On the Buchdahl Inequality for Spherically Symmetric Static Shells

Abstract

A classical result by Buchdahl [6] shows that for static solutions of the spherically symmetric Einstein equations, the ADM mass M and the area radius R of the boundary of the body, obey the inequality 2M/R ≤ 8/9. The proof of this inequality rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl’s hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R ₀, R ₁], R ₀ > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1. We show a Buchdahl type inequality for shells which are thin; given an $\epsilon < 1/4$ there is a κ > 0 such that 2M/R ₁ ≤ 1 − κ when $R_1/R_0 \leq 1 + \epsilon$. It is also shown that for a sequence of solutions such that R ₁/R ₀ → 1, the limit supremum of 2M/R ₁ of the sequence is bounded by ((2Ω + 1)² − 1)/(2Ω + 1)². In particular if Ω = 1, which is the case for Vlasov matter, the bound is 8/9. The latter result is motivated by numerical simulations [3] which indicate that for non-isotropic shells of Vlasov matter 2M/R ₁ ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R ₁/R ₀ → 1. In [1] a sequence of shells of Vlasov matter is constructed with the properties that R ₁/R ₀ → 1, and that 2M/R ₁ equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in [1] the Vlasov equation is important.