On the small redshift limit of steady states of the spherically symmetric Einstein-Vlasov system and their stability

Abstract

Families of steady states of the spherically symmetric Einstein-Vlasov system are constructed, which are parametrized by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov-Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analyzed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.