Sharp bounds on $2m\mathrm{/}r$ of general spherically symmetric static objects

In 1959 Buchdahl \cite{Bu} obtained the inequality $2M\mathrm{/}R\le 8\mathrm{/}9$ 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 $\mathit{\text{any}}$ static solution of the spherically symmetric Einstein equations for which the energy density $\rho \ge 0,$ and the radial- and tangential pressures $p\ge 0$ and $p_T,$ satisfy $p+2p_T\le \mathrm{\Omega }\rho ,\mathrm{\Omega }>0,$ and we show that $$\underset{r>0}{\sup }\frac{2m(r)}{r}\le \frac{(1+2\mathrm{\Omega }{)}^2-1}{(1+2\mathrm{\Omega }{)}^2},$$ where $m$ is the quasi-local mass, so that in particular $M=m(R)\mathrm{.}$ We also show that the inequality is sharp. Note that when $\mathrm{\Omega }=1$ the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with $p\ge 0$ which satisfies the dominant energy condition satisfies the hypotheses with $\mathrm{\Omega }=3.$