Uniqueness of static, isotropic low-pressure solutions of the Einstein-Vlasov system

TL;DR

This paper demonstrates that static, isotropic low-pressure solutions of the Einstein-Vlasov system are unique under certain conditions, extending a known uniqueness theorem from fluid models to Vlasov matter.

Contribution

It shows how isotropic Vlasov matter can be modeled as a perfect fluid with a barotropic equation of state, enabling the application of a known uniqueness theorem to these solutions.

Findings

Uniqueness applies to shallow gravitational potential wells.

Constructed an example of a unique static solution.

Numerical evidence suggests non-uniqueness for deep potential wells.

Abstract

Due to R. Beig and W. Simon (1990) there is a uniqueness theorem for static solutions of the Einstein-Euler system which applies to fluid models whose equation of state fulfills certain conditions. In this article it is shown that this uniqueness theorem can be applied to isotropic Vlasov matter, if the gravitational potential well is shallow. To this end we first show how isotropic Vlasov matter can be described as a perfect fluid giving rise to a barotropic equation of state. This 'Vlasov' equation of state is investigated and it is shown analytically that the requirements of the uniqueness theorem are met for shallow potential wells. Finally the regime of shallow gravitational potential is investigated by numerical means. An example for a unique static solution is constructed and it is compared to astrophysical objects like globular clusters. Finally we find numerical indications that solutions with deep potential wells are not unique.