An investigation of the Buchdahl inequality for spherically symmetric static shells

A classical result by Buchdahl [9] shows that a class of static spherically symmetric solutions of the Einstein equations obey the inequality 2M/R ≤ 8/9, where M is the total ADM mass and R the area radius of the body. Buchdahl's proof rests on the hypotheses that the energy density is non-increasing outwards and that the pressure is isotropic. In this work neither of Buchdahl's hypotheses are assumed. We consider non-isotropic spherically symmetric shells, supported in [R0, R1], R0 > 0, of matter models for which the energy density ρ ≥ 0, and the radial- and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1. Note that this inequality holds with Ω = 3 for any matter model which satisfies the dominant energy condition. We show a Buchdahl type inequality for shells which are thin; given an ε < 1/4 there is a κ > 0 such that 2M/R1 ≤ 1 - κ when R1/R0 1 + ε. It is also shown that for a sequence of solutions such that R1/R0 → 1, the limit supremum of 2M/R1 of the sequence is bounded by ((2Ω + 1)2 - 1)/(2Ω + 1)2 (which equals 8/9 if Ω = 1). We emphasize that no field equations for the matter are used for this result. However, in the second part we consider the Einstein-Vlasov system and use the matter field equation to construct a family of static solutions with the property that R1/R0 → 1. We also show that for this sequence the value 8/9 of 2M/R1 is attained in the limit (note that Ω = 1 for Vlasov matter). The energy density of this sequence get more and more peaked, which should be contrasted to the solution for which 2M/R = 8/9 in Buchdahl's original work where ρ is constant and p blows up at r = 0. Clearly, this solution cannot satisfy the inequality p = q ≤ Ωρ, in particular it violates the dominant energy condition