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 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 rho >= 0, and the radial- and tangential pressures p >= 0 and q, satisfy p + q <= Omega rho, Omega >= 1. We show a Buchdahl type inequality for shells which are thin; given an is an element of < 1/4 there is a k > 0 such that 2M/R-1 <= 1 - k when R-1/R-0 <= 1 + is an element of. 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 ((2 Omega + 1)(2) - 1)/(2 Omega + 1)(2). In particular if Omega = 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.