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 $2M\mathrm{/}R\le 8\mathrm{/}9.$ 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 $\rho \ge 0,$ and the radial- and tangential pressures $p\ge 0$ and $q,$ satisfy $p+q\le \mathrm{\Omega }\rho ,\mathrm{\Omega }\ge 1.$ We show a Buchdahl type inequality for shells which are thin; given an $ϵ<1\mathrm{/}4$ there is a $\kappa >0$ such that $2M\mathrm{/}{R}_1\le 1-\kappa$ when $R_1/R_0\leq 1+\epsilon.$Itisalsoshownthatforasequenceofsolutionssuchthat$R_1/R_0\to 1,$thelimitsupremumof$2M/R_1$ofthesequenceisboundedby$((2\Omega+1)^2-1)/(2\Omega+1)^2.$Inparticularif$\Omega=1,$ which is the case for Vlasov matter, the boumd is $8\mathrm{/}9.$ The latter result is motivated by numerical simulations \cite{AR2} which indicate that for non-isotropic shells of Vlasov matter $2M\mathrm{/}{R}_1\le 8\mathrm{/}9,$ and moreover, that the value 8/9 is approached for shells with $R_1\mathrm{/}{R}_0\to 1$. In \cite{An2} a sequence of shells of Vlasov matter is constructed with the properties that $R_1\mathrm{/}{R}_0\to 1,$ and that $2M\mathrm{/}{R}_1$ 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.