The Einstein-Vlasov sytem/Kinetic theory

TL;DR

This paper reviews theorems on the global behavior of solutions to the Einstein-Vlasov system, a kinetic matter model coupled with Einstein's equations, highlighting its unique properties compared to other matter models.

Contribution

It provides a comprehensive overview of mathematical results on the Einstein-Vlasov system's solutions, emphasizing the importance of kinetic theory in curved spacetime.

Findings

Many theorems on global properties of solutions have been established since 1990.

Most results are unique to the Vlasov matter model, not applicable to fluid models.

The paper introduces kinetic theory in both curved and non-curved spacetimes.

Abstract

The main purpose of this article is to guide the reader 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 where the main focus has been on nonrelativistic- and special relativistic physics, e.g. 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 (e.g. fluid models). The first part of this paper gives an introduction 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 in order to get a good comprehension of kinetic theory in general relativity.