Bounds on M/R for Charged Objects with positive Cosmological constant

We consider charged spherically symmetric static solutions of the Einstein-Maxwell equations with a positive cosmological constant $Λ$. If $r$ denotes the area radius, $m_g$ and $q$ the gravitational mass and charge of a sphere with area radius $r$ respectively, we find that for any solution which satisfies the condition $p+2p_{\perp}\leq ρ,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, $ρ\geq 0$ is the energy density, and for which $0\leq \frac{q^2}{r^2}+Λr^2\leq 1,$ the inequality $\frac{m_g}{r} \leq 2/9+\frac{q^2}{3r^2}-\frac{Λr^2}{3}+2/9\sqrt{1+\frac{3q^2}{r^2}+3Λr^2}$ holds. We also investigate the issue of sharpness, and we show that the inequality is sharp in a few cases but generally this question is open.

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