On the Buchdahl inequality for spherically symmetric static shells

Details On the Buchdahl inequality for spherically symmetric static shells On the Buchdahl inequality for spherically symmetric static shells

2006-05-17

arxiv.org/abs/gr-qc/0605097v2

Commun. Math. Phys. 274:399-408,2007

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, Latex. Second statement in Theorem 1 corrected

Scientific paper

10.1007/s00220-007-0283-6

A classical result by Buchdahl \cite{Bu1} shows that for static solutions of the spherically symmetric Einstein-matter system, the total ADM mass M and the area radius R of the boundary of the body, obey the inequality $2M/R\leq 8/9.$ 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 $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1.$ We show a Buchdahl type inequality for shells which are thin; given an $\epsilon<1/4$ there is a $\kappa>0$ such that $2M/R_1\leq 1-\kappa$ when $R_1/R_0\leq 1+\epsilon.$ It is also shown that for a sequence of solutions such that $R_1/R_0\to 1,$ the limit supremum of $2M/R_1$ of the sequence is bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In particular if $\Omega=1,$ which is the case for Vlasov matter, the boumd is $8/9.$ The latter result is motivated by numerical simulations \cite{AR2} which indicate that for non-isotropic shells of Vlasov matter $2M/R_1\leq 8/9,$ and moreover, that the value 8/9 is approached for shells with $R_1/R_0\to 1$. In \cite{An2} a sequence of shells of Vlasov matter is constructed with the properties that $R_1/R_0\to 1,$ 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 \cite{An2} the Vlasov equation is important.

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