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Sharp bounds on $$2m/r$$ of general spherically symmetric static objects. (English) Zbl 1151.83014
Summary: In 1959 H. A. Buchdahl [Phys. Rev., II. Ser. 116, 1027–1034 (1959; Zbl 0092.20802)] obtained the inequality $$2M/R\leq 8/9$$ under the assumptions that the energy density is non-increasing outwards and that the pressure is isotropic. Here $$M$$ is the ADM mass and $$R$$ the area radius of the boundary of the static body. The assumptions used to derive the Buchdahl inequality are very restrictive and for instance neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider any static solution of the spherically symmetric Einstein equations for which the energy density $$\rho\geq 0$$, and the radial and tangential pressures $$p\geq 0$$ and $$p_T$$ satisfy $$p+2p_T\leq \Omega\rho$$, $$\Omega>0$$, and we show that
$\sup_{r>0} \frac{2m(r)}{r}\leq \frac{(1+2\Omega)^2-1} {(1+2\Omega)^2}.$
where $$m$$ is the quasi-local mass, so that in particular $$M=m(R)$$. We also show that the inequality is sharp under these assumptions. Note that when $$\Omega=1$$ the original bound by Buchdahl is recovered. The assumptions on the matter model are very general and in particular any model with $$p\geq 0$$ which satisfies the dominant energy condition satisfies the hypotheses with $$\Omega=3$$.

##### MSC:
 83C15 Exact solutions to problems in general relativity and gravitational theory 83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.) 83E05 Geometrodynamics and the holographic principle
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