The Einstein–Vlasov System/Kinetic Theory The Einstein–Vlasov System/Kinetic Theory Håkan Andrèasson Department of Mathematics Chalmers University of Technology S-412 96 Göteborg, Sweden http://www.math.chalmers.se/~hand Abstract The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein–Vlasov system. This system couples Einstein’s equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on nonrelativistic and special relativistic physics, i.e. to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein–Vlasov system. Since then many theorems on global properties of solutions to this system have been established. The Vlasov equation describes matter phenomenologically, and it should be stressed that most of the theorems presented in this article are not presently known for other such matter models (i.e. fluid models). This paper gives introductions to kinetic theory in non-curved spacetimes and then the Einstein–Vlasov system is introduced. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity. 1 Introduction to Kinetic Theory In general relativity, kinetic theory has been used relatively rarely to model phenomenological matter in comparison to fluid models. From a mathematical point of view, however, there are fundamental advantages to using a kinetic description. In non-curved spacetimes kinetic theory has been studied intensively as a mathematical subject during several decades, and it has also played an important role from an engineering point of view. In the first part of this introduction, we will review kinetic theory in non-curved spacetimes and we will consider mainly the special relativistic case, but mathematical results in the nonrelativistic case will also be discussed. We believe that a good understanding of kinetic theory in non-curved spacetimes is fundamental to good comprehension of kinetic theory in general relativity. Moreover, it is often the case that mathematical methods used to treat the Einstein–Vlasov system are carried over from methods developed in the special relativistic or nonrelativistic case. The purpose of kinetic theory is to model the time evolution of a collection of particles. The particles may be entirely different objects depending on the physical situation. For instance, the particles are atoms and molecules in a neutral gas or electrons and ions in a plasma. In stellar dynamics the particles are stars and in a cosmological case they are galaxies or even clusters of galaxies. Mathematical models of particle systems are most frequently described by kinetic or fluid equations. A characteristic feature of kinetic theory is that its models are statistical and the particle systems are described by distribution functions f = f ( t , x , p ) , which represent the density of particles with given spacetime position ( t , x ) ∈ ℝ × ℝ 3 and momentum p ∈ ℝ 3 . A distribution function contains a wealth of information, and macroscopical quantities are easily calculated from this function. In a fluid model the quantities that describe the system do not depend on the momentum p but only on the spacetime point ( t , x ) . A choice of model is usually made with regard to the physical properties of interest for the system or with regard to numerical considerations. It should be mentioned that a fluid model that is too naive may give rise to shell-crossing singularities, which are unphysical. In a kinetic description such phenomena are ruled out. The time evolution of the system is determined by the interactions between the particles which depend on the physical situation. For instance, the driving mechanism for the time evolution of a neutral gas is the collision between particles (the relativistic Boltzmann equation). For a plasma the interaction is through the electric charges (the Vlasov–Maxwell system), and in the stellar and cosmological cases the interaction is gravitational (the Einstein–Vlasov system). Of course, combinations of interaction processes are also considered but in many situations one of them is strongly dominating and the weaker processes are neglected. 1.1 The relativistic Boltzmann equation Consider a collection of neutral particles in Minkowski spacetime. Let the signature of the metric be ( − , + , + , + ) , let all the particles have rest mass m = 1 , and normalize the speed of light c to one. The four-momentum of a particle is denoted by p a , a = 0 , 1 , 2 , 3 . Since all particles have equal rest mass, the four-momentum for each particle is restricted to the mass shell p a p a = − m 2 = − 1 . Thus, by denoting the three-momentum by p ∈ ℝ 3 , p a may be written p a = ( − p 0 , p ) , where | p | is the usual Euclidean length of p and p 0 = 2 1 + | p | 2 is the energy of a particle with three-momentum p . The relativistic velocity of a particle with momentum p is denoted by p ^ and is given by p ^ = p 2 1 + | p | 2 . (1) Note that | p ^ | < 1 = c . The relativistic Boltzmann equation models the spacetime behaviour of the one-particle distribution function f = f ( t , x , p ) , and it has the form ( ∂ t + p p 0 ⋅ ∇ x ) f = Q ( f , f ) , (2) where the relativistic collision operator Q ( f , g ) is defined by Q ( f , g ) = ∫ ℝ 3 ∫ S 2 k ( p , q , ω ) × (3) × [ f ( p + a ( p , q , ω ) ω ) g ( q − a ( p , q , ω ) ω ) − f ( p ) g ( q ) ] d p d ω . (Note that g = f in (2)). Here d ω is the element of surface area on S 2 and k ( p , q , ω ) is the scattering kernel, which depends on the scattering crosssection in the interaction process. See [23] for a discussion about the scattering kernel. The function a ( p , q , ω ) results from the collision mechanics. If two particles, with momentum p and q respectively, collide elastically (no energy loss) with scattering angle ω ∈ S 2 , their momenta will change, p → p ′ and q → q ′ . The relation between p , q and p ′ , q ′ is p ′ = p + a ( p , q , ω ) ω , q ′ = q − a ( p , q , ω ) ω , (4) where a ( p , q , ω ) = 2 ( p 0 + q 0 ) p 0 q 0 ( ω ⋅ ( q ^ − p ^ ) ) ( p 0 + q 0 ) 2 − ( ω ⋅ ( p + q ) ) 2 . (5) This relation is a consequence of four-momentum conservation, p a + q a = p a ′ + q a ′ , or equivalently p 0 + q 0 = p 0 ′ + q 0 ′ , (6) p + q = p ′ + q ′ . (7) These are the conservation equations for relativistic particle dynamics. In the classical case these equations read | p | 2 + | q | 2 = | p ′ | 2 + | q ′ | 2 , (8) p + q = p ′ + q ′ . (9) The function a ( p , q , ω ) is the distance between p and p ′ ( q and q ′ ), and the analogue function in the Newtonian case has the form a c l ( p , q , ω ) = ω ⋅ ( q − p ) . (10) By inserting a c l in place of a in (2) we obtain the classical Boltzmann collision operator (disregarding the scattering kernel, which is also different). The main result concerning the existence of solutions to the classical Boltzmann equation is a theorem by DiPerna and Lions [25] that proves existence, but not uniqueness, of renormalized solutions (i.e. solutions in a weak sense, which are even more general than distributional solutions). An analogous result holds in the relativistic case, as was shown by Dudyǹsky and Ekiel-Jezewska [26] . Regarding classical solutions, Illner and Shinbrot [46] have shown global existence of solutions to the nonrelativistic Boltzmann equation for small initial data (close to vacuum). At present there is no analogous result for the relativistic Boltzmann equation and this must be regarded as an interesting open problem. When the data are close to equilibrium (see below), global existence of classical solutions has been proved by Glassey and Strauss [36] in the relativistic case and by Ukai [87] in the nonrelativistic case (see also [84] ). The collision operator Q ( f , g ) may be written in an obvious way as Q ( f , g ) = Q + ( f , g ) − Q − ( f , g ) , where Q + and Q − are called the gain and loss term respectively. In [2] it is proved that given f ∈ L 2 ( ℝ 3 ) and g ∈ L 1 ( ℝ 3 ) with f , g ≥ 0 , then ∥ Q + ( f , g ) ∥ H 1 ( ℝ 3 ) ≤ C ∥ f ∥ L 2 ( ℝ 3 ) ∥ g ∥ L 1 ( ℝ 3 ) , (11) under some technical requirements on the scattering kernel. Here H s is the usual Sobolev space. This regularizing result was first proved by P.L. Lions [49] in the classical situation. The proof relies on the theory of Fourier integral operators and on the method of stationary phase, and requires a careful analysis of the collision geometry, which is very different in the relativistic case. The regularizing theorem has many applications. An important application is to prove that solutions tend to equilibrium for large times. More precisely, Lions used the regularizing theorem to prove that solutions to the (classical) Boltzmann equation, with periodic boundary conditions, converge in L 1 to a global Maxwellian, M = e − α | p | 2 + β ⋅ p + γ with α , γ ∈ R , α > 0 , β ∈ ℝ 3 , as time goes to infinity. This result had first been obtained by Arkeryd [8] by using non-standard analysis. It should be pointed out that the convergence takes place through a sequence of times tending to infinity and it is not known whether the limit is unique or depends on the sequence. In the relativistic situation, the analogous question of convergence to a relativistic Maxwellian, or a Jüttner equilibrium solution, J = e − α 2 1 + | p | 2 + β ⋅ p + γ , α , β and γ as above, with α > | β | , had been studied by Glassey and Strauss [36] [37] . In the periodic case they proved convergence in a variety of function spaces for initial data close to a Jüttner solution. Having obtained the regularizing theorem for the relativistic gain term, it is a straightforward task to follow the method of Lions and prove convergence to a local Jüttner solution for arbitrary data (satisfying the natural bounds of finite energy and entropy) that is periodic in the space variables. In [2] it is next proved that the local Jüttner solution must be a global one, due to the periodicity of the solution. For more information on the relativistic Boltzmann equation on Minkowski space we refer to [29] [23] [86] and in the nonrelativistic case we refer to the excellent review paper by Villani [88] and the books [29] [16] . 1.2 The Vlasov–Maxwell and Vlasov–Poisson systems Let us consider a collisionless plasma, which is a collection of particles for which collisions are relatively rare and the interaction is through their charges. We assume below that the plasma consists only of one type of particle, whereas the results below also hold for plasmas with several particle species. The particle rest mass is normalized to one. In the kinetic framework, the most general set of equations for modelling a collisionless plasma is the relativistic Vlasov–Maxwell system: ∂ t f + v ^ ⋅ ∇ x f + ( E ( t , x ) + v ^ × B ( t , x ) ) ⋅ ∇ v f = 0 , (12) ∂ t E + j = c ∇ × B , ∇ ⋅ E = ρ , (13) ∂ t B = − c ∇ × E , ∇ ⋅ B = 0 . (14) The notation follows the one already introduced with the exception that the momenta are now denoted by v instead of p . This has become a standard notation in this field. E and B are the electric and magnetic fields, and v ^ is the relativistic velocity v ^ = v 2 1 + | v | 2 / c 2 , (15) where c is the speed of light. The charge density ρ and current j are given by ρ = ∫ ℝ 3 f d v , j = ∫ ℝ 3 v ^ f d v . (16) Equation (12) is the relativistic Vlasov equation and (13, 14) are the Maxwell equations. A special case in three dimensions is obtained by considering spherically symmetric initial data. For such data it can be shown that the solution will also be spherically symmetric, and that the magnetic field has to be constant. The Maxwell equation ∇ × E = − ∂ t B then implies that the electric field is the gradient of a potential φ . Hence, in the spherically symmetric case the relativistic Vlasov–Maxwell system takes the form ∂ t f + v ^ ⋅ ∇ x f + β E ( t , x ) ⋅ ∇ v f = 0 , (17) E = ∇ φ , Δ φ = ρ . (18) Here β = 1 , and the constant magnetic field has been set to zero, since a constant field has no significance in this discussion. This system makes sense for any initial data, without symmetry constraints, and is called the relativistic Vlasov–Poisson equation. Another special case of interest is the classical limit, obtained by letting c → ∞ in (12, 13, 14), yielding: ∂ t f + v ⋅ ∇ x f + β E ( t , x ) ⋅ ∇ v f = 0 , (19) E = ∇ φ , Δ φ = ρ , (20) where β = 1 . See Schaeffer [80] for a rigorous derivation of this result. This is the (nonrelativistic) Vlasov–Poisson equation, and β = 1 corresponds to repulsive forces (the plasma case). Taking β = − 1 means attractive forces and the Vlasov–Poisson equation is then a model for a Newtonian self-gravitating system. One of the fundamental problems in kinetic theory is to find out whether or not spontaneous shock formations will develop in a collisionless gas, i.e. whether solutions to any of the equations above will remain smooth for all time, given smooth initial data. If the initial data are small this problem has an affirmative solution in all cases considered above (see [30] [35] [10] [11] ). For initial data unrestricted in size the picture is more involved. In order to obtain smooth solutions globally in time, the main issue is to control the support of the momenta Q ( t ) : = sup { | v | : ∃ ( s , x ) ∈ [ 0 , t ] × ℝ 3 such that f ( s , x , v ) ≠ 0 } , (21) i.e. to bound Q ( t ) by a continuous function so that Q ( t ) will not blow up in finite time. That such a control is sufficient for obtaining global existence of smooth solutions follows from well-known results in the different cases (see [34] [42] [12] [30] ). For the full three-dimensional relativistic Vlasov– Maxwell system, this important problem of establishing whether or not solutions will remain smooth for all time is open. In two space and three momentum dimensions, Glassey and Schaeffer [31] have shown that Q ( t ) can be controlled, which thus yields global existence of smooth solutions in that case (see also [32] ). The relativistic and nonrelativistic Vlasov–Poisson equations are very similar in form. In particular, the equation for the field is identical in the two cases. However, the mathematical results concerning the two systems are very different. In the nonrelativistic case Batt [12] gave an affirmative solution 1977 in the case of spherically symmetric data. Pfaffelmoser [55] (see also Schaeffer [82] ) was the first one to give a proof for general smooth data. He obtained the bound Q ( t ) ≤ C ( 1 + t ) ( 51 + δ ) / 11 , where δ > 0 could be taken arbitrarily small. This bound was later improved by different authors. The sharpest bound valid for β = 1 and β = − 1 has been given by Horst [43] and reads Q ( t ) ≤ C ( 1 + t ) log ( 2 + t ) . In the case of repulsive forces ( β = 1 ) Rein [59] has found the sharpest estimate by using a new identity for the Vlasov–Poisson equation, discovered independently by Illner and Rein [45] and by Perthame [54] . Rein’s estimate reads Q ( t ) ≤ C ( 1 + t ) 2 / 3 . Independently and about the same time as Pfaffelmoser gave his proof, Lions and Perthame [50] used a different method for proving global existence. To some extent their method seems to be more generally applicable to attack problems similar to the Vlasov–Poisson equation but which are still quite different (see [3] , [48] ). On the other hand, their method does not give such strong growth estimates on Q ( t ) as described above. For the relativistic Vlasov–Poisson equation (with β = 1 ), Glassey and Schaeffer [30] showed that if the data are spherically symmetric, Q ( t ) can be controlled, which is analogous to the result by Batt mentioned above (we also mention that the cylindrical case has been considered in [33] ). If β = − 1 , it was also shown in [30] that blow-up occurs in finite time for spherically symmetric data with negative total energy. This system, however, is unphysical in the sense that it is not a special case of the Einstein–Vlasov system. Quite surprisingly, for general smooth initial data none of the techniques discussed above for the nonrelativistic Vlasov–Poisson equation apply in the relativistic case. This fact is annoying since it has been suggested that an understanding of this equation may be necessary for understanding the three-dimensional relativistic Vlasov–Maxwell equation. However, the relativistic Vlasov–Poisson equation lacks the Lorentz invariance; it is a hybrid of a classical Galilei invariant field equation and a relativistic transport equation (17). Only for spherical symmetric data is the equation a fundamental physical equation. The classical Vlasov–Poisson equation is on the other hand Galilean invariant. In [1] a different equation for the field is introduced that is observer independent among Lorentz observers. By coupling this equation for the field to the relativistic Vlasov equation, the function Q ( t ) may be controlled as shown in [1] . This is an indication that the transformation properties are important in studying existence of smooth solutions (the situation is less subtle for weak solutions, where energy estimates and averaging are the main tools, see [44] and [24] ). Hence, it is unclear whether or not the relativistic Vlasov–Poisson equation will play a central role in the understanding of the Lorentz invariant relativistic Vlasov–Maxwell equation. We refer to the book by Glassey [29] for more information on the relativistic Vlasov–Maxwell system and the Vlasov–Poisson equation. 1.3 The Einstein–Vlasov system In this section we will consider a self-gravitating collisionless gas and we will write down the Einstein–Vlasov system and describe its general mathematical features. Our presentation follows to a large extent the one by Rendall in [76] . We also refer to Ehlers [27] and Stewart [85] for more background on kinetic theory in general relativity. Let M be a four-dimensional manifold and let g a b be a metric with Lorentz signature ( − , + , + , + ) so that ( M , g a b ) is a spacetime. We use the abstract index notation, which means that g a b is a geometric object and not the components of a tensor. See [89] for a discussion on this notation. The metric is assumed to be time-orientable so that there is a distinction between future and past directed vectors. The worldline of a particle with non-zero rest mass m is a timelike curve and the unit future-directed tangent vector v a to this curve is the four-velocity of the particle. The four-momentum p a is given by m v a . We assume that all particles have equal rest mass m and we normalize so that m = 1 . One can also consider massless particles but we will rarely discuss this case. The possible values of the four-momentum are all future-directed unit timelike vectors and they constitute a hypersurface P in the tangent bundle T M , which is called the mass shell. The distribution function f that we introduced in the previous sections is a non-negative function on P . Since we are considering a collisionless gas, the particles travel along geodesics in spacetime. The Vlasov equation is an equation for f that exactly expresses this fact. To get an explicit expression for this equation we introduce local coordinates on the mass shell. We choose local coordinates on M such that the hypersurfaces t = x 0 = constant are spacelike so that t is a time coordinate and x j , j = 1 , 2 , 3 , are spatial coordinates (letters in the beginning of the alphabet always take values 0 , 1 , 2 , 3 and letters in the middle take 1 , 2 , 3 ). A timelike vector is future directed if and only if its zero component is positive. Local coordinates on P can then be taken as x a together with the spatial components of the four-momentum p a in these coordinates. The Vlasov equation then reads ∂ t f + p j p 0 ∂ x j f − 1 p 0 Γ a b j p a p b ∂ p j f = 0 . (22) Here a , b = 0 , 1 , 2 , 3 and j = 1 , 2 , 3 , and Γ a b j are the Christoffel symbols. It is understood that p 0 is expressed in terms of p j and the metric g a b using the relation g a b p a p b = − 1 (recall that m = 1 ). In a fixed spacetime the Vlasov equation is a linear hyperbolic equation for f and we can solve it by solving the characteristic system, d X i / d s = P i P 0 , (23) d P i / d s = − Γ a b i P a P b P 0 . (24) In terms of initial data f 0 the solution to the Vlasov equation can be written as f ( x a , p i ) = f 0 ( X i ( 0 , x a , p i ) , P i ( 0 , x a , p i ) ) , (25) where X i ( s , x a , p i ) and P i ( s , x a , p i ) solve (23) and (24), and where X i ( t , x a , p i ) = x i and P i ( t , x a , p i ) = p i . In order to write down the Einstein–Vlasov system we need to define the energy-momentum tensor T a b in terms of f and g a b . In the coordinates ( x a , p a ) on P we define T a b = − ∫ ℝ 3 f p a p b | g | 1 / 2 d p 1 d p 2 d p 3 p 0 , where as usual p a = g a b p b and | g | denotes the absolute value of the determinant of g . Equation (22) together with Einstein’s equations G a b : = R a b − 1 2 R g a b = 8 π T a b then form the Einstein–Vlasov system. Here G a b is the Einstein tensor, R a b the Ricci tensor and R is the scalar curvature. We also define the particle current density N a = − ∫ ℝ 3 f p a | g | 1 / 2 d p 1 d p 2 d p 3 p 0 . Using normal coordinates based at a given point and assuming that f is compactly supported it is not hard to see that T a b is divergence-free which is a necessary compatability condition since G a b is divergence-free by the Bianchi identities. A computation in normal coordinates also shows that N a is divergence-free, which expresses the fact that the number of particles is conserved. The definitions of T a b and N a immediately give us a number of inequalities. If V a is a future directed timelike or null vector then we have N a V a ≤ 0 with equality if and only if f = 0 at the given point. Hence N a is always future directed timelike if there are particles at that point. Moreover, if V a and W a are future directed timelike vectors then T a b V a W b ≥ 0 , which is the dominant energy condition. If X a is a spacelike vector then T a b X a X b ≥ 0 . This is called the non-negative pressure condition. These last two conditions together with the Einstein equations imply that R a b V a V b ≥ 0 for any timelike vector V a , which is the strong energy condition. That the energy conditions hold for Vlasov matter is one reason that the Vlasov equation defines a well-behaved matter model in general relativity. Another reason is the well-posedness theorem by Choquet-Bruhat for the Einstein–Vlasov system that we will state below. Before stating that theorem we will first discuss the initial conditions imposed. The data in the Cauchy problem for the Einstein–Vlasov system consist of the induced Riemannian metric g i j on the initial hypersurface S , the second fundamental form k i j of S and matter data f 0 . The relations between a given initial data set ( g i j , k i j ) on a three-dimensional manifold S and the metric g i j on the spacetime manifold is that there exists an embedding ψ of S into the spacetime such that the induced metric and second fundamental form of ψ ( S ) coincide with the result of transporting ( g i j , k i j ) with ψ . For the relation of the distribution functions f and f 0 we have to note that f is defined on the mass shell. The initial condition imposed is that the restriction of f to the part of the mass shell over ψ ( S ) should be equal to f 0 ∘ ( ψ − 1 , d ( ψ ) − 1 ) ∘ φ , where φ sends each point of the mass shell over ψ ( S ) to its orthogonal projection onto the tangent space to ψ ( S ) . An initial data set for the Einstein–Vlasov system must satisfy the constraint equations, which read R − k i j k i j + ( t r k ) 2 = 16 π ρ , (26) ∇ i k l i − ∇ l ( t r k ) = 8 π j l . (27) Here ρ = T a b n a n b and j a = − h a b T b c n c , where n a is the future directed unit normal vector to the initial hypersurface and h a b = g a b + n a n b is the orthogonal projection onto the tangent space to the initial hypersurface. In terms of f 0 we can express ρ and j l by ( j a satisfies n a j a = 0 so it can naturally be identified with a vector intrinsic to S ) ρ = ∫ ℝ 3 f p a p a | ( 3 ) g | 1 / 2 d p 1 d p 2 d p 3 1 + p j p j , (28) j l = ∫ ℝ 3 f p l | ( 3 ) g | 1 / 2 d p 1 d p 2 d p 3 . (29) Here | ( 3 ) g | is the determinant of the induced Riemannian metric on S . We can now state the local existence theorem by Choquet-Bruhat [17] for the Einstein–Vlasov system. Theorem 1 Let S be a 3-dimensional manifold, g i j a smooth Riemannian metric on S , k i j a smooth symmetric tensor on S and f 0 a smooth non-negative function of compact support on the tangent bundle T S of S . Suppose that these objects satisfy the constraint equations (26) and (27). Then there exists a smooth spacetime ( M , g a b ) , a smooth distribution function f on the mass shell of this spacetime, and a smooth embedding ψ of S into M which induces the given initial data on S such that g a b and f satisfy the Einstein–Vlasov system and ψ ( S ) is a Cauchy surface. Moreover, given any other spacetime ( M ′ , g ′ a b ) , distribution function f ′ and embedding ψ ′ satisfying these conditions, there exists a diffeomorphism χ from an open neighbourhood of ψ ( S ) in M to an open neighbourhood of ψ ′ ( S ) in M ′ which satisfies χ ∘ ψ = ψ ′ and carries g a b and f to g ′ a b and f ′ , respectively. In this context we also mention that local existence has been proved for the Einstein–Maxwell–Boltzmann system [9] and for the Yang–Mills–Vlasov system [18] . A main theme in the following sections is to discuss special cases for which the local existence theorem can be extended to a global one. There are interesting situations when this can be achieved, and such global existence theorems are not known for Einstein’s equations coupled to other forms of phenomenological matter models, i.e. fluid models (see, however, [21] ). In this context it should be stressed that the results in the previous sections show that the mathematical understanding of kinetic equations on a flat background space is well-developed. On the other hand the mathematical understanding of fluid equations on a flat background space (also in the absence of a Newtonian gravitational field) is not that well-understood. It would be desirable to have a better mathematical understanding of these equations in the absence of gravity before coupling them to Einstein’s equations. This suggests that the Vlasov equation is natural as matter model in mathematical general relativity. 2 Global Existence Theorems for the Einstein– Vlasov System In general relativity two classes of initial data are distinguished. If an isolated body is studied, the data are called asymptotically flat. The initial hypersurface is topologically ℝ 3 and (since far away from the body one expects spacetime to be approximately flat) appropriate fall off conditions are imposed. Roughly, a smooth data set ( g i j , k i j , f 0 ) on ℝ 3 is said to be asymptotically flat if there exist global coordinates x i such that as | x | tends to infinity the components g i j in these coordinates tend to δ i j , the components k i j tend to zero, f 0 has compact support and certain weighted Sobolev norms of g i j − δ i j and k i j are finite (see [76] ). The symmetry classes that admit asymptotical flatness are few. The important ones are spherically symmetric and axially symmetric spacetimes. One can also consider a case in which spacetime is asymptotically flat except in one direction, namely cylindrical spacetimes. Regarding global existence questions, only spherically symmetric spacetimes have been considered for the Einstein–Vlasov system in the asymptotically flat case. Spacetimes that possess a compact Cauchy hypersurface are called cosmological spacetimes, and data are accordingly given on a compact 3-manifold. In this case the whole universe is modelled and not only an isolated body. In contrast to the asymptotically flat case, cosmological spacetimes admit a large number of symmetry classes. This gives one the possibility to study interesting special cases for which the difficulties of the full Einstein equations are strongly reduced. We will discuss below cases for which the spacetime is characterized by the dimension of its isometry group together with the dimension of the orbit of the isometry group. 2.1 Spherically symmetric spacetimes The study of the global properties of solutions to the spherically symmetric Einstein–Vlasov system was initiated by Rein and Rendall in 1990. They chose to work in coordinates where the metric takes the form d s 2 = − e 2 μ ( t , r ) d t 2 + e 2 λ ( t , r ) d r 2 + r 2 ( d θ 2 + sin 2 θ d φ 2 ) , where t ∈ ℝ , r ≥ 0 , θ ∈ [ 0 , π ] , φ ∈ [ 0 , 2 π ] . These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions lim r → ∞ λ ( t , r ) = lim r → ∞ μ ( t , r ) = 0 , ∀ t ≥ 0 . A regular centre is also required and is guaranteed by the boundary condition λ ( t , 0 ) = 0 . With x = ( r sin φ cos θ , r sin φ sin θ , r cos φ ) as spatial coordinate and v j = p j + ( e λ − 1 ) x ⋅ p r x j r as momentum coordinates the, Einstein–Vlasov system reads ∂ t f + e μ − λ v 2 1 + v 2 ⋅ ∂ x f − ( λ t x ⋅ v r + e μ − λ μ r 2 1 + v 2 ) x r ⋅ ∂ v f = 0 , (30) e − 2 λ ( 2 r λ r − 1 ) + 1 = 8 π r 2 ρ , (31) e − 2 λ ( 2 r μ r + 1 ) − 1 = 8 π r 2 p . (32) The matter quantities are defined by ρ ( t , x ) = ∫ ℝ 3 2 1 + | v | 2 f ( t , x , v ) d v , (33) p ( t , x ) = ∫ ℝ 3 ( x ⋅ v r ) 2 f ( t , x , v ) d v 2 1 + | v | 2 . (34) Let us point out that this system is not the full Einstein–Vlasov system. The remaining field equations, however, can be derived from these equations. See [62] and the erratum for more details. Let the square of the angular momentum be denoted by L , i.e. L : = | x | 2 | v | 2 − ( x ⋅ v ) 2 . A consequence of spherical symmetry is that angular momentum is conserved along the characteristics of (30). Introducing the variable w = x ⋅ v r , the Vlasov equation for f = f ( t , r , w , L ) becomes ∂ t f + e μ − λ w E ∂ r f