Sharp bounds on $2m/r$ of general spherically symmetric static objects

Hakan Andreasson

doi:10.1016/j.jde.2008.05.010

paper

Sharp bounds on $2 m / r$ of general spherically symmetric static objects

arXiv:gr-qc/0702137 · doi:10.1016/j.jde.2008.05.010

Abstract

In 1959 Buchdahl \cite{Bu} obtained the inequality $2 M / R \leq 8/9$ 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 e.g. neither of them hold in a simple soap bubble. In this work we remove both of these assumptions and consider \textit{any} static solution of the spherically symmetric Einstein equations for which the energy density $ρ \geq 0,$ and the radial- and tangential pressures $p \geq 0$ and $p_{T},$ satisfy $p + 2 p_{T} \leq Ω ρ, Ω > 0,$ and we show that $r > 0 sup \frac{2 m ( r )}{r} \leq \frac{( 1 + 2Ω ) ^{2} - 1}{( 1 + 2Ω ) ^{2}},$ where $m$ is the quasi-local mass, so that in particular $M = m (R) .$ We also show that the inequality is sharp. Note that when $Ω = 1$ 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 $Ω = 3.$

28 pages, typos corrected and the analogy in section 4 is presented slightly differently

Sharp bounds on $2 m / r$ of general spherically symmetric static objects

Abstract

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Cited by in corpus (84)

Sharp bounds on 2m/r of general spherically symmetric static objects

Abstract

References in corpus (5)

Cited by in corpus (84)

Sharp bounds on $2 m / r$ of general spherically symmetric static objects