On the Buchdahl inequality for spherically symmetric static shells

A classical result by Buchdahl [6] shows that for static solutions of the spherically symmetric Einstein-matter system, the total 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 [R0, R1], R0 > 0, of matter models for which the energy density ρ ≥ 0, and the radialand 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 ǫ < 1/4 there is a κ > 0 such that 2M/R1 ≤ 1−κ when R1/R0 ≤ 1+ǫ. It is also shown that for a sequence of solutions such that R1/R0 → 1, the limit supremum of 2M/R1 of the sequence is bounded by ((2Ω+ 1) − 1)/(2Ω+ 1). In particular if Ω = 1, which is the case for Vlasov matter, the boumd is 8/9. The latter result is motivated by numerical simulations [3] which indicate that for non-isotropic shells of Vlasov matter 2M/R1 ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R1/R0 → 1. In [1] a sequence of shells of Vlasov matter is constructed with the properties that R1/R0 → 1, and that 2M/R1 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.