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 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.
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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. Consider a collection of neutral particles in Minkowski spacetime. See [ 23 ] for a discussion about the scattering kernel. These are the conservation equations for relativistic particle dynamics.
In the classical case these equations read.
By inserting a cl 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.
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 ].
Here H s is the usual Sobolev space. This regularizing result was first proved by P. 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,.
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. 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 ]. 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:. 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. 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. Hence, in the spherically symmetric case the relativistic Vlasov-Maxwell system takes the form.
This system makes sense for any initial data, without symmetry constraints, and is called the relativistic Vlasov-Poisson equation. See Schaeffer [ 80 ] for a rigorous derivation of this result. 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.
If the initial data are small this problem has an affrmative 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. 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 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. This bound was later improved by different authors. 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. 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 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. 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.
We use the abstract index notation, which means that g ab 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 mv a. 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 TM , 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.
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. In a fixed spacetime the Vlasov equation is a linear hyperbolic equation for f and we can solve it by solving the characteristic system,.
In terms of initial data f 0 the solution to the Vlasov equation can be written as. In order to write down the Einstein-Vlasov system we need to define the energy-momentum tensor T ab in terms of f and g ab. In the coordinates x a , p a on P we define.