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

Journal article, 2008

In 1959 Buchdahl [H.A. Buchdahl, General relativistic fluid spheres, Phys. Rev. 116 (1959) 1027-1034] obtained the inequality 2 M / 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 + 2 pT ≤ Ω ρ, Ω > 0, and we show thatunder(sup, r > 0) frac(2 m (r), r) ≤ frac((1 + 2 Ω)2 - 1, (1 + 2 Ω)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.