The Einstein-Vlasov System/Kinetic Theory

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 non-relativistic 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. 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 a good comprehension of kinetic theory in general relativity.

The formation of black holes in spherically symmetric gravitational collapse

“…It is moreover well-known that the (DEC) condition is satisfied for Vlasov matter; see [1,Sec. 1.4].…”

“…The system (1.1), (1.4), (1.5) is the Einstein-Vlasov system in general coordinates. For an introduction to relativistic kinetic theory and the Einstein-Vlasov system we refer to [1] and [33]. If, for comparison, the matter is to be described as a perfect fluid with density R, four-velocity field U α , and pressure P , then the matter evolution equations are the Euler equations…”

On the existence, structure and stability of static and stationary solutions of the Einstein-Vlasov system

“…For a general introduction to relativistic kinetic theory and the Einstein-Vlasov system I refer to [3] and the notation below follows the one given in [3].…”

“…If a steady state has support in [R 0 , R 1 ], then the ADM mass M of the configuration is M = m(R 1 ), where m(r) is the quasi local mass, cf. [3]. Schwarzschild asked already in 1916 the question: How large can 2M/R possibly be?…”

“…The equations become much more complicated by going beyond spherical symmetry and only solutions close to spherically symmetric ones have been analyzed. This review is to a large extent an extract from the the section on static and stationary solutions in the review article [3]. …”