The Einstein-Vlasov system/Kinetic theory

TL;DR

This paper reviews theorems on the global behavior of solutions to the Einstein-Vlasov system, a coupling of Einstein's equations with kinetic matter models, highlighting recent mathematical advances since 1990.

Contribution

It provides a comprehensive guide to existing theorems on the Einstein-Vlasov system's solutions and emphasizes the importance of kinetic theory in curved spacetime contexts.

Findings

Many theorems on global properties have been established since 1990.

Most theorems 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 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, {\it 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 ({\it 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.