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Sharp bounds on $2m/r$ of general spherically symmetric static objects

In 1959 Buchdahl \cite{Bu} obtained the inequality $2M/R\leq 8/9$ under the assumptions that the energy density is non-increasing outwards and that the pressure is isotropic. Here $M$ is the ADM mass and $R$ the area radius of the boundary of the static body. The assumptions used to derive the Buchdahl inequality are very restrictive and e.g. neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider \textit{any} static solution of the spherically symmetric Einstein equations for which the energy density $ρ\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $p_T,$ satisfy $p+2p_T\leqΩρ, Ω>0,$ and we show that $$\sup_{r>0}\frac{2m(r)}{r}\leq \frac{(1+2Ω)^2-1}{(1+2Ω)^2},$$ where $m$ is the quasi-local mass, so that in particular $M=m(R).$ We also show that the inequality is sharp. Note that when $Ω=1$ the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with $p\geq 0$ which satisfies the dominant energy condition satisfies the hypotheses with $Ω=3.$