but were not definitive since the calculation broke down too soon. Today Pretorius showed impressive numerical results where the calculation runs much longer and reveals surprising phenomena. The diagnostic quantity which is plotted is the radius of the apparent horizon as a function of the extra spatial coordinate and time. To start with some spherical lumps appear in the string and these become more and more pronounced with the part of the string between the lumps becoming thinner and thinner. Being thinner corresponds to a smaller mass and increased susceptibility to the Gregory-Laflamme instability. As time passes new lumps appear and grow to a certain size. The number of these grows larger and larger and it may be that this process takes place infinitely often within a finite time interval. The pictures were beautiful but rather frightening in their complexity. It was mentioned that relations have been made between this process and the Rayleigh-Plateau instability which occurs in the break-up of a stream of water into individual drops.

Mixmaster in motion

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 -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 ) can occur as -limit sets. Apart from three exceptional points (the Taub points) each point of the Kasner circle is the -limit set of a unique solution of Bianchi type II. The -limit set of that solution of Bianchi type II is also a point of the Kasner circle, call it . This defines a mapping from the complement of the Taub points in to , 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 -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 then the result is restricted to the range . 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 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 . 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 is not true. There is an analogue of the Hartman-Grobman theorem in the 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.