On Static Shells and the Buchdahl Inequality for the Spherically Symmetric Einstein-Vlasov System

Abstract

In a previous work [1] matter models such that the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 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 ((2Ω + 1)² − 1)/(2Ω + 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 ((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 [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.