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On the Buchdahl inequality for spherically symmetric static shells

A classical result by Buchdahl \cite{Bu1} 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 $2M/R\leq 8/9.$ 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&#39;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 $ρ\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leqΩρ, Ω\geq 1.$ We show a Buchdahl type inequality for shells which are thin; given an $ε<1/4$ there is a $κ>0$ such that $2M/R_1\leq 1-κ$ when $R_1/R_0\leq 1+ε.$ It is also shown that for a sequence of solutions such that $R_1/R_0\to 1,$ 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 boumd is $8/9.$ The latter result is motivated by numerical simulations \cite{AR2} which indicate that for non-isotropic shells of Vlasov matter $2M/R_1\leq 8/9,$ and moreover, that the value 8/9 is approached for shells with $R_1/R_0\to 1$. In \cite{An2} a sequence of shells of Vlasov matter is constructed with the properties that $R_1/R_0\to 1,$ 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 \cite{An2} the Vlasov equation is important.