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Title of Journal: Commun Math Phys

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Abbravation: Communications in Mathematical Physics

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Springer-Verlag

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10.1002/anie.200903104

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1432-0916

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On Static Shells and the Buchdahl Inequality for t

Authors: Håkan Andréasson
Publish Date: 2007/06/20
Volume: 274, Issue: 2, Pages: 409-425
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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 + 12 − 1/2Ω + 12 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 + 12 − 1/2Ω + 12 = 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

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