On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system

Abstract

In a previous work \cite{An1} matter models such that the energy density $ρ\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leqΩρ, Ω\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2Ω+1)^2-1)/(2Ω+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2Ω+1)^2-1)/(2Ω+1)^2=8/9,$ since $Ω=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.