1 Introduction to Kinetic Theory

In general relativity, kinetic theory has been used relatively sparsely to model phenomenological matter in comparison to fluid models, although interest has increased in recent years. From a mathematical point of view 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.

The main purpose of this review paper is to discuss mathematical results for the Einstein–Vlasov system. However, in the first part of this introduction, we review kinetic theory in non-curved spacetimes and focus on the special-relativistic case, although some results in the non-relativistic case will also be mentioned. The reason that we focus on the relativistic case is not only that it is more closely related to the main theme in this review, but also that the literature on relativistic kinetic theory is very sparse in comparison to the non-relativistic case, in particular concerning the relativistic and non-relativistic Boltzmann equation. 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 non-relativistic 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 astrophysics the particles are stars, 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 density functions f = f(t,x,p), which represent the density of particles with given spacetime position (t,x) ∈ ℝ × ℝ3 and momentum 3 p ∈ ℝ. A density function contains a wealth of information, and macroscopic 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 too naive fluid model 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 Boltzmann equation). For a plasma the interaction is through the electromagnetic field produced by the charges (the Vlasov–Maxwell system), and in astrophysics the interaction is gravitational (the Vlasov–Poisson system and 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
 1.2 The Vlasov–Maxwell and Vlasov–Poisson systems
 1.3 The Nordström–Vlasov system

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