Sharp bounds on [formula omitted] of general spherically symmetric static objects

Abstract

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 p T satisfy p + 2 p T ⩽ Ω ρ , Ω > 0 , and we show that sup r > 0 2 m ( r ) r ⩽ ( 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 .