On the Buchdahl Inequality for Spherically Symmetric Static Shells |
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Authors: | Håkan Andréasson |
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Institution: | 1.Mathematical Sciences,Chalmers and G?teborg University,G?teborg,Sweden |
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Abstract: | A classical result by Buchdahl 6] shows that for static solutions of the spherically symmetric Einstein equations, the ADM
mass M and the area radius R of the boundary of the body, obey the inequality 2M/R ≤ 8/9. The proof of this inequality 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 R
0, R
1], R
0 > 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. We show a Buchdahl type inequality for shells which are thin; given an there is a κ > 0 such that 2M/R
1 ≤ 1 − κ when . It is also shown that for a sequence of solutions such that R
1/R
0 → 1, the limit supremum of 2M/R
1 of the sequence is bounded by ((2Ω + 1)2 − 1)/(2Ω + 1)2. In particular if Ω = 1, which is the case for Vlasov matter, the bound is 8/9. The latter result is motivated by numerical
simulations 3] which indicate that for non-isotropic shells of Vlasov matter 2M/R
1 ≤ 8/9, and moreover, that the value 8/9 is approached for shells with R
1/R
0 → 1. In 1] a sequence of shells of Vlasov matter is constructed with the properties that R
1/R
0 → 1, and that 2M/R
1 equals 8/9 in the limit. We emphasize that in the present paper no field equations for the matter are used, whereas in 1]
the Vlasov equation is important. |
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Keywords: | |
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