On static shells and the buchdahl inequality for the spherically symmetric einstein-vlasov system

In a previous work [1] matter models such that the energy density rho >= 0, and the radial- and tangential pressures p >= 0 and q, satisfy p + q >= Omega rho, Omega >= 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) -> 1, then the limit supremum of 2M/R (1) was shown to be bounded by ((2 Omega + 1)(2) - 1)/(2 Omega + 1)(2). In this paper we show that the hypothesis that R (1)/R (0) -> 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 ((2 Omega + 1)(2) - 1)/(2 Omega + 1)(2) = 8/9, since Omega = 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 [3], 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.