On Static Shells and the Buchdahl Inequality for the Spherically Symmetric EinsteinVlasov System
Abstract
In a previous work [1] matter models such that the energy density ρ ≥ 0, and the radial and tangential pressures p ≥ 0 and q, satisfy p + q ≤ Ωρ, Ω ≥ 1, were considered in the context of Buchdahl’s inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, [R _{0}, R _{1}], R _{0} > 0, satisfies R _{1}/R _{0} < 1/4. Moreover, given a sequence of solutions such that R _{1}/R _{0} → 1, then the limit supremum of 2M/R _{1} was shown to be bounded by ((2Ω + 1)^{2}  1)/(2Ω + 1)^{2}. In this paper we show that the hypothesis that R _{1}/R _{0} → 1, can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric EinsteinVlasov system with this property. We also prove that for this sequence not only the limit supremum of 2M/R _{1} is bounded, but that the limit is ((2Ω + 1)^{2}  1)/(2Ω + 1)^{2} = 8/9, since Ω = 1 for Vlasov matter. Thus, static shells of Vlasov matter can have 2M/R _{1} arbitrary close to 8/9, which is interesting in view of [3], where numerical evidence is presented that 8/9 is an upper bound of 2M/R _{1} of any static solution of the spherically symmetric EinsteinVlasov system.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 2007
 DOI:
 10.1007/s0022000702854
 arXiv:
 arXiv:grqc/0605151
 Bibcode:
 2007CMaPh.274..409A
 Keywords:

 General Relativity and Quantum Cosmology;
 Mathematical Physics
 EPrint:
 20 pages, Latex