Archive for the ‘general relativity’ Category

Shock waves, part 3

April 6, 2010

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

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

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

Asymptotic discrete self-similarity in cosmological models

January 12, 2010

In the previous post I discussed some ideas from the theory of dynamical systems and mentioned some applications of these ideas to models arising from chemistry and biology. Here I will explain an example where they can be applied to mathematical cosmology. In cosmology it is common to study spatially homogeneous solutions of the Einstein equations with matter modelled by a perfect fluid with linear equation of state p=(\gamma-1)\rho. In a spatially homogeneous solution there is a preferred foliation by spacelike hypersurfaces defined by the orbits of the symmetry group. An important simplifying assumption is that the four-velocity u^\alpha of the fluid is orthogonal to these hypersurfaces. If this assumption is not satisfied the fluid is said to be tilted and the dynamics of the models becomes significantly more complicated. Physically tilt means that in a frame of reference where the energy density is spatially constant there is a bulk motion of the fluid.

Starting with a paper of Coley and Hervik it has been observed that solutions of the Einstein equations with tilted fluid can show dynamical behaviour which is qualitatively different from anything which is possible in the absence of tilt. The approach used combines analytical and numerical techniques. I have been aware of this work and found it interesting for a long time but I now see that I had not gone into it deeply enough to get a real understanding of what is going on there. This dynamics involves periodic solutions and Hopf bifurcations, which is why it is related to the previous post. The interesting new phenomena occur for certain Bianchi types of class B, namely types {\rm IV}, {\rm VI} {}_h and {\rm VII} {}_h. Some background information on this terminology can be found here. The relevant dynamical regime is that where the spacetime expands forever. Many of the basic quantities tend to zero at late time and to get a less degenerate description it is useful to introduce a dynamical system for certain dimensionless quantities. When this is done a stationary solution of the dimensionless system corresponds to a continuously self-similar solution of the original Einstein-Euler system and a periodic solution of the dimensionless system corresponds to a discretely self-similar solution of the original system. The latter type of solution will be of particular interest in what follows.

In the classes of solutions considered here there are stationary solutions of the dimensionless systems which play the role of late time attractors in most cases. The solutions are asymptotically self-similar at late times. There is, however, a small exceptional region called the loophole where no stable self-similar solutions are available. In that case the asymptotic behaviour is more complicated. The dimensionless variable \Omega corresponding to the energy density of the fluid tends to zero at late times. Thus in a sense the solution is converging to a vacuum solution. However certain variables describing the fluid retain a non-trivial dynamics so that in a sense the fluid leaves a trace on the vacuum solution. More precisely, in the limit the spacetime geometry converges in a suitable sense to that of a solution of the vacuum Einstein equations while the fluid variables describing the tilt converge to a test fluid on that background. A test fluid means that while the fluid satisfies the Euler equations its energy-momentum tensor does not make a contribution to the Einstein equations. The limiting dynamical system is defined on the unit ball in R^3 with the unit sphere being an invariant submanifold. On the unit sphere the test fluid can be thought of as null dust, where the four-velocity is null. The parameter \gamma in the equation of state vanishes completely from the restrictions of the equations for the tilt variables to the boundary sphere. In terms of the dimensionless dynamical system the solutions with test fluid are not solutions of that system but solutions of a smooth extension of that system to part of the boundary of the physical region. The restriction of the extended system to that part of the boundary will be called the asymptotic system.

For some values of the parameters the asymptotic system has periodic solutions. These can be of different types. Probably the simplest case is that described in Theorem 4.3 of gr-qc/0409106. It concerns Bianchi type {\rm VII}{}_h. There the system on the boundary sphere has precisely two stationary points which are hyperbolic sources. Thus it follows immediately by Poincaré-Bendixson theory that there exists a periodic solution. The uniqueness of this periodic solution which is also part of the assertion of the theorem requires a more complicated proof. A Hopf bifurcation occurs explicitly in the treatment of Bianchi type {\rm VI}{}_{-\frac19} in arXiv:0706.3184 where the authors give the expression for the first Lyapunov number in this case.

The cosmic no hair theorem

October 15, 2009

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.

The Newtonian limit of general relativity

October 14, 2009

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 \lambda, roughly corresponding to c^{-2} where c is the speed of light. The Newtonian limit is then the limit \lambda\to 0. What I proved in 1994 was the existence of families which are continuous in \lambda at \lambda=0. It is also interesting to know how smooth the family is at \lambda=0. The derivatives, when they exist, define higher order approximations to general relativity called the post-Newtonian approximations (nPN for n 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.

Formation of black holes in vacuum, part 2

September 22, 2009

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 C_0. 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 C_0 which is taken to be a level hypersurface of a function \underline{u}. This function agrees with the affine distance (suitably normalized) on C_0. A function u 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 u and \underline{u} 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.

The Mixmaster model

June 24, 2009

The Mixmaster model arises in the study of solutions of the vacuum Einstein equations which are spatially homogeneous. To say that a solution is spatially homogeneous means that it is invariant under a group of symmetries whose orbits are three-dimensional and spacelike. This implies that it can be described by functions depending only on time and that the Einstein equations reduce to a system of ordinary differential equations. The details of these equations depend on the group defining the symmetry. In all but one case this group can be taken to be three-dimensional. Then classifying the symmetry type essentially comes down to classifying all three-dimensional Lie algebras.This was done by Bianchi in the 1890’s. He introduced types I to IX and this terminology is used in the study of the Einstein equations to this day. Types VI and VII are actually one-parameter families of non-isomorphic Lie algebras and are therefore denoted by VI{}_h and VII{}_h, where h is the parameter. The most general Bianchi types (in the sense that the solutions depend on the largest number of parameters) are types VIII, IX and VI{}_{-\frac19}. All Bianchi types contain solutions with initial singularities and the three types just listed exhibit complicated oscillatory behaviour in the approach to the singularity. The case which has been studied most is Bianchi IX and this was christened the Mixmaster model by Charles Misner in 1969. He named it after a kind of food mixer which was popular at the time. There has been a great deal of heuristic and numerical work on the dynamics of this system of ODE. Rigorous mathematical results have been relatively rare. The most notable results in this direction are those arising from the PhD thesis of Hans Ringström, some of which will now be described.

It is useful in the study of Bianchi models to distinguish between the Lie algebras which are unimodular (trace of the structure constants zero) and the rest. These are called Class A and Class B respectively. All models of Class A can be incorporated into a single dynamical system. The dynamical systems for types VIII and IX occupy open subsets of the state space and the other types are represented on lower dimensional submanifolds on the boundaries of these open sets. Class A consists of the types I, II, VI{}_0, VII{}_0, VIII and IX. It was shown by Ringström that the limit points of solutions of Bianchi type IX in the approach to the singularity (technically the \alpha-limit points) are of type I and II. For almost all initial data it was proved that the approach to the singularity is oscillatory in the sense that there are at least three distinct \alpha-limit points. More details can be found in this paper. The wider context of these results and aspects of the history of the problem are discussed in the paper ‘Mixmaster: Fact and Belief‘ by Mark Heinzle and Claes Uggla.

Recently there has been renewed interest in these problems. A project which I have started in cooperation with the research group of Bernold Fiedler has generated a lot of activity. The central idea is to bring together knowledge about the concrete application with sophisticated techniques from the theory of dynamical systems. What are the problems to be solved? A first question is whether the direct analogues of the Bianchi IX results hold for type VIII. This has not been proved and there are serious reasons for doubting if it is even true. The key issue, at least at our present state of understanding is whether Bianchi VIII solutions may have \alpha-limit points of type VI{}_0. For type VI{}_{-\frac19} almost nothing is known about the corresponding questions on a rigorous level.

Misner’s original motivation for introducing the Mixmaster model was to find an explanation for the observed isotropy of the universe. Whether this explanation can work depends on the nature of the causal structure of the geometry near the singularity. These days it is usual to explain the isotropy of the universe on the basis of inflation and so Misner’s proposal no longer has the same direct physical significance. The mathematical problem remains and has lost none of its intrinsic fascination. In this context it is desirable to understand the detailed structure of the \alpha-limit sets of generic and non-generic solutions. In particular there are three special points in the phase space, the flat Kasner solutions. Whether Misner’s mechanism is effective depends on how much time solutions spend close to these special points. Quantifying this requires a very detailed understanding of the asymptotics of solutions near the singularity. This issue is under intensive investigation. I hope to write about the results here when they are ready to come into the public domain.

These questions are of interest for the understanding of solutions far beyond the homogeneous case. This has to do with the picture of general singularities in solutions of the Einstein equations developed by Belinkii, Khalatnikov and Lifshitz, which is one of the most outstanding challenges in mathematical relativity.

Cosmological perturbation theory, part 2

June 16, 2009

In a previous post which I wrote several months ago I promised some more information about the work which Paul Allen and I have been doing on cosmological perturbation theory. Next Thursday I will give a talk on the subject at a conference in Lisbon. This work has been delayed by other commitments but now we posted a paper as arXiv:0906.2517. Here I will explain some of the results. In this paper we study the equation which describes so-called scalar perturbations. I will not attempt to explain the name but instead just say that these are the type of perturbations which cosmologists use to describe processes like the formation of the distribution of galaxies. In the simplest case these are described by the equation \Phi''+\frac{6(1+w)}{1+3w}\frac1{\eta}\Phi'=w\Delta\Phi. Here \Phi is a real valued function on {\rm R}\times T^3 where T^3 is the three-dimensional torus. The prime denotes a derivative with respect to a time coordinate \eta and \Delta is the Laplacian. The constant parameter w comes from the equation of state of the fluid, assumed to be of the form p=w\rho.

The aim of the paper is to obtain information about the asymptotics of general solutions of this equation in the limits t\to 0 and t\to\infty. It may be noted that this equation bears a certain resemblance to the polarized Gowdy equation. In fact we are able to import a number of techniques which have been used in the study of the Gowdy equations to understand the solutions of the equation above. The solutions can be parametrized by certain asymptotic data in each asymptotic regime. For the limit t\to 0 this data consists of two free functions which are coefficients in an asymptotic expansion of the form \sum_i\Phi_i(\eta) \zeta_i(t). For the limit t\to\infty it turns out to be useful to distinguish between solutions which are constant on the hypersurfaces of constant \eta and those whose integral over the torus is zero for each fixed \eta. Here I restrict consideration to the latter. It then turns out that after rescaling by a suitable power of \eta the solution looks like a solution of the flat space wave equation plus a remainder term. Thus in fact the asymptotics in each direction bears a strong qualitative resemblance to the asymptotics in the corresponding direction for solutions of the polarized Gowdy equations.

The paper also proves results about more general equations of state. Of course in that case the equation above is replaced by a more complicated one. It is assumed that in the appropriate limit the equation of state is the sum of a linear term and an expression which has an asymptotic expansion in powers of the energy density \epsilon which are negligible with respect to linear terms in the given regime. In many cases the modification does not make much of a difference and the leading order asymptotics is the same as in the corresponding linear case. An exception is the late time behaviour when the coefficient of the linear term vanishes so that the leading term in f(\epsilon) in the limit \epsilon\to 0 is proportional to \epsilon^{1+\sigma} for some positive \sigma. There is a bifurcation at \sigma=\frac13. When \sigma is smaller than this value the asymptotics is similar to that in the linear case. It does however happen that in the leading term the time coordinate in the solution of the flat space wave equation has to be distorted. For \sigma>\frac13 there is a more radical change in the asymptotics. In that case the behaviour looks more like what happens in the limit t\to 0. Waves which continue to propagate for ever are replaced by waves which freeze.In a sense the natural time coordinate for describing the dynamics is one which brings infinity to a finite value. This is reminiscent of the behaviour of the gravitational field in perturbed de Sitter spacetimes.

Respecting the matter in general relativity

April 25, 2009

General relativity is a theory which describes space, time and the gravitational field in terms of a Lorentzian metric g_{\alpha\beta}. A complete understanding of the gravitational field requires an understanding of the matter sources which generate it. In the Einstein equations, G_{\alpha\beta}=8\pi T_{\alpha\beta}, the left hand side depends only on g_{\alpha\beta} and is a feature of the geometry alone. On the other hand the right hand side, the energy-momentum tensor, depends not only on the metric but also on some matter fields. The right hand side of Einstein’s equations seems to have suffered from bad press coverage from an early stage. Einstein himself is often quoted as having said that the left hand side of his equations is made of marble while the right hand side is made of wood. I do not have a source for this quote – if anyone reading this does I would be grateful to hear about it. In this post I want to suggest treating that humble right hand side with more respect. If I lived in a palace made of marble with beautiful wooden furniture then I might be more impressed by the marble than by the wood. I would nevertheless do my best to prevent little boys from carving their initials into the furniture with penknives or the cat (much as I love cats) from using it as an accessory for the care of its claws.

The left hand side of the Einstein equations is universal within general relativity – it is always the same, no matter which type of physical situation is to be described. On the other hand the nature of the matter fields depends very much on what physical situation is to be described and what aspects of it are to be included in the description. It is necessary to make a choice of matter model. What is remarkable is that there is a large variety of choices which, in conjunction with the Einstein equations, lead to a consistent closed system of equations which bears no traces of the fact that other physical effects have been omitted. In fact there are three related choices which have to be made to set up the mathematical model in any given case. The first is the matter fields themselves – what kind of geometrical objects are they? The second is the expression which defines the energy-momentum tensor in terms of the matter fields and the metric. The third is the system of equations of motion which describe the dynamics of the matter. Note that in general the energy-momentum tensor depends explicitly on the metric. It is not possible to define an energy-momentum tensor unless the spacetime geometry is given. The same is true in the case of the equations of motion of the matter. They also contain the metric explicitly. Without the metric even the nature of the matter fields themselves can become ambiguous. Which positions should we choose for the indices of a tensor occurring in the description of the matter fields? From a physical point of view it is clear why the metric is necessary in so many ways. The mathematical model must be given a physical interpretation which involves the consideration of measurements. In the absence of a given geometry there is no way to talk about measurements.

When a matter model has been chosen the basic equations which are to be solved are the Einstein-matter equations, i.e the Einstein equations coupled to the equations of motion for the chosen type of matter. The unknowns are the metric and the basic matter fields. For any reasonable choice of matter fields the energy-momentum tensor has zero divergence as a consequence of the equations of motion. However the equations of motion in general contain more information than the divergence-free property of the energy-momentum tensor. For more discussion of these things together with examples see Chapter 3 of my book. I emphasize that solving the equations describing the physical situation within the given model means solving both the Einstein equations and the equations of motion of the matter. This is too often neglected in the literature. A particular danger occurs when the solutions under consideration are of low regularity. If the Einstein equations do not make sense pointwise then it should be checked that they hold in the sense of distributions. For solutions which lack regularity on a hypersurface this is expressed in the junction conditions and it is common in the literature to check that they hold. The equations of motion should also be satisfied in the sense of distributions and this is often ignored. When I use the phrase ‘in the sense of distributions’ here this is just a shorthand since the equations are nonlinear. The correct statement is that it is necessary to think carefully about the sense in which the equations are satisfied.

An example may help to make the importance of the issue clear. At the GR12 conference in Boulder in 1989 there was a heated discussion of the question, whether colliding plane waves can give rise to spontaneous creation of matter. (I emphasize that this discussion was in a purely classical context. Quantum theory was not being taken into account.) This kind of creation of matter sounds ridiculous from a physical point of view. Nevertheless people exhibited ‘solutions’ which showed this type of effect. Their mistake was that they had only verified those things which I said above were usual in the literature. They had not considered whether the equations of motion of matter were satisfied. If the equations of motion are ignored, it is not surprising that arbitrary things can happen.

In defence of cosmic censorship

February 17, 2009

Today Helmut Friedrich told me about an article by Pankaj Joshi in the February 2009 issue of Scientific American on the subject of cosmic censorship. I consider this article one-sided and consequently misleading and I find it sad that it appeared in such a prominent place. Here I present an alternative view.

General relativity predicts the occurrence of spacetime singularities where the gravitational force and/or matter density become unboundedly large, implying that known physical laws break down. If a singularity of this type can influence distant regions – this is referred to as a naked singularity – then the predictability, and hence the viability, of general relativity is undermined. Thus there is a strong motivation for ruling out this type of scenario. The cosmic censorship hypothesis of Roger Penrose suggests that the dynamics of the theory contrive to hide the effects of singularities from faraway observers – this is the cosmic censor. In fact there are two variants of this, called weak and strong cosmic censorship. In the first case the singularities are hidden because they are covered by the event horizon of a black hole while in the second they are made invisible to any observer outside the singularity.

The basic equations of general relativity are the Einstein equations which describe the gravitational field coupled with matter equations which describe the sources of the gravitational field. There are mathematical proofs of the existence of singularities in general relativity under rather general physically reasonable conditions. These stem from the singularity theorems of Penrose and Hawking which appeared starting in the mid 1960’s. It is not a priori clear how to define a singularity in this context but a mathematically clear definition has emerged which is useful and widely accepted. It is formulated in terms of geodesic incompleteness. The mathematics entering the proofs of these theorems is differential geometry and ordinary differential equations. The input is less than the full Einstein-matter equations with a definite choice of matter model – rather only certain inequalities on the energy-momentum tensor, the right hand side of the Einstein equations, are required. These inequalities are the so-called energy conditions.

In contrast there is no proof or disproof of cosmic censorship and even finding a precise formulation is difficult. There a couple of concrete obstacles involved:

1. There are very special classes of solutions known, for instance some with high symmetry, which do contain naked singularities. To avoid these a genericity assumption is required.

2. There are certain matter models with pathological properties which can lead to singularities even in the absence of gravity. The most notorious of these is dust, a fluid without pressure. A formulation of cosmic censorship which has a chance of being true must include some restriction on the matter model.

I like to hope that cosmic censorship is true. I strongly believe that it is impossible to prove it using differential geometry and ordinary differential equations alone. A proof will require taking the Einstein-matter equations seriously as a system of partial differential equations. It so happens that proving results about solutions of partial differential equations is much harder than proving analogous results about ordinary differential equations. Thus it is not surprising that progress in proving cosmic censorship has been limited, even taking the optimistic view that it is true. Another difficulty is that relevant parts of PDE theory simply do not belong to the usual mathematical repertoire of people working in general relativity.

Now I come back to the article of Joshi. The author appears rather fond of naked singularities and has mentioned a lot of work tending to support their existence. My aim here is not to give a detailed criticism of the work he cites. Instead I will describe important work providing support for cosmic censorship which is completely ignored in the article. As already mentioned, there is no proof of cosmic censorship. The character of the results I will mention here is that they concern model problems (with some assumptions on symmetry and the choice of matter model) where the mechanisms which are invoked to make cosmic censorship work can be seen in action. The first mechanism is genericity. I give two examples. The first is the spherically symmetric scalar field, as studied by Christodoulou (Ann. Math. 149, 183 (1999)). He showed that in this model naked singularities do occur but that they are eliminated by a genericity assumption. The second is the case of vacuum Gowdy spacetimes where a conceptually similar result was obtained by Ringström (to appear in Ann. Math., see The second mechanism is the influence of the choice of matter model. Gerhard Rein, Jack Schaeffer and I showed that a large class of naked singularities (the non-central ones) in solutions of the Einstein-dust equations can be removed by replacing dust by collisionless matter. There are many other papers tending to support cosmic censorship. Here I have just mentioned some which illustrate important aspects well. For a valuable account of the development of ideas about cosmic censorship I recommend the Prologue to the recent book of Christodoulou “The formation of black holes in general relativity”. A preprint version is available as gr-qc/0805.3880.

The jury is still out on cosmic censorship. I think that the appropriate expert witnesses who might be called to open the way to a just verdict are the specialists in partial differential equations.

Electrifying Gowdy

January 27, 2009

In a previous post I wrote about the polarized Gowdy equations. If the condition of polarization is dropped the full Gowdy equations are obtained. They form a system of two semilinear wave equations:

\displaystyle P_{tt}+t^{-1}P_t-P_{\theta\theta}=e^{2P}(Q_t^2-Q_\theta^2),

\displaystyle Q_{tt}+t^{-1}Q_t-Q_{\theta\theta}=-2(P_tQ_t-P_\theta Q_\theta).

These equations are very similar to those for a wave map from two-dimensional Minkowski space to the hyperbolic plane with metric dP^2+e^{2P}dQ^2 with the difference of the singular term involving t^{-1}. (It fact it is known that these equations can be given a formulation in terms of a wave map. The relevant wave map is on the domain with metric -dt^2+d\theta^2+t^2 dx^2 and is asssumed to be independent of the additional coordinate x.) The boundary conditions are as in the polarized case. Given initial data for t=t_0 there exists a unique solution on the time interval (0,\infty), as was proved by Vincent Moncrief in 1981.

The asymptotics for t\to 0 is significantly more complicated than in the polarized case. There is a large class of solutions with rather simple asymptotics. Their asymptotic form in the limit t\to 0 is:

P(t,\theta)=-k(\theta)\log t+\phi(\theta)+\ldots,Q(t,\theta)=q(\theta)+t^{2k(\theta)}\psi(\theta)+\ldots.

In 1998 Satyanad Kichenassamy and I proved the existence of solutions of this type for prescribed functions k, \phi, q and \psi, subject to the condition 0<k(\theta)<1. I will say no more about he positivity condition on k here but instead concentrate on the inequality k<1. The quantity k is sometimes called the asymptotic velocity and the condition k<1 the low velocity condition. It turns out that this is not just a technical restriction. It has been shown by Hans Ringström that the quantity -tP_t(t,\theta) always has a limit as t\to 0. Call its modulus the asymptotic velocity v_\infty (\theta). In the low velocity case described above it coincides with k(\theta). More generally, when it is allowed to exceed one other phenomena occur. They are associated to the phenomenon of spikes. At a low velocity point P_\theta/P converges to a smooth limit as t\to 0. A spike is a value of \theta where P_\theta/P becomes unbounded. In general v_\infty has a discontinuity at a point of this type. These phenomena have been analysed in detail under a genericity assumption by Ringström. He used these results in his proof of strong cosmic censorship for Gowdy spacetimes. There are also subtle phenomena which occur in the limit t\to\infty. (For more information about many of the results mentioned in this post see the papers on Ringström’s web page.)

The Gowdy spacetimes are solutions of the Einstein vacuum equations. They can be generalized by adding an electromagnetic field. Recently Ernesto Nungesser completed his diploma thesis under my supervision on the subject of these electromagnetic generalizations of Gowdy solutions. He and I just produced a paper where the results of his thesis are described and extended somewhat. One key element of this work is that there is a class of polarized ‘electro-Gowdy’ solutions which can be defined by symmetry conditions. Another is that there is a change of variables which transforms the polarized electro-Gowdy equations to the full (i.e. non-polarized) Gowdy equations. This allows the extensive results available on Gowdy solutions to be applied to the electromagnetic case. On the analytical level nothing happens but the geometrical interpretation of the variables is different in the two cases. It turns out that in this way strong cosmic censorship can be proved for polarized electro-Gowdy spacetimes. As discussed in the paper, much less is known in the non-polarized electro-Gowdy case .