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

In 1959 Buchdahl [H.A. Buchdahl, General relativistic fluid spheres, Phys. Rev. 116 (1959) 1027-1034] obtained the inequality 2M/R⩽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 for instance neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider any static solution of the spherically symmetric Einstein equations for which the energy density ρ⩾0, and the radial and tangential pressures p⩾0 and pT satisfy p+2pT⩽Ωρ, Ω>0, and we show thatsupr>02m(r)r⩽(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 under these assumptions. 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⩾0 which satisfies the dominant energy condition satisfies the hypotheses with Ω=3.