Sharp Bounds on 2m/r of General Spherically Symmetric Static Objects

In 1959 Buchdahl [13] 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 eg. 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 p T , satisfy p + 2p T ≤ Ωρ, Ω > 0, and we show that sup r>0 2m(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. 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.